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Economic Quarterly—Volume 94, Number 4—Fall 2008—Pages 301–309
Introduction to the New
Keynesian Phillips Curve
Andreas Hornstein
I
n most industrialized economies inflation tends to be pro-cyclical; that is,
inflation is high during times of high economic activity. When economic
activity is measured by the unemployment rate this statistical relationship
is known as the Phillips curve. The Phillips curve is sometimes viewed as a
menu for monetary policymakers, that is, they can choose between high inflation and low unemployment or low inflation and high unemployment. But
this interpretation of the Phillips curve assumes that the relationship between
unemployment and inflation is structural and will not break down once a policymaker attempts to exploit the perceived tradeoff. After the high inflation
episodes experienced by many economies in the 1970s, this structural interpretation of the Phillips curve was discredited. Yet, after a period of low inflation
in the 1980s and early 1990s, economists have again worked on a structural interpretation of the Phillips curve. This New Keynesian Phillips curve (NKPC)
assumes the presence of nominal price rigidities. In this special issue of the
Economic Quarterly, we publish four surveys on the history of the Phillips
curve, the structural estimation of the New Keynesian Phillips curve, and the
policy implications of the nominal rigidities underlying the New Keynesian
Phillips curve.
The Phillips Curve and U.S. Economic Policy
Robert King surveys the evolution of the Phillips curve itself and its usage
in U.S. economic policymaking from the 1960s to the mid-1990s. He first
describes how, in the 1960s, the Phillips curve became an integral part of U.S.
macroeconomic policy in its pursuit of low unemployment rates. A stylized
version of the Phillips curve that emerges from this period relates current
The views expressed do not necessarily reflect those of the Federal Reserve Bank of Richmond
or the Federal Reserve System.
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inflation, π , to the current unemployment rate, u, and lagged inflation,
γ i π t−i − βut .
πt =
i≥1
Similar to other elements of the then-standard Keynesian IS-LM macromodel,
economists would tell stories that motivated the Phillips curve but the Phillips
curve was not derived from an explicit theory. Furthermore, the estimated
parameters were taken as structural, in particular as invariant to policy interventions. In the late 1960s, Phelps (1968) and Friedman (1968) interpreted
the Phillips curve as arising from search and information frictions in labor
markets, and they argued that the relation between a real variable such as
unemployment and nominal inflation was based on misperceptions about inflation on the part of the public. Phelps proposed an expectations-augmented
Phillips curve,
π t − ρπ et = −βut ,
where π e denotes expected inflation. If, as Phelps and Friedman argued,
ρ = 1, then a tradeoff between inflation and unemployment exists only to the
extent that actual inflation deviates from expected inflation. At the time, inflation expectations were modeled as adaptive, that is, a geometric distributed
lag of past actual inflation. In this case, for a constant actual inflation rate
the expected inflation rate would eventually converge to the actual inflation
rate and the unemployment rate would settle down at its natural rate. Thus,
there is no long-run tradeoff between inflation and unemployment. Although
Phelps and Friedman’s argument originally represented a minority view in
the profession, the argument became more widely accepted in the 1970s after
periods of high inflation and unemployment.
Accounting for the instability of the Phillips curve in the 1970s had
lasting effects on the way macroeconomic analysis was done and continues to be done today. First, since expectations play a crucial role in the
expectations-augmented Phillips curve, it seemed necessary not to resort to
some arbitrary assumption on the expectations mechanism. For this purpose,
macroeconomists started to assume that expectations are rational. By this we
mean that expectations are such that they do not lead to systematic mistakes
given the available information. Sargent and Wallace (1975) used the idea
of rational expectations in an otherwise standard IS-LM macromodel with
an expectations-augmented Phillips curve to argue that systematic monetary
policy actions do not systematically affect unemployment or output. Second,
macroeconomists not only started to work with model-consistent expectations in otherwise ad hoc models, but they started to study the optimal choices
of economic agents in explicitly specified environments agents; that is, they
started to study macroeconomic questions using the tools of general equilibrium analysis. The seminal work was Lucas’ (1972) formal analysis of the
A. Hornstein: Introduction
303
Phelps-Friedman Phillips curve in an environment where agents had difficulty
sorting out their own relative price shocks from aggregate price level shocks.
King describes how, at the end of the 1970s after years of persistently high
inflation and high unemployment, monetary policymakers moved to lower the
inflation rate. At that time, the debate centered on the perceived cost (in terms
of elevated unemployment) associated with a reduction of the inflation rate.
On the one hand, proponents of the more standard Phillips curve argued that
these costs would be substantial. On the other hand, proponents of a rational
expectations-augmented Phillips curve argued that the costs could be quite
low, especially if the low inflation policy was credible to the public. In the
end, the Federal Reserve under Paul Volcker reduced inflation over a relatively
short time period at some cost, but not as high a cost as predicted by standard
Phillips curves. For the remainder of the 1980s and the early 1990s, the Federal
Reserve under Alan Greenspan further lowered average inflation and, in the
process, strengthened its credibility for continued low inflation policies. King
ends his survey in the mid-1990s when the Federal Reserve Board’s monetary policy model incorporated an expectations-augmented Phillips curve with
elements of rational expectations, and the Federal Open Market Committee
debated the desirability of a target for low long-run inflation and what that
target should be.
The New Keynesian Phillips Curve
At the time that U.S. inflation started to decline in the 1980s there was a
resurgence of interest in business cycle analysis. Continuing the general equilibrium program in macroeconomics started with Lucas (1972), real business
cycle analysis developed quantitative models of the aggregate economy based
on the stochastic neoclassical growth model, e.g., Kydland and Prescott (1982)
or Long and Plosser (1983). Using simulation studies, one could show that
these models were able to mimic the U.S. business cycle in terms of the statistical properties of the time series of a limited number of aggregate variables
(output, consumption, investment, and employment). As the name indicates,
real business cycle theory addressed the behavior of quantities and relative
prices over the business cycle, implicitly assuming that money is neutral.
Working on the assumption that money is not neutral, economists in the mid1990s then started to introduce nominal price rigidities into these models,
now also known as Dynamic Stochastic General Equilibrium (DSGE) models. From this research program emerged the New Keynesian Phillips curve
that relates actual and expected inflation not to the unemployment rate but to a
measure of aggregate marginal cost. The second and third paper in this issue
discuss the estimation of the structural parameters of the NKPC.
Once one assumes that nominal prices do not continuously adjust to
clear markets, one has to decide how these prices are set in the first place.
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Almost all of the work on nominal price rigidities has answered this question
using the framework of monopolistic competition, which assumes that the
product whose price has to be determined is produced by a profit-maximizing
monopoly. There may be imperfect substitutes for the monopolist’s product;
that is, the demand for the product depends not only on its own price but also
on the prices of the substitutes. When the monopolist decides on his own price
he will, however, take these other prices as given, hence the term monopolistic competition. A monopolist that can continuously adjust his nominal price
will set the price to equate contemporaneous marginal revenue and marginal
cost and the price will be a markup over marginal cost. Compare this with
flexible prices in perfectly competitive markets where the price and marginal
cost are equated. If nominal prices cannot be continuously readjusted, then
the monopolist will choose the current nominal price such that he equates the
expected present value of marginal revenue and marginal cost over the time
that the price remains fixed.
The model of an individual monopolistically competitive producer is then
typically embedded into a general equilibrium model with a large number
of these producers, e.g., Blanchard and Kiyotaki (1987). These producers
are identical except for the time when they can adjust their nominal price.
Various mechanisms for price adjustment have been proposed; most assume
that the opportunity for price adjustment is exogenously given. One popular
modeling technique is a Calvo-type price adjustment where, each period, a
firm gets to adjust its price with some probability that is fixed over time. Using
Calvo-type price adjustment, Woodford (2003) shows that the aggregation of
the linearized optimal price adjustment rules for the individual firms yields an
expression in current and expected future inflation and a measure of aggregate
marginal cost, mc,
π t = γ f Et π t+1 + λmct + ξ t .
This is the structural NKPC where γ f and λ are functions of structural parameters, including the probability of price adjustment, α, and ξ t is a random
variable. The random disturbance is often interpreted as an exogenous shock
to the firms’ markup. Solving this difference equation forward, one can see
that current and expected future marginal cost are driving today’s inflation.
For most measures of inflation and what could be considered reasonable
measures of marginal cost, inflation tends to be more persistent than marginal
cost. Since marginal cost “drives” inflation in the basic NKPC, this makes it
hard for the model to match the data. Economists have, therefore, modified
the basic NKPC by introducing “rule of thumb” price adjusters or firms that
simply index their price to the aggregate inflation rate, e.g., Galı́ and Gertler
(1999). These assumptions lead to the inclusion of lagged inflation,
π t = γ b π t−1 + γ f Et π t+1 + λmct + ξ t ,
A. Hornstein: Introduction
305
and, therefore, make the NKPC a hybrid of the basic NKPC and more standard Phillips curves. The coefficients γ b , γ f , and λ are again functions of
structural parameters. The ability of monetary policy to control inflation with
a NKPC depends on the relative magnitudes of these coefficients. Loosely
speaking, monetary policy affects inflation through its effects on marginal
cost. Thus, the smaller the coefficient on marginal cost, the less impact monetary policy will have on inflation. In the extreme case when λ = 0, inflation
evolves independently of monetary policy and whatever else happens in the
rest of economy. How “costly” it is to reduce inflation depends on the relative
magnitude of the coefficients on past and future inflation, γ b and γ f . If the
coefficient on lagged inflation is large, then inflation is mostly driven by its
own past and policy actions might affect inflation only with a long time lag. In
order to evaluate the effectiveness of monetary policy actions we, therefore,
need estimates of these parameters.
Single-Equation Estimation of the NKPC
In the second paper of this issue, James Nason and Gregor Smith survey
the estimation of the parameters of the NKPC using only the NKPC itself.
Single-equation estimation of the NKPC parameters is appealing because it
does not require any assumption on how the rest of the economy should be
specified. Yet standard ordinary least squares estimation of the NKPC is not
applicable since expected inflation in the NKPC is an endogenous variable
that is correlated with the error term of the estimation equation. Consistent
parameter estimates can still be obtained through the use of the General Method
of Moments (GMM) technique, which in turn requires instrumental variables
that are correlated with expected inflation but uncorrelated with the other
variables in the NKPC.
Nason and Smith report that, in general, estimated parameters for the
hybrid NKPC are consistent with prior restrictions. For example, estimated
price adjustment probabilities are between zero and one. They also find that the
coefficient on expected future inflation tends to be larger than the coefficient
on lagged inflation. This suggests that monetary policy can affect inflation in
the short term. Nason and Smith also discuss the finding that the estimated
coefficient on marginal cost tends to be small and barely significant. This is
bad news for the NKPC as a model of inflation and for monetary policy.
The ambiguous evidence on the marginal cost coefficient may be related to
weak identification through weak instrumental variables in the GMM estimation. Instrumental variables are essentially used to forecast expected inflation
independent of the other variables in the NKPC. For an instrumental variable
to serve its purpose it has to be correlated with expected future inflation and
it should not be correlated with marginal cost and current and lagged inflation. But as Nason and Smith point out, past empirical work on inflation has
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shown that lagged inflation tends to be a good forecast of future inflation and
it is difficult to improve on that forecast. This suggests that the instrumental
variables in the GMM procedure are quite weak. Nason and Smith then show
that after one takes into account that we have weak instruments, the evidence
in favor of the NKPC is weakened or the NKPC is rejected outright.
System Estimation of the NKPC
In the third paper of this issue, Frank Schorfheide surveys system methods to
estimate the parameters of the NKPC. For this approach one specifies a more
or less complete model of the aggregate economy, a DSGE model, and then
identifies the structural parameters from the restrictions that the equilibrium
process imposes on the moments of a set of observable variables.
Using a simple example, Schorfheide interprets the various identification
schemes used in the literature. He explains why it may not be possible to
obtain consistent parameter estimates using single-equation methods. System
methods on the other hand can obtain consistent parameter estimates through
the imposition of prior constraints on elements of the DSGE model other than
the NKPC. Essentially these prior restrictions allow one to identify exogenous shocks that may serve as instruments for the NKPC. As an example,
Schorfheide points to the procedure of identifying monetary policy shocks
from the restriction that the public cannot respond to contemporaneous monetary policy shocks. Schorfheide also suggests that it may not be possible
to identify the coefficient on lagged inflation in the NKPC if one allows for
serially correlated markup shocks. Indeed, single-equation estimates of the
NKPC identify γ b through the implicit prior restriction that the markup shock
is i.i.d. This lack of identification affects the evaluation of policy effectiveness
if it also implies that the coefficient on future inflation is not identified.
Schorfheide then surveys papers that estimate the NKPC as part of a more
complete DSGE model. Most of this empirical work uses data on output, inflation, and a nominal interest rate. Marginal cost in the NKPC is then treated
as a latent variable that is constructed from the observable variables and the
equilibrium relationships implied by the DSGE model. But some empirical
work also includes measures of marginal cost in the set of observable variables. Schorfheide observes that the range for the estimated coefficients on
marginal cost in the NKPC is much larger when marginal cost is a latent variable. The range of estimated NKPC coefficients on marginal cost becomes
much closer to that obtained from single-equation estimations once observations on marginal cost are included. Thus, with marginal cost as a latent
variable, features of the DSGE model that are different from the NKPC can
become much more important for the determination of the NKPC marginal
cost coefficient. As is apparent from the work of Krause, López-Salido, and
Lubik (2008), the implied process for the latent marginal cost variable is then
A. Hornstein: Introduction
307
very different from the process of various measures of marginal cost used in
the literature.
In general, the literature review suggests that there is no consensus on
the magnitude and role of nominal rigidities in the estimated price-setting
process. Furthermore, introducing additional nominal rigidities in the wagesetting process affects the estimates for nominal rigidities in the price-setting
process, that is, the NKPC. It also appears as if the relative role of nominal
price and wage rigidities is not identified from the data.
Policy Implications of Nominal Price Rigidities
In the final paper of this issue, Stephanie Schmitt-Grohé and Martı́n Uribe discuss the implications of nominal price rigidities for optimal monetary policy.
They first ask how the presence of nominal price rigidities affects the design
of optimal policy when fiscal and monetary policy are jointly determined.
They then go on to study if simple policy rules such as the Taylor rule can get
the economy close to the optimal policy outcome. They find that with small
amounts of nominal price rigidities, optimal policy involves price stability,
i.e., it tightly stabilizes inflation at zero, and that simple rules that exclusively
focus on deviations from price stability get the economy very close to the
optimum.
These results provide a nice contrast between optimal monetary policy in
environments with and without nominal rigidities. When nominal prices are
flexible and there is a well-defined demand for real balances, a zero nominal
interest and, hence, deflation minimize the welfare costs from holding money.
Furthermore, if in a stochastic environment fiscal policy has to use distortionary taxes to finance given expenditures, mean zero unanticipated changes
in the inflation rate represent lump-sum taxes and are an efficient way to raise
revenues. Thus, optimal policy leads to low and volatile inflation. In contrast
with nominal rigidities, deviations from price stability introduce relative price
distortions among the monopolistically competitive producers and make production inefficient. Schmitt-Grohé and Uribe argue that in environments that
contain both a well-defined demand for real money and nominal rigidities,
even small amounts of nominal rigidities imply that price stability is optimal.
This is a useful result since the surveys of Nason and Smith and Schorfheide
provide some evidence for the presence of nominal rigidities, but also show
that there is no agreement on how substantial nominal rigidities are.
Optimal policies that determine fiscal and monetary policies jointly can be
quite complicated, yet Schmitt-Grohé and Uribe show that simple policy rules
involve only minor welfare losses relative to the optimal policy. These simple
rules are modeled on the Taylor rule that has the nominal interest responding to
deviations of inflation and output from their targets with some dependence on
past interest rates. It turns out that a simple rule that aggressively targets price
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stability involves only minimal welfare losses relative to the optimal policy,
and that a response to deviations of output from trend significantly decreases
welfare. An open question remains why most monetary policymakers prefer
to target some positive inflation rate rather than price stability with a zero
inflation rate.
Conclusion
The surveys in this special issue show that discussions of the Phillips curve
have been at the core of monetary policymaking since the 1960s. Our understanding of what underlies the correlation between unemployment and the
inflation rate and what that means for monetary policymaking has changed
over the years. At first, many economists and policymakers took the statistical
relationship as a fixed menu of choices between inflation and unemployment
and targeted relatively low unemployment outcomes. From the period of high
inflation and high unemployment in the 1970s, economists emerged believing that there is no inflation-unemployment tradeoff that remains invariant to
policy interventions, and policymakers agreed that the objective of monetary
policy should be low and stable inflation. Finally, in the 1990s, economists
again started to study the inflation-output tradeoff using the new techniques
developed in macroeconomics in the 1970s and 1980s, rational expectations
and explicit quantitative general equilibrium models of the aggregate economy. This research program gave rise to the NKPC, which is based on the
maintained assumption of nominal price rigidities. As is apparent from the
surveys in this issue, there is some support for the NKPC in aggregate data, but
there is no agreement on the extent of nominal price rigidities in the aggregate
economy. Furthermore, one should be aware that not all macroeconomists
agree that nominal rigidities are relevant for an understanding of the aggregate economy, e.g., see Williamson (2008) or Chari, Kehoe, and McGrattan
(2009) for a skeptical view on this research program. To be sure, research
on the relationship between unemployment and inflation will remain an active area in macroeconomics for anyone with an interest in applied monetary
economics.
REFERENCES
Blanchard, Olivier Jean, and Nobuhiro Kiyotaki. 1987. “Monopolistic
Competition and the Effects of Aggregate Demand.” American
Economic Review 77 (September): 647–66.
A. Hornstein: Introduction
309
Chari, V.V., Patrick J. Kehoe, and Ellen R. McGrattan. 2009. “New
Keynesian Models: Not Yet Useful for Policy Analysis.” American
Economic Journal: Macroeconomics 1 (January): 242–66.
Friedman, Milton. 1968. “The Role of Monetary Policy.” American
Economic Review 58 (March): 1–17.
Galı́, Jordi, and Mark Gertler. 1999. “Inflation Dynamics: A Structural
Econometric Analysis.” Journal of Monetary Economics 44 (October):
195–222.
Krause, Michael, David López-Salido, and Thomas Lubik. 2008. “Inflation
Dynamics with Search Frictions: A Structural Econometric Analysis.”
Journal of Monetary Economics 55 (July): 892–916.
Kydland, Finn E., and Edward C. Prescott. 1982. “Time to Build and
Aggregate Fluctuations.” Econometrica 50 (November): 1345–70.
Long, John B., Jr., and Charles I. Plosser. 1983. “Real Business Cycles.”
Journal of Political Economy 91 (February): 39–69.
Lucas, Robert E., Jr. 1972. “Expectations and the Neutrality of Money.”
Journal of Economic Theory 4 (April): 103–24.
Phelps, Edmund S. 1968. “Phillips Curve, Expectations of Inflation, and
Optimal Inflation over Time.” Economica 34: 254–81.
Sargent, Thomas J., and Neil Wallace. 1975. “‘Rational’ Expectations, the
Optimal Monetary Instrument, and the Optimal Money Supply Rule.”
Journal of Political Economy 83 (April): 241–54.
Williamson, Stephen D. 2008. “New Keynesian Economics: A Monetary
Perspective.” Federal Reserve Bank of Richmond Economic Quarterly
94 (Summer): 197–218.
Woodford, Michael. 2003. Interest and Prices. Princeton, N.J.: Princeton
University Press.
Economic Quarterly—Volume 94, Number 4—Fall 2008—Pages 311–359
The Phillips Curve and U.S.
Macroeconomic Policy:
Snapshots, 1958–1996
Robert G. King
T
he curve first drawn by A.W. Phillips in 1958, highlighting a negative
relationship between wage inflation and unemployment, figured
prominently in the theory and practice of macroeconomic policy during
1958–1996.
Within a decade of Phillips’ analysis, the idea of a relatively stable longrun tradeoff between price inflation and unemployment was firmly built into
policy analysis in the United States and other countries. Such a long-run
tradeoff was at the core of most prominent macroeconometric models as of
1969.
Over the ensuing decade, the United States and other countries experienced stagflation, a simultaneous rise of unemployment and inflation, which
threw the consensus about the long-run Phillips curve into disarray. By the
end of the 1970s, inflation was historically high—near 10 percent—and poised
to rise further. Economists and policymakers stressed the role of shifting expectations of inflation and differed widely on the costliness of reducing inflation, in part based on alternative views of the manner in which expectations
were formed. In the early 1980s, the Federal Reserve System undertook
an unwinding of inflation, producing a multiyear interval in which inflation
fell substantially and permanently while unemployment rose substantially but
temporarily. Although costly, the disinflation process involved lower unemployment losses than predicted by consensus macroeconomists, as rational
expectations analysts had suggested that it would.
The author is affiliated with Boston University, National Bureau of Economic Research, and
the Federal Reserve Bank of Richmond. Thanks to Michael Dotsey, Andreas Hornstein, Mark
Watson, and Alexander Wolman for valuable comments. Devin Reilly helped with the figures.
The views expressed herein are the author’s and not necessarily those of the Federal Reserve
Bank of Richmond or the Federal Reserve System. E-mail: rking@bu.edu.
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By 1996, the central bank of the United States had constructed a largescale rational expectations model, without any long-run tradeoff, which it
began to use to evaluate alternative policy scenarios. Monetary policymakers
at that time accepted the idea that there was no long-run tradeoff at, and above,
the then-prevailing price inflation rate of 3 percent. Yet many felt that there
were important tradeoffs over short-run horizons, diversely defined, and some
saw long-run tradeoffs near zero price inflation.
This article reviews the evolving role of the Phillips curve as an element
of macroeconomic policy during 1958–1996, as well as academic and central bank research on it, via a series of snapshots over this roughly 40-year
period. In conducting the research summarized in this article, my motivation
is to better understand the mindset about the tradeoff between inflation and
unemployment over an important period of U.S. history with an eye toward
ultimately better understanding the joint behavior of the Federal Reserve and
the U.S. economy during that period. Diverse research in macroeconomics—
notably Sargent (1999), Orphanides (2003), and Primiceri (2006)—has sought
an explanation of inflation’s role in the behavior of a central bank that has an
imperfect understanding of the operation of the private economy. The perceived nature of the Phillips curve plays an important role in these analyses,
so that my reading of U.S. history may provide input into future work along
these lines. I draw upon two distinct and complementary sources of information, published articles and documents of the Federal Open Market Committee
(FOMC), to trace the evolving interpretation of the tradeoff over this roughly
40-year period.
The discussion is divided into six sections that follow this introduction.
Section 1 provides a quick overview of the U.S. experience with price inflation and unemployment during 1958–1996. As the objective of this article is
to provide a description of how policymakers’ visions of the Phillips curve
may have evolved during this time, resulting from empirical and theoretical
developments, it is useful to have these series in mind as we proceed. Section
2 describes the birth of the Phillips curve as a policy tool, highlighting three
core contributions: Phillips’ original analysis of U.K. data, Samuelson and
Solow’s (1961) estimates of the curve on U.S. data and their depiction of it as
a menu for policy choice, and the econometric analysis by Klein et al. (1961)
and Sargan (1964) of the interrelationship between wage inflation, price inflation, and unemployment, which formed the background for wage and price
blocks of macroeconomic policy models. Section 3 depicts the battle against
unemployment that the United States waged during the 1962–1968 period and
its relationship to the Phillips curve in then-prominent macroeconomic policy
models. Section 4 discusses the breakdown of the empirical Phillips curve
during 1969–1979, a period including intervals of stagflation in which unemployment and inflation rose together, and theoretical criticisms of the Phillips
curve as a structural macroeconomic relation. Section 5 indicates the role of
R. G. King: Phillips Curve Snapshots
313
the Phillips curve during the unwinding of inflation in the United States during
1980 through 1986. Section 6 concerns several aspects of policy modeling and
policy targeting in 1996 as the United States returned to a sustained interval
of relatively low inflation. Section 7 concludes.
1.
INFLATION AND UNEMPLOYMENT, 1958–1996
Since my discussion focuses on studies of inflation and unemployment that
were written during 1958 through 1996, it seems useful to start by providing
information on U.S. inflation and unemployment over that historical period,
augmented by a few initial years, as in Figure 1. As measured by the yearover-year percentage change in the gross domestic product deflator, inflation
averaged just under 4 percent, starting and ending the 1955–1996 interval at
about 2.5 percent. Inflation twice exceeded 10 percent, in 1974–75 and 1981.
The unemployment rate averaged 6 percent, starting and ending the sample
period near 5 percent. Recession intervals, as dated by the National Bureau
of Economic Research (NBER), are highlighted by the shaded lines in Figure
1.
My snapshots of the Phillips curve and its role in macroeconomic policy
are usefully divided into five periods.
• The formative years in which the initial studies were conducted, 1955–
1961. During this interval, there were two recessions (August 1957
through April 1958 and April 1960 through February 1961), each of
which was marked by declining inflation and rising unemployment.
• The battle against unemployment from 1962 through 1968 during which
unemployment fell substantially, with inflation being at first quiescent
and then rising substantially toward the end of the period.
• The breakdown of the Phillips curve empirically and intellectually
came from 1969 through 1979. In this period, there were two recessions. During December 1969–November 1970, both inflation and
unemployment rose but there was a brief decline in inflation within the
recession. During November 1973 through March 1975, inflation and
unemployment both rose dramatically. This period was a tumultuous
one, marked by departure from gold standard, wage and price controls,
energy shocks, as well as difficult political and social events.
• The unwinding of inflation took place during 1980 through 1985, with
a substantial reduction in inflation accompanied by a sustained period
of unemployment.
• In the aftermath, 1986–1996, the Phillips curve assumed a new form
in monetary policy models and monetary policy discussions.
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Figure 1 Inflation and Unemployment, 1955–1996
Inflation
Unemployment
12
10
8
6
4
2
0
1955
1960
1965
Martin
(1951-1970)
1970
1975
Burns
1980
Miller
1985
Volcker
(1970-1978) (1978-1979) (1979-1987)
1990
1995
2000
Greenspan
(1987-2000)
Notes: The inflation rate is the year-over-year change in the gross domestic product
(GDP) deflator, the unemployment rate is the civilian unemployment rate, quarterly averages of monthly figures. All data from the Federal Reserve Economic Database (FRED)
at the Federal Reserve Bank of St. Louis. The shaded grey areas are NBER recessions, while the vertical black lines separate the tenures of the various Federal Reserve
chairman whose names appear below the horizontal axis.
2. THE FORMATIVE YEARS
Figure 2 is the dominant image from Phillips’ initial article: a scatter plot of
measures of wage inflation and unemployment in the United Kingdom over
1861–1913 supplemented by a convex curve estimated by a simple statistical procedure. During the 1960s, U.S. macroeconomic policy analysis and
models were based on a central inference from this figure, which was that
a permanent rise in inflation would be a necessary cost of permanently reducing unemployment. However, as background to that period, it is useful
for us to understand how the Phillips curve was estimated initially, how it
crossed the Atlantic, and how it was modified so that it could be imported into
macroeconomic policy models.
R. G. King: Phillips Curve Snapshots
315
Figure 2 Phillips’ Curve
10
8
Wage Inflation
6
4
2
0
-2
-4
0
1
2
3
4
5
6
7
8
9
10
11
Unemployment
Notes: This is Figure 1 from Phillips (1958), displaying the relationship between unemployment and wage inflation over 1861–1913. The dots represent annual observations,
while the crosses represent trade cycle averages.
The Original Study
Phillips (1958) described the objective of his study as follows: “to see whether
statistical evidence supports the hypothesis that the rate of change of money
wage rates in the United Kingdom can be explained by the level of unemployment and the rate of change of unemployment, except in or immediately after
those years in which there was a very rapid rise in import prices, and if so to
form some quantitative estimate of the relation between unemployment and
the rate of change of money wage rates.”1 He began with the study of inflation
and unemployment over multiyear periods, which he called trade cycles, and
then he assembled these intervals into the overall curve that bears his name.
Trade cycles and the Phillips curve
The celebrated trade-off curve was derived by a complicated procedure. First,
Phillips explored the behavior of a measure of wage change and unemployment
1 Phillips (1958, 284).
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over a series of historical United Kingdom “trade cycles,” an alternative label
for the sort of business cycles that Burns and Mitchell (1946) had identified
for the United States.2 The cycle for 1868–1879 is shown in Panel A of Figure
3. It begins with several years of falling unemployment and rising wage
inflation, then an interval of rapidly declining wage inflation and modestly
rising unemployment, then a number of years of wage declines accompanied
by substantially increasing unemployment. Over the course of this cycle,
there was an initial interval (1868–1872) during which inflation rose by about
10 percent, while unemployment dropped by about 5 percent. Then, from
1872–1875, there was a period of sharply declining inflation accompanied by
modestly rising unemployment. Finally, from 1876–1879, there was a period
of negative inflation (−1 to −3 percent per year) coupled with dramatically
rising unemployment.
Phillips’ identification of the tradeoff between inflation and unemployment did not rely on the shape of the cyclical pattern over the course of this
and other individual trade cycles. Instead, the wage inflation and unemployment observations over the 1868–1879 trade cycle were averaged by Phillips
to produce one of the “+” points in Figure 2, with the long-run curve adjusted so that it fit through these cycle averages.3 The curve, fitted to six “+”
points, contained three free parameters and implied that very low values of
unemployment would lead to very high inflation, while very high values of
unemployment would lead to very low inflation.4
Thus, the Phillips curve was based on average inflation and unemployment
observations over the course of trade cycles of varying lengths. Although it
was sometimes criticized as capturing short-run relations, Phillips’ procedure
contained significant lower frequency information. Yet, these cycle averages
were drawn from the period when the United Kingdom was on the gold standard so that there were limits to the extent of price inflation or deflation.
Exploration of subsequent periods
After estimating the long-run curve on 1861–1913 data, Phillips then examined
the extent to which the subsequent behavior of wage inflation and unemployment could be understood using the curve.
2 Phillips’ annual wage inflation observations are effectively a two-period average of the inflation rate in the future year and the current year.
3 To explain the cyclical pattern around the long-run curve, Phillips developed a theory in
which wage inflation was affected negatively both by the rate of change and level of unemployment.
That part of his analysis was less broadly taken up by subsequent researchers, although there was
a significant literature on “Phillips loops” during the 1970s.
4 As in Phillips, letting y be the wage inflation and x be unemployment, the fitted curve took
the form y = −.9 + 9.638x −1.394 , with the paremeters selected by a combination of least-squares
and trial-and-error (Phillips 1958, 285).
R. G. King: Phillips Curve Snapshots
317
Figure 3 Trade Cycles and the Phillips Curve
Panel A
10
Curve fitted to 1861--1913 data
72
8
71
6
73
4
70
2
69
74
0
75
79
68
76
-2
77
78
-4
0
1
2
3
4
5
6
7
8
9
10
11
Panel B
Curve fitted to 1861--1913 data
Panel C
11
51
10
Curve fitted to 1861--1913 data
9
8
56
7
55
52
6
57
5
50
54
4
48
53
3
2
49
1
0
0
1
2
3
4
5
Notes: A key part of Phillips’ analysis was to study the behavior of unemployment and
wage inflation over various trade cycles, with three early cycles shown in this figure.
For each, Phillips computed a cycle average, which was then used as one of the central
data points through which he drew the long-run curve shown in Figure 1 (+). Panel A
is Figure 3 from Phillips (1958); Panel B is Figure 9 from Phillips (1958); Panel C is
Figure 10 from Phillips (1958). In all panels, the horizontal axis is unemployment and
the vertical axis is wage inflation, as in Figure 2.
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Looking at 1913–1948 as shown in Panel B of Figure 3, Phillips concluded
that the general approach worked well. In particular, the trade cycle of 1929–
1937 fit his general pattern, but he puzzled somewhat over the relatively high
rates of inflation in 1935–1937. Other parts of the 1913–1948 interval fit in
less well. As potential explanations of the behavior of wage inflation during
both the First World War and the subsequent deflation to return the pound
to its pre-war parity, Phillips discussed the potential importance of cost-ofliving changes (effects of price inflation on wage-setting) as contributing to
wage inflation in the wartime period and to wage deflation during the postWWI interval). The sharp declines in nominal wage rates in 1921 and 1922, as
Britain returned to pre-war parity with gold, stand out dramatically in Panel B
of Figure 3. Although these points lie far from the curve based on 1861–1913
data, they are dramatic outliers that nevertheless show a negative comovement
of inflation and unemployment in line with Phillips’ general ideas.
Phillips also explored the consistency of the period following the Second
World War, 1948–1957, with his long-run curve. During this period, U.K.
unemployment was at a remarkably low level (between 1 and 2 percent) and
inflation varied widely, as shown in Panel C of Figure 3. Phillips commented
on several aspects of this interval. First, he noted that a governmental policy
of wage restraint was in place in 1948 and apparently temporarily retarded
wage adjustments. Second, he noted that the direction of the trade-cycle
“loop” had reversed from the earlier period, which he suggested might be
due to a lag in the wage-setting process. Third, he used this period to show
how his curve could be used to partition wage inflation into a “demand-pull”
component, associated with variation in unemployment along the curve and
other factors, which induced departures from the curve. After looking at retail
price inflation during this period, Phillips suggested that some of the wage
inflation observations, such as the 1948 value that lies well above the curve,
could have arisen from “cost-push” considerations in which workers bargained
aggressively for higher nominal wages. However, Phillips concluded that the
post-WWII period was broadly consistent with the curve fit to the 1861–1913
data.
U.S. Background to a Vast Experiment
In 1960, Paul Samuelson and Robert Solow examined U.S. data on the rate of
change in average hourly earnings in manufacturing and the annual average
data on unemployment over an unspecified sample period, which is most likely
1890 through the late 1950s.5 Their empirical analysis most closely resembles
5 Samuelson and Solow (1960) indicate that their study is based on the data of Rees, which
is most likely his 1961 monograph on real wages in manufacturing, where the earliest data is
1890.
R. G. King: Phillips Curve Snapshots
319
Figure 4 Phillips Curves of Samuelson and Solow
Panel A
32
28
24
Wage Inflation
20
16
12
8
4
0
-4
-8
-12
-14
0
2
4
6
8
10
12
14
16
18
20
22
24
Unemployment
Panel B
11
10
9
8
Price Inflation
7
6
5
B
4
3
2
1
A
0
1
-1
Notes:
Figure
points
Solow
2
3
4
5
6
7
8
9
Unemployment
Panel A shows annual unemployment and wage inflation in U.S. data (this is
1 from Samuelson and Solow [1961]), with their figure notes indicating that circled
are for “recent years.” Panel B shows the trade-off curve that Samuelson and
discussed for the United States (this is Figure 2 in their article).
the 1861–1913 Phillips analysis that we have just looked at, but the overall
association was looser, as dramatically displayed in Panel A of Figure 4.
Like Phillips, Samuelson and Solow looked at sub-samples, noting that
money wages rose or failed to fall during the high unemployment era of 1933
to 1941, which they suggested might be due to the workings of the New
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Deal. Further, they noted that the World War I period also failed to fit into the
expected pattern, in line with Phillips’ findings discussed above.
Overall, though, Samuelson and Solow argued that “the bulk of the observations—the period between the turn of the century and the first war, the
decade between the end of that war and the Great Depression, and the most
recent ten or twelve years—all show a rather consistent pattern. Wage rates
do tend to rise when the labor market is tight, and the tighter the faster.” They
noted with interest that “the relation, such as it is, has shifted upward slightly
but noticeably in the forties and the fifties.” In the early years, before and
after the first war, “manufacturing wages seem to stabilize absolutely when
4 or 5 percent of the labor force is unemployed; and wage increases equal to
the productivity increase of 2 to 3 percent per year is the normal pattern at
about 3 percent unemployment” and described this finding as “not so terribly
different” from Phillips’ results. In the later years, 1946–1959, Samuelson
and Solow judged that it “would take more like 8 percent unemployment to
keep money wages from rising and that they would rise at 2 to 3 percent per
year with 5 or 6 percent of the labor force unemployed.”6 It is these later years
that Samuelson and Solow circled in Panel B of Figure 4.
To describe the policy implications of their findings, Samuelson and Solow
(1961) drew a version of the Phillips curve as representing tradeoffs between
price inflation and unemployment. Essentially, this involved using the idea
that price inflation and wage inflation were different mainly by the growth
of labor productivity, suggesting an implicit model of relatively quick passthrough from wages to prices.
To obtain price stability under the assumption that real wages would grow
at 2.5 percent per year, they suggested that the American economy would have
to experience a 5 to 6 percent rate of unemployment (this option is marked
as point A in Panel B of Figure 4). By contrast, they suggested that “in order
to achieve the nonperfectionist’s goal of 3 percent unemployment, the price
index might have to rise by 4 to 5 percent per year” (this option is marked as
point B).7
Seeking to understand whether inflation originated from cost-push or
demand-pull factors, Samuelson and Solow described a “vast experiment”
in which “by deliberate policy one engineered a sizeable reduction in demand” so as to explore the effects on unemployment and inflation. Although
they were not explicit about the mechanism, they likely shared the prevailing
Keynesian view of the time that fiscal and other policies that cut aggregate
demand would first increase unemployment, with higher unemployment then
reducing wage and price inflation. One interpretation of the subsequent 30
6 All quotations in this paragraph are from Samuelson and Solow (1961, 189).
7 Samuelson and Solow (1961, 192).
R. G. King: Phillips Curve Snapshots
321
years of U.S. history is that versions of such experiments, with both increases
and decreases in demand, were repeatedly undertaken.8
The Samuelson and Solow analysis led to a detailed research program of
estimating the long-run tradeoff between inflation and unemployment in the
United States. Given that interpretation and the subsequent development of
macroeconometric models, it is interesting to note that Samuelson and Solow
(1961) included a foreshadowing of future critiques of the long-run tradeoff:
“aside from the usual warning that these are simply our best guesses, we must
give another caution. All of our discussion has been phrased in short-run
terms, dealing with what might happen in the next few years. It would be
wrong, though, to think that our menu (Figure 4B) that relates obtainable
price and unemployment during the next few years will maintain its shape in
the longer run.” They pointed to two reasons for potential instability—one
was that “wage and other expectations” might shift the position of the Phillips
curve and the other was that “institutional reforms” including product and
labor market regulations or direct wage and price controls might shift the
American Phillips curve downward and to the left.9 Both expectations and
wage-price controls were to play an important role in the subsequent history
of the Phillips curve in the United States and other countries.
Wages, Prices, and Lags
Macroeconomic models along Keynesian lines first aimed at capturing the dynamics of aggregate demand. Thus, for example, the Duesenberry, Eckstein,
and Fromm (1960) simulation study of the U.S. economy in recession used a
quarterly econometric model with 14 equations governing aggregate demand:
it contained neither a monetary sector nor a wage-price block. That is, the
interaction between shocks and the components of aggregate demand was
viewed as first order for understanding the behavior of the U.S. economy in a
recession, with implications for wages and prices or their influences taken as
less important. Fiscal policy measures rather than monetary policy measures
were introduced in many studies of the time, reflecting a professional focus
on fiscal rather than monetary policy tools.
