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Authorized for public release by the FOMC Secretariat on 1/12/2024
BOARD OF GOVERNORS OF THE FEDERAL RESERVE SYSTEM
DIVISION OF MONETARY AFFAIRS
FOMC SECRETARIAT
Date:
June 5, 2018
To:
Research Directors
From:
Matthew M. Luecke
Subject: Supporting Documents for DSGE Models Update
The attached documents support the update on the projections of the DSGE
models. Note that the Chicago Fed’s DSGE documentation has not changed since
the last quarter, so their guide from December is being redistributed.
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The Current Outlook in EDO:
June 2018 FOMC Meeting
Class II FOMC – Restricted (FR)
Diego Vilán∗
June 1, 2018
1
The EDO Forecast from 2018 to 2020
The EDO model’s forecast is conditional on data through the first quarter of 2018 and on a preliminary Tealbook forecast for the second quarter of 2018.
Real GDP growth is about 2.6 percent, on average, over the projection horizon, somewhat below
its long-run value of 3 percent. Inflation reaches the Committee’s 2 percent objective in the third
quarter of 2020. Below-trend real GDP growth is driven by the slow fading of risk premium shocks
and the waning effects of the currently accommodative stance of monetary policy. On the nominal
side, for a number of years wages have been below the level consistent with the models wage Phillips
curve, holding down marginal cost and depressing inflation over that time period. The persistence
of these wage shocks accounts for most of the weakness in inflation over the forecast. By contrast,
the rise in inflation at the beginning of 2018 is attributed by the model to transient shocks to the
price Phillips curve, which largely dissipate by the first half of 2019.
The output gap is currently estimated to be −0.7 percent, and the economy slowly reaches its
full potential by the end of 2020. The real natural rate of interest is estimated to be around 0.9
percent in the third quarter of 2018 and hovers around 1.7 percent thereafter until the end of 2020,
0.4 percentage point below its steady-state value of 