Yet, after these first stages, U.K. and U.S. modelbuilders introduced a block
of equations for wages and prices, stimulated in part by the work of Phillips
(1958). The monograph by Klein et al. (1961) reports on a multiyear project
to construct quarterly U.K. data and to estimate an econometric model with a
8 All quotations in this paragraph are from Samuelson and Solow (1961, 191).
9 All quotations in this paragraph are from Samuelson and Solow (1961) page 193 except
for the final one, which is from page 194.
322
Federal Reserve Bank of Richmond Economic Quarterly
wage-price sector. These authors found that distributed lags were important
in the wage and price equations.10
In contrast to the specifications of Klein et al. (1961), one notable element
of Sargan’s (1964) investigation was that he required that his equations display
homogeneity, so that the absolute levels of wages and prices were not important
for model properties. Sargan, therefore, studied a wage equation of the form
Wt − Wt−1 = λ(Wt−1 − Pt−1 ) + βut−1 + γ t + ξ ft + φ(Pt−1 − Pt−4 ), (1)
where Wt is the log nominal wage rate in quarter t, Pt is the log nominal price
level, ut is the unemployment rate, and ft is a measure of the political party in
power. He divided the analysis of this equation into two components. First,
an equation that described wage changes as deriving from deviations from an
equilibrium real wage,
Wt − Wt−1 = λ[Wt−1 − Pt−1 − wt−1 ],
(2)
and a specification for the equilibrium real wage
β
γ
φ
ξ
wt−1 = [ ut−1 + t + ft + (Pt−1 − Pt−4 )].
(3)
λ
λ
λ
λ
This is simply an algebraic decomposition, but Sargan (1964) was insistent that
the elements of the equilibrium wage process made internal sense, for example
requiring that the coefficient γλ is interpretable as the effect of productivity
growth on the real wage. He also interpreted the parameter λ as a speed of
adjustment toward the equilibrium.11
Following the work of Klein et al., Sargan also estimated a price equation that linked prices to wages. Sargan explored measures of productivity,
demand, and relative input costs as additional determinants of prices. Combining the wage and price equations, Sargan was able to trace out dynamic
consequences of changes in the unemployment rate on wages and prices.
These were influenced by the strength of the equilibrating tendencies (λ) and
the influence of the price terms (φ) from the wage equation. For example,
even if there were no lags of wages in the wage equation, there still could be
indirect effects coming from the presence of price lags.
10 These results echoed the earlier findings of Fisher (1926) who had, in fact, invented the
concept of a distributed lag for the purpose of empirical analysis of inflation and interest rates.
More generally, the estimation of wage-price blocks has provided the basis for many advances in
time series econometrics. In particular, Sargan (1964) used the wage-price block of Klein et al.
(1961) as the basis for an investigation that was the starting point for the so-called London School
of Economics (LSE) approach to econometric dynamics. For a recent study of the UK Phillips
curve, using Sargan’s work as its starting point, see Castle and Hendry (Forthcoming).
11 Sargan also investigated generalization of the first specification to allow for additional lags
e ] + J
of wage changes, Wt − Wt−1 = λ[Wt−1 − Pt−1 − wt−1
j =1 δ j (Wt−j − Wt−j −1 ) to enrich
this dynamic adjustment process toward equilibrium.
R. G. King: Phillips Curve Snapshots
323
Sargan (1964) concluded that there was a long-run tradeoff between wage
inflation and unemployment, but that there were also lengthy average lags so
that changes in unemployment and other variables would take several years to
be fully reflected in wage inflation. These broad properties were widely built
into Keynesian macroeconometric models, as wage and price sectors were
added to the initial aggregate demand constructions.
3. THE BATTLE AGAINST UNEMPLOYMENT
The 1962 Economic Report of the President was the first prepared by the
Kennedy Council of Economic Advisors (CEA), which was eager to implement “The New Economics” originating in the work of Keynes.12 The 1962
Report discussed the origins of unemployment in labor market frictions and
in aggregate demand conditions, concluding that the “objective of maximum
employment” would have to use policies aimed principally at labor market
conditions. The 1962 Report argued that “in the existing economic circumstances, an unemployment rate of about 4 percent is a reasonable and prudent
full employment target for stabilization policy,” further stressing that additional policy interventions to reduce structural unemployment would make it
possible to further reduce that target. At the same time, the Report built the
case that the macroeconomic conditions of the late 1950s and early 1960s had
led to an “output gap” of between 4 and 10 percent. As shown in the top panel
of Figure 5, the output gap was the difference between actual output and a
smooth trend line, based on an assumed level and rate of growth of capacity.
Based on the work of a young economist at the CEA (Okun 1962), unemployment was linked to the output gap, so that a 2-percentage-point higher
unemployment rate was related to an output gap of 5 percent. That is, the
Report built in an Okun’s Law coefficient of 2.5 to produce the second panel
of Figure 5. While the “Phillips curve tradeoff” is now frequently discussed in
terms of inflation and the output gap using some version of Okun’s Law, this
article will maintain the original linkage between inflation and unemployment
as its focus.
In fact, the Kennedy-Johnson administration did deliver a substantial decline in unemployment, as a look back at Figure 1 confirms. In keeping with
the tenor of the times, in which a package of fiscal, structural, and monetary
policies was viewed as necessary for and capable of producing this decline, the
present article will not seek to separately identify the contributions of different
types of policies. Histories of the period, such as Hetzel (2008, chapters 6
and 7), stress the coordination of fiscal and monetary decisionmaking, so that
such an identification could be quite subtle.
12 Walter Heller was the chairman of the Council of Economic Advisors from 1961–64, with
the other members being Kermit Gordon and James Tobin. The terminology “new economics” was
widely used at the time and apparently dates back to a 1947 volume by Seymour Harris.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 5 Unemployment and Output Gap in the 1961 CEA Report
Billions of Dollars (Ratio Scale)
a
Panel A: Gross National Product in 1961 Prices
600
550
Potential
500
b
Gap
Actual
450
400
1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963
Panel B: Unemployment and the Output Gap
10
8
5
6
0
4
-5
2
1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963
Percent
Percent
GNP Gap as Percent of Potential (Left Scale)
c
Unemployment Rate (Right Scale)
0
Notes: Panel A is the CEA’s potential, actual, and output gap decomposition, while
Panel B shows the link between the output gap and the unemployment rate. The CEA
economists attached notes as follows: a Seasonally adjusted annual rates; b potential output based on a 3 21 percent trend line through the middle of 1955; c unemployment as a
percent of civilian labor force, seasonally adjusted; A, B, and C represent GNP in the
middle of 1963, assuming an unemployment rate of 4 percent, 5 percent, and 6 percent, respectively. They listed their sources as: Department of Commerce, Department
of Labor, and Council of Economic Advisors.
R. G. King: Phillips Curve Snapshots
325
The 1962 Report did note that “the economy last experienced 4 percent
unemployment in the period May 1955–August 1957. . . . During this period,
wages and prices rose at rates which impaired the competitiveness of some
U.S. goods in world markets. However, there is good reason to conclude
that upward pressures of this magnitude are not a permanent and systematic
feature of our economy when it is operating in the neighborhood of 4 percent
unemployment.” Looking back at Figure 1, the reader will notice that the
inflation rate rose by several percentage points during the 1955–1957 period
alluded to in the CEA report, while unemployment averaged about 4 percent.
By the late 1960s, some version of the long-run Phillips curve tradeoff
had become a cornerstone of economic policy. It entered centrally in macroeconomic models and more ephemerally in macroeconomic reports.13
Macroeconomic Models
In fall 1970, the Federal Reserve System sponsored a major conference on
“The Econometrics of Price Determination,” which contained a wide range
of studies and later appeared in 1972 as a volume edited by Otto Eckstein.
Drawn from one of these studies, Figure 6 displays the long-run relationship
between price inflation and unemployment as of 1969 within three prominent
macroeconomic models of the sort used by the U.S. private sector for forecasting purposes, by the executive branch of the U.S. government, and by the
U.S. central bank. This figure is reproduced from the Hymans (1972) survey
of the price dynamics within the Office of Business Economics (OBE) model
used by the executive branch, the Federal Reserve-MIT-Penn (FMP) model
used by the central bank, and the DHL-III model developed by Hymans and
Shapiro at the University of Michigan.
As Hymans explains (1972, 313), this figure was produced by taking
the wage-price block of the various models and evaluating these equations
at alternative unemployment rates. One first finds the long-run inflation rate
when the unemployment rate is constant at, say 5 percent, and then one finds
the long-run inflation rate at 4 percent and so on.
By and large, these estimates of the long-run relationship accord well
with that portrayed by Samuelson and Solow (1961) and reproduced as Panel
B of Figure 4 in this article. Further, the increase in the inflation rate from
about 1 percent in 1960–61 to about 4.5 percent in 1968–69 is particularly
well captured by the FMP and DHL models. Although each long-run Phillips
curve is nonlinear and there are differences in models, the “average” tradeoff
13 As, for example, in the “Phillips plot without a Phillips curve” of the 1969 Report, discussed further below. Presumably, the economists at the CEA were not too interested in taking
“credit” for the effect of low unemployment on inflation, while the economists at the Federal
Reserve Bank (FRB) had a model that featured the tradeoff and could not escape the connection.
326
Federal Reserve Bank of Richmond Economic Quarterly
Annual Rate of Price Change (Percent)
Figure 6 The Long-Run Tradeoff in Major Macroeconomic Models
Global Unemployment Rate (Percent)
Notes: The long-run inflation and unemployment tradeoff in three prominent macroeconometric models as of 1970. The figure also shows actual unemployment and inflation
(annual averages) during 1960–1969.
Source: Hymans (1972).
over this range is that lowering unemployment by 2.5 percent (from 6 to 3.5)
costs about 3.5 percent in terms of inflation (from 1 percent to 4.5 percent).
Smaller changes in inflation and unemployment feature a roughly one-for-one
tradeoff.
The dynamics of wages and prices were studied by many authors under a
variety of assumptions within the FMP and other large models. For example,
R. G. King: Phillips Curve Snapshots
327
de Menil and Enzler (1972) considered the effect of changing the unemployment rate from 4 percent to 5 percent in the FMP model: The results of their
investigations are shown in Figure 7. The economy is assumed to be in an
initial steady state with price inflation of 3.4 percent per year (wage inflation
is just over 6 percent) and unemployment of 4 percent. Then, unemployment
increased to 5 percent at date 1 and at all future dates. The inflation rate declines to 2.7 percent after four quarters, to 2.2 percent after three years and to
1.9 percent after five years. Compensation per man-hour (the wage measure)
drops from 6 percent to 4.9 percent after a year, to 4.6 percent after three
years, and to 4.4 percent after five years. The more rapid response of wage
inflation is related to the fact that unemployment affects wages immediately,
with effects of wages on prices occurring only with a distributed lag.14
Overall, the short-run Phillips curve reported by de Menil and Enzler
(1972) is flatter than the long-run one that they report (essentially that for the
FMP in Figure 6): a 1 percent increase in unemployment brings about a .7
percent change after a year’s time, but a 1.5 percent decline in inflation after five
year’s time. From the standpoint of the econometric modelers of the time, this
was a natural result of the lags in the wage-price components of their model,
built in along Klein-Sargan lines. Looking at Figure 6, economists such as
de Menil and Enzler likely saw a consistency with the dynamic specification
of the macroeconomic policy model: the historical inflation rate initially lies
below the long-run Phillips curve in the early 1960s during the start of the
transition to lower unemployment.
Macroeconomic Reports
The 1969 Economic Report of the President was the last report of the KennedyJohnson era and was prepared under the leadership of Arthur Okun. With
inflation rising, early 1968 saw a new cabinet-level Committee on Price Stability charged to recommend actions to contain inflation. President Johnson’s
introductory remarks in the Report distinguished between “roads to avoid” and
“roads to reducing inflation.” The roads to avoid were an “overdose of fiscal
and monetary restraint” or “mandatory wage and price controls.” The “roads
to follow” included a combined fiscal and monetary program—including a
continuation of the 1968 tax surcharge—as a “first line of defense,” but also
14 Note that de Menil and Enzler’s (1972) experiment assumes an immediate and permanent
change in unemployment induced by macroeconomic policies. Given the nature of the wage-price
block in the FMP model, it was conceptually feasible to simply change unemployment and trace
out the implications for wage and price inflation. However, since changes in fiscal and monetary
instruments had only a gradual effect on aggregate demand and unemployment within the FMP
and similar models, responses to more standard policy changes were more complicated and had
gradual effects on both inflation and unemployment.
328
Federal Reserve Bank of Richmond Economic Quarterly
Figure 7 The Dynamic Tradeoff in the FMP Model circa 1970
6
5
Compensation per Manhour
4
3
Nonfarm Price Deflator
2
0
1
3 5
7
9 11 13 15 17 19 21 23 25 27 29
Notes: The vertical axis measures annualized percentage rates of change of the wage and
price series. The horizontal axis is in quarters of a year.
Source: de Menil and Enzler (1972).
voluntary cooperation in wage and price setting to aid the process of reducing
inflation.
The 1969 Report portrayed the U.S. economy as running at or slightly
above potential output, with associated unemployment in the neighborhood
of 4 percent. As portrayed in Figure 8, the potential output series resembles
that presented in the 1962 Report (Figure 5), but the detailed notes make
clear that potential output grew at 3.5 percent over 1955–1962; at 3.75 percent
over 1963–1965; and at 4 percent over 1966–1968. Thus, an acceleration of
potential output growth was necessary to fit together the 1962 Report’s view of
4 percent as the unemployment target, which finally was hit in the latter years
of the Kennedy-Johnson era, with the behavior of output during those years.
With more modest potential output, there would have been a very negative
output gap during the final years of the Kennedy-Johnson era, whereas it is
only slightly negative in Figure 8.
The 1969 Report also featured a Phillips scatter plot, from 1954–1968,
highlighting the historical relationship between “price performance and
R. G. King: Phillips Curve Snapshots
329
Figure 8 Unemployment and Output Gap in the 1961 CEA Report
Billions of Dollars (Ratio Scale)
Panel A: Gross National Product in 1958 Prices
b
a
Potential
Gap
Actual
Panel B: Unemployment and the Output Gap
Percent
Percent
GNP Gap as Percent of Potential (Left Scale)
Unemployment Ratec (Right Scale)
Notes: Panel A shows the CEA’s potential, actual, and output gap decomposition, while
Panel B shows the link between the output gap and the unemployment rate. The CEA
economists attached notes as follows: a Seasonally adjusted annual rates;b potential output
is a trend line of 3 21 percent through the middle of 1955 to 1962 IV, 3 41 percent from
1962 IV to 1965, and 4 percent from 1965 IV to 1968 IV; c unemployment as a percent of
civilian labor force, seasonally adjusted. They list as sources: Department of Commerce,
Department of Labor, and Council of Economic Advisors.
unemployment” during the Kennedy-Johnson years, although no trade-off
curve was displayed. The decline in unemployment from the 5.5 percent level
of 1963 to the 3.5 percent level of 1968 was associated with a rise in inflation
from 1.5 percent to close to 4 percent. The Report’s accompanying discussion
330
Federal Reserve Bank of Richmond Economic Quarterly
of the historical record stresses the importance of a wage-price spiral arising
from demand growth in excess of capacity growth. It argued that “once such
a spiral starts, it becomes increasingly difficult to arrest, even after productive
capacity has caught up with demand and the initial pressures have largely
subsided.”
Thus, the Report’s explicit stand is that the U.S. economy in 1969 was operating close to capacity, with a rate of unemployment that was not necessarily
inflationary, a viewpoint that echoes the appraisal in the 1962 Report. The
rise in inflation over the eight years of the Kennedy-Johnson administration
is, therefore, implicitly portrayed as arising from the effects of a wage and
price spiral, not a purposeful movement along a long-run Phillips curve.
4.
BREAKDOWN, 1969–1979
The breakdown of the consensus concerning the long-run Phillips curve
involved a major revision of macroeconomic theory along with an unusual
pattern of inflation and unemployment, with these intertwined developments
reinforcing each other.
Macroeconomic Theory
As the Phillips curve played an increasing role in macroeconomic models,
and as inflation rose during the mid-1960s, economists began to take a harder
look at its theoretical underpinnings. The implications of new models were
then compared to unemployment and inflation data, with results that sparked
a major empirical controversy and a revolution in macroeconomic modeling.
The natural rate hypothesis
In the late 1960s, Milton Friedman and Edmund Phelps made separate arguments about why the long-run Phillips curve should be vertical. Friedman
(1968) began from a vision of the labor market in which real wages and employment (or unemployment) were jointly determined in response to local and
aggregate conditions of supply and demand. His natural rate of unemployment was “the level that would be ground out by the Walrasian system of
general equilibrium equations, provided there is imbedded in them the actual
structural characteristics of the labor and commodity markets, including market imperfections, stochastic variability in demands and supplies, the cost of
gathering information about job vacancies and labor availabilities, the costs
of mobility, and so on.” He then imagined a situation in which firms offered
workers nominal rather than real wages, with workers evaluating labor supply
opportunities based on their best estimate of the purchasing power of those
wages. With a known path for the price level, this calculation is easy for
R. G. King: Phillips Curve Snapshots
331
workers, but it becomes harder when the price level is changing. Friedman
(1968) imagined the central bank increasing the growth of the money supply
and stimulating the demand for the final product. To hire additional workers, firms would offer higher nominal wages. Faced with higher nominal
wages that were interpreted as higher real wages, workers would supply more
hours and potential workers would accept more jobs. So, it was possible for
Friedman’s model to reproduce a Phillips curve of sorts. However, if workers
correctly understood that the general level of prices was increasing as a result
of a monetary expansion, there would be no real effects: The rate of inflation
and the rate of wage growth would jointly neutralize the effect of a higher rate
of monetary growth, leaving real activity unaffected.
Phelps (1967) analyzed the problem in more Keynesian terms, based on
a specification of a price equation of the following type:15
Pt − Pt−1 = π t = β(ut − u∗ ) + π et ,
(4)
where u∗ is the “natural rate of unemployment,” in Friedman’s terminology, and π e is the expected rate of inflation. Phelps argued for this sort
of “expectations-augmented Phillips curve” specification on grounds similar
to those of Friedman that we have already discussed: labor suppliers should
make their decisions on real, not nominal grounds.
Further, Phelps studied this inflation equation under the assumption of
adaptive expectations,
π et = θ π t−1 + (1 − θ )π et−1 ,
(5)
where 0 < θ < 1 governs the weight placed on recent information in forming
expectations. This specification implies that if inflation were maintained at
any constant level, π , then expected inflation would ultimately catch up to it
since the sum of coefficients in the distributed lag representation of expected
inflation,
π et = θ
∞
(1 − θ )j π t−j −1 ,
j =0
is equal to one.
The “expectations-augmented Phillips curve” means that unemployment
would be low (u < u∗ ) only if agents are surprised. Thus, a policy of maintaining low unemployment requires consistent underforecasting of inflation,
π < π e . But low unemployment could only be brought about by raising
15 There are several cosmetic differences with Phelps’ (1967, equation 3) specification. First,
Phelps worked in continuous time while the text equation is in discrete time. Second, Phelps’
specification is in terms of a general utilization variable rather than unemployment. Third, Phelps
worked with the expected return on money, which is the negative of the expected inflation rate.
Fourth, Phelps employed a nonlinear (convex) specification of the link from utilization to inflation
rather than a linear one as in the text.
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Federal Reserve Bank of Richmond Economic Quarterly
the inflation rate to a higher and higher level, with expectations always lagging behind because of the adaptive mechanism (5). Hence, the view of
Friedman and Phelps became known as the “accelerationist hypothesis” in
some quarters.
Phelps also stressed that the accelerationist model meant that a temporary period of low unemployment would bring about a permanently higher
rate of inflation. He recognized that this led to an important new dynamic
element in policy design, relative to traditional work that had stressed the
Phillips curve as a stable “menu of policy choice” for the long run. Generally, a period of temporarily high unemployment would be necessary to
permanently reduce inflation and some basic economic mechanisms—such as
a momentary social planner objective that attributes increasing cost to high
unemployment—making it desirable to smooth the adjustment process.
Overall, the natural rate/accelerationist hypothesis moved the worries of
Samuelson and Solow (1961) about the effects of expectations on the Phillips
curve from second-order to first-order status.
Tests of Solow and Gordon
From the perspective of wage and price adjustment equations of the form
prominent in Keynesian theoretical and empirical models, the arguments of
Friedman and Phelps suggested that there were important omitted expectational terms. Solow (1969) and Gordon (1970) devised tests to determine
whether there was a long-run tradeoff between inflation and unemployment.
To capture the spirit of these tests, consider a wage equation along the KleinSargan lines, such as (1) above. As above, the nominal wage at a given date
will be Wt and the real wage will be wt = Wt − Pt . However, the inflation terms in (1) are replaced by expected inflation, π et , with a coefficient α
attached,
∗
) + β(ut − u∗t ) + απ et ;
(6)
Wt = λ(wt−1 − wt−1
other time-varying terms will be omitted for simplicity.16
The tests of Solow and Gordon made diverse use of price and wage equations, but the essential features can be simply described using this expression. First, in the wage equation above, the Friedman-Phelps conclusion
obtains if α = 1 since this is simply a restriction on expected real wages and
unemployment,
∗
wt − wt−1 = λ(wt−1 − wt−1
) + β(ut − u∗t ) − (π t − π et ).
(7)
There is, thus, no influence of inflation if expectations are correct and no
tradeoff between real and nominal variables. Solow and Gordon proposed
16 In this expression, w ∗ would be the “natural” real wage, similar to the natural rate of
unemployment, u∗ .
R. G. King: Phillips Curve Snapshots
333
to directly estimate the parameter α and to evaluate the accelerationist view
by testing whether α differed significantly from unity. Second, this test is
challenging to implement because expectations are unobservable. However,
if expectations are formed adaptively, as in (5), then it is possible to conduct
the test. Solow (1969) estimated parameters such as α for a range of different
values of θ , while Gordon (1970) used a more general distributed lag but maintained the requirement that the coefficients summed to unity. This sum of the
coefficients restriction was rationalized by the Phelpsian thought experiment,
comparing a zero inflation steady state to a positive inflation steady state at
rate π : If the sum of coefficients is one, then expected inflation is π e = 0 in
the first case and π e = π in the second case.
All of their diverse estimates suggested values of α that were positive, but
significantly less than one. Thus, an aggregate demand policy that lowered
unemployment in a sustained manner would create rising inflation over time,
as expectations increased, but it would not ultimately be unsustainable.
Rational expectations critique
Sargent (1971) and Lucas (1972a) criticized the tests of Solow and Gordon by
invoking two arguments. First, Sargent and Lucas insisted that expectations
formation should be rational along the lines of Muth (1961). Second, they
constructed example economies in which the accelerationist position was exactly correct, but in which an econometrician using the methods of Solow and
Gordon would reach an incorrect conclusion.
A valuable example arises when inflation (π ) has a persistent (x) and
temporary (η) component, so that it is generated according to
πt
xt
= xt + η t
= ρxt−1 + et ,
where ηt and et are serially uncorrelated, zero mean random variables and
|ρ| < 1.
As analysis along the lines of Muth (1960) determines, rational expectations then are formed according to
Et−1 π t = ρ[θ π t−1 + (1 − θ )Et−2 π t−1 ] = ρθ
∞
(1 − θ )j π t−j −1 ,
(8)
j =0
with 0 < θ < 1. This is broadly the same form as the adaptive expectations
formula above, except that the distributed lag now is multiplied by ρ, which
captures the degree of persistence of inflation.17
17 The inflation process implies π = ρπ
t
t−1 + et + ηt − ρηt−1 . A “Wold representation”
is π t = ρπ t−1 + at − ρ(1 − θ )at−1 , with at a forecast error for π t relative to its own past
history. These two processes are observationally equivalent if they imply the same restrictions on
334
Federal Reserve Bank of Richmond Economic Quarterly
In particular, suppose that inflation is somewhat persistent but stationary
so that 0 < ρ < 1, and that there is no effect of inflation on the real wage,
i.e., α = 1. Then, an econometrician constructing a measure of expectations
∞
(1−θ )j π t−j −1 for any θ < 1 = ρπ t−1 would face data generated
π t−1 = θ
j =0
according to
∗
Wt − Wt−1 = γ (wt−1 − wt−1
) + β(ut − u∗t ) + ρ
π t−1 ,
(9)
and would conclude that α = ρ. That is, the econometrician would estimate
that there was a long-run tradeoff, α < 1, even though none, in fact, existed.
The critique played an important role in the evolution of macroeconomic
modeling. Lucas (1972b) built a small-scale general equilibrium macroeconomic model with a short-run Phillips curve arising without any long-run
tradeoff, providing an analytical interpretation of Friedman’s (1968) suggestion about the nature of the link between inflation and real activity. Lucas
(1976) expanded the critique of the policy invariance of parameters within
1970s macroeconometric models into a general challenge, noting that similar
difficulties were contained in the consumption function and the investment
function, as well as the wage-price block. He stressed that rational expectations and dynamic optimization, which seemed useful as basic postulates
for model construction, inevitably led to such problems. A major revolution
in econometric model construction ensued, leading central banks around the
world to develop new models for forecasting and policy analysis during the
1990s, as we discuss in Section 6.
Shifting unemployment-inflation tradeoffs
What is wrong with the sum of coefficients restriction employed by Solow
and Gordon? Sargent (1971) notes that inflation through the mid-1960s did
not display much serial correlation, so it is not well forecasted by a moving
average with weights that sum to unity. Intuitively, in an economy in which
there are never any permanent changes in inflation, rational expectations are
not designed to guard against this possibility.
Yet, as inflation rose and stayed high through the 1970s, empirical modelbuilders found estimates of parameter values drifting toward one and were
the variance and first-order autocorrelation of π t − ρπ t−1 . (All other autocovariances are zero.)
Using the Wold representation, the forecast is
Et−1 π t
=
=
ρπ t−1 − ρ(1 − θ )at−1 = ρπ t−1 − ρ(1 − θ )(π t−1 − Et−2 π t )
ρ[θ π t−1 + (1 − θ )Et−2 π t−1 ]
as reported in the text. The covariance restrictions imply var(at ) = var(ηt ) + 1 2 var(et ) and
1+ρ
var(η )
θ = 1 − var(at ) .
t
R. G. King: Phillips Curve Snapshots
335
more open to accelerationist models of the Phillips curve. I have searched
for, but have not found, a 1970s study that succinctly shows this drift in
coefficients. McCallum (1994) calculates the sum of coefficients implied by
a fifth-order autoregression for inflation estimated over various periods: it is
about one-third in 1966–67, two-thirds in 1968–70, and between .88 and 1.02
for 1973–1980. McCallum’s exercise indicates why econometric modelers
found increasing evidence for the accelerationist hypothesis as the evidence
from the 1970s was added.
Up a derivative: NAIRU models
As inflation rose during the mid-1970s, the wage and price blocks of standard
macroeconometric models were augmented in various ways. One route was to
include expectations terms explicitly but to make these expectations respond
sluggishly to macroeconomic conditions. Another, arguably initially more
popular strategy originates in the work of Modigliani and Papademos (1975).
These authors argued that empirical models of price and wage dynamics would
have better success if they related changes in inflation measures to levels
of real variables. Modigliani and Papademos viewed inflation as likely to
accelerate if unemployment was low relative to a benchmark value and as
likely to decelerate if unemployment was high relative to a benchmark. They
argued that this feature, as indicated by points A and C in Figure 9, was far
more important for policy analysis than the question of whether the longrun Phillips curve had a negative slope (as in P P´) or was vertical (as in
F F´).18 Further, they stressed that “the shading of an area on either side of
NAIRU indicates both uncertainty about the exact location of NAIRU and the
implausibility that any single unemployment rate separates accelerating and
slowing inflation.”
The NAIRU model has been commonly captured by a simple model of
the form,
π t = π t−1 + β(ut − un ),
(10)
a specification closely related to that of Phelps (equation [4]), but with past
inflation replacing expected inflation. Moving up a derivative allowed for a
continuation of empirical research on price and wage inflation, which investigated the consequences adding lags of inflation and unemployment as well
as adding shift variables to the basic NAIRU model. Thus, such empirical
investigations built in a particular assumption on the “accelerationist hypothesis” that differed from the earlier research following Phillips, but continued
to study many similar questions. However, uncertainty about the location of
18 In the original unnumbered figure early in the Modigliani and Papademos (1975) article,
the non-accelerating inflation rate of unemployment is marked “NIRU,” but Figure 9 follows the
now-standard acronym. It also corrects a typo in the labelling of the vertical axis.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 9 The NAIRU Substitute for the Original Phillips Curve
F
Change in Inflation Rate
P P'
A
C
2.0
B
NAIRU
F
P1
1
Unemployment
Notes: Modigliani and Papademos (1975) posited a relationship between unemployment
and the change in the inflation rate. They stressed that there was a range of unemployment rates (gray area) over which the effect on inflation was quite uncertain. In the
longer run, they did not take a firm stand on whether there was a tradeoff between the
level of unemployment and the acceleration of inflation (the curve P P´) or not (the curve
F F´).
the NAIRU, as Modigliani and Papademos had suggested, led to challenges
in the application of this approach in forecasting and policy analysis.19
Macroeconomic Policy
The Nixon administration came into office in 1969 with the aim of reducing
inflation via a combination of orthodox fiscal and monetary methods, but
sought to do so without prolonging the recession that the country was then
experiencing.
19 Staiger, Stock, and Watson (1997) document the considerable uncertainty surrounding estimates of the NAIRU.
R. G. King: Phillips Curve Snapshots
337
Gradualism
The Nixon administration initially embarked on a course of “policy gradualism” in an attempt to reduce inflation while maintaining real activity and
unemployment at relatively constant levels. In a mid-course appraisal, Poole
(1970) provides a useful definition of such policies: “The prescription of gradualism involves the maintenance of firm but mild restraint until the objectives
of anti-inflationary policy are realized. Real output is to be maintained somewhat below potential until the rate of inflation declines to an acceptable level.”
The policies outlined in the 1970 Economic Report of the President, produced
under the leadership of CEA chairman Paul McCracken, contained both fiscal
and monetary components designed to generate a modest reduction in output
for the purpose of reducing inflation.20
Rising understanding of the importance of expectations
Arthur Burns became head of the Federal Reserve System in early 1970,
replacing William McChesney Martin, who had served since 1951. During
1969, Martin had undertaken restrictive monetary policy to reduce inflation,
indicating that “expectations of inflation are deeply imbedded. . . . A slowing
in expansion that is widely expected to be temporary is not likely to be enough
to eradicate such expectations” and “a credibility gap has developed over our
capacity and willingness to maintain restraint.”21 The phrase “credibility gap”
had a particularly harsh ring to it, even as part of a self-criticism, as it had been
widely used to describe the Vietnam policies of the Johnson administration:
The Tet offensive of September 1968 had convinced many that there was
no credibility to the administration’s previous upbeat forecasts for military
success or its description of the offensive as a disastrous defeat for the Viet
Cong. Martin believed that unemployment was unsustainably low and that
economic growth would need to slow to reduce inflation, which was at about
5 percent during late 1968 when Richard Nixon was elected president. But
the restrictive monetary policy of 1969–1970 started by Martin, envisioned as
part of a gradualist strategy by McCracken, and continued by Burns, resulted
only in a slowing of inflation, but not a major decline, during the recession of
December 1969–November 1970.
As Hetzel (1998) stresses, Burns had a long-standing belief that expectations were important for inflation, writing in the late 1950s that: “One of the
main factors in the inflation that we have had since the end of World War II is
20 Gradualist policies were advocated by a range of economists and policymakers. On this
dimension, the monetarist economists of the Shadow Open Market Committee (SOMC) shared some
of the reservations of their Keynesian counterparts. See Meltzer (1980) for a discussion of the
SOMC perspective on gradualism.
21 Martin’s February 26, 1969, testimony to Congress, quoted in Hetzel (2008, 75).
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Federal Reserve Bank of Richmond Economic Quarterly
that many consumers, businessmen, and trade union leaders expected prices
to rise and therefore acted in ways that helped to bring about this result.”22
After taking over the Fed, he made the case that it was possible to side-step
the Phillips curve through the imposition of wage and price controls, which
he believed would exert a substantial effect on expectations.23
Incomes policies
By the middle of 1971, the Nixon administration lost patience with gradualism,
imposing a 90-day temporary freeze on prices on August 15, 1971, and ending
the convertibility of the dollar into gold. The temporary freeze evolved into
a multiyear incomes policy with various phases differing in intensity and
coverage. The 1973 Economic Report of the President noted that “1972 was
the first full year in American history that comprehensive wage and price
controls were in effect when the economy was not dominated by war or its
immediate aftermath” (page 51). The Report indicated that there had been
three purposes for the controls (page 53). First, the controls were intended
to directly affect the rate of inflation, lower the probability of its increase,
and raise the probability of its decline. Second, the controls were aimed at
“reducing the fear that the rate of inflation would rise or not decline further.”
Third, the controls were designed to “strengthen the forces for expansion
in the private economy and to free the Government to use a more expansive
policy.” In describing the conditions in the spring and summer of 1971 that led
to the imposition of the controls, the Report specifically discussed “anxiety”
about increasing inflation as holding back consumer spending and rising longterm interest rates that “may have signalled rising inflationary expectations.”
Overall, a key motivation for the controls was to affect expectations of inflation
and their incorporation into price and wage setting.
Many analysts see the incomes policy period as involving expansionary
monetary policy—as in the third Report point above—with fiscal policy under
the Nixon team and monetary policy under Burns producing an economic
expansion. Unemployment hovered in the 6 percent range through 1971,
dropping to 5 percent by the end of 1972 as Nixon won a landslide re-election.
Inflation, according to the gross domestic product deflator shown in Figure 1,
had fallen from 5 percent in 1971 to 4 percent in 1972, but it then rose to a 7
percent annual rate by the end of 1973.
22 Burns 1957, p. 71, quoted in Hetzel (1998).
23 Burns had played a leading role in the Council of Economic Advisors during the
Eisenhower administration and knew Nixon well. As chairman of the Fed, he played an important role in administration policy more broadly, including the mid-August 1971 meetings at
Camp David that formulated major changes in Nixon administration economic policies.
R. G. King: Phillips Curve Snapshots
339
Skepticism about government goals
By the time of the publication of the 1973 Economic Report of the
President, many economists were becoming more skeptical about both the
long-run Phillips curve and, more specifically, about whether government
plans were consistent with the available information on the historical linkage between inflation and unemployment. The review of the 1973 Report by
the mainstream economist Carl Christ provided one clear presentation of this
skepticism. He noted that every economic report contains an overall statement
of objectives by the President followed by a more detailed and nuanced report
by his economic advisors.
An initial quote from Christ’s review summarized the condition of the
previous year for us: “Mr. Nixon’s report begins with a review of the good
things about 1972: a 7 21 percent rise in real output, a reduction of the inflation
rate (measured by the consumer price index) to about 3 percent from about 6
percent in 1969, and a reduction of the unemployment rate to 5.1 percent in
December 1972 from 6 percent in December 1971.”
The 1973 goals of the Nixon administration were summarized by Christ
using a series of quotations from Nixon’s message: “Output and incomes
should expand. Both the unemployment rate and the rate of inflation should
be reduced further, and realistic confidence must be created that neither need
rise again. The prospects for achieving these goals in 1973 are bright—if we
behave with reasonable prudence and foresight” (p. 4); “We must prepare for
the end of wage and price controls. . . ” (p. 6). Christ notes that this buoyant
optimism on inflation and unemployment is “reminiscent of Mr. Nixon’s
statement in January 1969, shortly after taking office, that he would reduce
inflation without increasing unemployment and without imposing wage and
price controls.”
Christ then proceeded to argue that the optimism was unwarranted: “The
evidence strongly suggests it is not possible for the American economy,
structured as it has been since World War II, to achieve simultaneously unemployment rates that remain at 4.75 percent or less, and consumer price
increases that remain at 2.4 percent a year or less, without wage or price controls. In the 25 years since consumer prices leveled off at the end of World
War II, this has been achieved in only 4 years: 1952, 1953, 1955, and 1965. . . .
In those same 25 years, the average unemployment rate was 4.8 percent, and
the average increase in consumer prices was 2.4 percent a year.”
Figure 10 was produced by Christ using data from the 1972 Report. Notice that Christ’s argument is not that the long-run Phillips curve is vertical,
although that view is not inconsistent with his figure. Instead, it is that public
policy goals ought to be consistent with the available evidence and that unemployment far below the 1972 level of 5 percent and inflation below the 1972
level of 3 percent did not seem consistent with U.S. experience.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 10 Christ’s Summary of U.S. Inflation and Unemployment
Experience
69
6%
51
50
70
5%
68
4%
66
Inflation
56
2%
65
1%
59 63
60
58
64 62
62
53
71
57
67
48
3%
61
55
0
1%
2%
3%
4%
6%
7%
54
-1%
49
-2%
Unemployment
Notes: Using data from the 1973 Annual Report, Christ showed that the announced administration objectives of inflation below 3 percent and unemployment below 5 percent
did not seem consistent with prior U.S. experience.
Christ’s warning was a timely one: The remainder of the Nixon-Ford
administration saw the onset of stagflation, as a look back at Figure 1 reminds
us. The United States was not to see 5 percent unemployment and 3 percent
inflation at any time in the next two decades.
Humphrey-Hawkins
While Christ may have been skeptical about the internal consistency of the
Economic Report of the President in 1973, the debates over the legislation
put forward by Representative Augustus Hawkins and Senator Hubert H.
Humphrey five years later illustrated that other elements of government and
society continued to seek very low unemployment.
As initially passed by the House in March 1978, the Full Employment and
Balanced GrowthAct specified the goal of lowering the unemployment rate to 4
R. G. King: Phillips Curve Snapshots
341
percent for all working age individuals and 3 percent for all individuals over the
age of 20. Early drafts of the House bill had mandated that the government be
an “employer of last resort” for the long-term unemployed, but this provision
was dropped, while the focus on unemployment was maintained. The national
unemployment goal was to be reached within five years and the bill called for
cooperation between the executive branch, Congress, and the Federal Reserve
Board in working toward the specified target. It also specified that the President
should submit an annual economic report to Congress including numerical
goals for employment, unemployment, and inflation, as well as some other
macroeconomic indicators. Amendments to add budget balance as a goal at
the five-year horizon and to include an inflation goal of 3 percent at that time
were defeated. The bill evolved substantially in order to gain Senate approval.
In the process, inflation objectives were reinstated. It was signed into law by
President Jimmy Carter on October 27, 1978.
The Full Employment and Balanced Growth Act in final form established
national goals of full employment, growth in production, price stability, and
balance of trade and public sector budgets. More specifically, it specified that
by 1983, unemployment rates should be no more than 3 percent for persons
aged 20 or over and no more than 4 percent for persons aged 16 or over.
Inflation should be no more than 4 percent by 1983 and 0 by 1988. Thus, in
its nonbinding goals, it displays the same tendencies that Christ identified in
the Economic Report of the President.
While these goals were nonbinding, the Humphrey-Hawkins Act did require that the Federal Reserve Board of Governors transmit a report to Congress
twice a year outlining its monetary policy.
5.
UNWINDING INFLATION
By the late 1970s, a wide range of economists and politicians were becoming
concerned about high inflation and recommending disinflation. However,
economists and politicians differed widely on the costs of reducing inflation.
Forecasting the Costs of Disinflation
Surveying six estimates of “macroeconomic Phillips curves,” Okun (1978)
found that the experience of the 1970s had led to the abandonment of the
long-run Phillips curve. Yet, he also stressed that “while they are all essentially accelerationist, implying no long-run tradeoff between inflation and
unemployment, they all point to a very costly short-run tradeoff.” Thinking
about the Phelpsian question of how fast unemployment should be raised from
a situation of initially low capacity utilization, in terms of the consequences for
long-run inflation, he calculated “for an extra percentage point of unemployment maintained for a year, the estimated reduction in the ultimate inflation
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Federal Reserve Bank of Richmond Economic Quarterly
rate.” Comparing the various studies, he found that this disinflation gain for
a given amount of unemployment ranged between 16 and 21 of a percent, with
an average estimate of 0.3 percent.
To put Okun’s numbers in a specific context, consider the unemployment
cost attached to a 5 percent reduction in long-run inflation. The estimates
reported by Okun meant that this cost would be between 10 = [5/(1/2)] and
30 = [5/(1/6)] “point years of unemployment,” with an average estimate of
16.7[5/.3]. That is, if the cost of eliminating a 5 percent inflation was spread
evenly over four years, then each year would see an unemployment rate that
was between 2.5 and 7.5 percent above the natural rate with a mean estimate
of over 4 percent.
It is now more standard to discuss disinflation costs as a ratio of point
years of unemployment arising from a one percent change in inflation, which
is called the “sacrifice ratio.”24 Okun’s estimates were that the sacrifice ratio
was in the range of 2 to 6, with a mean of 3.3 in that each percentage point
reduction in inflation would involve very major economic costs.
Put another way, by the late 1970s, policymakers may have abandoned
the long-run Phillips curve in the face of evidence and theory. But most major
econometric models continued to maintain a tradeoff over horizons of four or
more years, as originally described by Samuelson and Solow. Just as there
had been a protracted period of low unemployment as inflation had risen, so
too did Okun envision a protracted period of high unemployment as inflation
was reduced.
The perceived severity of a potential reduction in inflation is perhaps best
illustrated in an excerpt from James Tobin’s (1980) review of stabilization
policies at the close of the first decade of the Brooking’s Panel. To put the
excerpt in context, Tobin’s review described the accelerationist hypothesis as
having been a core part of macroeconomics for the better part of the previous
decade. Tobin wrote that it was broadly recognized that “inflation accelerates
at high employment rates because tight markets systematically and repeatedly
generate wage and price increases in addition to those already incorporated in
expectations and historical patterns. At low utilization rates, inflation decelerates, but probably at an asymmetrically slow pace. At the Phelps-Friedman
“natural rate of unemployment,” the degrees of resource utilization and market tightness generate no net wage and price pressures up or down and are
consistent with accustomed and expected paths, whether stable prices or any
other inflation rate. The consensus view accepted the notion of a nonaccelerating inflation rate of unemployment (NAIRU) as a practical constraint on
24 The sacrifice ratio is now perhaps more commonly described as the output gap.
R. G. King: Phillips Curve Snapshots
343
policy, even though some of its adherents would not identify NAIRU as full,
equilibrium, or optimum employment.”25
To put the potential costs of a disinflation in front of his audience, Tobin
used a very simple inflation model with a NAIRU of 6 percent unemployment
and assumed that the economy originated from an inflation rate of 10 percent.
As displayed in Figure 11, he studied a gradual disinflation in which the central
bank reduced the growth rate of nominal aggregate demand smoothly so that
it falls by 1 percent each year for 10 years. Thus, after a decade, the conditions
for price stability are met from the aggregate demand side. Tobin also assumed
that the expectations term was an eight-quarter, backward-looking average of
recent inflation rates. Tobin stressed that the result was “not a prediction!. . . but
a cautionary tale. The simulation is a reference path, against which policymakers must weigh their hunches that the assumed policy, applied resolutely
and irrevocably, would bring speedier and less costly results.” The cautionary
tale of Figure 11 involves an initial decade in which unemployment looks to
average about 8.5 percent, 2.5 percent higher than its equilibrium value, so
that the sacrifice ratio during this period is about 2.5 since inflation is being
reduced by 10 percent. So, while the tale was cautionary, the message was
consistent with the range of Okun’s sacrifice ratio estimates and, hence, meant
to depict some potential consequences of disinflation.
There were some skeptics. William Fellner (1976) viewed the government policies of the 1970s as sharply inconsistent with the objective of bringing about low inflation, echoing Christ’s 1973 concerns. Fellner argued that
households and firms would be similarly skeptical and that the disinflation
process was costly in part because of the imperfect credibility of policies,
so he endorsed a policy of gradualism like that which Tobin explored in his
simulation coupled with strong announcements about future policy intentions.
However, economists like Tobin were quite skeptical about the practical importance of this line of argument, while accepting the basic logical point that
expectations effects could mitigate some of the output losses associated with
his gradualist simulation. Considering the benefits of preannounced stabilization plan credibility, Tobin (1980) wrote: “The question is how much.
One obvious problem is that a long-run policy commitment can never be irrevocable, especially in a democracy. Important economic groups will not
find it wholly credible, and some will use political power to relax or reverse
the policy. Even assuming credibility and understanding by private agents,
25 Thus, Tobin uses the “natural rate” and the “NAIRU” interchangeably when it comes to
the analysis of inflation. However, many Keynesian economists did not want to assume that the
level of real activity consistent with constant inflation was an efficient level. As discussed above,
Friedman had written that the natural rate was to include “the actual structural characteristics of the
labor and commodity markets, including market imperfections, stochastic variability in demands and
supplies, the cost of gathering information about job vacancies and labor availabilities, the costs
of mobility, and so on,” but these conditions had sometimes been ignored in the debate over the
efficiency and inevitability of the natural rate.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 11 Tobin’s Cautionary Tale
10
1980
1981
1982
8
1983
1984
6
1985
1986
Inflation
4
1987
2
1996
0
1995
1997 1998
1988
1999
2000
1989
Equilibrium
1990
1994
-2
1991
1993
1992
-4
4
5
6
7
8
9
10
11
Unemployment
Notes: A simulation study of the unemployment and inflation implications of a gradual
reduction in the rate of growth of the money supply, as described in more detail in Tobin
(1980).
their responses are problematic. In the decentralized but imperfectly competitive U.S. economy, wage and price decisions are not synchronized but staggered. It is hard to predict how individual firms, employees, and unions will
translate a threatening macroeconomic scenario into their own demand curves.
If each group worries a lot about its relative status, each group will decide that
the best strategy is to disinflate very little.” Thus, Tobin (1980) argued that
it would be “recklessly imprudent to lock the economy into a monetary disinflation without auxiliary incomes policies. The purpose of these policies
would be to engineer directly a deceleration of wages and prices consistent
with the gradual slowdown of dollar spending.” In contrast to Fellner’s case
for gradualism, rational expectations theorists like Sargent began to explore
actual disinflation experiences, using the lessons of the models that they had
developed in the mid-1970s. In particular, in his “Ends of Four Big Inflations,”
circulated no later than spring 1981, Sargent argued that dramatic, sustained
R. G. King: Phillips Curve Snapshots
345
anti-inflation policies could bring about reductions in inflation with relatively
low unemployment costs, as long as such policy changes were credible and
that their dramatic nature enhanced their credibility.
The Volcker Disinflation
Paul Volcker assumed the chairmanship of the Federal Reserve System (FRS)
in August 1979. Looking back at Figure 1, we can see that inflation was
substantially reduced, while there was a lengthy period of high unemployment.
As cataloged in many discussions, the Federal Reserve made a high-profile
announcement of a shift to monetary targeting in October 1979 in the face of
rapidly rising inflation; there were two recessions during the period, one short
and relatively mild, one lengthy and severe, and the inflation rate had declined
dramatically by 1984.
There are many questions about the Phillips curve during this important
historical period, but our focus in this section will be limited to two. First,
how did the unemployment cost of the actual disinflation line up with the suggestions of Okun and others? Second, how did the Federal Reserve perceive
the menu of policy choice during this period?
The unemployment cost
Mankiw (2002, 369–701) calculates the unemployment cost of the Volcker
disinflation under the assumption that there was a 6 percent natural rate of
unemployment during 1982–1985, with the inflation rate falling from 9.7
percent in 1981 to 3 percent in 1985. His annual average unemployment
numbers were 9.5 percent in 1982 and 1983, 7.4 percent in 1984, and 7.1
percent in 1985 so that there was a total cyclical unemployment cost of 9.5
percent of unemployment. One can argue about details of this calculation, for
example with whether the disinflation should be viewed as starting in 1980
or in 1981, about the natural rate of unemployment, and so on. But it is a
reference textbook calculation familiar to many: The sacrifice ratio during the
Volcker disinflation is estimated by Mankiw to be about 9.5/6.7 = 1.5.
Mankiw’s sacrifice ratio is about one-half of that which Okun suggested
on the basis of his mean estimate and lies below the low end of the range in the
studies that he reviewed. The 6.7 percent decline in the inflation rate should
have had the effect of raising the unemployment rate by a total of 22 percent
over the period according to the average estimate, 13.4 percent according to
the low estimate, and over 40 percent for the high estimate. Put another way,
the mean estimate implies that unemployment should have been higher by
more than 5 percent over each year of a four-year disinflation period.
Some have suggested that this lower cost was due to increased credibility of the Fed and its disinflationary policies under Volcker; others have
346
Federal Reserve Bank of Richmond Economic Quarterly
suggested that the cost was largely due to the central bank’s imperfect credibility.26 However, as McCallum (1984) points out, the testing of hypotheses
about credibility is subtle because the measures of private expectations about
policy that must be constructed are more involved than in standard rational
expectations models.
The Fed’s Perceived Unemployment Cost
One key question about this disinflation period is “How did the Federal
Reserve System view the tradeoff between inflation and unemployment?” The
developments reviewed above, in which the nature of the tradeoff was subject
to substantial controversy and evaluated in an evolving model, makes this a
particularly interesting question. It is also not an easy question to answer, as
any central bank is a large organization with many differing viewpoints and
its policymakers do not file survey answers about their perceived tradeoffs.
However, the question can be answered in part because the HumphreyHawkins legislation requires testimony by the FRS chairman twice a year, in
late January or early February and again in July. For the six FOMC meetings each year, the research staff under Volcker prepared a basic forecast
of the economy’s developments within the “Green Book” under a particular
benchmark set of policy assumptions. For the FOMC meetings that precede the chairman’s testimony, the staff also prepared a set of alternative
policy options within the “Blue Book” of which Table 1 provides examples
at two FOMC meetings. In both cases, then, the alternative strategies were
framed in terms of growth rates for the M1 concept of money and were based
on the Board’s quarterly macroeconometric (MPS) model along with staff
judgemental adjustments, so that they reflected the effects of pre-existing economic conditions as well as alternative paths of policy variables.27 Projections
26 To my mind, the role of imperfect credibility in the Phillips curve in U.S. history remains
an open, essential area of research at present. Goodfriend and King (2005) argue that Volcker’s
actions were based significantly on his perception that his policy actions were not perfectly credible
and that the nature of the disinflation dynamics was also substantially influenced by imperfect
credibility of policy.
27 The Federal Reserve Bank of New York (1998, 123) describes the preparation of “blue
book” material, from which the Table 1 entries are taken, as follows: “The blue book provides the
Board staff’s view of recent and prospective developments related to the behavior of interest rates,
bank reserves, and money. The blue books written for the February and July meetings contain
two extra sections to assist the Committee in its preparation for the Humphrey-Hawkins testimony.
The first of these sections provides longer-term simulations, covering the next five or six years.
One of these simulations represents a judgmental baseline, while two or three alternative forecasts
use a Board staff econometric model to derive the deviations from the judgmental baseline under
different policy approaches. Typically, at least two scenarios are explored: one incorporates a
policy path that is designed to bring economic activity and employment close to their perceived
long-run potential paths fairly quickly, and another is intended to achieve a more rapid approach to
stable prices. The section also offers estimates of how different assumptions about such factors as
fiscal policy, the equilibrium unemployment rate, or the speed of adjustment to changed inflationary
expectations would affect the predicted outcome.”
R. G. King: Phillips Curve Snapshots
347
Table 1 FRB Economic Projections Associated with Alternative
Monetary Growth Strategies
January 1980
Money Growth
Inflation
Unemployment
Strategy
1
2
=1−2
1
2
=1−2
1
2
=1−2
1980
6.0
4.5
−1.5
9.5
9.1
−0.4
8.1
8.4
0.7
1981
6.0
4.5
−1.5
8.7
8.2
−0.5
8.9
10.1
1.2
1982
6.0
4.5
−1.5
7.7
6.8
−0.9
9.3
11.6
2.3
1983
3.5
5.0
−1.5
5.1
5.4
−0.3
9.1
8.2
0.9
12.5
8.3
4.2
1984
3.0
4.5
−1.5
4.2
5.3
−0.9
8.9
6.9
2.0
11.0
8.4
2.6
January 1982
Money Growth
Inflation
Unemployment
T-Bill
Strategy
1
2
=2−1
1
2
=2−1
1
2
=2−1
1
2
=2−1
1982
4.0
5.5
−1.5
6.4
6.5
−0.1
9.3
9.0
0.3
13.0
9.7
3.3
Notes: These table items are drawn from the Federal Reserve Blue Books “Monetary
Aggregates and Money Market Conditions,” January 4, 1980, page 11, and “Monetary
Policy Alternatives,” January 29, 1982, page 7.
under alternative strategies during the Volcker deflation also took into account
forecasted developments in fiscal policy, which were being revamped by the
Reagan administration at the time.
The first meeting is the January 1980 FOMC session, at which the benchmark strategy (called strategy 1 in this meeting) was for 6 percent money
growth over each of three years: 1980, 1981, and 1982. Under this benchmark policy, as can be seen by reading across the relevant row of the table,
the forecast was that U.S. inflation would gradually decline from 9.5 percent
in 1980 to 7.7 percent in 1982, but that there would be high and rising unemployment in each year (8.1 percent in 1980, 8.8 percent in 1981, and 9.3
percent in 1982). In this general sense, the forecast incorporated a Phillips
curve but one that depended in a complex manner on initial conditions and
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Federal Reserve Bank of Richmond Economic Quarterly
shocks. However, the staff also forecasted unemployment and inflation under
an alternative policy, 4.5 percent money growth. Hence, it is possible to use
the difference in forecasts to gain a sharper sense of the tradeoff under alternative monetary policies. The forecast differences are listed as the row in
the table. The January 1980 FOMC meeting corresponded with the onset of
a recession, as later dated by NBER researchers.
The second meeting is the January 1982 FOMC session, at which the
benchmark strategy (again called strategy 1 in this meeting) was for a gradually
declining path of money growth: 4.0 percent in 1982, 3.5 percent in 1983,
and 3.0 percent in 1984. Again, the FRS staff forecasted declining inflation
(from 6.4 in 1982 to 4.2 in 1984) but this time with declining unemployment
(from 9.3 percent in 1982 to 8.9 percent in 1984). The perceived Phillips curve
was less evident in these forecasts, but an opportunity to appraise its nature is
afforded by the fact that the staff also prepared forecasts under the assumption
of higher money growth (strategy 2).
There are a number of aspects of the benchmark policy projections that
are notable. First, in each case, strategy 1 is the assumption under which the
Federal Reserve staff made its “Green Book” forecast for inflation and real
activity for the coming years. In both 1980 and 1982, inflation was expected to
decline by about two percentage points under the benchmark forecast. Second,
in both 1980 and 1982, the projections implied that a policy change (lowering
money growth by 1.5 percent for three years) would have no effect on inflation
within the first year. Third, looking out two years after such a policy change,
the alternative Blue Book policy scenarios suggested substantial effects on
both unemployment and inflation of changing monetary policy. A shift from
strategy 1 to strategy 2 in 1980 was projected to produce a .9 percent decline
in inflation in 1982 and a 2.3 percent increase in unemployment in 1982. Seen
in terms of a “menu of policy choice,” the unemployment cost in two years of
a 1 percent reduction in inflation was 2.3 percent. A 1982 shift from strategy
1 to strategy 2 was predicted to have the same effect on inflation at a two-year
horizon, at an unemployment of 2.0 percent.
Proceeding further, we can set a lower bound on the perceived unemployment cost of disinflation by cumulating the “deltas” over the three-year period
and viewing the result as the first part of a transition to a 1.5 percent lower
inflation rate.28 From that standpoint, in 1980 and 1982, the Fed’s perception
was that the first three years of restrictive monetary policy would cost about 4
point years of unemployment to lower the inflation rate by 1.5 percent. Thus,
a 6.7 percent decline in the inflation rate was perceived to cost no less than 18
28 It is a lower bound because the staff projection would certainly have viewed unemployment
as remaining high beyond the three-year projection in the Blue Book; the unemployment
is
highest in the third year.
R. G. King: Phillips Curve Snapshots
349
point years of unemployment and it could have been a good bit higher once
fourth-year costs were included.
Overall, in 1980 and 1982, it seems that the Fed’s perception was that the
disinflation would be at least twice as costly as the cyclical unemployment
that was actually experienced. The perceived cost was not too much different
across these years, although it was modestly smaller in 1982, and it was in
line with the consensus estimates of Okun (1978). Accordingly, it does not
seem that the Fed undertook the disinflation because its research staff, at least,
believed that the costs would be small.
The Consolidation of Disinflation Gains
The reduction in inflation during the early 1980s had to be followed up by
a lengthy period of inflation fighting, as discussed in Goodfriend (1993). In
particular, upon taking over as chairman in 1987, Alan Greenspan had to fight
a series of inflation scares. Yet, by 1994–1995 it seemed that the United States
had settled into a period of low inflation (about 3 percent) and low unemployment (about 5 percent), essentially returning to conditions that resembled
those in the mid-1950s, where we started our Phillips curve documentary.
6.
IN THE AFTERMATH
The year 1996 saw two novel developments on the Phillips curve front, which
are closing snapshots: the completion of version 1 of a new quarterly Federal
Reserve Board macroeconometric model of the United States and an explicit
discussion of Phillips curve tradeoffs by the FOMC.
The Model
The structure and results of large-scale models are notoriously difficult to
convey in a compact and coherent manner. In that regard, the 1996 “Guide to
FRB/US: A Macroeconomic Model of the United States,” edited by Brayton
and Tinsley, is a remarkable document. It provides the reader with a clear
model-building vision and a set of clean experiments that can be used to learn
about the model.
The wage-price block of the new model combines the sort of forwardlooking price-setting and wage-setting specifications that are standard in modern macroeconomic analysis, with a set of gradual adjustment specifications
of the variety that applied econometricians have found useful for fitting data
since the days of Fisher, Klein, and Sargan. The specific modeling is in the
tradition of the approach to time series econometrics initiated by Sargan and
refined by Hendry and others.
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Federal Reserve Bank of Richmond Economic Quarterly
For example, the price-level specification of the model contains a long-run
relationship that makes
Pt∗ = .98 ∗ (Wt − at ) + .02 ∗ Pt − .003ut ,
f
so that the “equilibrium” price level strongly depends on the gap between the
log nominal wage rate (W ) and productivity (a) with weaker effects from the
f
nominal energy fuel price (Pt ), consistent with factor share data. Further,
the unemployment rate has a small effect, via an effect on the desired markup
(the units of measurement imply that a 1 percent increase in unemployment
lowers the desired markup by .3 percent). The adjustment dynamics indicate
that inflation is high as the price level adjusts gradually toward this target level,
via
Pt − Pt−1 = .10(P∗ t−1 − Pt−1 ) + .57lags2 [Pt−1 − Pt−2 ]
∗
− Pt∗ ]e .
+.43leads∞ [Pt+1
∗
That is, there is a gradual elimination of .1 of the gap (Pt−1
−Pt−1 ) each quarter,
some additional backward-looking adjustment terms with substantial weight,
and variations in the expected target. The requirement that the structural
lead and lag coefficients sum to one, along with similar restrictions in the
companion wage equation, means that the FRB/US model features no longrun tradeoff between inflation and unemployment.
Thus, the new model represented a blend of the Klein-Sargan approach,
with a new macroeconomic theory that stresses expectational elements of pricing and other behavior. The new FRB/US model also had common elements
with a set of small, fully articulated dynamic models then being developed
in academia (King and Wolman [1996] and Yun [1996]), which were early
examples of the types of new macroeconomic models explored elsewhere in
this Economic Quarterly issue.
The frictions in the model are substantial, as Brayton and Tinsley (1996)
make clear, in that they apply to changes as well as levels. The associated
distributed lags and leads are lengthy, averaging 3.3 quarters for unanticipated
shocks. Hence, there is a short-run Phillips curve in the model that involves
dynamics over many quarters. Figure 12 shows the response to a permanent
decline in the inflation rate within the FRB/US model, essentially obtained by
shifting down the constant term in an interest rate rule along Taylor (1993)
lines.
The FRB/US model can be solved under alternative assumptions about
expectation formation, with rational expectations being one specification and
a modern version of adaptive expectations being the other. More precisely, the
second specification is expectations based on a vector-autoregression
estimated from a model’s data for a subset of just three of that model’s variables. The model implies a gradual disinflation process as a result of the lags
R. G. King: Phillips Curve Snapshots
351
Figure 12 Credible Disinflation Dynamics in the FRB/US Model
0.0
-0.2
Output Gap
Consumer Price Inflation
0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
2
4
6
8
Years
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
0
2
4
6
8
Years
Ten-Year Government Bond Rate
Nominal Federal Funds Rate
0
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
0
2
4
6
8
Years
0
2
4
6
8
Years
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
Notes: The FRB/US model can be used to calculate the implications of a permanent shift
down in the target inflation rate, with results that are reported in Brayton and Tinsley
(1996) and displayed above. The model can be solved under full rational expectations
(solid line) or under a simpler procedure of vector-autoregression (VAR) expectations.
in the price and wage specification, but the transition to the new lower inflation
rate is completed within about two years, although the real consequences are
present for several years. Overall, since the reported simulations are in terms
of an output gap, they cannot be directly compared to those considered above.
Yet a back-of-the-envelope calculation suggests that they are quite major but
smaller than those experienced in the Volcker period or in the MPS model
discussed above. To see why, suppose that we summarize the figure as indicating that an upper bound on the output loss is a .5 percent output gap on
average for 6 years; this is a 3 percent cumulative output loss. Suppose also
that we use the same Okun’s Law coefficient of 2.5 that links unemployment
to output gaps as in the 1962 Economic Report. Then, the unemployment cost
of a permanent disinflation is about 1.2 point years of unemployment for each
point of inflation, a number that is in line with ratios reported by Brayton,
Tinsley, and collaborators. The FRB/US team also reported that imperfect
352
Federal Reserve Bank of Richmond Economic Quarterly
credibility of monetary policy actions can more than double the unemployment cost of a disinflation.
A notable feature of the disinflation simulation displayed in Figure 12 is
that monetary policy initially works heavily through expectational channels.
On impact, at date 0, the long-term bond rate drops dramatically (say, 80 basis
points), while the federal funds rate moves by much less and only averages
about a 40-basis-point decline over the first year. By contrast, the MPS model
simulations of a lower money growth rate displayed nominal interest rate
increases by an average of over 300 basis points for the first three years as
shown in Table 1. Thus, the FRB/US model differs importantly from its MPS
predecessor in terms of other areas, notably the term structure of interest rates,
in ways that are important for monetary policy.
The Background to the Meeting
In 1996, the FOMC conducted a remarkable discussion of its long-run policy
goals, stimulated by earlier calls for an increased emphasis on price stability
by some of its members, as well as the adoption of inflation-targeting systems
by other countries around the world. Notably, at a January 1995 meeting,
Al Broaddus, the then-president of the Richmond Fed, had called within the
FOMC for a system of inflation reports each year to accompany the Fed chairman’s Humphrey-Hawkins testimony.29 Broaddus’ suggestion was opposed
by FRB Governor Janet Yellen in January 1995, but the FOMC had agreed
to continue the discussion in the context of future meetings that preceded the
Humphrey-Hawkins testimony.
When the FOMC met in January 1996, the U.S. economy had been experiencing low inflation and strong macroeconomic activity for some time. In the
first quarter of 1996, inflation was running at about 2 percent per year, with unemployment in the neighborhood of 5.5 percent. Since 1980, the United States
had experienced the major decline in inflation described in the last section,
during which unemployment had ranged over 10 percent in the last quarter of
1982 and the last two quarters of 1983. In the last year of Volcker’s chairmanship and during the first few years of Greenspan’s, a rise in inflation had taken
place—from the 2 percent range in 1986 to about 4 percent in 1990—which
had been accompanied by a decline in unemployment.30 Subsequently, during 1991 and 1992, there had been a rise in unemployment while inflation fell
back in the 2 percent range. Most recently, from mid-1992 through the end
29 Broaddus (2004) used his proposals at this meeting as one of three examples of his use
of macroeconomic principles in practical monetary policy discussion.
30 At the time of Greenspan’s appointment in August of 1987, inflation was at 2.8 percent,
while unemployment was 5.8 percent.
R. G. King: Phillips Curve Snapshots
353
of 1995, there had been about 2 percent inflation, while unemployment was
between 5.5 and 6 percent.
These developments are shown in Figure 13, which is a Phillips-style
plot of unemployment and inflation during 1980 through 1996. Observations
during the Volcker period are marked with a circle (o) and those during the
Greenspan period are marked with a diamond (♦). This figure captures the
background to the FOMC’s 1996 discussion. The major disinflation is the
first half-loop: an interval of declining inflation and rising unemployment
between 1980 and mid-1983, followed by an interval of declining inflation
and declining unemployment with inflation reaching the 4 percent range by
the second quarter of 1984. Subsequently, there was a year in which inflation
fell in the 2 percent range, with little accompanying change in unemployment.
The increase in inflation between mid-1985 through 1989 was followed by a
decline in inflation to the 2 percent range during late 1991 and early 1993,
accompanied by increases in unemployment. In late 1993 through early 1995,
unemployment fell sharply, with little change in inflation. Thus, the lateVolcker and early-Greenspan years trace out a full clockwise loop, after the
disinflation of 1980–1984. The negative association between inflation and
unemployment during the first stages of each of these three episodes (one
of increasing inflation and one of decreasing inflation) corresponds to periods
that FOMC members would have viewed as reflecting the phenomena isolated
by Phillips.
The Meeting
At the time of the January 1996 meeting, there were two important economic
conditions that occupied the FOMC’s attention. First, there was a sense that
key aspects of the U.S. economy were changing, with the possibility of a “New
Economy” based on computer and communications advances. Second, and
most important for the meeting, the inflation rate for personal consumption
expenditures was running at about a 3 percent rate and its “core” component—
that stripped of food, energy, and other volatile price components—was running at about 2.5 percent, but staff forecasts suggested that it was poised to
rise to the 3 percent range as well. The strong real growth in the economy,
coupled with a decline in unemployment to the range of 5.5 percent, had led
some FOMC members to express concerns about inflation.
In detailed prepared remarks, Governor Janet Yellen discussed a costbenefit approach to determining the optimal long-run rate of inflation and the
transition path. She noted that the Board’s new model indicated a cost of
2.5 point years of unemployment for every 1 percent decline in the long-run
inflation rate, under imperfect credibility. To warrant a reduction in inflation,
she argued that such a cost of permanently lower inflation had to be less than
the discounted value of a stream of future benefits. However, Yellen also
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 13 Inflation and Unemployment, 1978Q4–1996Q4
10
9
Volcker
Greenspan
1981.III
1979.IV
8
Inflation
7
6
1982.IV
5
1984.II
4
1988.II
3
2
1986.IV
1996.I
1992.IV
1
5
6
7
8
9
10
11
Unemployment
Notes: The inflation rate is the year-over-year change in the gross domestic product
(GDP) deflator, the unemployment rate is the civilian unemployment rate, quarterly averages of monthly figures. All data are from the Federal Reserve Economic Database
(FRED) at the Federal Reserve Bank of St. Louis.
returned to a theme suggested by Phillips’ original research, which was that
there could be particular costs to low rates of inflation. Citing research by
Akerlof, Dickens, and Perry (1996) which argued that worker-resistance to
nominal pay cuts produced a long-run Phillips curve with a negative slope at
low rates of inflation,Yellen also argued for a positive rate of long-run inflation
“to grease the wheels of the labor market.”
As they considered the appropriate long-run rate of inflation, the FOMC
decisionmakers took into account their perceived transition costs, their sense
of the benefits from permanently low inflation, and their sense of the costs
of permanently low inflation. There was diversity in the views reflected in
the statements of various members on each of these topics. But, as Broaddus
noted, there was a consensus that the long-run inflation rate should not be
higher than the current level of 3 percent. Broaddus and then-Cleveland Fed
President Jerry Jordan stressed the importance of explicit public discussion of
R. G. King: Phillips Curve Snapshots
355
inflation objectives as a means of enhancing Fed credibility and thus lowering
the cost of further reductions in inflation.
The FOMC discussed how to define price stability as an objective of monetary policy. Greenspan suggested that “price stability is that state in which
expected changes in the general price level do not effectively alter business
or household decisions,” but Yellen challenged him to translate that general
statement into a specific numerical value. He responded that “the number
is zero, if inflation is properly measured.” Yellen said that she preferred 2
percent “imperfectly measured.”
The FOMC settled on 2 percent inflation as an interim goal, with a policy
of deliberately moving toward that lower level. Presumably, some members
viewed it as the natural first step toward a lower ultimate inflation objective,
while others thought of it as an end point. On the second day of the twoday meeting, Greenspan cautioned the committee that the 2 percent objective
was included within “the highly confidential nature of what we talk about at
an FOMC meeting.” He noted that “the discussion we had yesterday was
exceptionally interesting and important” but warned that “if the 2 percent
inflation figure gets out of this room, it is going to create more problems for
us than I think any of you might anticipate.” He did not elaborate on whether
he was concerned about market or political reactions to the inflation goal.
7.
SUMMARY AND CONCLUSIONS
With a series of snapshots over a nearly 40-year period, this article has reviewed
the evolution of the Phillips curve in macroeconomic policy analysis in the
United States. During this period, U.S. inflation rose dramatically, initially
during a decade of glittering economic performance and then further during an
interval of stagflation. The reversal of inflation beginning in the early 1980s
was associated with a major recession, although perhaps not as large a one as
policymakers and economists had feared.
The rise and fall of inflation brought about a major change in the style
of macroeconometric models that were used to evaluate policy choices. The
earliest versions of these models featured a substantial long-run tradeoff consistent with the findings of Phillips over a near-century of U.K. data. The
subsequent evolution of models first involved altering their wage-price block
so that there was no long-run tradeoff and then, later, a more comprehensive rational expectations revision that included forward-looking wage and
price-setting structured so that there was no long-run tradeoff.
More generally, the rise and fall of inflation led monetary policymakers to
place greater weight on the role of expectations in governing macroeconomic
activity, with central banks working to extract information in long-term interest
rates about market expectations of inflation. Toward the end of the historical
period examined here, the Federal Reserve System had decided to maintain a
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Federal Reserve Bank of Richmond Economic Quarterly
goal of a low, but positive rate of inflation. Yet, it also chose not to communicate
that long-run target directly to the public. The decision to choose a positive rate
of inflation was traced, in part, to a concern about the transitory unemployment
costs of moving to a zero rate of inflation and in part to a concern about high
long-run costs of low inflation, in the spirit of Phillips’ analysis.
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Economic Quarterly—Volume 94, Number 4—Fall 2008—Pages 361–395
The New Keynesian
Phillips Curve: Lessons
From Single-Equation
Econometric Estimation
James M. Nason and Gregor W. Smith
T
he last decade has seen a renewed interest in the Phillips curve that
might be an odd awakening for a macroeconomic Rip van Winkle
from the 1980s or even the 1990s. Wasn’t the Phillips curve tradition
discredited by the oil prices shocks of the 1970s or by theoretical critiques of
Friedman, Phelps, Lucas, and Sargent? It turns out that the New Keynesian
Phillips curve (NKPC) is consistent with both the theoretical demands of
modern macroeconomics and some key statistical properties of inflation. In
fact, the NKPC can take a sufficient number of guises to accommodate a wide
range of perspectives on inflation.
The NKPC originated in descriptions of price setting by firms that possess
market power. For example, Rotemberg (1982) describes how a monopolist
sets prices if it faces a cost of adjustment that rises with the scale of the
price change. He shows that prices then gradually track a target price and
also depend on expected, future price targets. Calvo (1983) instead describes
firms that are monopolistic competitors. They change their prices periodically.
Knowing that some time may pass before they next set prices, firms anticipate
future cost and demand conditions, as well as current ones, in setting their
price. Also, the staggering or nonsynchronization of price setting by firms
James M. Nason is a policy advisor and research economist at the Federal Reserve Bank of
Atlanta. Gregor W. Smith is a professor of economics at Queen’s University in Canada. The
authors thank the Social Sciences and Humanities Research Council of Canada and the Bank
of Canada Research Fellowship program for support of this research. The views in this paper
represent those of the authors and not the Bank of Canada, the Federal Reserve Bank of
Richmond, the Federal Reserve Bank of Atlanta, the Federal Reserve System, or any of the
staff. The authors thank Nikolay Gospodinov for the use of his GMM code; Ellis Tallman,
Ricardo Nunes, and Frank Schorfheide for helpful suggestions; and Andreas Hornstein and
Thomas Lubik for detailed comments that greatly improved this paper.
362
Federal Reserve Bank of Richmond Economic Quarterly
creates an aggregate stickiness: The aggregate price level will react only
partially on impact to an economy-wide shock, such as an unexpected change
in monetary policy.
These theoretical models link prices to a targeted real variable such as a
markup on the costs faced by the price-setting firm. Therefore, they also relate
the change in prices over time (i.e., the inflation rate) to real variables. So it
is natural to label them as Phillips curves. In fact, there are a range of setups
called NKPCs that vary depending on (a) the information and price-setting
behavior attributed to firms and (b) the measure of costs or demand that firms
are assumed to target. Whether a specific version of the NKPC fits inflation
data has implications for our understanding of recent macroeconomic history
and for the design of good policy. For example, the parameters of the NKPC
influence how monetary policy ideally should respond to external shocks.
Schmitt-Grohé and Uribe, in this issue, make this connection clear. (King,
also in this issue, describes the uses of an older Phillips curve tradition in
policymaking.)
Yet, putting the NKPC to use for policy analysis requires that it has a good
econometric track record in describing actual inflation dynamics. In this article
we review this record using single-equation statistical methods that study the
NKPC on its own. These methods stand in contrast to approaches that place the
NKPC in larger economic models, sometimes referred to as systems methods,
which are reviewed by Schorfheide in this issue. A disadvantage of singleequation methods is that they do not make use of everything known about
the economy (e.g., the monetary policy regime), so they generally do not
provide the greatest statistical precision. Their advantage is that they allow
us to be agnostic about the rest of the economy, and so their findings remain
valid and will not be affected by misspecification of other parts of a larger
macroeconomic model.
This article asks the following questions: How can we estimate the NKPC
and what do we find when we do so for the United States? Are its parameters
stable over time and well-identified? Is there a relation between inflation and
real activity? Do we reach similar conclusions about the NKPC regardless of
the way in which we measure inflation, forecast future inflation, or model the
costs or output gap that inflation tracks?
We focus on marginal cost as the real activity variable in the NKPC.
We find that the single-equation statistical evidence for this relationship is
mixed. Since 1955 there does seem to be a stable NKPC for the United States
with positive parameter values as we would expect from economic theory.
But our confidence intervals for these parameter values are somewhat wide,
the findings depend on how we model expected future inflation, and further
research is needed on the best way to represent the marginal cost variable
to which price changes react. Before outlining the methods and findings,
J. M. Nason and G. W. Smith: Single-Equation Estimation
363
though, we begin by introducing the specific NKPC that we will estimate and
the inflation history it aims to explain.
1.
FOUNDATIONS AND INTERPRETATIONS
The New Keynesian Phillips curve arises from a description of staggered price
setting, which is then linearized for ease of study. The result is an equation in
which the inflation rate, π t , depends on the expected inflation rate next period,
Et π t+1 , and a measure of marginal costs, denoted xt :
π t = γ f Et π t+1 + λxt .
Iterating this NKPC difference equation forward gives inflation as the
present value of future marginal costs:
πt = λ
∞
γ if Et xt+i .
i=0
This present-value relation shows that firms consider both their current marginal costs, xt , and their expectations or forecasts of future costs when adjusting prices.
Lacker and Weinberg (2007) describe the history and derivation of this
key building block of New Keynesian macroeconomic models. Dennis (2007)
outlines a range of environments that can underpin a hybrid NKPC. Calvo’s
(1983) specific price-setting model is only one of several possible microfoundations for the NKPC. In a Calvo-based NKPC, a fraction, θ, of firms cannot
change prices in a given period. Firms also have a discount factor, β. The
reduced-form parameters of the NKPC, γ f and λ, are related to these two
underlying pricing parameters according to
(1 − θ )(1 − βθ )
.
θ
Because β is a discount factor, both it and γ f must range between zero and
one. The same holds for θ because it represents the fraction of firms unable
to move prices at any moment.
Many estimates of the NKPC find that lagged inflation helps to explain
current inflation. We report much the same in this article. This has suggested
to some economists that a better fit to inflation history can be obtained with
this equation:
γ f = β,
λ=
π t = γ b π t−1 + γ f Et π t+1 + λxt ,
which Galı́ and Gertler (1999) call the hybrid NKPC. They develop their
NKPC by modifying Calvo’s (1983) description of price-setting decisions. In
this case, a fraction, ω, of firms can change prices, but do not choose this
option. Define φ = θ + ω[1 − θ (1 − β)]. Then the mapping between these
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Federal Reserve Bank of Richmond Economic Quarterly
structural parameters and the reduced-form parameters is
(1 − ω)(1 − θ )(1 − βθ )
ω
βθ
, λ=
.
γb = , γf =
φ
φ
φ
Galı́ and Gertler (1999) note that this mapping between the structural, pricesetting parameters ω, θ , and β and the reduced-form hybrid NKPC parameters
γ b , γ f , and λ is unique if the former set of parameters lie between zero and
one.
This form of the NKPC also is consistent with the incomplete indexing
model of Woodford (2003). He assumes that those firms that cannot optimally
alter their prices instead index to a fraction of the lagged inflation rate. This
feature makes current inflation depend on lagged inflation, and so provides an
alternative interpretation of the hybrid NKPC. Christiano, Eichenbaum, and
Evans (2005), among others, study the implications of full indexation, which
is equivalent to the restriction γ b + γ f = 1 in the Galı́-Gertler hybrid NKPC.
Like the original NKPC, the hybrid version can be rewritten in presentvalue form:
∞
1 k
λ
π t = δ 1 π t−1 +
Et xt+k ,
δ 2 γ f k=0 δ 2
where δ 1 and δ 2 are stable and unstable roots, respectively, of the characteristic
equation in the lag operator L:
1
γ L
−L−1 +
− b = 0.
γf
γf
The present-value version of the hybrid NKPC shows that inflation persistence
can arise from the influence of lagged inflation or the slow evolution of the
present value of marginal costs.
The NKPC is often derived by log-linearizing a typical firm’s price-setting
rule around a mean zero inflation rate. Ascari (2004) shows that non-zero mean
inflation can affect the response of inflation to current and future marginal cost
in the NKPC. However, we follow much of the empirical NKPC literature and
demean the data.
Cogley and Sbordone (forthcoming) build on Ascari’s work, among others, by log-linearizing the NKPC around time-varying trend inflation. This
procedure assumes that inflation is nonstationary to obtain a NKPC with timevarying coefficients even though the underlying Calvo-pricing parameters are
constant. The resulting NKPC is purely forward-looking, assigning no role
to lagged inflation. Cogley and Sbordone estimate the structural coefficients
of their NKPC using a vector autoregression. Hornstein (2007) assesses the
implications of this approach for the stability of the NKPC. Since these studies
use system estimators, we omit them from our review.
In sum, the hybrid NKPC is consistent with various pricing or information
schemes. This suggests a focus on the reduced-form coefficients λ, γ b , and
J. M. Nason and G. W. Smith: Single-Equation Estimation
365
γ f (rather than the structural price-setting parameters ω, θ, and β), which is
our emphasis in this article. We also use single-equation estimators to explore
the fit of the hybrid NKPC to U.S. data. After all, obtaining a good fit for the
hybrid NKPC is a necessary first step in attributing a monetary transmission
mechanism to staggered price setting by firms.
Economists have not yet reached a consensus on two key questions concerning the NKPC parameters. First, what is the mixture of forward (γ f ) and
backward (γ b ) weights? If γ f is large, events in the future (including changes
in monetary policy) can influence the current inflation rate. If, instead, γ b is
large, inflation has considerable inertia independent of any slow movements
in the cost variable. Such inertia affects the design of monetary policy (again,
see Uribe and Schmitt-Grohé). Woodford (2007) reviews several explanations
for inflation inertia and also discusses whether it could be stable over time.
Second, can we identify a significant λ̂, the coefficient on marginal costs?
In this case, identification simply refers to measuring a partial correlation
coefficient in historical data rather than the possibility of misspecification
(i.e., whether this coefficient necessarily measures the theoretical parameter
studied in New Keynesian models.) Finding a significant value is a sine qua
non for empirical work with the NKPC. If we cannot find a way to represent
a price-setting target, we cannot hope to identify the adjustment process of
inflation to its target. Consequently, much of the research on estimating the
NKPC involves exploring the x-variable or how to measure marginal costs.
Before looking at formal econometric methods, let us look at the data.
Figure 1 plots the U.S. inflation rate (the black line) and a measure of marginal
cost (the gray line) from 1955:1 to 2007:4. We measure the inflation rate, π t , as
the quarter-to-quarter, annualized growth rate in the U.S. implicit GDP deflator
(GDPDEF from FRED at the Federal Reserve Bank of St. Louis). Marginal cost,
xt , is the update of the series on real unit labor costs used by Galı́ and Gertler
(1999) and Sbordone (2002). It is given by 1.0765 times the logarithm of
nominal unit labor costs in the nonfarm business sector (the ratio of COMPNFB
to OPHNFB from FRED) divided by the implicit GDP deflator. Both series
have fallen since 1980, which in itself provides some statistical support for
the idea that inflation tracks marginal cost. There also are some obvious
divergences, for example around 2000–2001. However, it is possible that
these occurred because inflation was tracking expected future marginal cost
(as in the present-value model) or because it was linked to lagged inflation (as
in the hybrid NKPC). We next describe the statistical tools economists have
used to see if either of these explanations fits the facts.
2.
ESTIMATION
The fundamental challenge with estimating the parameters of the hybrid NKPC
is that expected inflation cannot be directly observed. The most popular
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Federal Reserve Bank of Richmond Economic Quarterly
0.25
12
10
0.20
8
0.15
6
0.10
4
π
0.05
2
0
0.00
1960
1970
1980
1990
2000
Real Unit Labor Costs (Marginal Cost)
Quarterly Inflation Rate (GDP Deflator)
Figure 1 U.S. Inflation and Marginal Cost
Year
econometric method for dealing with this issue begins from the properties
of forecasts. To see how this works, let us label the information used by pricesetters to forecast inflation by It , so their forecast is E[π t+1 |It ]. Economists
do not observe this forecast, but it enters the NKPC and influences the current inflation rate. Denote by E[π t+1 |zt ] an econometric forecast that uses
some variables, zt , to predict next period’s inflation rate, π t+1 . Suppose that
zt is a subset of the information available to price setters. To construct our
econometric forecast, we simply regress actual inflation on our set of variables
(sometimes called instruments), zt , like this:
π t+1 = bzt + t+1 ,
so that our forecast is simply the fitted value
E[π t+1 |zt ] = b̂zt .
By construction it is uncorrelated with the residual term ˆ t+1 .
A key principle of forecasts (or rational expectations) is the law of iterated
expectations. According to that law, our econometric prediction of pricesetters’ forecast of inflation is simply our forecast. Symbolically,
E E[π t+1 |It ]|zt = E[π t+1 |zt ].
The idea is that our effort to predict what someone with better information
will forecast simply gives us our own best forecast. With the law of iterated
expectations in hand, we can also imagine regressing the unknown forecast
on zt :
J. M. Nason and G. W. Smith: Single-Equation Estimation
E π t+1 | It
367
= E π t+1 | It | zt + ηt
= E π t+1 | zt + ηt ,
in which the residual, ηt , also is uncorrelated with the econometric forecast.
The econometric forecast does not use all the information available to price
setters when they construct forecasts, so it does not capture all the variation in
their forecasts. Put differently, the unobserved, economic forecast has some
added variation that appears in ηt .
With this statistical reasoning behind us, the hybrid NKPC can be rewritten:
πt
= γ b π t−1 + γ f E π t+1 | It + λxt
= γ b π t−1 + γ f E π t+1 | zt + ηt + λxt
= γ b π t−1 + γ f E π t+1 | zt + λxt + γ f ηt .
This is an econometric equation that can be used to estimate the parameters
by least squares, for we now have measurements of the three variables on the
right-hand side of the equation. In fact, this two-step procedure—forecast
using predictors zt , then substitute and apply least squares—is just two-stage
least squares, familiar from econometrics textbooks. Provided that we include
the other hybrid NKPC explanatory variables, π t−1 and xt , in the list of firststage regressors, the error term will be uncorrelated with them, too, and so
least-squares will be valid in the second stage. (In contrast, simply estimating
the NKPC by least squares, using π t+1 in place of E[π t+1 ], yields inconsistent
estimates of the parameters, because π t+1 is correlated with the residual, ηt .)
Two-stage least squares, in turn, is a special case of a method known as generalized instrumental variables, or generalized method-of-moments (GMM)
estimation. To see how this works, take the hybrid NKPC and write it as
follows:
π t − γ b π t−1 − γ f E[π t+1 |It ] − λxt = 0.
Then imagine forecasting this entire combination of variables:
E π t − γ b π t−1 − γ f E[π t+1 |It ] − λxt |zt
E π t − γ b π t−1 − γ f π t+1 − λxt |zt
=
= 0,
where we again have used the law of iterated expectations to replace the
unobserved market forecast with our own econometric one. The last part of
this equation is the basis for numerous studies of the NKPC. Simply put, there
should be no predictable departures from the inflation dynamics implied by
the hybrid NKPC or, equivalently, the residuals should have a mean of zero
and be uncorrelated with predictor variables zt . These properties allow for a
diagnostic test of the NKPC.
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Federal Reserve Bank of Richmond Economic Quarterly
Finding estimates of the hybrid NKPC parameters proceeds as follows.
We collect data on inflation and marginal costs. Then we make a list of widely
available information, zt , that could include current and lagged values of these
same variables, as well as other macroeconomic indicators such as interest
rates. These instrumental variables (also known simply as instruments), zt ,
must have two key properties. First, they must be at least as numerous as the
parameters of the model (which here number three) and provide independent
sources of variation (i.e., they cannot be perfectly correlated with one another).
Intuitively, to measure three effects on inflation there must be at least three
independent pieces of information or exogenous variables. Second, they must
be uncorrelated with forecast errors that appear as residuals in the hybrid
NKPC. Instruments with these properties are called valid.
Next, we use some econometric software to adjust the economic parameters {γ b , γ f , λ} so that departures from the hybrid NKPC are uncorrelated
with zt , and so are unpredictable. The criterion guiding the adjustment is that
the moment conditions that consist of the cross-products of the NKPC residuals with the instruments should be as close to zero as possible. Whenever we
have at least three valid instruments in the set zt , we can identify and solve for
values of the three hybrid NKPC parameters using this criterion. In practice,
the algorithm attempts this task by squaring the deviations of the moment
conditions from zero and then minimizing the weighted square of this list of
deviations. Cochrane (2001, chapters 10–11) provides a lucid introduction to
GMM.
We make brief technical digressions on two details of GMM estimation. First, the distance of moment conditions (i.e., forecasts of departures
from the hybrid NKPC) from zero is measured relative to their sampling
variability, just as with any statistic. For GMM this involves calculating a
heteroskedasticity-and-autocorrelation-consistent (HAC) covariance matrix.
This article employs either a Newey and West (1994) or an Andrews (1991)
quadratic-spectral HAC estimator with automatic lag-length selection. Second, some authors note that the way the hybrid NKPC is written matters for
its estimation. For example, multiplying it by the Calvo parameter, φ, might
seem to make it easier to estimate φ as the weight on π t and ω as the weight on
π t−1 . In this article, we use the continuously updated (CU-)GMM estimator
of Hansen, Heaton, and Yaron (1996), from which estimates of the hybrid
NKPC are independent of any normalization applied to it.
There are many macroeconomic indicators that could be included in zt .
We need to place at least three macroeconomic variables in zt so that the three
parameters of the hybrid NKPC are just-identified, in the jargon of econometrics. The parameters are said to be overidentified when four or more
macroeconomic variables are included in zt . It turns out that this possibility
provides a test of the validity of the NKPC. According to econometric theory,
any instrument set should yield the same coefficients except for some random
J. M. Nason and G. W. Smith: Single-Equation Estimation
369
sampling error. So by estimating with various sets of instruments and comparing the findings (or seeing if the NKPC departures are close to zero even
when we use a long list of instruments) we can test whether the NKPC really
holds or not. This diagnostic procedure is called a test of overidentifying restrictions. Informally, we refer to an NKPC that passes this test as fitting the
data.
In practice, most researchers have used lagged macroeconomic variables
as instruments. To see why, recall that an error term, γ f ηt , arises in the
estimating equation. Recall, however, that by including E[π t+1 |zt ] we in fact
are trying to represent the regressor E[π t+1 |It ] (a forecast of inflation made in
the current period), not π t+1 . So one can think of the econometric equation as
containing an error term dated t that reflects the difference between these two
measures. Moreover, some economists have argued that there are unobserved
cost shocks (components of xt ) that also can underlie an error term in the
NKPC. Recall that a key property of instruments is that they be uncorrelated
with the error term. Many researchers studying the NKPC, therefore, have
used only lagged variables as instruments, labelled zt−1 , to try to ensure that
this property holds.
Another way to think of this approach is that using instrumental variables
is a classic way of dealing with the problem of a regressor that is subject
to measurement error. If the marginal cost series, xt , is measured with error,
then including xt as an instrument will lead to the attenuation bias (bias toward
zero) familiar in this errors-in-variables problem. Using lagged instruments
can avoid this bias.
Next, we present examples of GMM estimation. Instruments include
lagged values of inflation and marginal costs. In addition, we also present
results with a longer list of instruments. This list includes the term spread
of the five-year Treasury bond over the 90-day Treasury bill, ts, which is a
natural candidate for forecasting inflation. Table 1 gives the complete list of
variables and instruments, with the symbols used to represent them and the
sources for these data. Different econometricians might measure the output
gap, yt , differently. We used linear detrending of real per capita GDP to
produce the output gap as an instrument, but the findings are very similar if
we use other possible measures of the output gap as an instrument instead.
The list of instruments is inspired by Galı́ and Gertler (1999). They use four
lags of this list of six variables for their quarterly 1960–1997 sample. We
adopt the same list, though updated to 2007, so that the reader can compare
our findings to theirs.
Table 2 reports CU-GMM estimates of the hybrid NKPC on the 1955:1–
2007:4 sample. The first column lists the instrument set, zt . The next columns
list estimates of the reduced-form parameters γ̂ b , γ̂ f , and λ̂ over their standard
errors, followed by the structural Calvo-pricing estimates ω̂, θ̂ , and β̂ over
their standard errors. The final column gives a test statistic, denoted J , for the
370
Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Measuring Variables and Instruments
Label
π
x
Definition
Inflation rate, implicit GDP deflator
Log labor share of income
y
Linearly detrended
Log per capita real GDP
Five-year Treasury constant-maturity
interest rate minus 90-day Treasury bill
rate, quarterly average
Wage inflation
PPI commodity price inflation
ts
wi
cp
Code/Source
GDPDEF/FRED
1.0765 ln[COMPNFB/OPHNFB]
– ln[GDPDEF]/FRED
ln[GDPC96/CNP16OV]/FRED
GS5 – TB3MS/FRED
ln[COMPNFB×HOANBS]/FRED
BLS
hypothesis that the overidentifying restrictions hold. The hypothesis implies
the same parameters apply for any instrument set. We include the J statistic,
along with its degrees of freedom (df ), over its p-value. The J statistic is
asymptotically distributed χ 2 with df equal to the number of overidentifying
restrictions (the number of instruments minus the number of parameters).
Fixing the number of overidentifying restrictions, a larger J statistic yields a
smaller p-value, which indicates that the residual is predictable and constitutes
a rejection of the hybrid NKPC.
The top row of Table 2 presents CU-GMM estimates based on four instruments. With three hybrid NKPC parameters to estimate, this gives one
overidentifying restriction. Besides lagged inflation, the set of instruments in
the first row contains only lags of marginal cost, xt . Between the first and
last rows of Table 2, we add instruments one by one from the longer Galı́ and
Gertler (1999) list. The penultimate row of Table 2 includes the entire set of
24 instruments used by Galı́ and Gertler. (The last row presents least-squares
estimates, discussed in Section 4.)
Table 2 shows that the hybrid NKPC estimates vary with the set of instruments. We return to that finding in Section 4. Meanwhile, once we include
lags of marginal cost and of the output gap, y, we find that γ b and γ f are significant, positive fractions, with the weight on expected future inflation much
greater than the weight on lagged inflation (see the second and sixth rows).
Estimates based on these instrument sets indicate that there is little inflation
inertia. These estimates also reveal a significant, positive impact of real unit
labor costs, measured by λ̂, but the scale of the inflation response is small.
Sbordone (2002) and Eichenbaum and Fisher (2007) show that if firms have
firm-specific capital it can lead to a low response to a cost shock, i.e., a small
value for λ.
1.14
(0.58)
0.68
(0.10)
1.25
(0.44)
0.81
(0.20)
1.00
(0.19)
0.73
(0.05)
0.75
(0.03)
−0.11
(0.50)
0.29
(0.10)
−0.20
(0.38)
0.17
(0.18)
0.02
(0.17)
0.25
(0.05)
0.25
(0.01)
π t−1 , xt−1 , xt−2 ,
xt−3
π t−1 , xt−1 , xt−2 ,
xt−3 , yt−1 , yt−2
π t−1 , xt−1 , xt−2 ,
xt−3 , tst−1 , tst−2
π t−1 , xt−1 , xt−2 ,
xt−3 , wit−1 , wit−2
π t−1 , xt−1 , xt−2 ,
xt−3 , cpt−1 , cpt−2
(π , x, y, ts, wi, cp)t−1 , ...
(π, x, y, ts, wi, cp)t−4
π t−1 , π t+1 , xt
(NLLS)
2.65
(1.33)
0.63
(0.31)
1.07
(0.53)
0.89
(0.45)
0.17
(0.82)
0.67
(0.35)
1.41
(1.00)
100 ×λ̂
(se)
0.27
(0.01)
0.29
(0.08)
0.02
(0.16)
0.18
(0.24)
−0.15
(0.23)
0.36
(0.17)
−0.08
(0.36)
ω̂
(se)
0.81
(0.06)
0.92
(0.02)
0.88
(0.03)
0.90
(0.02)
0.89
(0.02)
0.91
(0.03)
0.88
(0.03)
θ̂
(se)
0.99
(0.06)
0.95
(0.03)
1.02
(0.05)
0.97
(0.06)
1.04
(0.04)
0.94
(0.05)
1.03
(0.06)
β̂
(se)
17.08(21)
(0.71)
1.45(3)
(0.69)
2.26(3)
(0.52)
1.05(3)
(0.79)
3.42(3)
(0.33)
0.30(1)
(0.58)
J (df )
(p)
Notes: Data are de-meaned prior to estimation. The estimator is CU-GMM, except the last case is NLLS, and all use a
Newey-West HAC correction and automatic plug-in lag length.
γ̂ f
(se)
γ̂ b
(se)
Instruments (zt )
E π t − γ b π t−1 − γ f π t+1 − λxt | zt =0
Table 2 U.S. New Keynesian Phillips Curve, 1955:1–2007:4
J. M. Nason and G. W. Smith: Single-Equation Estimation
371
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Federal Reserve Bank of Richmond Economic Quarterly
The second and sixth rows of Table 2 also contain estimates of the structural Calvo-pricing parameters that are positive fractions, significant, and in
keeping with the theory. These estimates yield a discount factor of around
0.95 and indicate that about 90 percent of firms are unable to change prices in
a given quarter. Of the 10 percent that can change prices, about a third decide
against it.
Additional support for the NKPC is provided by the J -test statistics in the
final column of Table 2. Across all instrument sets, we obtain a large p-value
associated with this statistic, so that the overidentifying restrictions cannot be
rejected.
3.
STABILITY
So far, the updated empirical evidence supports the hybrid NKPC. However, as mentioned in the introduction, it also is natural to ask whether the
hybrid NKPC parameters are stable over time. To test their stability, we
divide the entire sample at a given quarter, called a break date, into two separate subsamples. We consider all possible break dates between 1963:1 and
1999:4, which trims 15 percent from the beginning and end of the entire
sample. For each date in this range, we estimate the NKPC twice: first
from 1955:1 to the break date quarter and second from one quarter beyond
the break date to 2007:4. All CU-GMM estimates employ the instrument
vector {π t−1 , xt−1 , xt−2 , yt−1 , yt−2 , wit−1 , wit−2 }, which mimics that of Galı́,
Gertler, and López-Salido (2005) minus {π t−2 , π t−3 , π t−4 }.
Figure 2 presents the results. The three panels plot estimates of the
reduced-form parameters γ b , γ f , and λ, respectively, for break dates from
1963:1 to 1999:4. In each panel, the black line graphs estimates from the
sample beginning in 1955:1 and ending at the break date shown, while the
gray line graphs estimates from a sample beginning at the break date plus one
quarter to 2007:4. For any date, the vertical distance between the two lines
gives the difference between estimates from the “before” and “after” samples.
The figure shows variation in the estimates that is limited for break dates since
1980, but noticeable for earlier break dates, particularly for γ̂ b and for λ̂. The
coefficient γ̂ b ranges from 0.29 to 0.40 estimating on the before sample and
from 0.27 to 0.35 estimating on the after sample. For γ̂ f , the corresponding
ranges are from 0.64 to 0.73 and from 0.60 to 0.69, respectively. A glance
at the vertical axes shows that these estimates confirm the earlier finding that
the coefficient on expected future inflation, γ̂ f , exceeds the coefficient on
lagged inflation, γ̂ b . Estimates of 100 × λ̂ range from −1.31 to 1.03 on the
before sample and from 0.08 to 0.87 on the after sample. However, ignoring
estimates from the 1960s break dates constrains 100 × λ̂ to range from about
zero to around one.
J. M. Nason and G. W. Smith: Single-Equation Estimation
373
Figure 2 Parameter Stability
γ
0.40
0.38
1955Q1
0.36
0.34
0.32
0.30
0.28
2007Q4
0.26
1970
1980
2000
1990
γ
0.74
0.72
1955Q1
0.70
0.68
0.66
0.64
0.62
2007Q4
0.60
1970
1980
1990
2000
λ
0.010
0.005
0.000
2007Q4
-0.005
-0.010
1955Q1
-0.015
1970
1980
1990
2000
Next, we test one-by-one whether any of the three parameters {γ b , γ f , λ}
changes significantly from the first time period to the second time period. The
method developed byAndrews (1993, 2003) allows a statistical test along these
lines without pre-supposing knowledge of the exact date at which a break or
shift in a parameter value took place. Following this method, we calculate the
Wald test for the hypothesis that there is a significant difference between the
“before” and “after” estimates for γ b , γ f , and λ over 1963–1999. We record
the maximal value for each of these Wald statistics. Andrews (2003) gives
critical values for this test statistic, while Hansen (1997) provides a method
for computing p-values. For γ b , the test statistic is 1.50 with a p-value of
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Federal Reserve Bank of Richmond Economic Quarterly
0.91; for γ f , the test statistic is 1.41 with a p-value of 0.92; and for λ, the
test statistic is 3.30 with a p-value of 0.50. Since these p-values are far above
conventional levels of statistical significance such as 0.05 or 0.10, the tests
fail to reject the hypothesis that the parameters are stable.
In summary, our tests suggest that the reduced-form hybrid NKPC parameters are stable. This result is striking because the behavior of inflation
has changed over time. For example, inflation was on average lower and less
volatile after the mid-1980s. Yet despite this change in the statistical properties of inflation, the link between inflation and marginal cost has remained
stable. The stability of this relationship is striking because it suggests a flat
hybrid NKPC—with the same relatively low slope λ̂—during the business
cycles of the 1970s and the Great Moderation and disinflation that took hold
in the mid-1980s. Whatever the sources of this Great Moderation in inflation,
the single-equation stability tests suggest that they acted through real activity
as measured by marginal costs.
4. WEAK IDENTIFICATION?
Section 2 referred to the need to find valid instruments in order to use singleequation methods. Instruments must satisfy two statistical criteria. First,
they must be as numerous as the parameters and must help predict or forecast
π t+1 so that a projection based on them can be reasonably substituted for the
unobserved forecast on the right-hand side of the Phillips curve. Second, they
must be uncorrelated with the error term in the econometric equation, just like
any regressor.
Unfortunately, these two criteria sometimes can conflict. To see how this
can come about, recall that researchers often have used lagged instruments,
zt−1 . The rationale for this choice is that these past outcomes must be exogenous and, therefore, uncorrelated with unobserved shocks to today’s inflation
rate, thus satisfying the second criterion. But now satisfying the first criterion
can be challenging. The researcher needs to find at least one variable, zt−1 , that
helps forecast π t+1 . Also, the list of instruments must include something other
than the other two variables that enter the hybrid NKPC, xt and π t−1 . That
is because the constructed forecast Eπ t+1 |zt−1 has to exhibit some variation
independent of xt and π t−1 . Otherwise, there will be no possibility to measure
or identify separately the effects of π t−1 , xt , and Et π t+1 on current inflation.
We want to identify these three effects on current inflation so, logically, we
need an inflation forecast that sometimes varies separately from π t−1 and xt .
Seen in this way, the problem of finding instruments is recast as the problem of trying to forecast inflation but with a twist. The statistical challenge is
to predict next quarter’s inflation rate, π t+1 , but without using this quarter’s
inflation rate, π t (because it is the variable we seek to explain on the lefthand side of the hybrid NKPC), or last quarter’s inflation rate, π t−1 , or this
J. M. Nason and G. W. Smith: Single-Equation Estimation
375
quarter’s costs or aggregate demand, xt (because they appear separately on the
right-hand side of the hybrid NKPC). Forecasting inflation is difficult, even
without one hand tied behind one’s back in this way. The statistical studies by
Stock and Watson (1999, 2007) and Ang, Bekaert, and Wei (2007) show that
it is challenging to find a stable relationship that can be used to forecast U.S.
inflation, especially over the past 15–20 years. Perhaps competent central
bankers can take some credit for creating a low, stable inflation rate that has
not displayed persistent swings or cycles, but that outcome inherently makes
it difficult to isolate an inflation forecast that differs from current or lagged
inflation.
The hybrid NKPC provides another perspective on how to forecast π t+1 .
We lead the present-value version of the hybrid NKPC forward by one time
period and forecast to obtain
∞
1 k
λ
Et xt+1+k .
Et π t+1 = δ 1 π t +
δ 2 γ f k=0 δ 2
Next, suppose that xt can be forecasted only from its own, lagged value. Suppose that marginal costs follow a first-order autoregression, with coefficient
ρ, so that its multistep forecast is
Et xt+1+k = ρ 1+k xt .
Combining the last two equations gives the forecasting equation
Et π t+1 = δ 1 π t +
λρ
xt .
γ f (δ 2 − ρ)
In this case, the three reduced-form parameters cannot be identified by
GMM because there is no source of variation in Et π t+1 other than π t and xt
(which already are included in the hybrid NKPC). Nason and Smith (2008)
show that if xt is an autonomous pth-order autoregression, p must equal 2
to just identify and be greater than 2 to overidentify the three hybrid NKPC
parameters using GMM. In other words, higher-order dynamics are needed
for identification if xt is predicted from its own past. That is why the first row
of Table 2 includes several lags of marginal cost.
Nason and Smith (2008) also show that setting the NKPC in a broader,
New Keynesian model does not suggest sources of identification for singleequation estimation. They show analytically the problem facing an econometrician who tries to estimate the NKPC by GMM in a textbook world where
the hybrid NKPC combines with a dynamic I S curve and a Taylor rule. It
turns out that there may be no valid instruments available. The logic is that
the econometrician must lag instruments to make sure that they are uncorrelated with the residual in the NKPC equation. But lagging them enough to
satisfy that criterion for instrumental variables also makes them irrelevant for
forecasting π t+1 .
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Federal Reserve Bank of Richmond Economic Quarterly
We dwell on the challenges of forecasting inflation because of another
statistical issue. For instrumental variables (GMM) estimation to be informative, it turns out that we need a significant amount of predictability. Imagine
reconstructing a forecast equation by regressing π t+1 on zt−1 . The F -statistic
for the joint significance of the variables zt−1 in this regression must be above
some threshold in order for the full GMM estimation of the hybrid NKPC to
yield meaningful results. If this F -statistic, or inflation predictability, is too
low, then the econometrician is said to be using weak instruments. In that case,
the subsequent estimates of the hybrid NKPC parameters will be imprecisely
estimated (possess large standard errors). Also, hypothesis tests may have the
wrong size (probability of type I error); for example, they may not reject often
enough. These problems will persist even in large samples.
Another symptom of the syndrome of weak identification is that estimates
may vary a great deal with changes to the instrument set. Two economists with
the same hybrid NKPC may obtain disparate parameter estimates when they
employ different, but apparently equally admissible, instrument sets, zt−1 . In
Table 2, this sensitivity is apparent in the GMM estimates of the reducedform and structural hybrid NKPC parameters that are grounded on different
combinations of the Galı́-Gertler instruments. Since different researchers have
tended to apply different instrument sets, weak identification might help to
explain the current lack of consensus on parameter estimates of the hybrid
NKPC.
Recall from Section 2 that least-squares estimation of the NKPC yields
inconsistent estimates; we cannot represent Et π t+1 simply by replacing it with
the actual value, π t+1 . Another useful result from research on weak instruments is that instrumental-variables estimates converge to least-squares estimates as the econometrician adds more and more weak instruments. The last
row of Table 2 shows what can happen in that case by reporting least-squares
estimates. With the exception of the larger value for λ̂, the least-squares estimates are similar to those in some previous rows. That similarity shows that
finding these plausible values for the coefficients does not necessarily imply
they have a sound statistical basis. And it raises the possibility that the GMM
estimates are only weakly identified.
Ma (2002), Mavroeidis (2005), Dufour, Khalaf, and Kichian (2006), and
Nason and Smith (2008) draw attention to the pitfalls of weak identification in
GMM estimation of the hybrid NKPC. One response to this issue has been to
reformulate the hybrid NKPC so that it involves fewer parameters. The idea
is simply that by trying to measure a shorter list of effects, the investigator
might have greater success in precisely measuring them. For example, one
could set γ b = 0 and so work with the original NKPC rather than the hybrid
version. In that case, π t−1 also would become available as an instrument.
A number of investigators—including Henry and Pagan (2004) and Rudd
and Whelan (2006)—suggest restricting the reduced-form, hybrid NKPC
J. M. Nason and G. W. Smith: Single-Equation Estimation
377
parameters so that γ b + γ f = 1. Imposing this restriction helps with identification by reducing the number of coefficients to be estimated by one. It turns
out, though, that this restriction is inconsistent with one popular interpretation
of the hybrid NKPC parameters, namely that they reflect an underlying Calvotype model of staggered pricing. To show this, we use the earlier equations
that Galı́ and Gertler (1999) outline to connect the hybrid NKPC parameters to
those of the Calvo pricing model, namely ω, θ , and β. The proposed restriction
gives
γb + γf =
ω βθ
+
= 1,
φ
φ
where φ = θ + ω[1 − θ (1 − β)]. Some algebra reveals that this restriction
implies that the fraction of firms that can change prices but choose not to is
ω = 1. However, this extreme result forces the reduced-form parameter on
marginal costs,
λ=
(1 − ω)(1 − θ )(1 − βθ )
,
φ
to equal zero. Although Galı́ and Gertler point out that β = 1 also is consistent
with γ b + γ f = 1, often this restriction is imposed without recourse to
calibrating the firm’s discount factor to one.
More generally, restricting the hybrid NKPC parameters can be problematic because we want to test hypotheses about all relevant values. We
next explore statistical methods that apply even if identification is weak. An
econometrician also can test the hybrid NKPC parameters (and compute their
confidence intervals) using methods that are robust to weak identification, i.e.,
that remain valid whether the instruments are weak or not.
Many of these robust methods are based on a 60-year-old statistical insight
from Anderson and Rubin (1949). Here is their idea, as applied to the hybrid
NKPC. Rewrite the equation by taking the future value of inflation to the lefthand side (without forecasting it) and by adding some list of other variables,
ut , on the right-hand side:
π t − γ f 0 π t+1 = γ b π t−1 + λxt + δut .
To create this composite variable on the left-hand side of the equation, we
need to choose a value for γ f , labelled γ f 0 . We cannot use this regression
to estimate that value. But it can be used to test any value for this weight on
expected future inflation. To test the hypothesis that γ f = γ f 0 , we simply
perform a traditional F -test of the hypothesis that δ = 0 so that the auxiliary variables, ut , are insignificant. The logic is that if we happen to select
the correct value for γ f , then the three explanatory variables in the hybrid
NKPC will reproduce the time-series pattern in inflation π t , and no systematic
pattern in the residuals will be detected by including other macroeconomic
variables, ut .
378
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Anderson-Rubin Tests
10
0.6
p
0.5
0.4
6
F
0.3
4
0.2
2
0.1
0
0.0
p-Value
F-Statistic
8
0.5
1.0
1.5
0.0
2.0
γ
To illustrate the Anderson-Rubin (AR) test, we collect auxiliary variables
that include the 90-day Treasury bill interest rate, rt (again, a natural variable
to consider in forecasting inflation), as well as extra lags of inflation and unit
labor costs. The complete list is: ut = {rt , rt−1 , xt−1 , π t−2 }. The sample
period is 1955:1–2007:4. We run the regression on a fine grid of values of
γ f 0 between 0 and 2. For each such value we record the F -statistic associated
with the restriction that none of the variables in ut enters the equation, and we
calculate the corresponding p-value by locating the statistic in the F (4, 204)
distribution. Figure 3 graphs the candidate values, γ f 0 , on the horizontal axis
and the F -statistics (the solid black line) and their p-values (the dashed gray
line) on the two vertical axes.
Figure 3 shows that the AR test rejects the restrictions for low values of
the weight on expected future inflation and also for high values. In particular,
when γ f 0 is less than 0.5 or greater than 1.5, the F -statistics are high and the
p-values are low. This means that δ̂ is far from zero, the auxiliary variables,
ut , enter the equation, and so the candidate values of γ f 0 can be rejected. The
test does not reject at intermediate values of γ f 0 . The F -statistic reaches its
minimum and the associated p-value its maximum for γ f 0 around 1.0.
We already know that Table 2 has CU-GMM estimates of γ f that are a
large positive fraction (though the estimate depends on the instrument set)
with a small standard error. Moreover, the J test did not reject the overidentifying restrictions. So what is gained from the AR approach? The answer is
that tests in Table 2 may have been affected by weak identification, whereas
statistics in Figure 3 apply whether identification is weak or not. To illustrate
the effect of this robust method on inference, note that the range of values
for which the F -statistics in Figure 3 fall below the α-percent critical value
of the F -distribution (equivalently the p-values lie above α) constitutes the
J. M. Nason and G. W. Smith: Single-Equation Estimation
379
1 − α-percent confidence interval for γ f . In this case, the 90 percent confidence interval is (0.66, 1.62) and the 95 percent confidence interval is (0.61,
1.78). For comparison, the traditional, asymptotic confidence intervals for γ f
from the GMM estimates in the second-to-last row of Table 2 are (0.65, 0.81)
at the 90 percent level and (0.63, 0.83) at the 95 percent level. These intervals
understate the uncertainty, compared with the intervals that are robust to weak
instruments. The AR test suggests a positive value for γ f , but considerable
uncertainty or imprecision remains, and values greater than 1 are possible.
How to draw inference with weak instruments is an active area of research
by statisticians. Excellent surveys of inference under weak identification are
provided by Dufour (2003) and Andrews and Stock (2006). The AR test
assumes xt is exogenous, whereas some more recent methods allow it to be
endogenous. These methods allow tests of all the NKPC parameters, whereas
we have focused only on γ f . One important finding in this research is that
the AR test also may lack power, especially when there is overidentification.
In other words, it may fail to reject a false, assumed value γ f 0 and so give
too wide a confidence interval. This outcome is particularly likely if there are
many auxiliary variables, ut , and some are irrelevant as instruments.
Using identification-robust methods, Ma (2002) finds large confidence
sets for the hybrid NKPC parameters, which suggests that they are weakly
identified. Dufour, Khalaf, and Kichian (2006) apply the AR test and some
more recent tests to the United States for a 1970–1997 sample. They too find
wide confidence sets. Nason and Smith (2008) reject the hybrid NKPC for the
United States—by finding empty confidence intervals—when testing either
reduced-form parameters or the underlying ones ω, θ , and β. For no value of
γ f 0 does the hybrid NKPC produce unpredictable residuals, so the confidence
intervals are empty. (They use a slightly different definition of xt , described
in Section 6.) Kleibergen and Mavroeidis (2008) use identification-robust
methods and conclude that γ f > γ b . However, they find wide confidence
intervals, especially for γ b and for λ, where the confidence interval includes
zero. They also apply a stability test devised by Caner (2007) that is robust to
weak identification. This test suggests that the NKPC experienced a structural
break around 1984 and subsequently became flatter. Overall, methods that are
robust to weak identification suggest more skepticism about the NKPC than
do traditional econometric tools. They reveal considerable uncertainty about
the NKPC parameters or, in some cases, reject all reasonable values.
One way to gain power in tests like these (or to find more precise estimates of the hybrid NKPC parameters) is to utilize more information on
the inflation forecasting equation and the evolution of the exogenous variable
xt , a conclusion that directs us to consider systems of econometric equations
that set the hybrid NKPC within a broader economic/statistical model. In
these systems, researchers supplement the hybrid NKPC either with (a) an
explicit, statistical forecasting model that recognizes that xt is most likely
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Federal Reserve Bank of Richmond Economic Quarterly
endogenous, or (b) additional equations like a policy rule and dynamic I S
curve so as to form a coherent New Keynesian model. Either of these approaches can potentially provide more precision at the cost of introducing bias
if the added assumptions are misspecified. Studies that use forecasting models (vector autoregressions) include those of Fuhrer (1997), Sbordone (2002,
2005), Kurmann (2005, 2007), and Rudd and Whelan (2005a, 2005b, 2006),
while Lindé (2005) uses a three-equation New Keynesian model. On the
other hand, Galı́, Gertler, and López-Salido (2005) review these approaches
and conclude that GMM estimation remains informative. Schorfheide’s article in this issue provides a complete review of systems estimation of the hybrid
NKPC.
5.
FORECAST SURVEY DATA
As we have noted, many of the statistical challenges with estimating and testing the NKPC arise because inflation expectations cannot be directly observed.
There is an alternative to constructing these forecasts with instrumental variables, though, and that is simply to ask some people what they expect the
inflation rate to be in the next quarter. The Federal Reserve Bank of Philadelphia does just this in its Survey of Professional Forecasters (SPF). There are
other measures of actual forecasts, but they tend to belong to forecasters either with (potentially) more information (in the case of the Federal Reserve’s
Greenbook forecasts) or different information (in the case of the Michigan
household survey) than we might attribute to a typical, price-setting firm.
These issues have helped to make the SPF the most widely used data source
in this context. Another reason to favor the survey-based measures is that they
are in real time. Unlike our typical, instrumental-variables estimates, their
construction does not involve estimation with any data reported subsequent
to the date of the forecast. Roberts (1995), Orphanides and Williams (2002,
2005), Adam and Padula (2003), Dufour, Khalaf, and Kichian (2006), Zhang,
Osborn, and Kim (2008), and Brissimis and Magginas (2008) use forecasts to
estimate the NKPC.
Next, we see what happens when we use the median forecast from the SPF
in our estimator. The series on expected inflation, labeled π st+1 , is the median
of the one-quarter-ahead forecasts of the GDP deflator growth rate quarter-toquarter at annual rates, dpgdp3 from the SPF file MedianGrowth.xls, and
is available for 1968:4–2007:4. In fact, the SPF survey referred to the GNP
deflator until the end of 1991. This matters for the actual inflation rate, π t ,
used to estimate the hybrid NKPC when the median SPF inflation is equated
with expected inflation. In this case, we measure π t with GNPDEF from FRED
for the period prior to 1991.
As a benchmark, we present CU-GMM results similar to those in Table 2,
but with a 1969:1–2007:3 sample. The first row of Table 3 has these results.
J. M. Nason and G. W. Smith: Single-Equation Estimation
381
Table 3 Forecast Surveys in the U.S. NKPC, 1969:1–2007:3
Forecast
E π t+1 | zt
E π st+1 | zt
π st+1
(NLLS)
γ̂ b
(se)
γ̂ f
(se)
100×λ̂
(se)
ω̂
(se)
θ̂
(se)
β̂
(se)
J (df )
(p)
0.27
(0.07)
0.72
(0.07)
2.46
(1.66)
0.30
(0.10)
0.81
(0.05)
0.97
(0.06)
3.58
(0.73)
0.38
(0.12)
0.56
(0.17)
1.36
(4.56)
0.50
(0.15)
0.83
(0.24)
0.83
(0.23)
11.63
(0.07)
0.36
(0.03)
0.68
(0.03)
-0.14
(0.13)
0.56
(0.04)
0.91
(0.11)
1.16
(0.09)
Notes: Data are de-meaned prior to estimation. The estimator is CU-GMM with a
Newey-West HAC correction and automatic, plug-in lag length. The instrument vector is
π t−1 , xt−1 , xt−2 , wit−1 , wit−2 , yt−1 , yt−2 , cpt−1 , cpt−2 using a linearly detrended output gap.
The weight on future inflation is greater than the weight on past inflation,
and both are estimated precisely. The estimated values for the underlying
parameters ω, θ, and β, are similar to some of those found in Table 2. The J
test does not reject, but the coefficient on marginal costs, λ̂, while positive, is
statistically insignificant.
The second row of Table 3 lists results when the median survey value is
equated with expected inflation. We continue to estimate with CU-GMM to
allow for the possibility that this inflation expectations measure is contaminated with measurement error that is correlated with the survey measure. By
comparison, with the CU-GMM estimates in the top row, the weights on past
and future inflation tilt, with a larger weight on lagged inflation and a smaller
one on expected future inflation in the second row of Table 3. There is now no
significant role for marginal costs in explaining the inflation series (λ̂ is smaller
than its standard error) and the J test rejects the overidentifying restrictions
at conventional levels of significance.
Brissimis and Magginas (2008) perform a similar exercise but find that the
γ̂ f and γ̂ b weights tilt in the opposite direction, with a large weight on expected
future inflation and no statistically significant weight on lagged inflation. They
use the Bureau of Labor Statistics measure of the labor share of output as xt ,
whereas we use the adjusted Sbordone (2002) measure that also is adopted by
Galı́ and Gertler (1999). This sensitivity of the findings with forecast survey
data to the measure of marginal costs may show either that we need further
research on modeling marginal costs or that this is not a fruitful way to model
expectations in the hybrid NKPC.
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Finally, we replace the unobservable E[π t+1 |It ] with π st+1 and estimate
the hybrid NKPC by least squares. Taking this step does not mean assuming
these two series coincide. Instead, it yields consistent estimates whenever
the median survey is based on less information than that reflected in forecasts
driving actual inflation, so that
E[π t+1 |It ] = π st+1 + ηt ,
in which ηt is uncorrelated with π st+1 . In other words, it assumes that the
median forecast is an unbiased predictor of the broader-based inflation forecast
that influences the behavior of Calvo price setters.
The third row of Table 3 contains the results of least-squares estimation
with the median report from the SPF. The striking finding is that λ̂ is negative
so that real unit labor costs enter the equation with the wrong sign. However,
the point estimate is small and statistically insignificant. This finding can be
viewed as evidence against the use of the median survey measure. Perhaps
there is an errors-in-variables problem associated with this representation of
expected inflation. But it is not straightforward to explain a negative coefficient, albeit an insignificant one, which argues that this finding also can be
viewed as evidence against the hybrid NKPC.
A resolution to the question of how to represent expected inflation, that
is, with instrumental variables forecasts or survey forecasts, can be found by
including both. Smith (2007) and Nunes (2008) include a linear combination
of the two measures and ask which combination best explains current inflation.
The estimating equation becomes
π t = γ b π t−1 + γ f μE[π t+1 |zt ] + (1 − μ)π st+1 + λxt .
Nunes offers an economic interpretation of this mixture as reflecting pricesetters’ different forecasting methods. Smith instead has a purely statistical
interpretation. Either way, the evidence is that both measures matter. Their
estimates place a slightly greater weight on the survey measure than on the
econometric measure. The estimated hybrid NKPC parameters {γ̂ b , γ̂ f , λ̂} in
these studies resemble those in Table 2 and are consistent with theory.
6. WHAT DRIVES INFLATION?
Up to this point we have studied the hybrid NKPC in which inflation tracks
real unit labor costs. But several authors have argued that measures of the
output gap (i.e., the cyclical component of real GDP) are better explanatory
variables for inflation. This section sets both types of x-variables in the hybrid
NKPC to learn which might be most useful for explaining U.S. inflation. We
first describe the properties of nine different candidate x-variables. We then
use these series to estimate the hybrid NKPC.
We consider two measures of real unit labor costs. RULC1 is the Sbordone
(2002) measure described earlier in Table 1. RULC2 measures real unit labor
J. M. Nason and G. W. Smith: Single-Equation Estimation
383
cost as the cointegration relation of the logarithm of nominal unit labor cost
with the logarithm of the GDP deflator (allowing for an intercept and time
trend). The estimated cointegrating coefficient is 1.03. Nason and Slotsve
(2004) show that RULC2 is consistent with Calvo’s staggered pricing mechanism. Nason and Smith (2008) use this variable in their estimated hybrid
NKPC.
The first measure of the output gap, labeled CBO, is published by the Congressional Budget Office. The remaining measures are based on per capita
output. LT (QT) is the series of residuals from linearly (quadratic) detrending
per capita real GDP. Next, measures UC and BN are based on the unobservedcomponents model and Beveridge-Nelson decomposition, respectively. Both
of these measures treat real per capita output as the sum of a permanent component and a transitory component. These time-series models assume the permanent component of output is a random walk with drift while the transitory
component follows a second-order autoregression. The difference is that the
UC model imposes a zero correlation between innovations to the permanent
and transitory components. The BN decomposition estimates this correlation,
which is -0.97. Maximum likelihood estimation of the UC and BN models
is undertaken with the Kalman filter, and the associated output gap estimates
are filtered, not smoothed. The UC and BN output gap measures rely on the
work of Morley, Nelson, and Zivot (2003).
Measure BK is based on the Baxter and King (1999) bandpass filter. Since
the technical details for this implementation are not straightforward, we refer
statistically inclined readers to Harvey and Trimbur (2003). They show how to
estimate the BK cycle or output gap with the Kalman filter. Finally, measure
HP is the cycle that remains after applying the Hodrick and Prescott (1997)
filter to output growth, as implemented by Cogley and Nason (1995).
Figure 4 plots the nine (demeaned) measures. All series are shown since
1955 (omitting the volatile Korean war period), but the vertical scale varies
across the three panels. In the top panel the two measures of marginal cost
have different trends, but RULC1 tends to be dominated by low-frequency
movements. The middle panel shows the CBO output gap and the two deterministically detrended output gaps. These three generate more cycles than the
marginal cost measures and are also more volatile than RULC1 and RULC2.
The CBO, LT, and QT output gaps behave similarly except during the late
1960s and since 1999. The bottom panel of Figure 4 presents UC, BN, BK,
and HP measures of the output gap. The BN and BK output gaps have most of
their variation between two and four years per cycle, while relatively lowerfrequency fluctuations produce most of the variation in the UC and HP output
gaps. Volatility varies from one measure to another, with the UC and HP
output gaps exhibiting the most variance.
Nason and Smith (2008) show that being able to predict future xt can
be key for the viability of single-equation approaches to the NKPC. Recall
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 4 Marginal Costs and Output Gaps
Panel A: Real Unit Labor Costs
0.10
0.05
0.00
-0.05
-0.10
RULC1
RULC2
-0.15
1960
1970
1980
1990
2000
Panel B: CBO and Deterministically Detrended Output Gaps
0.10
0.05
0.00
-0.05
CBO
LT
QT
-0.10
1960
1970
1980
1990
2000
Panel C: Kalman Filter Estimated Output Gaps
0.1
0.0
UC
BN
BK
HP
-0.1
-0.2
1960
1970
1980
1990
2000
that (a) according to the hybrid NKPC, inflation is related to lagged inflation
and to the present value of current and future xt , and (b) finding instruments
involves predicting next quarter’s inflation rate, π t+1 . Combining these two
facts means that we must predict future values of xt in order to identify the
NKPC.
One possibility discussed in Section 4 is that xt can be forecasted using
its own lagged values. In that case, higher-order dynamics are needed for
identification. The idea that some complicated dynamics in xt help us learn
about the NKPC makes intuitive sense. If there are predictable movements
in these fundamentals, they should be matched by swings in inflation. The
J. M. Nason and G. W. Smith: Single-Equation Estimation
385
extent to which they are matched can shed light on whether the NKPC is a
good guide to inflation. If, instead, there are no predictable movements in xt
that inflation is theorized to be tracking, there will be no way to identify the
response of inflation.
We test for lag length in univariate autoregressions for each of the nine
x-variables using the Akaike information criterion, Hannan-Quinn information criterion, Schwarz or Bayesian information criterion, and likelihood ratio
(LR) test. The evidence is that for most of these series there are complicated
dynamics in which three to five lags contain forecasting information. The
two measures of unit labor costs, RULC1 and RULC2, and the BN output
gap appear to be exceptions, because LR tests suggest high-order dynamics
that reach 10 to 12 lags. These results suggest the RULCs and output gaps
have the requisite dynamics to overidentify the three structural price-setting
or reduced-form parameters of the hybrid NKPC.
Of course, our main reason for studying RULC1 and RULC2 or the output
gaps is to use them in the hybrid NKPC. Thus, the main goal of this section is
to estimate this NKPC with the nine x-measures. Table 4 contains the results.
When we estimate using RULC1 or RULC2 we find a small positive effect
of marginal costs on inflation. The significance of this effect depends on the
instrument set, as we documented in Section 2.
The remaining rows of Table 4 present NKPC estimates using the output
gaps. We find no significant role for any of these x-measures. The coefficient
on xt , λ̂, is small and negative (albeit statistically insignificant) for all the output
gaps. However, most economists would predict the opposite effect: a positive
output gap leading to a rise in prices. The other hybrid NKPC coefficients
also take surprising values that are difficult to interpret. The coefficient on
lagged inflation is negative, while the coefficient on expected future inflation
is greater than one. These coefficients on past and future inflation most likely
are affected by omitted-variables bias. Without some confidence in one’s
measure of the x-variable that inflation tracks in the NKPC, there cannot be
much confidence in estimates of inflation inertia or other properties of inflation
dynamics.
Investigators who work with an output gap might sometimes wonder
whether their findings depend on the specific filtering or detrending procedure
they use to measure this variable. We have used measures that are commonly
adopted and that have been used in forecasting or explaining inflation, yet
found no role for any of them. Our evidence suggests little support for the
idea that the output gap drives U.S. inflation.
Some recent studies work with inflation and output gaps and do find statistical links between them. Harvey (2007) adopts an unobserved-components
model of both inflation and output and finds a link between the cyclical components of the two series. Basistha and Nelson (2007) use the NKPC to define
or measure the output gap so that it fits into the NKPC by construction. But
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Federal Reserve Bank of Richmond Economic Quarterly
Table 4 U.S. New Keynesian Phillips Curve, 1955:1–2007:4
x-Measure
RULC1
RULC2
CBO
LT
QT
UC
BN
BK
HP
E π t − γ b π t−1 − γ f π t+1 − λxt | zt = 0
γ̂ f
100×λ̂
γ̂ b
(se)
(se)
(se)
0.28
0.68
0.56
(0.16)
(0.18)
(0.27)
0.34
0.61
0.70
(0.16)
(0.18)
(0.37)
−0.57
1.70
−3.90
(0.84)
(1.02)
(3.13)
−0.58
1.76
−2.90
(0.70)
(0.88)
(2.20)
−0.66
1.82
−2.60
(0.77)
(0.94)
(1.83)
−0.51
1.59
−0.97
(0.55)
(0.65)
(0.65)
−0.43
1.56
−1.78
(0.83)
(1.02)
(2.89)
−0.66
1.83
−2.15
(1.68)
(2.03)
(2.88)
−0.54
1.70
−1.84
(0.69)
(0.87)
(1.34)
J (df )
(p)
1.28(1)
(0.26)
1.64(1)
(0.20)
0.02(1)
(0.89)
0.02(1)
(0.90)
0.11(1)
(0.73)
0.80(1)
(0.37)
0.39(2)
(0.94)
0.34(3)
(0.84)
0.06(1)
(0.80)
Each equation includes a constant term and each instrument set includes a vector of ones.
Instruments are xt , ..., xt−J +1 and π t−1 , where J is the lag length estimated from the
AIC. Estimation is by CU-GMM with a quadratic-spectral kernal HAC estimator.
these studies do not test for the role of conventionally measured output gaps
in the standard NKPC. Conversely, there also is statistical work that, like ours,
questions the links between measures of the output gap and inflation. For
example, Orphanides and van Norden (2005) find that output gaps do not help
forecast inflation when both are measured realistically in real time (rather than
in revised data).
Marginal Costs Revisited
We have seen that the NKPC that uses the labor share to represent RULCs is
relatively successful empirically. This measure of costs is easy to construct
and has intuitive appeal. But some labor market arrangements imply that
this measure is misspecified, so some recent research augments this model of
marginal costs.
Macroeconomic models contain descriptions of the production technology
that firms use. Models that contain different technologies will predict different
ways to measure the marginal cost variable toward which firms adjust their
prices. In particular, if a firm faces other frictions besides the costs of adjusting
J. M. Nason and G. W. Smith: Single-Equation Estimation
387
Table 5 Augmenting the Labor Share
Study
Blanchard and Galı́ (2005)
Christiano, Eichenbaum, and Evans (2005)
Ravenna and Walsh (2006)
Chowdury, Hoffman, and Schabert (2006)
Krause, López-Salido, and Lubik (2008)
Coenen, Levin, and Christoffel (2007)
Batini, Jackson, and Nickell (2005) and
Guerrieri, Gust, and López-Salido (2008)
Real Rigidity
Real wage stickiness
Multiple
Financial friction
(cost channel)
Labor market search
Fixed and random
length Calvo pricing
Foreign competition
Factor Added
Unemployment rate
Interest rate
Interest rate
Hiring cost
Inflation target
Import prices
prices, those may affect how it sets prices. For example, imagine a firm that
must borrow from a bank to finance its wage bill. An increase in the interest
rate it pays then will act like a cost shock and affect how it prices its goods.
These additional frictions are sometimes called “real rigidities.” They can
include the financing constraint just mentioned, sticky real wages, or costs of
hiring new employees.
Table 5 lists several recent studies that augment the labor-share measure
of real unit labor costs with additional variables. Moreover, several of these
studies estimate the NKPC by GMM with the revised measures of xt and
find statistical support for the added terms or right-hand-side variables. Few
economists would argue that our model of firms’ costs should be chosen according to how well it explains inflation in the NKPC, and these studies also
examine other empirical evidence. But it is promising that a range of plausible
modifications have improved the fit of the NKPC (its success in passing tests
of overidentifying restrictions or stability tests) without significantly altering
the findings about forward-looking and backward-looking weights.
7.
MEASURES OF INFLATION
Conclusions about the NKPC also might depend on how the inflation rate is
measured. The statistics so far have been based on the GDP deflator, so it
seems natural to wonder whether they change if we measure inflation using
another index such as the consumer price index (CPI), the deflator for personal
consumption expenditure, or the producer price index. To check on the first of
these alternatives, we average the monthly CPI (all items, all urban consumers,
seasonally adjusted), CPIAUCSL from FRED to find the quarterly value, then
construct the inflation rate as the annualized, quarter-to-quarter growth rate in
percentage points.
Figure 5 shows this CPI inflation rate (the dashed gray line) and the inflation rate measured with the GDP deflator (the solid black line) used so far
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Federal Reserve Bank of Richmond Economic Quarterly
Quarterly Inflation Rate (Annualized)
Figure 5 Inflation in the GDP Deflator and CPI
CPI
15
10
5
GDP Deflator
0
1960
1970
1980
1990
2000
in this article. The figure shows a common, low-frequency cycle in the two
measures of quarterly inflation. But the CPI inflation rate is more volatile.
The only persistent difference between the two series occurred in the late
1970s when CPI inflation exceeded deflator inflation for several consecutive
quarters.
When we estimate the NKPC with CPI inflation and RULC1 and RULC2,
the results change modestly. The coefficient on lagged inflation, γ̂ b , is slightly
larger, and the coefficient on expected future inflation, γ̂ f , is slightly smaller.
The coefficient on marginal costs is smaller and is estimated less precisely.
Finally, when we combine the CPI inflation rate with the seven output gaps,
the results are quite negative for that approach, just as in the previous section.
Overall, we conclude that the evidence summarized so far does not depend
significantly on how the inflation rate is measured.
8.
INTERNATIONAL EVIDENCE
Researchers also have used single-equation methods to study the NKPC in
other countries. Galı́, Gertler, and López-Salido (2001) find the hybrid NKPC
fits well in quarterly Euro-area aggregate data for 1970–1998. As in the U.S.
data, γ̂ f > γ̂ b , λ̂ is statistically significant, and the J test does not reject
overidentifying restrictions. Neiss and Nelson (2005) compare estimates of
the NKPC for the United States, United Kingdom, and Australia. They also
propose a new measure of the output gap that statistically explains inflation as
well as measures of marginal costs. Leith and Malley (2007) estimate hybrid
NKPCs for the G7 countries for 1960–1999, while Rumler (2007) does so
for eight Euro-area countries for 1980–2003. Both of these studies discuss
the role of the terms of trade, in addition to the labor share, in measures of
marginal costs. They also report on differences in parameter estimates (such
as those measuring price stickiness or inflation inertia) across countries. The
J. M. Nason and G. W. Smith: Single-Equation Estimation
389
international evidence on the hybrid NKPC may provide a guide to reform
the measurement of marginal costs in open economies, in that the effects of
foreign trade may be easier to detect in small, open economies than in the
United States.
Batini, Jackson, and Nickell (2005) extend the model of marginal costs to
reflect the relative price of imports, varying markups, and costs of adjusting
employment. They arrive at a hybrid NKPC that nests the standard version and
improves on its fit for the United Kingdom for 1972–1999. Bårsden, Jansen,
and Nymoen (2004) also estimate more general statistical models of inflation,
for the Euro area. They find that the hybrid NKPC can be improved on in
terms of forecasting inflation even though it passes the J test.
Nason and Smith (2008) estimate the NKPC for the United Kingdom and
Canada and provide tests that are robust to weak instruments. As in U.S. data,
they find that the robust tests and traditional single-equation GMM estimation
give different messages. The robust tests provide little evidence of forwardlooking dynamics in these NKPCs. This international research thus conveys
a similar message to the work on U.S. data.
9.
CONCLUSION
This article outlines single-equation econometric methods for studying the
NKPC and offers a progress report on the empirical evidence. How successful
is the NKPC when estimated and tested on U.S. inflation? Enter the proverbial
two-handed economist. On the one hand, the hybrid NKPC estimated by
GMM on a quarterly 1955–2007 sample has coefficients that have signs and
sizes that accord with economic theory and are statistically significant. The
structural coefficients (ω̂, θ̂ , and β̂) are positive fractions, as are the reducedform coefficients on past inflation and expected future inflation (γ̂ b and γ̂ f ),
while the slope of the reduced-form Phillips curve (λ̂) is positive. The hybrid
NKPC also passes statistical tests based on the unpredictability of its residuals
(the J test) and its stability over time (the sup-Wald test). The findings are
not sensitive to alternative measures of inflation. Real unit labor costs are
much better at statistically explaining inflation than are a plethora of output
gap measures.
On the other hand, the t-statistic on real unit labor costs usually is not much
above two. This indicates that there is not a close relationship between inflation
and this measure of marginal costs. Estimates of the NKPC using surveys
of forecasts give very different coefficients from those using instrumentalvariables estimation. The confidence interval that is valid even with weak
instruments gives a wide range of possible values for the parameter on expected
future inflation. Moreover, other tests that are robust to weak identification
often yield unreasonable values for the other hybrid NKPC parameters or reject
the NKPC entirely. Our macroeconomic “Rip van Winkle,” accustomed to the
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Federal Reserve Bank of Richmond Economic Quarterly
evidence against the Phillips curve garnered during the 1970s and 1980s, might
find that the world has not changed much after all.
How will we learn more from single-equation methods? One promising
and active avenue of research focuses on measurement of the cost variable, xt ,
toward which prices adjust. Econometric tools for drawing inferences with
weak identification also continue to advance. And the simple accumulation
of macroeconomic data over time may help with precision, too.
Of course, systems methods of estimation also continue to be fruitful ways
to identify and estimate the NKPC. Another complement to traditional, singleequation methods is to look at microeconomic data from individual firms or
industries. Economists increasingly ask whether macroeconomic models of
price stickiness are consistent with data on how prices are adjusted at the
microlevel. It may be possible to measure cost shocks in microeconomic data
and estimate pricing equations at that level, too.
The NKPC continues to be a key building block for macroeconomic models that require a monetary transmission mechanism. Our econometric work
shows that marginal costs may be superior to many output gaps as a guide
to inflation. We also obtain GMM estimates that give an important role to
expected future inflation in explaining current inflation, while lagged inflation
receives less weight. But measuring the effect of expected, future inflation on
current inflation can be problematic because of weak instruments. Future research on this key response would be valuable because such forward-looking
effects continue to have implications for the design of good monetary policy.
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Economic Quarterly—Volume 94, Number 4—Fall 2008—Pages 397–433
DSGE Model-Based
Estimation of the New
Keynesian Phillips Curve
Frank Schorfheide
A
n important building block in modern dynamic stochastic general
equilibrium (DSGE) models is the price-setting equation for firms.
In models in which the adjustment of nominal prices is costly, this
equation links inflation to current and future expected real marginal costs
and is typically referred to as the New Keynesian Phillips curve (NKPC). Its
most popular incarnation can be derived from the assumption that firms face
quadratic nominal price adjustment costs (Rotemberg 1982) or that firms are
unable to re-optimize their prices with a certain probability in each period
(Calvo 1983). The Calvo model has a particular appeal because it generates
predictions about the frequency of price changes, which can be measured with
microeconomic data (Bils and Klenow 2004, Klenow and Kryvtsov 2008).
The slope of the NKPC is important for the propagation of shocks and determines the output-inflation tradeoff faced by policymakers. The Phillips curve
relationship can also be used to forecast inflation.
This article reviews estimates of NKPC parameters that have been obtained
by fitting fully specified DSGE models to U.S. data. By now, numerous
empirical papers estimate DSGE models with essentially the same NKPC
specification. In this literature, the Phillips curve implies that inflation can
be expressed as the discounted sum of expected future marginal costs, where
marginal costs equal the labor share. We document that the identification of
Ed Herbst and Maxym Kryshko provided excellent research assistance. I am thankful to
Michael Dotsey, Andreas Hornstein, Thomas Lubik, and James Nason for helpful comments
and suggestions. Support from the Federal Reserve Bank of Richmond and the National Science Foundation (Grant SES 0617803) is gratefully acknowledged. The views expressed in
this article do not necessarily reflect those of the Federal Reserve Bank of Richmond or the
Federal Reserve System. E-mail: schorf@ssc.upenn.edu.
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the Phillips curve coefficients is tenuous and no consensus about its slope and
the importance of lagged inflation has emerged from the empirical studies.
We begin by examining how the NKPC parameters are identified in a
DSGE model-based estimation. This is a difficult question. Many estimates
are based on a likelihood function, which is the model-implied probability distribution of a set of observables indexed by a parameter vector. The likelihood
function peaks at parameter values for which the model-implied autovariance function of a vector of macroeconomic time series matches the sample
autocovariance function. Unfortunately, this description is not particularly
illuminating. More intuitively, the NKPC parameters are estimated by a regression of inflation on the sum of discounted future expected marginal costs.
The likelihood function corrects the bias that arises from the endogeneity of
the marginal cost regressor. We show that if one simply uses ordinary leastsquares (OLS) to regress inflation on measures of expected marginal costs,
the slope coefficient is very close to zero. This finding is quite robust to the
choice of detrending method and marginal cost measure. Hence, much of
the variation in the estimates reported in the literature is due to the multitude
of endogeneity corrections that arise by fitting different DSGE models that
embody essentially the same Phillips curve specification.
The review of empirical studies distinguishes between papers in which
marginal costs are included in the observations and, hence, are directly used
in the estimation and studies that treat marginal costs as a latent variable. In
the latter case, NKPC estimates are more sensitive to the specification of the
households’ behavior, the conduct of monetary policy, and the law of motion
of the exogenous disturbances. Estimates of the slope of the Phillips curve lie
between 0 and 4. If the list of observables spans the labor share, then the slope
estimates fall into a much narrower range of 0.005 to 0.135. No consensus
has emerged with respect to the importance of lagged inflation in the Phillips
curve. We compare estimates of the relative movement of inflation and output
in response to a monetary policy shock, which captures an important tradeoff
for monetary policymakers. We find that the estimates in the studies that are
surveyed in this article range from 0.07 to 1.4. A value of 0.07 (1.4) implies that
a 1 percent increase in output due to a monetary policy shock is accompanied
by a quarter-to-quarter inflation rate of 7 (140) basis points.
The remainder of this paper is organized as follows. We discuss the
derivation of the NKPC as well as our concept of DSGE model-based estimation in Section 1. In Section 2, a simple DSGE model that can be solved
analytically is used to characterize various sources of NKPC parameter identification. Any particular DSGE model-based estimation might exploit some
or all of these sources of information. Section 3 provides empirical evidence
from least-squares regressions of inflation on the discounted sum of future
marginal costs as well as evidence from a vector autoregression (VAR) on the
relative movement of output and inflation in response to a monetary policy
F. Schorfheide: DSGE Model-Based NKPC Estimation
399
shock. We thereby characterize some features of the data that are important
for understanding the DSGE model-based parameter estimates reviewed in
Section 4. Finally, Section 5 concludes.
1.
PRELIMINARIES
This section begins with a brief description of the price-setting problem that
gives rise to a Phillips curve in New Keynesian DSGE models. We then
discuss some of the defining characteristics of DSGE model-based estimation
of NKPC parameters.
Price Setting in DSGE Models
New Keynesian DSGE models typically assume that production is carried
out by two types of firms: final good producers and intermediate goods
producers. The latter hire labor and capital services from the households
to produce a continuum of intermediate goods. The final good producers purchase the intermediate goods and bundle them into a single aggregate good
that can be used for consumption or investment. The intermediate goods are
imperfect substitutes and, hence, each producer faces a downward-sloping
demand curve. Price stickiness is introduced by assuming that it is costly to
change nominal prices. Rotemberg (1982) assumed that the price adjustment
costs are quadratic, whereas Calvo (1983) set forth a model of staggered price
setting in which the costs are either zero or infinite with fixed probabilities,
i.e., only a fraction of firms is able to change or, more precisely, re-optimize
prices.
Aggregating the optimal price-setting decisions of firms leads to the following expression for inflation in the price of the final good, referred to as the
New Keynesian Phillips curve:
t + ξ̃ t .
π̃ t = γ b π̃ t−1 + γ f Et π̃ t+1 + λMC
(1)
t is real marginal costs, and
Here
π t represents inflation, MC
ξ t is an
exogenous disturbance that is often called a mark-up shock. We use
zt to
denote percentage deviations of a variable, zt , from its steady state. The
coefficients γ b , γ f , and λ are functions of model-specific taste and technology
parameters. For instance, in Calvo’s (1983) model of price stickiness
γb =
ω
,
1 + βω
γf =
β
, and
1 + βω
λ=
(1 − ζ )(1 − ζ β)
,
ζ (1 + βω)
where β is the households’ discount factor and ζ is the probability that an
intermediate goods producer is unable to re-optimize its price in the current
period. In the derivation of (1), it was assumed that those firms that are unable
to re-optimize their prices either adjust their past price by the steady-state
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Federal Reserve Bank of Richmond Economic Quarterly
inflation rate or by lagged inflation. The parameter ω represents the fraction
of firms that indexes their prices to lagged inflation.
Assuming that β = 0.99, the sum of γ b and γ f is slightly less than 1
and the coefficient of lagged inflation lies between 0 (no dynamic indexation,
ω = 0) and 0.5 (full dynamic indexation, ω = 1). If ω = 0 and steadystate inflation is 0, then 1/(1 − ζ ) can be interpreted as the expected duration
between price changes. For instance, ζ = 23 implies that the expected duration
of a price set by an intermediate goods producer is three quarters, which leads
to a slope coefficient of λ = 0.167. On the other hand, if ζ = 78 , which means
that the duration of a price is eight quarters, then the NKPC is much flatter:
λ = 0.018.
Our survey of the empirical literature will focus on coefficient estimates
for γ b , γ f , and λ rather than the model-specific preference-and-technology
parameters. The slope, λ, determines the output-inflation tradeoff faced by
central banks and affects, for instance, the relative response of output and
inflation in response to an unanticipated monetary policy shock. A detailed
exposition of the role that the NKPC plays in the analysis of monetary policy
is provided in an article by Stephanie Schmitt-Grohé and Martı́n Uribe in this
issue. The coefficient γ b affects the persistence of inflation and, for instance,
the rate at which inflation effects of shocks to marginal costs die out. This
is an important parameter, particularly for central banks that pursue a policy
of inflation targeting. If we rearrange the terms in (1), such that expected
inflation appears on the left-hand side and all other terms on the right-hand
side, then the Phillips curve delivers a forecasting equation for inflation.
DSGE Model-Based Estimation
This article focuses on estimates of γ b , γ f , and λ that are obtained by exploiting the full structure of a model economy. Thus, we consider approaches
in which the researcher solves not only the decision problems of the firms
but also those of the other agents in the economy and imposes an equilibrium
concept. If the economy is subject to exogenous stochastic shocks, the DSGE
model generates a joint probability distribution for time series such as aggregate output, inflation, and interest rates. Suppose we generically denote the
vector of time, t, observables by xt and assume that the DSGE model has been
solved by log-linear approximation techniques. Then the equilibrium law of
motion takes the form of a vector autoregressive moving average (VARMA)
process of the form (omitting deterministic trend components)
xt = 1 xt−1 + . . . p xt−p + R t + 1 R t−1 + . . . q R t−q .
(2)
The matrices i , j , and R are complicated functions of the Phillips curve
parameters γ b , γ f , and λ, as well as the remaining DSGE model parameters,
F. Schorfheide: DSGE Model-Based NKPC Estimation
401
which we will summarize by the vector θ . The vector t stacks the innovations
to all exogenous stochastic disturbances and is often assumed to be normally
and independently distributed.
A natural approach of exploiting (2) is likelihood-based estimation. Maximum likelihood (ML) estimation of optimization-based rational expectations
models in macroeconomics dates back at least to Sargent (1989) and has been
widely applied in the DSGE model literature (e.g., Altug [1989], Leeper and
Sims [1994], and many of the papers reviewed in Section 4). The likelihood
function is defined as the joint density of the observables conditional on the
parameters, which can be derived from (2). Let Xt = {x1 , . . . , xt }, then
p(X T |γ b , γ f , λ, θ ) = p(x1 |γ b , γ f , λ, θ )
T
p(xt |Xt−1 , γ b , γ f , λ, θ ). (3)
t=2
The evaluation of the likelihood function typically requires the use of
numerical methods to solve for the equilibrium dynamics and to integrate
out unobserved elements from the joint distribution of the model variables
(see, for instance, An and Schorfheide [2007]). A numerical optimization
routine can then be used to find the maximum of the (log-)likelihood function.
The potential drawback of the ML approach is that identification problems can
make it difficult to find the maximum of the likelihood function and render
standard large sample approximations to the sampling distribution of the ML
estimator and likelihood ratio statistics inaccurate.
A popular alternative to the frequentist ML approach is Bayesian inference. Bayesian analysis tends to interpret the likelihood function as a density
function for the parameters given the data. Let p(γ b , γ f , λ, θ ) denote a prior
density for the DSGE model parameters. Bayesian inference is based on the
posterior distribution characterized by the density
p(γ b , γ f , λ, θ|X T ) =
p(XT |γ b , γ f , λ, θ )p(γ b , γ f , λ, θ )
.
p(XT |γ b , γ f , λ, θ )p(γ b , γ f , λ, θ )d(γ b , γ f , λ, θ )
(4)
Notice that the denominator does not depend on the parameters and simply
normalizes the posterior density so that it integrates to one. The controversial
ingredient in Bayesian inference is the prior density as it alters the shape of the
posterior, in particular if the likelihood function does not exhibit much curvature. On the upside, the prior allows the researcher to incorporate additional
information in the time series analysis that can help sharpen inference. Many
of the advantages of Bayesian inference in the context of DSGE model estimation are discussed in Lubik and Schorfheide (2006) and An and Schorfheide
(2007). The implementation of Bayesian inference typically relies on Markovchain Monte Carlo methods that allow the researcher to generate random draws
of the model parameters from their posterior distribution. These draws can
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then be transformed—one by one—into statistics of interest. Sample moments
computed from these draws provide good approximations to the corresponding
population moments of the posterior distribution.
Notwithstanding all the desirable statistical properties of likelihood-based
estimators, the mapping of particular features of the data into parameter estimates is not particularly transparent. Superficially, the likelihood function peaks at parameter values for which a weighted discrepancy between
DSGE model-implied autocovariances of xt and sample autocovariances is
minimized. The goal of the next section is to explore the extent to which
this matching of autocovariances can identify the parameters of the New
Keynesian Phillips curve.
2.
IDENTIFYING THE NKPC PARAMETERS
The identification of DSGE model parameters through likelihood-based methods tends to be a black box because the relationship between structural parameters and autocovariances or other reduced-form representations is highly
nonlinear. This section takes a look inside this black box to develop some
understanding about particular features of the DSGE model that contribute to
the identifiability of NKPC parameters. Rather than asking whether there is
enough variation in postwar data to estimate the NKPC parameters reliably,
for now we focus on sources of identification in infinite samples. In practice,
the estimation of a particular model might exploit several of these sources of
information simultaneously.
Since the Phillips curve provides a relationship between marginal costs
and inflation, the measurement of marginal costs is important for the identification of the NKPC parameters. A key feature of likelihood-based inference—as
opposed to the single-equation methods reviewed by James Nason and Gregor
Smith in this issue—is the exploitation of model-implied restrictions of contemporaneous correlations between variables, as well as the use of information
from impulse responses. In many instances, higher-order autocovariances of
inflation and marginal costs are an additional source of information.
While this section focuses on identifying the slope, λ, we also offer some
insights into identifying γ b and γ f . For now we assume that γ b = 0. In the
context of the Calvo model this assumption implies that the fraction, ω, of
firms that engage in dynamic indexation is zero. In this case, γ f = β. Since
β in a fully specified DSGE model is related to the steady-state real interest
rate, the coefficient γ f can be determined, for instance, by averaging interest
rate data, and its identification is not a concern. Under our simplifications, the
Phillips curve takes the form
t +
π t = βEt [
π t+1 ] + λMC
ξt.
(5)
F. Schorfheide: DSGE Model-Based NKPC Estimation
403
Solving this difference equation forward we find that today’s inflation is a
function of future expected marginal costs:
πt =
∞
t+j +
β j Et [λMC
ξ t+j ].
(6)
j =0
Observed Versus Latent Marginal Costs
The identification of λ crucially depends on whether real marginal costs are
t is directly obtreated as directly observable or as a latent variable. If MC
served and, hence, is an element of the vector xt in (2) and (4), then the main
obstacle to the identification of λ is the endogeneity problem caused by the
potential correlation between the mark-up shock, ξ t , and marginal costs. The
estimation of future expected marginal costs in (6) poses no real challenge
t+j ] can be obtained from the reduced-form representation asbecause Et [MC
sociated with the law of motion (2), which is always identifiable. The downside of including a direct measure of marginal costs in the set of observables is
that measurement errors pertaining to the marginal cost series can potentially
distort the inference about the NKPC parameters. Yet, identifying λ is more
tenuous if marginal costs are not included in the vector xt .
To make the discussion more concrete, imagine an economy in which
labor is the only factor of production and, in log-linear terms,
t = Z
t + H
t .
Y
Zt is an unobserved total factor productivity process and Ht is hours worked.
Marginal costs are given by
t = W
t − Z
t ,
MC
where Wt are wages. Moreover, suppose that the households’ instantaneous
utility function is of the form
1−1/τ
Ct
− φHt ,
1 − 1/τ
and μt denotes the marginal utility of consumption. Under these preferences
labor supply is infinitely elastic, the wage has to satisfy Wt = 1/μt , and the
−1/τ
marginal utility of consumption is given by μt = Ct
. Finally, assume that
output is entirely used for household consumption such that Ct = Yt . Then
we obtain the following link between marginal costs and output:
t = 1 (Y
t ).
t − τ Z
MC
τ
If the vector of observables, xt , contains output, wages, and hours worked,
then the marginal costs are directly observed because
t = lsh
t = W
t + H
t − Y
t .
MC
U (Ct , Ht ) =
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Federal Reserve Bank of Richmond Economic Quarterly
More generally, in models with Cobb-Douglas technology the vector xt spans
t , from the
marginal costs as long as one can construct the labor share, lsh
observables. If, however, the vector xt only contains observations on output in
addition to inflation and interest rates, then marginal costs are latent because
they depend on the observed output as well as the unobserved technology
t , and the unknown parameter, τ . Rewriting (5) in terms of inflation
process, Z
and output yields
λ
π t = βEt [
π t+1 ] + Y
t − Zt + ξ t .
τ
t exacerbates the endogeneity
Two challenges arise. First, the presence of Z
problem that arises in the NKPC estimation. Moreover, the coefficient associt in itself does not identify the original slope parameter, λ, since it
ated with Y
also depends on the utility function parameter, τ , which needs to be identified
from other equilibrium relationships.
In practice, likelihood-based estimation of DSGE models relies on the socalled state-space representation of the DSGE model, rather than the VARMA
representation in (2). Omitting deterministic trend components, the statespace representation takes the form
xt = Ast ,
st = B1 st−1 + B t ,
(7)
where xt is the vector of observables, st is a vector of latent variables, and the
matrices A, B1 , and B are functions of the DSGE model parameters. The
likelihood function associated with (7) can be computed with the Kalman filter. If the information in the vector xt does not span marginal costs directly,
then the Kalman filter constructs an estimate of the latent marginal costs (and
t , in our example) based on xt and the parameters λ and θ . To
technology, Z
the extent that the Kalman filter inference for the latent variables is sensitive
to the assumed law of motion of the unobserved exogenous processes, inference about the slope of the Phillips curve is also sensitive to these auxiliary
assumptions.
Identifying Information in Contemporaneous
Correlations
Fully-specified DSGE models impose strong restrictions on the contemporaneous interactions of macroeconomic variables. We will show in the context
of a simple example that these restrictions enter the likelihood function and
potentially provide important identifying information that is not used in the
single-equation approaches reviewed by James Nason and Gregor Smith in
this issue. For the remainder of Section 2 we adopt the convention that all
variables are measured in percentage deviations from a deterministic steady
state and omit tildes to simplify the notation.
F. Schorfheide: DSGE Model-Based NKPC Estimation
405
Consider the log-linear approximation of the Euler equation associated
with the households’ problem in the previous subsection:
Yt = Et [Yt+1 ] − τ (Rt − Et [π t+1 ]) + φ,t .
(8)
Rt −Et [π t+1 ] is the expected real return from holding a one-period nominal
bond. The parameter, τ , can be interpreted as the intertemporal substitution
elasticity of the household and φ,t is an exogenous preference shifter. To complete the model, we characterize monetary policy by an interest rate feedback
rule of the form
Rt = ψπ t + R,t ,
(9)
where R,t is a monetary policy shock.
We now substitute the marginal cost expression derived in the previous
subsection into the NKPC and obtain
λ
π t = βEt [π t+1 ] + (Yt − τ Zt ) + ξ t .
(10)
τ
Since the unobserved technology shock, Zt , and the mark-up shock, ξ t , affect
the equilibrium law of motion in a similar manner in this simple model, we set
Zt = 0 and let ξ t = ξ ,t . Moreover, we define κ = τλ and will direct our attention to the estimation of the output inflation tradeoff, κ, rather than λ. Thus,
we are essentially abstracting from the two additional difficulties that arise if
marginal costs are treated as a latent variable. Finally, it is assumed that the
three exogenous shocks, R,t , φ,t , and ξ ,t are independently and identically
distributed zero mean normal random variables with standard deviations σ R ,
σ φ , and σ ξ , respectively.
The linear rational expectations (LRE) model comprised of (8) to (10) can
be solved with standard methods such as the one described in Sims (2002). To
ensure that the LRE system has a unique stable solution, we impose ψ > 1,
which implies that the central bank raises the real interest rate in response to
an inflation rate that exceeds its steady-state level. Lubik and Schorfheide
(2004) show that the equilibrium law of motion for the three observables is of
the form
⎡
⎡
⎤
Yt
−τ
1
⎣ πt ⎦ =
⎣ −κτ
1 + κτ ψ
1
Rt
1
κ
κψ
⎤⎡
⎤
R,t
−τ ψ
1 ⎦ ⎣ φ,t ⎦ .
ψ
ξ ,t
(11)
Since our model lacks both endogenous and exogenous propagation mechanisms, output, inflation, and interest rates—the three variables observed by
the econometrician—are serially uncorrelated in equilibrium. Thus, all the
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Federal Reserve Bank of Richmond Economic Quarterly
information about the slope of the Phillips curve must come from the contemporaneous correlations among the three observables.
The single-equation approach to the estimation of the NKPC reviewed by
James Nason and Gregor Smith in this issue can be interpreted in two ways.
First, one can write the NKPC as a regression of the form
1
κ
1
1
π t − Yt − ηt+1 − ξ ,t = α 1 π t + α 2 Yt + residt+1 . (12)
β
β
β
β
Here we replaced the conditional expectation of inflation, Et [π t+1 ], by π t+1
and a forecast error ηt+1 = π t+1 −Et [π t+1 ]. The lack of serial correlation in
the equilibrium dynamics implies that least-squares estimates of α 1 and α 2
converge in probability to zero. Hence, based on a large sample, an econometrician concludes that the slope of the Phillips curve is zero. The estimation
of (12) with an instrumental variable estimator that tries to correct a potential
bias due to the correlation between Yt and ξ ,t is also bound to fail because
in equilibrium, lagged values of output and inflation are uncorrelated with the
regressors.
Alternatively, one can express the Phillips curve as a regression of the
form
π t+1 =
π t = α 1 Et [π t+1 ] + α 2 Yt + residt .
(13)
However, even if the econometrician realizes that Et [π t+1 ] = 0 and excludes
the expected inflation regressor, it is not possible to estimate the slope of
the Phillips curve consistently. The least-squares estimator of α 2 provides a
biased estimate of κ because of the correlation between output and the markup shock, which is subsumed in the residual. Instrumental variable estimation
is also uninformative because lagged endogenous variables are uncorrelated
with current output. Notice that this failure of single-equation estimation is not
directly apparent from (10). It is a consequence of the auxiliary assumptions
about the other sectors in the economy and the law of motion of the exogenous
disturbances. Nason and Smith (2008) show that the identification problems
associated with single-equation methods prevail, even if the DSGE model is
enriched with serially correlated exogenous disturbances.
DSGE model-based estimation of the Phillips curve parameters utilizes
the information in the contemporaneous relationship between output, inflation,
and interest rates.1 Let θ = [τ , ψ, σ R , σ φ , σ ξ ] and factorize the joint density
of the observables as
1 It is tempting to check identification by comparing the number of structural parameters to
the number of free parameters in the covariance matrix of Yt , π t , and Rt . In the DSGE model
described by the law of motion (11), the two parameter counts are equal to six. Unfortunately,
having at least as many estimable reduced-form parameters as structural parameters is neither sufficient for identification, nor does it provide interesting insights about the sources of identification.
F. Schorfheide: DSGE Model-Based NKPC Estimation
p(Yt , π t , Rt |κ, θ ) = p(Yt |κ, θ )p(π t |Yt , κ, θ )p(Rt |Yt , π t , θ ).
407
(14)
The first term represents the marginal density of output and the third term is
generated by the monetary policy rule. Key to understanding the DSGE modelbased estimation of κ is the second term, that is, the conditional distribution
of inflation given output. Since all the shocks are normally distributed,
π t |Yt ∼ N (E [π t | Yt ] , var[π t |Yt ])
and we can focus our attention on the conditional mean and variance.
We begin with the derivation of E[π t | Yt ]. Solving the Phillips curve
relationship forward as in (6) leads to
π t = κYt + ξ ,t .
(15)
Taking expectations conditional on Yt of the left-hand side and right-hand side
of (15) yields
E [π t | Yt ] = κYt + E ξ ,t | Yt .
Using (11) and the formula for the conditional moments of a joint normal
distribution,2 we obtain
E ξ ,t | Yt = μξ |y (θ )Yt = −
τ 2 ψ 2 σ 2ξ
1
Yt .
τ ψ τ 2 σ 2R + σ 2φ + τ 2 ψ 2 σ 2ξ
(16)
sh(σ 2y , ξ )
The conditional expectation depends on the intertemporal elasticity of substitution, the policy rule coefficient, and all the shock variances. Here sh(σ 2y , ξ )
is the fraction of the variance of output that is due to the mark-up shock, ξ ,t .
We now turn to the calculation of the conditional variance of inflation. Notice
that var[π t |Yt ] = var[ ξ ,t |Yt ]. Thus,
var[π t |Yt ] = σ 2ξ |y (κ, θ ) = σ 2ξ −
(τ ψσ 2ξ )2
(1 + κτ ψ)(τ 2 σ 2R + σ 2φ + τ 2 ψ 2 σ 2ξ )
.
We deduce that
p(π t |Yt , κ, θ )
∝ |σ 2ξ |y (κ, θ )|−1/2 exp
1
2
π t − [κ + μξ |y (θ )]Yt
, (17)
− 2
2σ ξ |y (κ, θ )
2 Suppose that X and Y are jointly normally distributed with means μ and μ , variances
x
y
−1
vxx and vyy , and covariance vxy ; then the conditional mean E[X | Y = y] = μx + vxy vyy
(y − μy )
2
and the conditional variance is var[X|Y = y] = vxx − vxy /vyy .
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Federal Reserve Bank of Richmond Economic Quarterly
where ∝ denotes proportionality.
We can draw several important conclusions from (17). First, the term μξ |y
given in (16) corrects for the endogeneity bias that arises in a regression of
inflation and marginal costs. Suppose we set ψ = 1.5, which is Taylor’s (1993)
value, assume that τ = 23 , which makes the agents slightly more risk-averse
than agents with log preferences, and assume that 20 percent of the variation
in output is due to mark-up or cost-push shocks. Then (16) implies that a
simple least-squares regression of inflation on marginal costs, i.e. output,
in our example model, would underestimate the slope, κ, by 0.2. Second,
(17) implies that knowledge of the conditional distribution of inflation given
output does not identify the slope of the Phillips curve. Moreover, the joint
distribution of output and inflation is also not sufficient, because the marginal
distribution of output only provides information about the variance of output,
σ 2y (κ, θ ), which is insufficient to disentangle the values of all the θ elements.
We will show below, however, that κ is identifiable with knowledge of the
monetary policy reaction function.
To summarize, our simple example has a number of startling implications.
First, a single-equation estimation based on (12) or (13) is unable to deliver
a consistent estimate of κ. Second, an OLS regression of inflation on the
sum of discounted future expected marginal costs generates a downwardbiased estimate of κ. The magnitude of the bias is a function of central bank
behavior, households’ preferences, and, more generally, the importance of
mark-up shocks for output fluctuations. Third, DSGE model-based estimation
is promising but might require a prior that is informative about other model
parameters, for instance those that control the law of motion of exogenous
shocks or the conduct of monetary policy. We will subsequently elaborate on
this last point.
Identifying Information in Impulse Response
Functions
If the DSGE model embodies enough restrictions to identify a structural shock
other than ξ t from the observables, then one can potentially infer the Phillips
curve slope from the impulse response function (IRF) associated with this
shock. Consider the model analyzed in the previous subsection. Suppose
that the policy rule coefficient, ψ, is known, which means that the sequence
of monetary policy shocks can be directly obtained from interest rate and
inflation data: R,t = Rt − ψπ t . Recall from (15) that the forward solution
of the Phillips curve takes the form
π t = κYt + ξ ,t .
We previously showed that the correlation between the mark-up shock, ξ ,t ,
and the regressor, Yt , creates an endogeneity problem that complicates the
F. Schorfheide: DSGE Model-Based NKPC Estimation
409
identification of κ. The monetary policy shock can serve as an instrumental
variable in the identification of κ. By assumption, the monetary policy shock
is uncorrelated with ξ ,t but correlated with the regressor Yt .
The argument can be formalized as follows. Suppose we factorize the
likelihood function into3
p(Yt , π t , Rt |κ, θ ) = p(Rt − ψπ t |κ, θ )p(Yt |Rt − ψπ t , κ, θ )
(18)
p(π t |Yt , Rt − ψπ t , κ, θ ).
Rt − ψπ t measures the monetary policy shock, R,t , and the first term corresponds to its density. The second factor captures the distribution of output
given the monetary policy shock. The third conditional density represents the
Phillips curve. From this factorization it is apparent that, in a linear Gaussian
environment, the following conditional expectations (we replace Rt − ψπ t by
R,t ) are identifiable:
E Yt| Rt − ψπ t , κ, θ = α 11 R,t and
E [π t | Rt − ψπ t , Yt ] = α 21 R,t + α 22 Yt ,
where α ij is a function of κ and θ . Since
∂Yt
= α 11 ,
∂ R,t
∂π t
= α 21 + α 22 α 11 ,
∂ R,t
it follows from (11) that κ is identified by the ratio of the output and inflation
response α 21 /α 11 + α 22 .
In our simple example the identification of the monetary policy shock depends on the assumed knowledge of the parameter ψ, which the reader might
find unconvincing. More interestingly, there are a number of papers that estimate DSGE models that are specified such that monetary policy shocks can be
identified from exclusion restrictions. Most notably, Rotemberg and Woodford
(1997), Christiano, Eichenbaum, and Evans (2005), and Boivin and Giannoni
(2006) consider models in which the private sector is unable to respond to
monetary policy shocks contemporaneously.4 In a Gaussian vector autoregressive system, this exclusion restriction is sufficient to identify monetary
policy shocks and the associated impulse response functions independently of
the DSGE model parameters.
3 The Jacobian associated with the transformation of [R − ψπ , Y , π ] into [R , Y , π ] is
t
t t
t
t t
t
equal to one. We maintain that θ is defined as θ = [τ , ψ, σ R , σ φ , σ ξ ] and, hence, includes ψ.
4 Rather than conducting likelihood-based inference, all three papers use an estimation method
that exclusively relies on the identification of model parameters from IRF dynamics. The structural
parameters are directly estimated by minimizing the discrepancy between the model-implied impulse
responses to a monetary policy shock and those obtained from estimating a structural VAR.
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Federal Reserve Bank of Richmond Economic Quarterly
Identifying Information in the Reduced-Form
Dynamics
The absence of equilibrium dynamics in (11) is clearly at odds with reality.
Aggregate output, inflation, and interest rates tend to exhibit fairly strong serial
correlation. This serial correlation opens up another avenue for identification
as lagged endogenous variables can serve as instruments to correct endogeneity biases. In fact, it is this serial correlation that single-equation approaches
rely on.
Suppose that the vector xt contains inflation, a measure of marginal costs
as well as other variables, denoted by zt : xt = [π t , MCt , zt ] . Moreover,
assume that the mark-up shock, ξ t , is independently distributed and that the
DSGE model-implied law of motion for xt has a VAR(1) representation:
xt = 1 (λ, θ )xt−1 + ut ,
where
ut = R(λ, θ ) t .
(19)
The matrices 1 and R are functions of the DSGE model parameters, the
vector t stacks the innovations to the exogenous driving processes of the
model economy, and ut can be interpreted as reduced-form, one-step-ahead
forecast errors. While the assumption that ξ t is serially uncorrelated is crucial
for the subsequent argument, the VAR(1) representation is not.
Define the selection vectors M1 and M2 such that M1 xt = π t and M2 xt =
MCt . Equation (15) implies that the slope of the Phillips curve has to solve
the following restriction:
M1 1 xt − λM2 (I − β1 )−1 1 xt = 0
for all xt .
(20)
Recall that under the assumption that ξ t is independently distributed, the forward solution of the Phillips curve takes the form
πt = λ
∞
β j Et [MCt+j ] + ξ t .
j =0
Thus, the first term in (20) can be interpreted as the one-step-ahead VAR
forecast of inflation. The second term in (20) corresponds to the one-stepahead forecast of the sum of discounted expected future marginal costs, scaled
by the Phillips curve slope. As long as ξ t is serially uncorrelated, the two
forecasts have to be identical. Notice that although it might be impossible to
uniquely determine λ and θ conditional on the VAR coefficient matrix 1 , 1
is always identifiable based on the autocovariances of xt , provided that E xt xt
is invertible: 1 =E xt−1 xt (E xt xt )−1 . Hence, provided that inflation is
serially correlated, the restriction (20) identifies λ.
Sbordone (2002, 2005) and Kurmann (2005, 2007) use (20) in conjunction
with reduced-form VAR estimates of to obtain estimates of the NKPC parameters. A system-based DSGE model estimation with serially uncorrelated
F. Schorfheide: DSGE Model-Based NKPC Estimation
411
mark-up shocks can be interpreted as simultaneously minimizing the discrepancy between an unrestricted, likelihood-based estimate of 1 and the DSGE
model-implied restriction function 1 (λ, θ ) and imposing the condition (20).
Identification of Backward-Looking Terms
Achieving identification becomes more difficult if we relax the restriction that
γ b = 0. Since insightful, analytical derivations are fairly complex, we offer
a heuristic argument and point to some empirical evidence. Three factors
contribute to the persistence of inflation: the backward-looking term γ b
π t−1 ,
the persistence of marginal costs, and the persistence of the mark-up shock,
ξ t . Roughly speaking, we can measure inflation and marginal cost persistence
from the data (provided observations on marginal costs are available). Hence,
the challenge is to disentangle the relative contribution of γ b and the mark-up
shock to the persistence of inflation. Del Negro and Schorfheide (2006, Figure
8) display plots of the joint posterior distribution of γ b and the autocorrelation
of a latent mark-up shock obtained from the estimation of a DSGE model that
is similar to the one developed by Smets and Wouters (2003). Not surprisingly,
there is a strong negative correlation, suggesting that without strong a priori
restrictions, it is difficult to measure the magnitude of γ b . One widely used a
priori restriction is the assumption that the mark-up shock is either absent or
serially uncorrelated.
3. A (CRUDE) LOOK AT U. S. DATA
Before reviewing the DSGE model-based NKPC estimates reported in the
literature, we will take a crude look at U.S. inflation, labor share, and output
data. In view of the analysis presented in Section 2, two potentially important sources of variation in DSGE model-based estimates are (1) detrending
methods for inflation data and marginal cost proxies and (2) endogeneity
corrections. Thus, in the first subsection we construct different measures
of steady-state deviations and compare the stochastic properties of the re t , and Y
t series. We established that the estimation of the
sulting
π t , MC
NKPC parameters amounts to a regression of inflation on future expected
marginal costs. This regression, hidden within a complicated likelihood function, is plagued by an endogeneity problem, which, according to the simple
model in Section 2, leads to a negative bias of least-squares estimates of the
Phillips curve slope. It turns out that these least-squares estimates are relatively
insensitive to data definitions (second subsection), which suggests that much
of the variation across empirical studies is attributable to differences in the
endogeneity correction.
We also showed that impulse response dynamics provide useful information about the NKPC coefficients. To the extent that a well-specified DSGE
412
Federal Reserve Bank of Richmond Economic Quarterly
model is comparable in fit to a more densely parameterized VAR, evidence
(reported in the third subsection) on the propagation of a monetary policy
shock can be helpful to understand DSGE model-based estimates of NKPC
parameters. Finally, the autocovariance restrictions exploited in the DSGE
model-based estimation tend to nest those used by Sbordone (2002) to construct a VAR-based minimum distance estimator. Hence, we briefly review
these minimum distance estimates in the fourth subsection.
Measures of Inflation and Marginal Costs
Most authors use the gross domestic product (GDP) deflator as a measure of
inflation when estimating New Keynesian DSGE models. Our subsequent
review focuses on estimates obtained with DSGE models in which marginal
costs equal the labor share. These estimates either include the labor share in
the vector of observables or treat marginal costs as a latent variable. In the
latter case, deviations of aggregate output from a trend or natural level are
implicitly used as a marginal cost proxy. To study the stochastic properties
of these series, we compile a small data set with quarterly U.S. observations.
The raw data are taken from Haver Analytics. Real output is obtained by
dividing the nominal series (GDP) by population 16 years and older and by
deflating using the chained-price GDP deflator. Inflation rates are defined as
log differences of the GDP deflator. The labor share is computed by dividing
total compensation of employees (obtained from the National Income and
Product Accounts) by nominal GDP. We take logs of real per capita output
and the labor share. Our sample ranges from 1960:Q1 to 2005:Q4.
We will consider three measures of
π t . First,
π (mean) is obtained by
subtracting the sample mean computed over the period 1960 to 2005 from the
GDP deflator inflation. This calculation assumes that the target inflation rate
has essentially stayed constant for the past 45 years. Second, we compute
separate means for the subsamples 1960–69, 1970–1982, and 1983–2005.
The break points are broadly consistent with the regime estimates obtained in
Schorfheide (2005). The resulting measure of inflation deviations is denoted
by
π (break) and reflects the view that the target inflation rate rose in the
1970s because policymakers perceived an exploitable long-run output inflation
tradeoff. Finally, we consider
π (HP), which can be interpreted as deviations
from a drifting target inflation rate.
We plot the inflation rate as well as the three versions of the target inflation
in Figure 1. It is apparent from the figure that views about target inflation
significantly affect the stochastic properties of
π t . For instance, the firstorder autocorrelations (see Table 1) are 0.88, 0.68, and 0.49 for
π (mean),
π
t as
(break), and
π (HP), respectively. The two panels of Figure 2 depict MC
approximated by output movements or measured by labor share fluctuations.
In models that treat marginal cost as a latent variable, the most common
F. Schorfheide: DSGE Model-Based NKPC Estimation
413
Figure 1 Inflation and Measures of Trend Inflation
14
Inflation Rate (A%)
12
HP Trend
Constant Mean
Mean with Breaks
10
8
6
4
2
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Notes: Inflation is measured as quarter-to-quarter changes in the log GDP deflator, scaled
by 400 to convert it into annualized percentages. The sample ranges from 1960:Q1 to
2005:Q4.
marginal cost proxies are given by linearly detrended output, output deviations
from a quadratic trend, and HP-filtered output. Since the potential output series
produced by the Congressional Budget Office closely resembles the HP trend,
we are not considering it separately.5 Panel A clearly indicates that output
deviations from a deterministic trend tend to be more volatile and persistent
than deviations from the HP trend, since the HP filter removes more of the
low frequency variation from the output series. Panel B shows time series
for labor share deviations from a constant mean and an HP trend. As before,
deviations from an HP trend tend to be smoother. First-order autocorrelations
for the marginal cost measures are reported in Table 1. They range from 0.7
(HP-filtered labor share) to 0.97 (linearly detrended output).
5 In some DSGE models, e.g., Schorfheide (2005), technology evolves according to a unit
root process and the output term that appears in the Phillips curve refers, strictly speaking, to
deviations from a latent stochastic trend. We do not consider this case in the regressions reported
in this section.
414
Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Measures of Marginal Cost Deviations
Panel A: Output Deviations from Trend
8
4
0
-4
Deviations from HP
Trend (%)
Deviations from
Quadratic Trend (%)
Deviations from Linear
Trend (%)
-8
-12
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Panel B: Labor Share Deviations from Trend
4
3
2
1
0
-1
-2
-3
Deviations from Mean (%)
Deviations from HP Trend (%)
-4
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Inflation and Marginal Cost Regressions
Under the assumptions that γ b = 0,
ξ t is serially uncorrelated, β = 0.993,
and marginal cost dynamics are well approximated by an AR(1) process with
coefficient ρ̂, one can express the forward solution of (6) as
F. Schorfheide: DSGE Model-Based NKPC Estimation
415
Table 1 Persistence of Marginal Cost and Inflation Measures
Series
π̃ (mean)
π̃ (break)
π̃ (HP)
Ỹ (linear trend)
Ỹ (quadratic trend)
Ỹ (HP)
l s̃h (mean)
l s̃h (HP)
AR(1)
0.88
0.68
0.49
0.97
0.96
0.85
0.94
0.70
Notes: The table reports AR(1) coefficient estimates based on a sample from 1960:Q1
to 2005:Q4.
1
1 − 0.993ρ̂ Y
1
t + ξ̃ t ,
lsh
1 − 0.993ρ̂ lsh
(21)
where lsh denotes the labor share. As in Section 2, the parameter, κ, confounds
the slope, λ, and the elasticity of marginal costs with respect to output. Least
squares regression results for (21) are summarized in Table 2. We report point
estimates of κ and λ in (12) and R 2 statistics in parenthesis for the full sample
as well as three subsamples: 1960–1969, 1970–1982, and 1983–2005.
Since there is no guarantee that the mean of inflation and marginal cost
deviations is zero in the subsamples, we also include an intercept in the regression. As in other studies, e.g., Rudd and Whelan (2007), we find that the
slope estimates and the R 2 statistics tend to be small. The largest estimate of
κ is 0.03, obtained by regressing demeaned inflation on the HP-filtered output
using the 1960–1969 subsample. If one regresses inflation on the labor share,
the largest slope estimate is 0.05, which is obtained by using an HP-filter on
both inflation and the labor share and restricting the sample to 1970–1982.
The median slope estimate reported in the table is 0.002. The R 2 values range
from nearly zero to 66 percent. If we assume that the target inflation rate has
shifted in early 1970 and 1982 and use the demeaned labor share as a measure
of marginal cost, then λ̂ = .003 and R 2 is 6 percent. The Durbin-Watson
statistics (not reported in the table) for the OLS regressions indicate that the
least-squares residuals have substantial positive serial correlation.
We draw two broad conclusions for the subsequent review of DSGE
model-based estimates. First, since the least-squares estimates range from
0 to 0.03 for κ and 0 to 0.05 for λ, any variation beyond this range is most
likely caused by the endogeneity correction. Second, for the Phillips curve to
πt = κ
t + ξ̃ t ,
Y
or
πt = λ
416
Table 2 Inflation and Marginal Cost Regressions
Inflation
π (break)
π (HP)
1960–2005
1E-4 (4E-4)
8E-4 (0.01)
.006 (.008)
0.01 (0.40)
0.04 (0.03)
5E-4 (0.02)
8E-4 (0.03)
0.01 (0.07)
.003 (0.06)
0.02 (0.02)
2E-4 (.005)
2E-4 (.006)
.004 (0.02)
.002 (0.03)
0.03 (0.08)
Sample Period
1960–1969
1970–1982
.002 (0.64)
−8E-4 (0.02)
.002 (0.65)
−8E-4 (0.01)
0.03 (0.35)
.009 (0.06)
0.01 (0.66)
.007 (0.03)
0.03 (0.03)
0.03 (0.04)
1983–2005
−.001 (0.08)
−.001 (0.05)
.003 (.008)
.002 (0.03)
.002 (.001)
same as
π (mean)
2E-4
2E-4
.007
.003
0.02
(0.03)
(0.03)
(0.12)
(0.16)
(0.07)
−4E-4
−4E-4
.002
0.01
0.05
(.007)
(.006)
(.005)
(0.17)
(0.15)
.001
.001
.007
5E-4
0.01
(0.11)
(0.11)
(0.07)
(.002)
(0.03)
t /(1 −
Notes: We are reporting coefficient estimates, α̂ 1 and R 2 , in parenthesis for a regression of the form
π t = α 0 + α 1 [MC
β̂ ρ̂ MC )], where β̂ = 0.993, ρ̂ MC is the first-order autocorrelation of the marginal cost measure, and the marginal cost measure
t.
t or lsh
is either Y
Federal Reserve Bank of Richmond Economic Quarterly
π (mean)
Marginal Cost
t
Measure MC
(lin trend)
Y
(quad trend)
Y
(HP)
Y
(mean)
lsh
(HP)
lsh
(lin trend)
Y
(quad trend)
Y
(HP)
Y
(mean)
lsh
(HP)
lsh
(lin trend)
Y
(quad trend)
Y
(HP)
Y
(mean)
lsh
(HP)
lsh
F. Schorfheide: DSGE Model-Based NKPC Estimation
417
capture the inflation persistence well, it has to be the case that lagged inflation
enters the NKPC, that the mark-up shock is fairly persistent, or that inflation
deviations are computed relative to a time-varying target inflation rate.
VAR-IRF Evidence
We explained in Section 2 that if the DSGE model imposes enough restrictions
to unambiguously identify, say, a monetary policy shock, then the response
of output and marginal costs to this shock provides useful information about
the NKPC parameters. To the extent that we would expect a well-specified
DSGE model to generate impulse responses that are similar to those obtained
from a structural VAR analysis, it is informative to examine prototypical VAR
responses to a monetary policy shock and to determine a range of NKPC
parameterizations that are consistent with these responses.
Under the assumption that lagged inflation does not enter the NKPC and
that marginal costs are proportional to output, the impulse responses to a
monetary policy shock have to satisfy
∞
∂
Ỹ
∂ π̃ t+h
t+h+j
=κ
β j Et+h
.
R
∂ Rt
∂
t
j =0
As in Section 2, we use κ to denote the slope of the Phillips curve with respect to
output. The parameter, κ, absorbs the elasticity of marginal costs with respect
to output into the definition of the slope. Suppose that the impulse responses
are monotonic and the output response decays approximately exponentially
at rate δ in response to a monetary policy shock. Then
∂ π̃ t+h
κ ∂ Ỹt+h
≈
.
R
1 − δβ ∂ Rt
∂ t
While a large literature exists (see Christiano, Eichenbaum, and Evans
[1999] and Stock and Watson [2001] for surveys) that uses structural VARs to
measure the effect of monetary policy shocks, we focus on a prominent recent
study by Christiano, Eichenbaum, and Evans (2005).
The authors estimate a nine-variable VAR using data on real GDP, real consumption, the GDP deflator, real investment, the real wage, labor
productivity, the federal funds rate, real profits, and the growth rate of M2.
Christiano, Eichenbaum, and Evans (2005) find that a 15-basis point (bp) drop
in the federal funds rate (quarterly percentage points) leads to a 5-bp increase
in the quarterly inflation rate after 12 quarters and a 50-bp increase of output after nine quarters.6 Hence, according to the mean impulse responses,
κ should be about 0.1 if we set the decay factor, δ, to zero and 0.05 if we
6 These numbers are approximate, based on Figure 1 in Christiano, Eichenbaum, and Evans
(2005).
418
Federal Reserve Bank of Richmond Economic Quarterly
set δ = 0.5. Suppose now that we ignore the dependence in the sampling
distribution of the impulse response function estimators and let δ = 0 again.
Combining the lower bound of the reported 95 percent confidence band of the
inflation response with the upper bound of the confidence band for the output
response suggests that κ could be as low as 0.01. Combining the upper bound
for the inflation response with the lower bound for the output response leads
to a value of κ = 0.5. If we consider the labor share instead of the output
response, we can obtain an estimate of λ instead of κ. Along the mean impulse
response estimated by Christiano, Eichenbaum, and Evans (2005), the labor
share appears to drop by about 25 bp, which for δ = 0 and δ = 0.5 leads to
values of λ = 0.2 and λ = 0.1, respectively.
Evidence from Inflation and Marginal Cost Dynamics
Several papers, e.g., Sbordone (2002, 2005) and Kurmann (2005, 2007), exploit the restriction (20) to construct minimum-distance estimates of the NKPC
parameters from the estimates of an unrestricted VAR that includes inflation
and a measure of marginal costs. Using data from 1951 to 2002 on the labor
share and inflation, Sbordone (2005) obtains an estimate of λ̂ = 0.025 in the
purely forward-looking specification, and λ̂ = 0.014 and γ̂ b = 0.18 if she allows lagged inflation to enter the NKPC. To the extent that the restriction (20)
is also embodied in a DSGE model likelihood function, the DSGE modelbased estimates of the NKPC parameters should be similar, provided that the
same measure of marginal costs is used, the mark-up shock is assumed to be
i.i.d., and the vector autoregressive approximation to the law of motion of the
estimated DSGE model resembles the unrestricted VAR estimates.
4.
REVIEW OF EMPIRICAL RESULTS
Broadly speaking, the empirical papers reviewed in this section fall into two
categories: either marginal costs are treated as a latent variable or the set of
observables spans the labor share and, hence, the model-implied measure of
marginal costs. Consider once again the simple model of Section 2 and let us
denote the labor share as lsh. Abstracting from inference about γ b and γ f , a
study that estimates λ in
λ
π̃ t = βEt [π̃ t+1 ] + Ỹt − Z̃t + ξ̃ t ,
τ
(22)
based on observations of π̃ t and Ỹt , falls in the first category. Identification of
λ in (22) is tenuous because the presence of Z̃t exacerbates the endogeneity
problem and the parameter, τ , has to be separately estimable from the observables for λ to be identifiable. On the upside, the use of (22) is more robust to
F. Schorfheide: DSGE Model-Based NKPC Estimation
419
the presence of measurement errors in the labor share (marginal cost) series.
For some of the papers that fall into the first category, we will report estimates
of the output coefficient, κ, which corresponds to τλ in the example, rather than
λ. A paper that estimates λ in
t + ξ̃ t ,
π̃ t = βEt [π̃ t+1 ] + λlsh
(23)
t , belongs to the second category.
with observations on π̃ t and lsh
Since the literature on estimated DSGE models is growing rapidly, we had
to strike a balance between scope and depth. This survey is limited to models in which firms’ price-setting equations are derived either under quadratic
adjustment costs or under the Calvo mechanism. Ongoing research explores
alternative sources of nominal rigidities that are not included in the subsequent
review, for instance, menu costs and state-dependent pricing models (Dotsey,
King, and Wolman 1999, Gertler and Leahy 2006), models with labor market search frictions (Gertler and Trigari 2006, Krause and Lubik 2007), and
models with information processing frictions (Sims 2003, Mackowiak and
Wiederholt 2007, Mankiw and Reis 2007, and Woodford 2008). Moreover,
we focus on models in which the labor share is the model-implied measure of
marginal costs.7
The numerical values reported in Tables 3–5 refer to point estimates that
are obtained with one of four methods. In addition to papers that use Bayesian8
and maximum likelihood methods as discussed in Section 1, we consider studies that estimate the DSGE model parameters by minimizing the discrepancy
between impulse responses implied by the DSGE model and those obtained
from the estimation of a structural VAR, or by minimizing the distance between sample moments obtained from U.S. data and DSGE model-implied
population moments. The remainder of this section is organized as follows.
We review estimates that are obtained by treating marginal costs as a latent
variable. We examine studies in which the authors treat marginal costs as
observable. For monetary policy analysis, the relationship between inflation
and output is at least as important as the relationship between inflation and
marginal costs. So we examine DSGE model-based estimates of the relative
movements of inflation and output in response to a monetary policy shock.
Finally, we discuss the role of wage stickiness.
7 Krause, López-Salido, and Lubik (2008) show that in a model with labor market search
frictions, marginal costs are also affected by the labor market tightness. However, empirically they
find that matching frictions in the labor market appear to affect the cyclical behavior of marginal
costs only slightly in terms of co-movement, persistence, and volatility.
8 Bayesian inference combines information contained in the likelihood function with prior
information to form posterior estimates. Since it is difficult to disentangle the contribution of
various sources of information ex post, we restrict our attention to the posterior estimates without
examining the priors that were used to generate these posteriors.
420
Federal Reserve Bank of Richmond Economic Quarterly
Latent Marginal Costs
Table 3 summarizes parameter estimates of a Phillips curve specification in
which marginal costs are replaced by output or a measure of the output gap:
π̃ t = γ b π̃ t−1 + γ f Et [π̃ t+1 ] + κ Ỹt + ξ̃ t ,
(24)
where ξ̃ t represents the latent variables that enter the NKPC in any particular
model. These estimates are obtained by fitting New Keynesian DSGE models
to observations of output, inflation, and interest rates. The models implicitly
share the following features: household preferences are linear in labor and
capital is not a factor of production. Estimates for κ range from values less
than 0.001 (Cho and Moreno 2006) to 4.15 (Canova forthcoming). While
the studies differ with respect to sample period as well as the detrending of
inflation and output, our least-squares analysis in Section 3 suggests that most
of the differences in κ̂ are probably due to the treatment of latent variables.
We showed that the likelihood function corrects for the endogeneity problem
that arises in a regression of inflation on future expected output due to the
correlation of the latent variables with expected output. This endogeneity
correction appears to be very sensitive to the assumed correlation among the
exogenous disturbances that enter the Phillips curve, the Euler equation, and
the monetary policy rule. Models in which the shocks in the Euler equation
and the Phillips curve are forced to be or allowed to be correlated tend to deliver
larger estimates of κ than models in which these disturbances are assumed to
be uncorrelated.9
We now turn to estimates of New Keynesian Phillips curves that are expressed in terms of marginal costs instead of output:
t + ξ̃ t .
π̃ t = γ b π̃ t−1 + γ f Et [π̃ t+1 ] + λMC
(25)
These estimates are reported in Table 4. Rabanal and Rubio-Ramı́rez
(2005) fit a canonical New Keynesian DSGE model without capital and habit
formation using a data set that contains, in addition to inflation, interest rates,
and detrended output, a measure of the real wage. For specifications in which
γ b is restricted to be zero, the authors obtain estimates of λ of about 0.015. If
γ b is estimated subject to the restriction that γ b + γ f = 0.99, the estimate of
9 Correlation arises either because the structural model implies that, say, the government
spending shock enters both the Phillips curve and the Euler equation (e.g., Schorfheide 2005),
or because authors attach reduced-form disturbances to the Phillips curve and the Euler equation
and assume that these disturbances are correlated (e.g., Lubik and Schorfheide 2004). In smallscale models, it is often “testable” whether the exogenous disturbances are correlated. In large
DSGE models, parameters associated with the endogenous propagation mechanism and auxiliary
parameters that generate correlation between exogenous disturbances are often not separately identifiable.
Study
No capital, no habit formation, output coefficient
Canova (forthcoming), Table 1
Cho and Moreno (2006), Table 2
Cho and Moreno (2006), Table 2
Cho and Moreno (2006), Table 2
Del Negro and Schorfheide (2004), Table 2
Del Negro and Schorfheide (2004), Table 2
Lindé (2005), Table 5
Lindé (2005), Table 5
Lubik and Schorfheide (2004), Table 3
Lubik and Schorfheide (2004), Table 3
Lubik and Schorfheide (2004), Table 3
Rotemberg and Woodford (1997), Page 321
Salemi (2006), Table 2
Salemi (2006), Table 2
Schorfheide (2005), Table 2
Schorfheide (2005), Table 2
Sample Period
in Phillips curve
1955:Q1–2002:Q1
1980:Q4–2000:Q1
1980:Q4–2000:Q1
1980:Q4–2000:Q1
1959:Q3–1979:Q2
1959:Q3–1979:Q2
1960:Q1–1997:Q4
1960:Q1–1997:Q4
1960:Q1–1979:Q2
1960:Q1–1979:Q2
1982:Q4–1997:Q4
1980:Q1–1995:Q2
1965:Q1–2001:Q4
1965:Q1–2001:Q4
1960:Q1–1997:Q4
1960:Q1–1997:Q4
π t−1
0.44
0.44
0.43
0.72
0.54
0.62
0.43
Et [π t+1 ]
0.98
0.56
0.56
0.57
1.00
1.00
0.28
0.46
.997
.997
.993
0.99
0.00
0.57
0.99
1.00
Yt
Method
4.150
0.001
0.001
0.000
0.314
0.249
0.048
0.048
0.770
0.750
0.580
0.024
0.055
0.003
0.370
0.360
Bayes
MLE
MLE
MLE
Bayes
Bayes
MLE
MLE
Bayes
Bayes
Bayes
IRF-MD
MLE
MLE
Bayes
Bayes
F. Schorfheide: DSGE Model-Based NKPC Estimation
Table 3 Published NKPC Estimates: Latent Labor Share (Part 1)
Notes: We are providing point estimates of the New Keynesian Phillips curve, π t = γ b π t−1 +γ f Et [π t+1 ]+κYt +ξ t , based on
the information provided in the cited studies. Estimation methods: MLE = maximum likelihood estimation; Bayes = Bayesian
analysis; and IRF-MD minimize discrepancy between impulse responses estimated with a structural VAR and those implied by
a DSGE model.
421
422
Federal Reserve Bank of Richmond Economic Quarterly
λ drops to 0.004, whereas the weight on lagged inflation in the Phillips curve
is 0.43.
The canonical three equation New Keynesian DSGE model that underlies,
for instance, the analysis in Rabanal and Rubio-Ramirez (2005) lacks persistent dynamics, which makes it difficult to capture the serial correlation in U.S.
data. The lack of persistence can be overcome in part by using household
preferences that exhibit habit formation, that is, the instantaneous utility is a
function of current consumption relative to some habit stock, which in turn
depends on past consumption. Habit formation enriches the endogenous propagation mechanism of the model and enhances the model’s ability to capture
the persistence in output and consumption. More importantly for us, it changes
the relationship between (observed) output and (latent) marginal costs. The
marginal utility of consumption, and thereby marginal costs, depends not just
on the current level of output, but also on past and expected future levels as
well as the parameters that determine the degree of habit formation. The estimates of λ reported in the second section of Table 4 range from 0.004 to
0.437.
If capital is treated as a variable input, marginal costs remain equal to the
labor share as long as the production function is of the Cobb-Douglas form.
However, if labor share observations are not used directly in the estimation,
the presence of capital affects the decomposition of marginal costs into an
observed and an unobserved component. In other words, if the DSGE model
is estimated based on observations of output, inflation, and interest rates,
introducing variable capital changes the stochastic properties of ξ̃ t in (24)
and the relationship between κ in (24) and λ in (25). The third section of
Table 4 reports NKPC estimates from six studies, ranging from 0.008 to 0.112.
Among these papers, Fernandez-Villaverde and Rubio-Ramirez (2007) allow
the parameters of the monetary policy rule and the parameters that determine
the degree of price and wage stickiness to vary over time. This allows the
authors to obtain a time series of the Phillips curve coefficient. If the slope
estimates of the Phillips curve are converted into the probability that a firm is
unable to change its price in a Calvo model (see Section 1), then the estimates
can be summarized as follows. Prices stayed constant for an average of four
quarters in the 1960s and 1970s, while inflation was relatively high and became
a bit more rigid after the Volcker disinflation. Based on a casual inspection
of the smoothed time series of the Phillips curve coefficients, λ appeared to
be, on average, around 0.06 before 1979 and subsequently dropped to 0.03.
The average estimate of γ b pre-1979 is about 0.35 and decreased to 0.3 after
1979. This pattern is broadly consistent with the notion that the NKPC is not
structural in the following sense: If a high target inflation rate makes it very
costly for firms not to change their prices—and, hence, more attractive to incur
the costs of adjusting the prices—we should observe a steeper Phillips curve
relationship.
Study
Sample Period
No capital, no habit formation
Rabanal and Rubio-Ramı́rez (2005), Table 2
1960:Q1–2001:Q4
Rabanal and Rubio-Ramı́rez (2005), Table 2
1960:Q1–2001:Q4
Rabanal and Rubio-Ramı́rez (2005), Table 2
1960:Q1–2001:Q4
Rabanal and Rubio-Ramı́rez (2005), Table 2
1960:Q1–2001:Q4
No capital, with habit formation
Andres, López-Salido, and Nelson (2004), Table 1
1980:Q1–1999:Q2
Andres, López-Salido, and Nelson (2005), Table 2
1979:Q3–2003:Q3
Boivin and Giannoni (2006), Table 2
1959:Q2–1979:Q2
Boivin and Giannoni (2006), Table 2
1979:Q3–2002:Q2
Galı́ and Rabanal (2004), Table 4
1948:Q1–2002:Q4
Lubik and Schorfheide (2006), Table 2
1983:Q1–2002:Q4
Milani (2007), Table 2
1960:Q1–2004:Q2
Models with capital
Bouakez, Cardia, and Ruge-Murcia (2005), Table 1
1960:Q1–2001:Q2
Bouakez, Cardia, and Ruge-Murcia (forthcoming), Table 4
1964:Q1–2002:Q4
Christensen and Dib (2008)
1979:Q3–2004:Q3
Fernández-Villaverde and Rubio-Ramı́rez (2007), Table 6.1
1955:Q1–2000:Q4
Fernández-Villaverde and Rubio-Ramı́rez (2007), Table 6.1
1955:Q1–1979:Q4
Fernández-Villaverde and Rubio-Ramı́rez (2007), Table 6.1
1980:Q1–2000:Q4
Laforte (2007), Table 3
1983:Q1–2003:Q1
Rabanal (2007), Table 2
1959:Q1–2004:Q4
π t−1
0.43
0.50
0.50
0.50
0.02
0.50
0.13
0.26
0.13
0.40
0.50
Et [π t+1 ]
MCt
Method
0.99
0.56
0.99
0.99
0.015
0.004
0.016
0.017
Bayes
Bayes
Bayes
Bayes
0.99
0.50
0.50
0.50
0.97
1.00
0.99
0.014
0.437
0.006
0.004
0.413
0.200
0.024
MLE
MLE
IRF-MD
IRF-MD
Bayes
Bayes
Bayes
0.50
1.00
.993
0.87
0.74
0.87
0.59
0.50
0.015
0.223
0.092
0.008
0.056
0.030
0.112
0.018
MLE
MD
MLE
Bayes
Bayes
Bayes
Bayes
Bayes
423
Notes: We are providing point estimates of the New Keynesian Phillips curve, π t = γ b π t−1 + γ f Et [π t+1 ] + λMCt + ξ t ,
based on the information provided in the cited studies. Estimation methods: MLE = maximum likelihood estimation; Bayes
= Bayesian analysis; IRF-MD minimize discrepancy between impulse responses estimated with a structural VAR and those
implied by a DSGE model; and MD minimize discrepancy between sample moments and DSGE model-implied moments.
F. Schorfheide: DSGE Model-Based NKPC Estimation
Table 4 Published NKPC Estimates: Latent Labor Share (Part 2)
424
Federal Reserve Bank of Richmond Economic Quarterly
Observed Marginal Costs
We now turn to the Bayesian estimation of New Keynesian DSGE models
based on a larger set of observables that spans the labor share and, hence,
marginal costs as they appear in the Phillips curve. Intuitively, the use of
labor share observations should lead to a sharper identification of λ. Table 5
summarizes empirical estimates from seven studies. Most estimates are based
on a variant of the Smets and Wouters (2003) model, which augments a DSGE
model by Christiano, Eichenbaum, and Evans (2005) with additional shocks to
make it amenable to likelihood-based estimation. Smets and Wouters (2005),
Levin et al. (2006), Del Negro et al. (2007), Smets and Wouters (2007), and
Justiniano and Primiceri (2008) obtain estimates of λ of 0.01, 0.03, 0.10, 0.02,
and 0.01, respectively. The estimates of the coefficient γ b on lagged inflation
are 0.25, 0.07, 0.43, 0.19, and 0.46, respectively. Compared to the numbers
reported in Tables 3 and 4, the variation across studies is much smaller.
Impulse Response Dynamics
Much of our previous discussion focused on the marginal cost coefficient in
the Phillips curve relationship. However, from a monetary policy perspective,
equally important is the output-inflation tradeoff in the estimated DSGE model.
This tradeoff not only depends on λ but also on the elasticity of marginal costs
with respect to output. Thus, we will examine the relative movements of output
and inflation in response to a monetary policy shock, that is, an unanticipated
deviation from the systematic component of the monetary rule. Of course,
these impulse responses do not merely depend on the slope of the NKPC,
they also depend on other aspects of the model, such as labor market frictions
and wage stickiness and the behavior of the central bank. Not all the papers
for which we have reported estimates of the NKPC parameters in Tables 3
to 4 present impulse response functions. Those that do typically represent
them in graphical form. The subsequent results are based on an inspection
of impulse response plots and are summarized in Table 6.10 We report the
magnitude of the peak responses of the interest rate, inflation rate, and the
output deviation from steady state. The interest rate response is measured in
annualized percentages; that is, an entry of 0.25 implies that the monetary
policy shock raises the interest rate 25 bp above its steady-state level. The
inflation rate is not annualized and represents a quarter-to-quarter difference
in the log price level, scaled by 100 to convert it into percentages. Output
deviations are also reported in percentages. Since the length of a period in a
DSGE model is typically assumed to be one quarter, in the context of the
10 In a number of studies, it turned out to be difficult to determine whether interest rates
and inflation rates are annualized. We tried to resolve this ambiguity.
Study
Avouyi-Dovi and Matheron (2007), Tables 3–4
Avouyi-Dovi and Matheron (2007), Tables 3–4
Christiano, Eichenbaum, and Evans (2005), Table 2
Del Negro et al. (2007), Table 1
Justiniano and Primiceri (2008), Table 1
Justiniano and Primiceri (2008), Table 1
Levin et al. (2006), Table 1
Smets and Wouters (2005), Table 1
Smets and Wouters (2007), Table 1A/B
Sample Period
1955:Q1–1979:Q2
1982:Q3–2002:Q4
1965:Q3–1995:Q3
1974:Q2–2004:Q1
1954:Q3–2004:Q4
1954:Q3–2004:Q4
1955:Q1–2001:Q4
1983:Q1–2002:Q2
1966:Q1–2004:Q4
π t−1
0.27
0.20
0.50
0.43
0.46
0.46
0.07
0.25
0.19
Et [π t+1 ]
0.73
0.80
0.50
0.57
0.54
0.54
0.92
0.74
0.82
MCt
0.008
0.010
0.135
0.100
0.007
0.005
0.033
0.007
0.020
Method
IRF-MD
IRF-MD
IRF-MD
Bayes
Bayes
Bayes
Bayes
Bayes
Bayes
Notes: We provide point estimates of the New Keynesian Phillips curve, π t = γ b π t−1 + γ f Et [π t+1 ] + λMCt + ξ t , based on
the information provided in the cited studies. Estimation methods: Bayes = Bayesian analysis; IRF-MD minimize discrepancy
between impulse responses estimated with a structural VAR and those implied by a DSGE model.
F. Schorfheide: DSGE Model-Based NKPC Estimation
Table 5 Published NKPC Estimates: Observed Labor Share
425
426
Federal Reserve Bank of Richmond Economic Quarterly
“back-of-the-envelope” calculation in Section 3, the ratio of the inflation and
output response, denoted by ∂π /∂y, would correspond to κ/(1 − δβ), where
δ is the factor at which the output response decays to zero.
In Table 6 we report the number of periods it takes for the responses to
reach their respective peaks, the ratio of the peak response of inflation and
output, and the estimate of κ̂ in the underlying model. Models without capital
and with little endogenous propagation typically generate monotonic impulse
response functions. For the models without capital, the relative responses of
inflation and output range from 0.07 to 2.00. Once capital is included and
the model is augmented by additional frictions, this range narrows to 0.08 to
0.17, which seems consistent with the VAR evidence provided by Christiano,
Eichenbaum, and Evans (2005). Comparing the estimates reported in Del
Negro et al. (2007) and Smets and Wouters (2007), it appears that these
tradeoffs can be obtained with quite differently priced Phillips curve slopes,
λ: 0.002 and 0.10.
Wage Versus Price Rigidity
This article has focused on estimates of the degree of price rigidity in New
Keynesian DSGE models. Many authors believe that inflexible wages are
another important source of nominal rigidities. In fact, the DSGE models
that are based on the work of Smets and Wouters (2003), and Christiano,
Eichenbaum, and Evans (2005) incorporate both price and wage stickiness.
Following work by Erceg, Henderson, and Levin (2000), in order to generate wage stickiness in DSGE models, one typically assumes that households
supply differentiated labor services that are aggregated by labor packers into
homogenous labor services. These homogeneous labor services are in turn
utilized by the intermediate goods-producing firms. Households act as monopolistically competitive suppliers and are subjected to a Calvo (1983) friction:
only a fraction of households is allowed to re-optimize nominal wage. To
clear the labor market ex post, one must assume that each household has to
satisfy the demand for its labor service at the posted price.
For a joint estimation of price and wage rigidity to be meaningful, the
set of observables needs to span inflation, labor share, and wages. The joint
dynamics of inflation and the labor share provide information about the price
Phillips curve, and the wage series, together with an implicit measure of the
marginal disutility of work, contains information about the degree of wage
stickiness. Del Negro and Schorfheide (2008) estimate a variant of the Smets
and Wouters (2003) under three priors that differ with respect to a priori
beliefs about nominal rigidities. The low rigidities prior assumes that the
price and wage Calvo parameters have a beta-distribution centered at 0.45
with a standard deviation of 0.10. The high rigidities prior is centered at 0.75
Table 6 Impulse Responses to a Monetary Policy Shock
Study
∂π /∂y
0.08
1.00
0.33
0.18
0.33
0.75
0.76
1.25
0.07
0.05
0.01
0.43
2.00
0.15
0.13
0.07
0.28
0.30
0.20
F. Schorfheide: DSGE Model-Based NKPC Estimation
Interest Rate
Inflation
Output
[Annualized %]
[Quarterly %]
[% Dev. from Trend]
Peak After x Q’s Peak After x Q’s Peak After x Q’s
No capital, no habit formation, marginal costs function of current output deviations
Cho and Moreno (2006), Figure 2
−0.80
0
0.01
9
0.12
9
Del Negro and Schorfheide (2004), Figure 2
−0.25
0
0.05
0
0.05
0
Ireland (2004a), Figure 1
−0.20
0
0.20
0
0.60
0
Ireland (2004b), Figure 1
−1.00
0
0.07
0
0.40
0
Ireland (2007), Figure 2
−0.40
0
0.10
0
0.30
0
Lubik and Schorfheide (2004), Figure 3
−0.70
0
0.12
0
0.16
0
Lubik and Schorfheide (2004), Figure 3
−0.60
0
0.17
0
0.16
0
Lubik and Schorfheide (2004), Figure 3
−0.60
0
0.20
0
0.16
0
Rotemberg and Woodford (1997), Figure 1
−0.80
0
0.03
2
0.38
2
Salemi (2006), Figure 3
−1.00
0
0.020
10
0.40
9
Salemi (2006), Figure 3
−1.00
0
0.002
8
0.30
8
No capital, with habit formation, marginal costs are function of current, past, and future output deviations
Andres, López-Salido, and Nelson (2004), Figure 2 −0.30
0
0.17
0
0.40
3
Andres, López-Salido, and Nelson (2005), Figure 1 −0.50
0
0.15
0
0.08
1
Andres, López-Salido, and Nelson (2005), Figure 1 −0.70
0
0.06
0
0.40
1
Boivin and Giannoni (2006), Figure 1
−1.00
0
0.14
6
1.10
4
Boivin and Giannoni (2006), Figure 1
−1.00
0
0.02
4
0.30
4
With capital, no direct observations on labor share
Christensen and Dib (2008), Figure 1
−0.48
2
0.14
2
0.50
2
Laforte (2007), Figure 2
−0.75
0
0.15
0
0.50
2
Rabanal (2007), Figure 4
−1.00
0
0.10
4
0.50
3
With capital, with direct observations on labor share
Christiano, Eichenbaum, and Evans (2005), Figure 1 −0.60
4
0.05
11
0.60
6
Del Negro et al. (2007), Figure 3
−1.10
0
0.05
2
0.33
2
Smets and Wouters (2005), Figure 5
−0.70
4
0.05
3
0.45
5
Smets and Wouters, (2007) Figure 6
−0.72
0
0.05
2
0.30
3
0.08
0.15
0.11
0.17
427
Notes: Based on the graphical information provided in the cited studies, we determined the peak responses for interest rates (annualized percentage points), inflation (quarter-to-quarter percentage points), and output (percentage deviations from trend/steady
state) to an unanticipated loosening of monetary policy.
428
Federal Reserve Bank of Richmond Economic Quarterly
with a standard deviation of 0.1. Finally, the agnostic prior is centered at 0.6
and is more diffuse—its standard deviation is 0.2.
Posterior inference based on these priors can be summarized as follows:
both under the agnostic and the low rigidities prior, the posterior estimate
of the wage stickiness is small. The Calvo parameter is around 0.25, which
means that the households re-optimize their wages, on average, every four
months. The estimated price stickiness translates into a value of λ of about
0.22. Under the high rigidities prior, the estimates of both the wage and the
price Calvo parameter turn out to be substantially larger, namely about 0.8.
Most interestingly, the time series fit of all three specifications is very similar,
yet the policy implications are quite different. The results presented in Del
Negro and Schorfheide (2008) suggest that the macro time series we typically
consider is not informative enough to precisely measure the degree of nominal
rigidity. This conclusion is consistent with the literature survey conducted in
this section: the variation of parameter estimates reported in the literature is
substantial. No clear consensus has emerged as of now.
5.
CONCLUSION
While the literature on DSGE model-based estimation of the NKPC is still
fairly young, a wide variety of results have been published in academic journals already. In most of these studies, the Phillips curve estimation is not a
goal but rather a byproduct of the empirical analysis. DSGE model-based
NKPC estimates tend to be fragile and sensitive to model specification and
data definitions, in particular if marginal costs are treated as a latent variable.
If the observations span the labor share, which is the model-implied measure
of marginal costs in the studies that we reviewed, then the slope estimates are
more stable. No consensus has emerged on the importance of lagged inflation
in the Phillips curve. Estimates are sensitive to detrending methods for inflation and assumptions about the autocovariance structure of the exogenous
disturbances in the DSGE model. Thus, from a policymaker’s perspective,
accounting for parameter and model uncertainty is important for prediction
and decision making.
We attempted to understand the identification of Phillips curve parameters in estimated DSGE models. Unlike single-estimation approaches, DSGE
model-based estimates are able to extract information about the structural
parameters from the contemporaneous correlations of output, inflation, interest rates, and other variables, as well as from impulse responses to structural
shocks that are identifiable based on exclusion restrictions hard-wired in model
specifications. Unfortunately, the data do not speak loudly and clearly to us,
and many DSGE models imply that if the model is “true,” it is difficult to
identify the NKPC parameters and the output-inflation tradeoff with only 20
to 40 years of observations.
F. Schorfheide: DSGE Model-Based NKPC Estimation
429
Identification in the context of simultaneous equations models is well
understood. To identify the slope of a supply curve we need variation in
exogenous demand shifters. Identification in DSGE models is much more
complicated. Variation in the data is created by unobserved shocks that in
most cases shift both demand and supply. Our reading of the early literature
on estimated DSGE models is that there was hope that the model-implied
cross-coefficient restrictions were so tight that identification was not a concern.
Over time the profession learned that, despite tight cross-equation restrictions,
identification should not be taken for granted, in particular in New Keynesian
DSGE models. While currently ongoing research is developing econometric
techniques to try to diagnose identification problems, it might be time to go
back to the drawing board and develop future DSGE models with parameter
identifiability in mind.
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Economic Quarterly—Volume 94, Number 4—Fall 2008—Pages 435–465
Policy Implications of the
New Keynesian Phillips
Curve
Stephanie Schmitt-Grohé and Martı́n Uribe
T
he theoretical framework within which optimal monetary policy was
studied before the arrival of the New Keynesian Phillips curve (NKPC),
but after economists had become comfortable using dynamic, optimizing, general equilibrium models and a welfare-maximizing criterion for
policy analysis, was one in which the central source of nominal nonneutrality
was a demand for money. At center stage in this literature was the role of
money as a medium of exchange (as in cash-in-advance models, money-inthe-utility-function models, or shopping-time models) or as a store of value (as
in overlapping-generations models). In the context of this family of models a
robust prescription for the optimal conduct of monetary policy is to set nominal interest rates to zero at all times and under all circumstances. This policy
implication, however, found no fertile ground in the boardrooms of central
banks around the world, where the optimality of zero nominal rates was dismissed as a theoretical oddity, with little relevance for actual central banking.
Thus, theory and practice of monetary policy were largely disconnected.
The early 1990s witnessed a profound shift in monetary economics away
from viewing the role of money primarily as a medium of exchange and
toward viewing money—sometimes exclusively—as a unit of account. A key
insight was that the mere assumption that product prices are quoted in units
of fiat money can give rise to a theory of price level determination, even if
money is physically nonexistent and even if fiscal policy is irrelevant for price
Stephanie Schmitt-Grohé is affiliated with Columbia University, CEPR, and NBER. She can
be reached at stephanie.schmittgrohe@columbia.edu. Martı́n Uribe is affiliated with Columbia
University and NBER. He can be reached at martin.uribe@columbia.edu. The views expressed
in this paper do not necessarily reflect those of the Federal Reserve Bank of Richmond or the
Federal Reserve System. The authors would like to thank Andreas Hornstein and Alexander
Wolman for many thoughtful comments.
436
Federal Reserve Bank of Richmond Economic Quarterly
level determination.1 This theoretical development was appealing to those
who regard modern payment systems as operating increasingly cashlessly.
At the same time, nominal rigidities in the form of sluggish adjustment of
product and factor prices gained prominence among academic economists.
The incorporation of sticky prices into dynamic stochastic general equilibrium
models gave rise to a policy tradeoff between output and inflation stabilization
that came to be known as the New Keynesian Phillips curve.
The inessential role that money balances play in the New Keynesian literature, along with the observed actual conduct of monetary policy in the United
States and elsewhere over the past 30 years, naturally shifted theoretical interest away from money growth rate rules and toward interest rate rules: In
the work of academic monetary economists, Milton Friedman’s celebrated
k-percent growth path for the money supply gave way to Taylor’s equally
influential interest rate feedback rule.
In this article, we survey recent advancements in the theory of optimal
monetary policy in models with a New Keynesian Phillips curve. Our survey
identifies a number of important lessons for the conduct of monetary policy.
First, optimal monetary policy is characterized by near price stability. This
policy implication is diametrically different from the one that obtains in models
in which the only nominal friction is a transactions demand for money. Second,
simple interest rate feedback rules that respond aggressively to price inflation
deliver near-optimal equilibrium allocations. Third, interest rate rules that
respond to deviations of output from trend may carry significant welfare costs.
Taken together, lessons one through three call for the adherence to an inflation
targeting objective. Fourth, the zero bound on nominal interest rates does not
appear to be a significant obstacle for the actual implementation of low and
stable inflation. Finally, product price stability emerges as the overriding goal
of monetary policy even in environments where not only goods prices but also
factor prices are sticky.
Before elaborating on the policy implications of the NKPC, we provide
some perspective by presenting a brief account of the state of the literature on
optimal monetary policy before the advent of the New Keynesian revolution.
1.
OPTIMAL MONETARY POLICY PRE-NKPC
Within the pre-NKPC framework, under quite general conditions, optimal
monetary policy calls for a zero opportunity cost of holding money, a result
known as the Friedman rule. In fiat money economies in which assets used
for transactions purposes do not earn interest, the opportunity cost of holding
money equals the nominal interest rate. Therefore, in the class of models
1 This is the case, for instance, when the monetary stance is active and the fiscal stance is
passive, which is the monetary/fiscal regime most commonly studied.
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
437
commonly used for policy analysis before the emergence of the NKPC, the
optimal monetary policy prescribed that the riskless nominal interest rate—the
return on federal funds, say—be set at zero at all times.
In the early literature, a demand for money is motivated in a variety of
ways, including a cash-in-advance constraint (Lucas 1982), money in the
utility function (Sidrauski 1967), a shopping-time technology (Kimbrough
1986), or a transactions-cost technology (Feenstra 1986). Regardless of how
a demand for money is introduced, the intuition for why the Friedman rule is
optimal in this class of model is straightforward: A zero nominal interest rate
maximizes holdings of a good—real money balances—that has a negligible
production cost. Another reason why the Friedman rule is optimal is that
a positive interest rate can distort the efficient allocation of resources. For
instance, in the cash-in-advance model with cash and credit goods, a positive
interest rate distorts the allocation of private spending across these two types of
goods. In models in which money ameliorates transaction costs or decreases
shopping time, a positive interest rate introduces a wedge in the consumptionleisure choice.
To illustrate the optimality of the Friedman rule, we augment a neoclassical
model with a transaction technology that is decreasing in real money holdings
and increasing in consumption spending. Specifically, consider an economy
populated by a large number of identical households. Each household has
preferences defined over processes of consumption and leisure and described
by the utility function
E0
∞
β t U (ct , ht ),
(1)
t=0
where ct denotes consumption, ht denotes labor effort, β ∈ (0, 1) denotes
the subjective discount factor, and E0 denotes the mathematical expectation
operator conditional on information available in period 0. The single period
utility function, U , is assumed to be increasing in consumption, decreasing in
effort, and strictly concave.
Final goods are produced using a production function, zt F (ht ), that takes
labor, ht , as the only factor input and is subject to an exogenous productivity
shock, zt .
A demand for real balances is introduced into the model by assuming
that money holdings, denoted Mt , facilitate consumption purchases. Specifically, consumption purchases are subject to a proportional transaction cost,
s(vt ), that is decreasing in the household’s money-to-consumption ratio, or
consumption-based money velocity,
Pt ct
vt =
,
(2)
Mt
where Pt denotes the nominal price of the consumption good in period t. The
transaction cost function, s(v), satisfies the following assumptions: (a) s(v)
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Federal Reserve Bank of Richmond Economic Quarterly
is nonnegative and twice continuously differentiable; (b) there exists a level
of velocity, v > 0, to which we refer as the satiation level of money, such
that s(v) = s (v) = 0; (c) (v − v)s (v) > 0 for v = v; and (d) 2s (v) +
vs (v) > 0 for all v ≥ v. Assumption (b) ensures that the Friedman rule (i.e.,
a zero nominal interest rate) need not be associated with an infinite demand
for money. It also implies that both the transaction cost and the distortion
it introduces vanish when the nominal interest rate is zero. Assumption (c)
guarantees that in equilibrium money velocity is always greater than or equal
to the satiation level. Assumption (d) ensures that the demand for money is
decreasing in the nominal interest rate.
Households are assumed have access to risk-free pure discount bonds,
denoted Bt . These bonds are assumed to carry a gross nominal interest rate
of Rt when held from period t to period t + 1. The flow budget constraint of
the household in period t is then given by
Pt ct [1 + s(vt )] + Pt τ Lt + Mt +
Bt
= Mt−1 + Bt−1 + Pt zt F (ht ),
Rt
(3)
where τ Lt denotes real lump sum taxes. In addition, it is assumed that the
household is subject to a borrowing limit that prevents it from engaging in
Ponzi-type schemes. The government is assumed to follow a fiscal policy
whereby it rebates any seigniorage income it receives from the creation of
money in a lump sum fashion to households.
A stationary competitive equilibrium can be shown to be a set of plans {ct ,
ht , vt }, satisfying the following three conditions:
vt2 s (vt ) =
−
Rt − 1
,
Rt
zt F (ht )
Uh (ct , ht )
=
, and
Uc (ct , ht )
1 + s(vt ) + vt s (vt )
[1 + s(vt )]ct = zt F (ht ),
(4)
(5)
(6)
given monetary policy {Rt }, with Rt ≥ 1, and the exogenous process {zt }.
The first equilibrium condition can be interpreted as a demand for money
or liquidity preference function. Given our maintained assumptions about
the transactions technology, s(vt ), the implied money demand function is
decreasing in the gross nominal interest rate, Rt . Further, our assumptions
imply that as the interest rate vanishes, or Rt approaches unity, the demand
for money reaches a finite maximum level given by ct /v. At this level of
money demand, households are able to perform transactions costlessly, as the
transactions cost, s(vt ), becomes nil. The second equilibrium condition shows
that a level of money velocity above the satiation level, v, or, equivalently, an
interest rate greater than zero, introduces a wedge between the marginal rate of
substitution of consumption for leisure and the marginal product of labor. This
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
439
wedge, given by 1 + s(vt ) + vt s (vt ), induces households to move away from
consumption and toward leisure. The wedge is increasing in the nominal
interest rate, implying that the larger is the nominal interest rate, the more
distorted is the consumption-leisure choice. The final equilibrium condition
states that a positive interest rate entails a resource loss in the amount of s(vt )ct .
This resource loss is increasing in the interest rate and vanishes only when the
nominal interest rate equals zero.
We wish to characterize optimal monetary policy under the assumption
that the government has the ability to commit to policy announcements. This
policy optimality concept is known as Ramsey optimality. In the context of
the present model, the Ramsey optimal monetary policy consists in choosing
the path of the nominal interest rate that is associated with the competitive
equilibrium that yields the highest level of welfare to households. Formally,
the Ramsey policy consists in choosing processes Rt , ct , ht , and vt to maximize
the household’s utility function given in equation (1) subject to the competitive
equilibrium conditions given by equations (4) through (6).
When one inspects the three equilibrium conditions above, it is clear that if
the policymaker sets the monetary policy instrument, which we take to be the
nominal interest rate, such that velocity is at the satiation level, (vt = v), then
the equilibrium conditions become identical to an economy without the money
demand friction, i.e., ct = zt F (ht ) and −Uh (ct , ht )/Uc (ct , ht ) = zt F (ht ).
Because the real allocation in the absence of the monetary friction is Pareto
optimal, the proposed monetary policy must be Ramsey optimal. By a Pareto
optimal allocation we mean a feasible real allocation (i.e., one satisfying ct =
zt F [ht ]) with the property that any other feasible allocation that makes at least
one agent better off also makes at least one agent worse off. It follows from
equation (4) that setting the nominal interest rate to zero (Rt = 1) ensures
that vt = v. For this reason, optimal monetary policy takes the form of a zero
nominal interest rate at all times.
Under the optimal monetary policy, the rate of change in the aggregate
price level varies over time. Because, to a first approximation, the nominal
interest rate equals the sum of the real interest rate and the expected rate of
inflation, and because under the optimal monetary policy the nominal interest
rate is held constant, the degree to which the inflation rate fluctuates depends
on the equilibrium variations in the real rate of interest. In general, optimal monetary policy in a model in which a role for monetary policy arises
solely from the presence of money demand is not characterized by inflation
stabilization.
A second important consequence of optimal monetary policy in the context
of the present model is that inflation is, on average, negative. This is because,
with a zero nominal interest rate, the inflation rate equals, on average, the
negative of the real rate of interest.
440
Federal Reserve Bank of Richmond Economic Quarterly
2. THE NKPC AND OPTIMAL MONETARY POLICY
The New Keynesian Phillips curve can be briefly defined as the dynamic
output-inflation tradeoff that arises in a dynamic general equilibrium model
populated by utility-maximizing households and profit-maximizing firms—
such as the one laid out in the previous section—and augmented with some
kind of rigidity in the adjustment of nominal product prices. The foundations
of the NKPC were laid by Calvo (1983) and Rotemberg (1982). Woodford
(1996) and Yun (1996) completed its development by introducing optimizing
behavior on the part of firms facing Calvo-type dynamic nominal rigidities.
The most important policy implication of models featuring a New
Keynesian Phillips curve is the optimality of price stability (see Goodfriend
and King [1997] for an early presentation of this result). We will discuss
the price stability result in a variety of theoretical models, including ones
with a realistic set of real and nominal rigidities, policy instruments and policy constraints, and sources of aggregate fluctuations. We start, however,
with the simplest structure within which the price stability result can be obtained. To this end, we strip the model presented in the previous section from
its money demand friction and instead introduce costs of adjusting nominal
product prices. In the resulting model, sticky prices represent the sole source
of nominal friction.
To introduce sticky prices into the model of the previous section, assume
that the consumption good, ct , is a composite good made of a continuum of
intermediate differentiated goods. The aggregator function is of the DixitStiglitz type. Each household/firm unit is the monopolistic producer of one
variety of intermediate goods. In turn, intermediate goods are produced using
a technology like the one given in the previous section. The household/firm
unit hires labor, h̃t , from a perfectly competitive market.
The demand faced by the household/firm unit for the intermediate input that it produces is of the form Yt d(P̃t /Pt ), where Yt denotes the level
of aggregate demand, which is taken as exogenous by the household/firm
unit; P̃t denotes the nominal price of the intermediate good produced by the
household/firm unit; and Pt is the price of the composite consumption good.
The demand function, d(·), is assumed to be decreasing in the relative price,
P̃t /Pt , and is assumed to satisfy d(1) = 1 and −d (1) ≡ η > 1, where η
denotes the price elasticity of demand for each individual variety of intermediate goods that prevails in a symmetric equilibrium. The restrictions on d(1)
and d (1) are necessary for the existence of a symmetric equilibrium. The
monopolist sets the price of the good it supplies, taking the level of aggregate
demand as given, and is constrained to satisfy demand at that price, that is,
zt F (h̃t ) ≥ Yt d(P̃t /Pt ).
Price adjustment is assumed to be sluggish, à la Rotemberg (1982). Specifically, the household/firm unit faces a resource cost of changing prices that is
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
quadratic in the inflation rate of the good it produces:
2
θ
P̃t
Price adjustment cost =
−1 .
2 P̃t−1
441
(7)
The parameter θ measures the degree of price stickiness. The higher is θ ,
the more sluggish is the adjustment of nominal prices. When θ equals zero,
prices are fully flexible. The flow budget constraint of the household/firm unit
in period t is then given by
⎡
2 ⎤
P̃t
⎣ P̃t Yt d P̃t − wt h̃t − θ
ct + τ Lt ≤ (1 − τ D
− 1 ⎦,
t )wt ht +
Pt
Pt
2 P̃t−1
where τ D
t denotes an income tax/subsidy rate. We introduce this fiscal instrument as a way to offset the distortions arising from the presence of monopolistic competition. We restrict attention to a stationary symmetric equilibrium
in which all household/firm units charge the same price for the intermediate
good they produce. Letting π t ≡ Pt /Pt−1 denote the gross rate of inflation,
the complete set of equilibrium conditions is then given by
!
wt
η−1
Uc (ct+1 , ht+1 )
ηct
−
+ βEt
π t+1 (π t+1 − 1),
π t (π t − 1) =
θ zt F (ht )
η
Uc (ct , ht )
(8)
−
Uh (ct , ht )
= (1 − τ D
t )wt , and
Uc (ct , ht )
(9)
θ
(10)
zt F (ht ) − (π t − 1)2 = ct .
2
The above three equations provide solutions for the equilibrium processes of
consumption, ct , hours, ht , and the real wage, wt , given processes for the rate
of inflation, π t , and for the tax rate, τ D
t , which we interpret to be outcomes of
the monetary and fiscal policies in place, respectively.
The first equilibrium condition, equation (8), represents the NKPC, to
which the current volume is devoted. It describes an equilibrium relationship between current inflation, π t , the current deviation of marginal cost,
wt /zt F (ht ), from marginal revenue, (η − 1)/η, and expected future inflation.
Under full price flexibility, the firm would always set marginal revenue equal
to marginal cost. However, in the presence of price adjustment costs, this
practice is costly. To smooth out price changes over time, firms set prices to
equate an average of current and future expected marginal costs to an average
of current and future expected marginal revenues. This optimal price-setting
behavior gives rise to a relation whereby, given expected inflation, current inflation is an increasing function of marginal costs. Intuitively, this relation is
steeper the more flexible are prices (i.e., the lower is θ ), the more competitive
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are product markets (i.e., the higher is η), and the higher is the current level
of demand (i.e., the larger is ct ). At the same time, given marginal cost, current inflation is increasing in expected future inflation. This is because, with
quadratic costs of changing nominal prices, a firm expecting higher inflation
in the future would like to smooth out the necessary price adjustments over
time by beginning to raise prices already in the current period.
We have derived the New Keynesian Phillips curve in the context of the
Rotemberg (1982) model of price stickiness. However, a similar relationship emerges under other models of nominal rigidity, such as those due to
Calvo (1983), Taylor (1993), Woodford (1996), and Yun (1996). For instance,
in the Calvo-Woodford-Yun model, price stickiness arises because firms are
assumed to receive an idiosyncratic random signal each period indicating
whether they are allowed to reoptimize their posted prices. A difference between the Rotemberg and the Calvo-Woodford-Yun models is that the latter
displays equilibrium price dispersion across firms even in the absence of aggregate uncertainty. However, up to first order, the NKPCs implied by the
Rotemberg and Calvo-Woodford-Yun models are identical. Indeed, much of
the literature on the NKPC focuses on a log-linear approximation of this key
relationship, as in equation (11).
The second equilibrium condition presented in equation (9) states that
the marginal rate of substitution of consumption for leisure is equated to the
after-tax real wage rate. The third equilibrium condition, equation (10), is
a resource constraint requiring that aggregate output net of price adjustment
costs equal private consumption.
It is straightforward to establish that, in this economy, the optimal monetary policy, that is, the policy that maximizes the welfare of the representative
household, is one in which the inflation rate is nil at all times. Formally, the
optimal monetary policy must be consistent with an equilibrium in which
πt = 1
for all t ≥ 0. This result holds exactly provided the fiscal authority subsidizes labor income to a point that fully offsets the distortion arising from
the existence of imperfect competition among intermediate goods producers.
Specifically, the income tax rate, τ D
t , must be set at a constant and negative
level given by
τD
t =
1
1−η
for all t ≥ 0.
To see that the proposed policy regime is optimal, we demonstrate that it
implies a set of equilibrium conditions that coincide with the one that arises
in an economy with fully flexible prices (θ = 0) and perfect competition in
product markets (η = ∞), such as the one analyzed in Section 1 in the absence
of a money-demand distortion. In effect, when π t = 1 for all t, equilibrium
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
443
condition (10) collapses to ct = zt F (ht ). In addition, under zero inflation
the NKPC (equation [8]) reduces to wt = zt F (ht )(η − 1)/η. Using this
expression along with the proposed optimal level for the income tax rate in
the equilibrium labor supply, equation (9), yields the efficiency conditions
−Uh (ct , ht )/Uc (ct , ht ) = zt F (ht ), which, together with the resource constraint ct = zt F (ht ), constitute the equilibrium conditions of a perfectly
competitive flexible-price economy. As we show in Section 1, the resource
allocation in this economy is Pareto optimal.
3. THE OPTIMAL INFLATION RATE
At this point, it is of interest to summarize and compare the results in this
section and in previous ones. We have shown that when prices are fully
flexible and the only nominal friction is a demand for money, then optimal
monetary policy takes the form of complete stabilization of the interest rate at
a value of zero (Rt = 1 for all t). We have also established that in a cashless
economy in which the only source of nominal friction is given by product
price stickiness, optimal monetary policy calls for full stabilization of the rate
of inflation at a value of zero (π t = 1 for all t). Under optimal policy in the
monetary flexible-price economy, inflation is time varying and equal to the
negative of the real interest rate on average, whereas in the cashless stickyprice economy, inflation is constant and equal to zero at all times. Also, in the
monetary flexible-price economy, optimal policy calls for a constant nominal
interest rate equal to zero at all times, whereas in the cashless sticky-price
economy, it calls for a time-varying nominal interest rate equal to the real
interest rate on average.
These results raise the question of what the characteristics of optimal
monetary policy are in a more realistic economic environment in which both
a demand for money and price stickiness coexist. In particular, in such an
environment a policy tradeoff emerges between the benefits of targeting zero
inflation—i.e., minimizing price-adjustment costs—and the benefits of deflating at the real rate of interest—i.e., minimizing the opportunity cost of holding
money. In the canonical economies with only one nominal friction studied in
this and previous sections, the characterization of the optimal rate of inflation
is relatively straightforward. As soon as both nominal frictions are incorporated jointly, it becomes impossible to determine the optimal rate of inflation
analytically. One is therefore forced to resort to numerical methods.
The resolution of the Friedman-rule-versus-price-stability tradeoff was
studied by, among others, Khan, King, and Wolman (2003) and Schmitt-Grohé
and Uribe (2004a, 2007b). As one would expect, when both the money demand
and sticky-price frictions are present, the optimal rate of inflation falls between
zero and the one called for by the Friedman rule. The question of interest,
however, is where exactly in this interval the optimal inflation rate lies. Khan,
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King, and Wolman find, in the context of a stylized model calibrated to match
aspects of money demand and price dynamics in the postwar United States,
that the optimal rate of inflation is −0.76 percent per year. By comparison, in
their model the Friedman rule is associated with a deflation rate of 2.93 percent
per year. Thus, in the study by Khan, King, and Wolman, the optimal policy
is closer to price stability than to the Friedman rule. Taking these numbers at
face value, one might conclude that price stickiness is the dominant friction
in shaping optimal monetary policy. However, Khan, King, and Wolman
(2003) and Schmitt-Grohé and Uribe (2004a, 2007b) show that the resolution
of the tradeoff is quite sensitive to plausible changes in the values taken by
the structural parameters of the model.
In Schmitt-Grohé and Uribe (2007b), we find that a striking characteristic of the optimal monetary regime is the high sensitivity of the welfaremaximizing rate of inflation with respect to the parameter governing the
degree of price stickiness for the range of values of this parameter that is
empirically relevant. The model underlying the analysis of Schmitt-Grohé
and Uribe (2007b) is a medium-scale model of the U.S. economy featuring, in
addition to money demand by households and sticky product prices, a number
of real and nominal rigidities including wage stickiness, a demand for money
by firms, habit formation, capital accumulation, variable capacity utilization,
and investment adjustment costs. The structural parameters of the model are
assigned values that are consistent with full- as well as limited-information
approaches to estimating this particular model.
In the Schmitt-Grohé and Uribe (2007b) model, the degree of price stickiness is captured by a parameter denoted α, measuring the probability that a
firm is not able to optimally set the price it charges in a particular quarter. The
average number of periods elapsed between two consecutive optimal price adjustments is given by 1/(1 − α). Available empirical estimates of the degree
of price rigidity using macroeconomic data vary from two to five quarters, or
α ∈ [0.5, 0.8]. For example, Christiano, Eichenbaum, and Evans (2005) estimate α to be 0.6. By contrast, Altig et al. (2005) estimate a marginal-cost-gap
coefficient in the Phillips curve that is consistent with a value of α of around
0.8. Both Christiano, Eichenbaum and Evans (2005) and Altig et al. (2005)
use an impulse-response matching technique to estimate the price-stickiness
parameter α. Bayesian estimates of this parameter include Del Negro et al.
(2004), Levin et al. (2006), and Smets and Wouters (2007), who report posterior means of 0.67, 0.83, and 0.66, respectively, and 90 percent posterior
probability intervals of (0.51,0.83), (0.81,0.86), and (0.56,0.74), respectively.
Recent empirical studies have documented the frequency of price changes
using microdata underlying the construction of the U.S. consumer price index.
These studies differ in the sample period considered, in the disaggregation
of the price data, and in the treatment of sales and stockouts. The median
frequency of price changes reported by Bils and Klenow (2004) is four to five
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
445
Figure 1 Price Stickiness, Fiscal Policy, and Optimal Inflation
0.0
ACEL
CEE
ACEL
0.5
1.0
1.5
2.0
CEE
2.5
-o-o- Optimal Distortionary
Taxes
Lump-Sum Taxes
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Notes: CEE and ACEL indicate, respectively, the values for the parameter, α, estimated
by Christiano, Eichenbaum, and Evans (2005) and Altig et al. (2005).
months, the one reported by Klenow and Kryvtsov (2005) is four to seven
months, and the one reported by Nakamura and Steinsson (2007) is eight to
11 months. However, there is no immediate interpretation of these frequency
estimates to the parameter, α, governing the degree of price stickiness in Calvostyle models of price staggering. Consider, for instance, the case of indexation.
In that case, even though firms change prices every period—implying the
highest possible frequency of price changes—prices themselves may be highly
sticky, for they may be only reoptimized at much lower frequencies.
Figure 1 displays with a solid line the relationship between the degree of
price stickiness, α, and the optimal rate of inflation in percent per year, π ,
implied by the model studied in Schmitt-Grohé and Uribe (2007b). When α
equals 0.5, the lower range of the available empirical evidence using macrodata, the optimal rate of inflation is −2.9 percent, which is the level called
for by the Friedman rule. For a value of α of 0.8, which is near the upper
range of the available empirical evidence using macrodata, the optimal level
of inflation rises to −0.4 percent, which is close to price stability.
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Federal Reserve Bank of Richmond Economic Quarterly
Besides the uncertainty surrounding the estimation of the degree of price
stickiness, a second aspect of the apparent difficulty in establishing reliably the
optimal long-run level of inflation has to do with the shape of the relationship
linking the degree of price stickiness to the optimal level of inflation. The
problem resides in the fact that, as is evident from Figure 1, this relationship
becomes significantly steep precisely for that range of values of α that is
empirically most compelling. It turns out that an important factor determining
the shape of the function relating the optimal level of inflation to the degree
of price stickiness is the underlying fiscal policy regime.
Fiscal considerations fundamentally change the long-run tradeoff between
price stability and the Friedman rule. To see this, we now consider an economy
where lump-sum taxes are unavailable (τ L = 0). Instead, the fiscal authority must finance government purchases by means of proportional capital and
labor income taxes. The social planner jointly sets monetary and fiscal policy
in a welfare-maximizing (i.e., Ramsey-optimal) fashion.2 Figure 1 displays
the relationship between the degree of price stickiness, α, and the optimal
rate of inflation, π. The solid line corresponds to the case discussed earlier
featuring lump-sum taxes. The dash-circled line corresponds to the economy
with optimally chosen distortionary income taxes. In stark contrast to what
happens under lump-sum taxation, under optimal distortionary taxation the
function linking π and α is flat and very close to zero for the entire range of
macrodata-based empirically plausible values of α, namely 0.5 to 0.8. In other
words, when taxes are distortionary and optimally determined, price stability
emerges as a prediction that is robust to the existing uncertainty about the
exact degree of price stickiness. Even if one focuses on the evidence of price
stickiness stemming from microdata, the model with distortionary Ramsey
taxation predicts an optimal long-run level of inflation that is much closer to
zero than to the level called for by the Friedman rule.
Our intuition for why price stability arises as a robust policy recommendation in the economy with optimally set distortionary taxation runs as follows.
Consider the economy with lump-sum taxation. Deviating from the Friedman
rule (by raising the inflation rate) has the benefit of reducing price adjustment
costs. Consider next the economy with optimally chosen income taxation and
no lump-sum taxes. In this economy, deviating from the Friedman rule still
provides the benefit of reducing price adjustment costs. However, in this economy, increasing inflation has the additional benefit of increasing seigniorage
revenue, thereby allowing the social planner to lower distortionary income tax
rates. Therefore, the Friedman rule versus price stability tradeoff is tilted in
favor of price stability.
2 The details of this environment are contained in Schmitt-Grohé and Uribe (2006). The
structure of this economy is identical to that studied in Schmitt-Grohé and Uribe (2007b), except
for the inclusion of fiscal policy.
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
447
It follows from this intuition that what is essential in inducing the optimality of price stability is that, on the margin, the fiscal authority trades off the
inflation tax for regular taxation. Indeed, it can be shown that if distortionary
tax rates are fixed, even if they are fixed at the level that is optimal in a world
without lump-sum taxes, and the fiscal authority has access to lump-sum taxes
on the margin, the optimal rate of inflation is much closer to the Friedman rule
than to zero. In this case, increasing inflation no longer has the benefit of reducing distortionary taxes. As a result, the Ramsey planner has less incentives
to inflate.
We close this section by drawing attention to the fact that, quite independently of the precise degree of price stickiness, the optimal inflation target
is below zero. In light of this robust result, it is puzzling that all countries
that self-classify as inflation targeters set inflation targets that are positive. In
effect, in the developed world inflation targets range between 2 and 4 percent
per year. Somewhat higher targets are observed across developing countries.
An argument often raised in defense of positive inflation targets is that negative inflation targets imply nominal interest rates that are dangerously close
to the zero lower bound on nominal interest rates and, hence, may impair the
central bank’s ability to conduct stabilization policy. In Schmitt-Grohé and
Uribe (2007b) we find, however, that this argument is of little relevance in the
context of the medium-scale estimated model within which we conduct policy
evaluation. The reason is that under the optimal policy regime, the mean of
the nominal interest rate is about 4.5 percent per year with a standard deviation of only 0.4 percent. This means that for the zero lower bound to pose
an obstacle to monetary stabilization policy, the economy must suffer from
an adverse shock that forces the interest rate to be more than ten standard
deviations below target. The likelihood of such an event is practically nil.
4. THE OPTIMAL VOLATILITY OF INFLATION
Two distinct branches of the existing literature on optimal monetary policy
deliver diametrically opposed policy recommendations concerning the cyclical behavior of prices and interest rates. One branch follows the theoretical
framework laid out in Lucas and Stokey (1983). It studies the joint determination of optimal fiscal and monetary policy in flexible-price environments with
perfect competition in product and factor markets. In this strand of the literature, the government’s problem consists of financing an exogenous stream of
public spending by choosing the least disruptive combination of inflation and
distortionary income taxes.
Calvo and Guidotti (1990, 1993) and Chari, Christiano, and Kehoe (1991)
characterize optimal monetary and fiscal policy in stochastic environments
with nominal nonstate-contingent government liabilities. A key result of these
papers is that it is optimal for the government to make the inflation rate highly
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volatile and serially uncorrelated. For instance, Schmitt-Grohé and Uribe
(2004b) show, in the context of a flexible-price model calibrated to the U.S.
economy, that under the optimal policy the inflation rate has a standard deviation of 7 percent per year and a serial correlation of −0.03. The intuition for
this result is that, under flexible prices, highly volatile and unforecastable inflation is nondistorting and at the same time carries the fiscal benefit of acting
as a lump-sum tax on private holdings of government-issued nominal assets.
The government is able to use surprise inflation as a nondistorting tax to the
extent that it has nominal, nonstate-contingent liabilities outstanding. Thus,
price changes play the role of a shock absorber of unexpected innovations in
the fiscal deficit. This “front-loading” of government revenues via inflationary
shocks allows the fiscal authority to keep income tax rates remarkably stable
over the business cycle.
However, as discussed in Section 2, the New Keynesian literature, aside
from emphasizing the role of price rigidities and market power, differs from
the earlier literature described above in two important ways. First, it assumes,
either explicitly or implicitly, that the government has access to (endogenous)
lump-sum taxes to finance its budget. An important implication of this assumption is that there is no need to use unanticipated inflation as a lump-sum
tax; regular lump-sum taxes take on this role. Second, the government is
assumed to be able to implement a production (or employment) subsidy to
eliminate the distortion introduced by the presence of monopoly power in
product and factor markets.
The key result of the New Keynesian literature, which we presented in
Sections 2 and 3, is that the optimal monetary policy features an inflation
rate that is zero or close to zero at all times (i.e., both the optimal mean and
volatility of inflation are near zero). The reason price stability is optimal in
environments of the type described there is that it minimizes (or completely
eliminates) the costs introduced by inflation under nominal rigidities.
Together, these two strands of research on optimal monetary policy leave
the monetary authority without a clear policy recommendation. Should the
central bank pursue policies that imply high or low inflation volatility? In
Schmitt-Grohé and Uribe (2004a), we analyze the resolution of this policy
dilemma by incorporating in a unified framework the essential elements of
the two approaches to optimal policy described above. Specifically, we build
a model that shares two elements with the earlier literature: (a) The only
source of regular taxation available to the government is distortionary income
taxes. As a result, the government cannot implement production subsidies
to undo distortions created by the presence of imperfect competition, and (b)
the government issues only nominal, one-period, nonstate-contingent bonds.
At the same time, the setup shares two important assumptions with the more
recent body of work on optimal monetary policy: (a) Product markets are
imperfectly competitive, and (b) product prices are assumed to be sticky and,
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
449
hence, the model features a New Keynesian Phillips curve. Schmitt-Grohé
and Uribe (2004a) introduce price stickiness as in the previous section by
assuming that firms face a convex cost of price adjustment (Rotemberg 1982).
In this environment, the government faces a tradeoff in choosing the path
of inflation. On the one hand, the government would like to use unexpected
inflation as a nondistorting tax on nominal wealth. In this way, the fiscal
authority could minimize variations in distortionary income taxes over the
business cycle. On the other hand, changes in the rate of inflation come at a
cost, for firms face nominal rigidities.
When price changes are brought about at a cost, it is natural to expect
that a benevolent government will try to implement policies consistent with a
more stable behavior of prices than when price changes are costless. However,
the quantitative effect of an empirically plausible degree of price rigidity on
optimal inflation volatility is not clear a priori. In Schmitt-Grohé and Uribe
(2004a), we show that for the degree of price stickiness estimated for the U.S.
economy, this tradeoff is overwhelmingly resolved in favor of price stability.
The Ramsey allocation features a dramatic drop in the standard deviation of
inflation from 7 percent per year under flexible prices to a mere 0.17 percent
per year when prices adjust sluggishly.3
Indeed, the impact of price stickiness on the optimal degree of inflation
volatility turns out to be much stronger than suggested by the numerical results
reported in the previous paragraph. Figure 2, taken from Schmitt-Grohé and
Uribe (2004a), shows that a minimum amount of price stickiness suffices to
make price stability the central goal of optimal policy. Specifically, when the
degree of price stickiness, embodied in the parameter θ (see equation [7]), is
assumed to be ten times smaller than the estimated value for the U.S. economy,
the optimal volatility of inflation is below 0.52 percent per year, 13 times
smaller than under full price flexibility.
A natural question elicited by Figure 2 is why even a modest degree of
price stickiness can turn undesirable the use of a seemingly powerful fiscal
instrument, such as large revaluations or devaluations of private real financial
wealth through surprise inflation. Our conjecture is that in the flexible-price
economy, the welfare gains of surprise inflations or deflations are very small.
Our intuition is as follows. Under flexible prices, it is optimal for the central
bank to keep the nominal interest rate constant over the business cycle. This
means that large surprise inflations must be as likely as large deflations, as
variations in real interest rates are small. In other words, inflation must have a
near-i.i.d. behavior. As a result, high inflation volatility cannot be used by the
Ramsey planner to reduce the average amount of resources to be collected via
distortionary income taxes, which would be a first-order effect. The volatility
3 This price stability result is robust to augmenting the model to allow for nominal rigidities
in wages and indexation in product or factor prices (Schmitt-Grohé and Uribe 2006, Table 6.5).
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Degree of Price Stickiness and Optimal Inflation Volatility
7
← Flexible Prices
6
Std. Dev. of π
5
4
3
2
1
← Baseline
0
0
2
4
6
8
10
Degree of Price Stickiness, θ
Notes: The parameter, θ , governs the cost of adjusting nominal prices as defined in
equation (7). Its baseline value is 4.4, in line with available empirical estimates. The
standard deviation of inflation is measured in percent per year.
of inflation primarily serves the purpose of smoothing the process of income
tax distortions—a second-order source of welfare losses—without affecting
their average level.
Another way to gain intuition for the dramatic decline in optimal
inflation volatility that occurs even at very modest levels of price stickiness
is to interpret price volatility as a way for the government to introduce real
state-contingent public debt. Under flexible prices, the government uses statecontingent changes in the price level as a nondistorting tax or transfer on private
holdings of government assets. In this way, nonstate-contingent nominal public debt becomes state-contingent in real terms. So, for example, in response
to an unexpected increase in government spending, the Ramsey planner does
not need to increase tax rates by much because by inflating away part of the
public debt he can ensure intertemporal budget balance. It is, therefore, clear
that introducing costly price adjustment is the same as if the government were
limited in its ability to issue real state-contingent debt. It follows that the larger
the welfare gain associated with the ability to issue real state-contingent public debt—as opposed to nonstate-contingent debt—the larger the amount of
price stickiness required to reduce the optimal degree of inflation volatility.
Aiyagari et al. (2002) show that indeed the level of welfare under the Ramsey
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
451
policy in an economy without real state-contingent public debt is virtually the
same as in an economy with state-contingent debt. Our finding that a small
amount of price stickiness is all it takes to bring the optimal volatility of inflation from a very high level to near zero is thus perfectly in line with the
finding of Aiyagari et al. (2002).
If this intuition is correct, then the behavior of tax rates and public debt
under sticky prices should resemble that implied by the Ramsey allocation in
economies without real state-contingent debt. Indeed, in financing the budget,
the Ramsey planner replaces front-loading with standard debt and tax instruments. For example, in response to an unexpected increase in government
spending, the planner does not generate a surprise increase in the price level.
Instead, he chooses to finance the increase in government purchases partly
through an increase in income tax rates and partly through an increase in public debt. The planner minimizes the tax distortion by spreading the required
tax increase over many periods. This tax-smoothing behavior induces nearrandom walk dynamics into the tax rate and public debt. By contrast, under
full price flexibility (i.e., when the government can create real-state-contingent
debt), tax rates and public debt inherit the stochastic process of the underlying
shocks.
An important conclusion of this analysis is, thus, that the Aiyagari et al.
(2002) result, namely, that optimal policy imposes a near-random walk behavior on taxes and debt, does not require the unrealistic assumption that the
government can issue only nonstate-contingent real debt. This result emerges
naturally in economies with nominally nonstate-contingent debt—clearly the
case of greatest empirical relevance—and a minimum amount of price rigidity.4 However, if government debt is assumed to be state contingent, the
presence of sticky prices may introduce no difference in the Ramsey real allocation, depending on the precise specification of the demand for money (see
Correia, Nicolini, and Teles 2008). The reason for this result is that, as shown
in Lucas and Stokey (1983), if government debt is state-contingent and prices
are fully flexible, the Ramsey allocation does not pin down the price level
uniquely. In this case, there is an infinite number of price-level processes (and
thus of money supply processes) that can be supported as Ramsey outcomes.
4 It is of interest to relate the near-random walk in taxes and debt that emerges as the optimal
policy outcome in a model featuring a New Keynesian Phillips curve with the celebrated taxsmoothing result of Barro (1979). In Barro’s formulation, the objective function of the government
is the expected present discounted value of squared deviations of tax rates from a target or desired
level. The government minimizes this objective function subject to a sequential budget constraint,
which is linear in debt and tax rates. The resulting solution resembles the random walk model of
consumption with taxes taking the place of consumption and public debt taking the place of private
debt. The analysis in Schmitt-Grohé and Uribe (2004a) departs from Barro’s ad hoc loss function
and replaces it with the utility function of the representative optimizing household inhabiting a
fully-articulated, dynamic, stochastic, general-equilibrium economy. In this environment, the random
walk result obtains from a more subtle channel, namely, the introduction of a miniscule amount
of nominal rigidity in product prices.
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Loosely speaking, the introduction of price stickiness simply “uses this degree
of freedom” to pin down the equilibrium process of the price level without
altering other aspects of the Ramsey solution.
5.
IMPLEMENTATION OF OPTIMAL POLICY
We established that in the simple New Keynesian model presented in Section
2, the optimal policy consists of setting the inflation rate equal to zero at all
times (π t = 1) and imposing a constant output subsidy (τ D
t = 1/[1 − η]).
The question we pursue in this section is how to implement the optimal policy.
Because central banks in the United States and elsewhere use the short-term
nominal interest rate as the monetary policy instrument, it is of empirical
interest to search for interest rate rules that implement the optimal allocation.
Using the Ramsey-Optimal Interest Rate Process as a
Feedback Rule
One might be tempted to believe that implementation of optimal policy is
trivial once the interest rate associated with the Ramsey equilibrium has been
found. Specifically, in the Ramsey equilibrium, the nominal interest rate can
be expressed as a function of the current state of the economy. Then, the
prescription would be simply to use this function as a policy rule in setting
the nominal interest rate at all dates and under all circumstances. It turns
out that conducting policy in this fashion would, in general, not deliver the
intended results. The reason is that although such a policy would be consistent
with the optimal equilibrium, it would at the same time open the door to
other (suboptimal) equilibria. It follows that the solution to the optimal policy
problem is mute with respect to the issue of implementation of such policy. To
see this, it is convenient to consider as an example a log-linear approximation
to the equilibrium conditions associated with the cashless, sticky-price model
presented in Section 2. It can be shown that the resulting linear system is
given by5
π̂ t = βEt π̂ t+1 + κ ĉt − γ ẑt
(11)
−σ ĉt = R̂t − σ Et ĉt+1 − Et π̂ t+1 ,
(12)
and
where σ , κ,and γ > 0 are parameters. Hatted variables denote percent deviations of the corresponding nonhatted variables from their respective values
5 The log-linearization is performed around the nonstochastic steady state of the Ramsey equi-
librium. In performing the linearization, we assume that the period utility function is separable in
consumption and hours and that the production function is linear in labor.
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
453
in the deterministic steady state of the Ramsey equilibrium. Equation (11)
results from combining equations (8), (9), and (10) and is typically referred to
as the New Keynesian Phillips curve.6 Equation (12) is a linearized version
of an Euler equation that prices nominally risk-free bonds, where Rt denotes
the gross nominal risk-free interest rate between periods t and t + 1. This
equation is typically referred to as the intertemporal IS equation.
Substituting the welfare-maximizing rate of inflation, π̂ t = 0, into the
intertemporal IS curve (12) implies that the nominal interest rate is given by
R̂t = σ Et (ĉt+1 − ĉt ), which states that under the optimal policy the nominal and real interest rates coincide. Suppose that the central bank adopted
this expression as a policy feedback rule for the nominal interest rate. The
question is whether this proposed rule implements the Ramsey equilibrium
uniquely. The answer to this question is no. To see why, consider a solution of the form π̂ t = t , where t is i.i.d. normal with mean zero and an
arbitrary standard deviation σ ≥ 0. Notice that for all positive values of
σ , the proposed solution for inflation is different from the optimal one. It
is straightforward to see that the proposed solution satisfies the intertemporal
IS equation (12). The solution for consumption can be read off the NKPC as
being ĉt = γ /κ ẑt + (1/κ) t . We have, therefore, constructed a competitive
equilibrium in which a nonfundamental source of uncertainty, embodied in the
random variable t , causes stochastic deviations of consumption and inflation
from their optimal paths. Notice that in this example there exists an infinite
number of different equilibria indexed by the parameter, σ , governing the
volatility of the nonfundamental shock, t .
One possible objection against the interest rate feedback rule proposed in
the previous paragraph is that it is cast in terms of the endogenous variable,
ct . In particular, one may wonder whether this endogeneity is responsible
for the inability of the proposed rule to implement the Ramsey equilibrium
uniquely. This concern is indeed unfounded. For, even if the interest rate
feedback rule were cast in terms of exogenous fundamental variables, the
failure of the strategy of using the Ramsey solution for Rt as an interest rate
feedback rule remains. Specifically, substituting the optimal rate of inflation,
π̂ t = 0, into the New Keynesian Phillips curve (11) yields ĉt = γ κ −1 ẑt .
In turn, substituting this expression into the intertemporal IS curve (12) implies that in the optimal equilibrium the nominal interest rate is given by
R̂t = r̂tn ≡ σ γ κ −1 Et (ẑt+1 − ẑt ). The variable rtn denotes the risk-free real (as
well as nominal) interest rate that prevails in the Ramsey optimal equilibrium
and is referred to as the natural rate of interest. Using this expression, equations (11)–(12) become a system of two linear stochastic difference equations
6 For detailed derivations of this expression, see, for instance, Woodford (2003). This linear
expression is the NKPC studied in the papers by Nason and Smith (2008) and Schorfheide (2008)
that appear in this issue.
454
Federal Reserve Bank of Richmond Economic Quarterly
in the endogenous variables π̂ t and ĉt . This system possesses one eigenvalue inside the unit circle and one outside. It follows that the solution for
inflation and consumption is of the form π̂ t+1 = ζ ππ π̂ t + ζ πz ẑt + t+1 and
ĉt = ζ cπ π̂ t +ζ cz ẑt , where π̂ 0 is arbitrary, t is a nonfundamental shock such as
the one introduced above, and the parameter ζ ππ is less than unity in absolute
value. Clearly, the competitive equilibrium that the proposed rule implements
displays persistent and stochastic deviation from the optimal solution. We
conclude that the use of the Ramsey-optimal interest rate process as a policy
feedback rule fails to implement the desired competitive equilibrium.
Can the Taylor Rule Implement Optimal Policy?
A Taylor-type rule is an interest rate feedback rule whereby the nominal interest
rate is set as an increasing linear function of inflation and deviations of real
output from trend, with an inflation coefficient greater than unity and an output
coefficient greater than zero. Formally, a Taylor-type interest rate rule can be
written as R̂t = α π π̂ t + α y ŷt , where α π > 1 and α y > 0 are parameters
and ŷt represents the percent deviation of real output from trend. Taylor’s
rule has been widely studied in monetary economics since the publication of
Taylor’s (1993) seminal article. It has been advocated as a desirable policy
specification and is considered by some to be a reasonable approximation
of actual monetary policy in the United States and many other developed
countries. For this reason, we now consider the question of whether a Taylor
rule can implement the optimal allocation. The answer is that, in general, an
interest rate feedback rule of the type proposed by Taylor is unable to support
the Ramsey optimal equilibrium. This issue was first analyzed by Woodford
(2001).
To establish whether the Taylor rule presented above can implement the
optimal allocation, we set the inflation rate at its optimal value of zero (π̂ t = 0)
and combine the intertemporal IS equation (12) with the Taylor rule. This
yields α y ĉt = σ (Et ĉt+1 − ĉt ). Now, replacing ĉt with its optimal value of
γ /κ ẑt , we obtain Et ẑt+1 = (1 + α y /σ )ẑt . This expression represents a contradiction because the productivity shock, ẑt , is assumed to be an exogenous
stationary process with a law of motion independent of the policy parameter,
α y , and the preference parameter, σ . We have established that the proposed
Taylor rule fails to implement the optimal equilibrium. One can show that
this result also obtains when the nominal interest rate is assumed to respond
to deviations of output from its natural level, which is defined as the level of
output associated with the optimal equilibrium and is given by ytn ≡ γ /κ ẑt .
However, the optimal allocation can indeed be implemented by a modified
Taylor rule of the form R̂t = r̂tn + α π π̂ t as long as α π > 1. In this rule, r̂tn
denotes the natural rate of interest defined earlier. The first term in the modified
Taylor rule makes the Ramsey allocation feasible as an equilibrium outcome.
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
455
The second term makes it unique. The key difference between the standard
and modified Taylor rules is that the latter features a time-varying intercept
that allows the nominal interest rate to accommodate movements in the real
interest rate one-for-one without requiring changes in the price level. More
generally, the optimal competitive equilibrium can be implemented via rules
of the form R̂t = r̂tn + α π π̂ t + α y (ŷt − ŷtn ), with policy parameters α π and
α y satisfying the restrictions imposed by the definition of a Taylor-type rule
given above.
Applying this type of rule can be quite impractical, for it would require
knowledge on the part of the central bank of current and expected future values
taken by all of the shocks that affect the real interest rate, as well as of the
function mapping such values to the natural rate of interest. This difficulty
raises the question of how close a less sophisticated interest rate rule would
get to implementing the optimal equilibrium. We turn to this issue next.
Optimal, Simple, and Implementable Rules
In this subsection, we analyze the ability of simple, implementable interest
rate rules to approximate the outcome of optimal policy. We draw from our
previous work (Schmitt-Grohé and Uribe 2007a), where we evaluate policy in
the context of a calibrated model of the U.S. business cycle featuring monopolistic competition, sticky prices in product markets, capital accumulation,
government purchases financed by lump-sum or distortionary taxes, and with
or without a transactional demand for money.7 In the model, business cycles
are driven by stochastic variations in the level of total factor productivity and
government consumption. We impose two requirements for an interest rate
rule to be implementable. First, the rule must deliver a unique rational expectations equilibrium. Second, it must induce nonnegative equilibrium dynamics
for the nominal interest rate. For an interest rule to be simple, we require that
the interest rate be set as a function of a small number of easily observable
macroeconomic indicators. Specifically, we study interest rate feedback rules
that respond to measures of inflation, output, and lagged values of the nominal
interest rate. The family of rules we consider is of the form
ln(Rt /R ∗ ) = α R ln(Rt−1 /R ∗ ) + α π Et ln(π t−i /π ∗ ) + α y Et ln(yt−i /y ∗ );
i = −1, 0, 1,
(13)
where y ∗ denotes the nonstochastic Ramsey steady-state level of aggregate
demand, and R ∗ , π ∗ , α R , α π , and α y are parameters. The index, i, can take
three values: 1, 0, and −1. When i = 1, we refer to the interest rate rule as
backward-looking, when i = 0 as contemporaneous, and when i = −1 as
7 See also Rotemberg and Woodford (1997).
456
Federal Reserve Bank of Richmond Economic Quarterly
Table 1 Evaluating Interest Rate Rules
Ramsey Policy
Optimized Rules
Contemporaneous (i = 0)
Smoothing
No Smoothing
Backward (i = 1)
Forward (i = −1)
Nonoptimized Rules
Taylor Rule (i = 0)
Simple Taylor Rule
Inflation Targeting
απ
αy
αR
Welfare Cost
σπ
σR
—
—
—
0.000
0.01
0.27
3
3
3
3
0.01
0.00
0.03
0.07
0.84
—
1.71
1.58
0.000
0.001
0.001
0.003
0.04
0.14
0.10
0.19
0.29
0.42
0.23
0.27
1.5
1.5
—
0.5
—
—
—
—
—
0.522
0.019
0.000
3.19
0.58
0.00
3.08
0.87
0.27
Notes: The interest rate rule is given by ln(Rt /R ∗ ) = α R ln(Rt−1 /R ∗ ) +
α π Et ln(π t−i /π ∗ ) + α y Et ln(yt−i /y ∗ ); i = −1, 0, 1. In the optimized rules, the policy parameters α π , α y ,and α R are restricted to lie in the interval [0, 3]. The welfare
cost is defined as the percentage decrease in the Ramsey-optimal consumption process
necessary to make the level of welfare under the Ramsey policy identical to that under
the evaluated policy. Thus, a positive figure indicates that welfare is higher under the
Ramsey policy than under the alternative policy. The standard deviation of inflation and
the nominal interest rate is measured in percent per year.
forward-looking. The optimal simple and implementable rule is the simple
and implementable rule that maximizes welfare of the representative agent.
Specifically, we characterize values of α π , α y , and α R that are associated with
the highest level of welfare of the representative agent within the family of
simple and implementable interest rate feedback rules defined by equation
(13). As a point of comparison for policy evaluation, we also compute the real
allocation associated with the Ramsey optimal policy.
The first row of Table 1 shows that under the Ramsey policy inflation is
virtually equal to zero at all times.8 The remaining rows of Table 1 report
policy evaluations. The welfare associated with each interest rate feedback
rule is compared to the level of welfare associated with the Ramsey-optimal
policy. Specifically, the welfare cost is defined as the fraction, in percentage
points, of the consumption stream an agent living in the Ramsey economy
would be willing to give up to be as well off as in an economy in which
8 In the deterministic steady state of the Ramsey economy, the inflation rate is zero. One may
wonder why, in an economy featuring sticky prices as the single nominal friction, the volatility
of inflation is not exactly equal to zero at all times under the Ramsey policy. The reason is
that we do not follow the standard practice of subsidizing factor inputs to eliminate the distortion
introduced by monopolistic competition in product markets. Introducing such a subsidy would
result in a constant Ramsey-optimal rate of inflation equal to zero.
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
457
monetary policy takes the form of the respective interest rate feedback rule
shown in the table.
We consider seven different monetary policies: Four constrained-optimal
interest rate feedback rules and three nonoptimized rules. In the constrainedoptimal rule labeled no-smoothing, we search over the policy coefficients, α π
and α y , keeping α R fixed at zero. The second constrained-optimal rule, labeled
smoothing in the table, allows for interest rate inertia by setting optimally all
three coefficients, α π , α y , and α R .
We find that the best no-smoothing interest rate rule calls for an aggressive
response to inflation and a mute response to output. The inflation coefficient of
the optimized rule takes the largest value allowed in our search, namely 3. The
optimized rule is quite effective as it delivers welfare levels remarkably close
to those achieved under the Ramsey policy. At the same time, the rule induces
a stable rate of inflation, a feature that also characterizes the Ramsey policy.
Taking together this finding and those obtained in the previous subsection, we
conclude that although a Taylor rule cannot exactly implement the Ramsey
allocation, it delivers outcomes that are so close to the optimum in welfare
terms that, for practical purposes, it can be regarded as implementing the
Ramsey allocation.
We next study a case in which the central bank can smooth interest rates
over time. Our numerical search yields that the optimal policy coefficients are
α π = 3, α y = 0.01, and α R = 0.84. The fact that the optimized rule features
substantial interest rate inertia means that the monetary authority reacts to
inflation much more aggressively in the long run than in the short run. The
finding that the interest rule is not superinertial (i.e., α R does not exceed
unity) means that the monetary authority is backward-looking. So, again, as
in the case without smoothing, optimal policy calls for a large response to
inflation deviations in order to stabilize the inflation rate and for no response
to deviations of output from the steady state. The welfare gain of allowing
for interest rate smoothing is insignificant. Taking the difference between the
welfare costs associated with the optimized rules with and without interest
rate smoothing reveals that agents would be willing to give up less than 0.001
percent of their consumption stream under the optimized rule with smoothing
to be as well off as under the optimized policy without smoothing.
The finding that allowing for optimal smoothing yields only negligible
welfare gains spurs us to investigate whether rules featuring suboptimal degrees of inertia or responsiveness to inflation can produce nonnegligible welfare losses at all. PanelA of Figure 3 shows that, provided the central bank does
not respond to output, α y = 0, varying α π and α R between 0 and 3 typically
leads to economically negligible welfare losses of less than 0.05 percent of
consumption. In the graph, crosses represent combinations of α π and α R that
are implementable and circles represent combinations that are implementable
458
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 The Cashless Economy
Panel A: Implementability and Welfare
a Y =0
3.0
Implementable Rule
Welfare Cost <0.05%
2.5
aR
2.0
1.5
1.0
0.5
0.0
0.5
1.0
2.0
1.5
ap
2.5
3.0
Panel B: The Importance of Not Responding to Output
0.25
u
Welfare Cost (l x 100)
0.20
0.15
0.10
0.05
0.00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ay
and that yield welfare costs less than 0.05 percent of consumption relative to
the Ramsey policy.
The blank area in the figure identifies α π and α R combinations that are
not implementable either because the equilibrium fails to be locally unique
or because the implied volatility of interest rates is too high. This is the case
for values of α π and α R such that the policy stance is passive in the long run,
απ
that is, 1−α
< 1. For these parameter combinations the equilibrium is not
R
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
459
locally unique. This finding is a generalization of the result that, when the
inflation coefficient is less than unity (α π < 1), the equilibrium is indeterminate, which obtains in the absence of interest rate smoothing (α R = 0). We
also note that the result that passive interest rate rules render the equilibrium
indeterminate is typically derived in the context of models that abstract from
capital accumulation. It is, therefore, reassuring that this particular abstraction
appears to be of no consequence for the finding that (long run) passive policy
is inconsistent with local uniqueness of the rational expectations equilibrium.
Similarly, we find that determinacy obtains for policies that are active in the
απ
long run, 1−α
> 1.
R
More importantly, Panel A of Figure 3 shows that virtually all parameterizations of the interest rate feedback rule that are implementable yield about
the same level of welfare as the Ramsey equilibrium. This finding suggests
a simple policy prescription, namely, that any policy parameter combination
that is irresponsive to output and active in the long run, is equally desirable
from a welfare point of view.
One possible reaction to the finding that implementability-preserving variations in α π and α R have little welfare consequences may be that in the class
of models we consider, welfare is flat in a large neighborhood around the
optimum parameter configuration, so that it does not really matter what the
government does. This turns out not to be the case. Recall that in the welfare calculations underlying Panel A of Figure 3, the response coefficient on
output, α y , was kept constant and equal to zero. Indeed, interest rate rules
that lean against the wind by raising the nominal interest rate when output is
above trend can be associated with sizable welfare costs. Panel B of Figure 3
illustrates the consequences of introducing a cyclical component to the interest
rate rule. It shows that the welfare costs of varying α y can be large, thereby
underlining the importance of not responding to output. The figure shows the
welfare cost of deviating from the optimal output coefficient (α y ≈ 0) while
keeping the inflation coefficient of the interest rate rule at its optimal value
(α π = 3) and not allowing for interest rate smoothing (α R = 0). Welfare costs
are monotonically increasing in α y . When α y = 0.7, the welfare cost is over
0.2 percent of the consumption stream associated with the Ramsey policy.
This is a significant figure in the realm of policy evaluation at business-cycle
frequency.9 This finding suggests that bad policy can have significant welfare
costs in our model and that policy mistakes are committed when policymakers
are unable to resist the temptation to respond to output fluctuations.
9 A similar result obtains if one allows for interest rate smoothing with α taking its optiR
mized value of 0.84.
460
Federal Reserve Bank of Richmond Economic Quarterly
It follows that sound monetary policy calls for sticking to the basics of
responding to inflation alone.10 This point is conveyed with remarkable simplicity by comparing the welfare consequences of a simple interest rate rule
that responds only to inflation with a coefficient of 1.5 to those of a standard
Taylor rule that responds to inflation as well as output with coefficients 1.5 and
0.5, respectively. Table 1 shows that the Taylor rule that responds to output
is significantly welfare inferior to the simple interest rate rule that responds
solely to inflation. Specifically, the welfare cost of responding to output is
about half a percentage point of consumption.11
The Ramsey-optimal monetary policy implies near complete inflation stabilization (see Table 1). It is reasonable to conjecture, therefore, that inflation
targeting, interpreted to be any monetary policy capable of bringing about zero
inflation at all times (π t = 1 for all t), would induce business cycles virtually
identical to those associated with the Ramsey policy. We confirm this conjecture by computing the welfare cost associated with inflation targeting. The
welfare cost of targeting inflation relative to the Ramsey policy is virtually nil.
An important issue in monetary policy is determining what measures of
inflation and aggregate activity the central bank should respond to. In particular, a question that has received considerable attention among academic
economists and policymakers is whether the monetary authority should respond to past, current, or expected future values of output and inflation.
Here we address this question by computing optimal backward- and forwardlooking interest rate rules. That is, in equation (13) we let i take the values −1
and +1. Table 1 shows that there are no welfare gains from targeting expected
future values of inflation and output as opposed to current or lagged values of
these macroeconomic indicators. Also, a muted response to output continues
to be optimal under backward- or forward-looking rules.
Under a forward-looking rule without smoothing (α R = 0), the rational
expectations equilibrium is indeterminate for all values of the inflation and
output coefficients in the interval [0,3]. This result is in line with that obtained
by Carlstrom and Fuerst (2005). These authors consider an environment similar to ours and characterize determinacy of equilibrium for interest rate rules
that depend only on the rate of inflation. Our results extend the findings of
Carlstrom and Fuerst to the case in which output enters in the feedback rule.
We close this section by noting that most of the results presented here,
extend to a model economy with a much richer battery of nominal and real
rigidities. In Schmitt-Grohé and Uribe (2007b), we consider an economy
featuring four real rigidities: habit formation, variable capacity utilization,
10 Other authors have also argued that countercyclical interest rate policy may be undesirable
(e.g., Ireland 1997 and Rotemberg and Woodford 1997).
11 The simple interest rate rule that responds solely to inflation is implementable, whereas
the standard Taylor rule is not, because it implies too high a volatility of nominal interest rates.
S. Schmitt-Grohé and M. Uribe: Policy Implications of the NKPC
461
investment adjustment costs, and monopolistic competition in product and
labor markets. The economy in that study also includes four nominal frictions, namely, sticky prices, sticky wages, money demand by households, and
money demand by firms. Finally, the model features a more realistic shock
structure that includes permanent stochastic variations in total factor productivity, permanent stochastic variations in the relative price of investment, and
stationary stochastic variations in government spending. The values assigned
to the structural parameters are based on existing econometric estimations of
the model. These studies, in turn, argue that the model explains satisfactorily
observed short-term fluctuations in the postwar United States. We find that
the Ramsey policy calls for stabilizing price inflation. More importantly, a
simple interest rate rule that responds only to inflation (with mute responses
to wage inflation or output) attains a level of welfare remarkably close to that
associated with the Ramsey optimal equilibrium.
6.
CONCLUSION
In this article, we present a selective account of recent developments on the
policy implications of the New Keynesian Phillips curve. The main lesson
derived from our analysis is that price stability emerges as a robust policy prescription in models with product price rigidities. In fact, a minimum amount
of price stickiness suffices to make inflation stabilization the overriding goal
of monetary policy.
The desirability of price stability obtains in several variations of the standard New Keynesian framework that include expanding the set of nominal
and real rigidities to allow for government spending financed by distortionary
taxes, a transactional demand for money by households and firms, nominal
wage rigidity, habit formation, variable capacity utilization, and investment
adjustment costs.
A second important message that emerges is that a simple interest rate
feedback rule that responds aggressively only to a measure of consumer price
inflation delivers outcomes that are remarkably close to the Ramsey optimal equilibrium. In particular, to emulate optimal monetary policy it is not
necessary that in setting the nominal rate the monetary authority respond to
deviations of output from trend or past values of the interest rate itself. In
this sense, the policy implications of the NKPC identified in this survey are
consistent with a pure inflation targeting objective.
We have left out a number of important issues in the theory of inflation
stabilization. For example, we limit attention to monetary policy under commitment. There is an active literature exploring the policy implications of the
NKPC when the government is unable to commit to future actions. A central
theme in this literature is to ascertain whether lack of commitment gives rise to
an optimal inflation bias. A second omission in the present analysis concerns
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Federal Reserve Bank of Richmond Economic Quarterly
models with asymmetric costs of price adjustment. Here again, the central
question is whether in the presence of downwardly rigid prices or wages, the
policymaker should pursue a positive inflation target. Finally, our article does
not discuss the recent literature on optimal monetary policy in models with
credit constraints. An important focus of this literature is whether this type of
friction introduces reasons for the central bank to respond to financial variables
in setting the short-term interest rate.
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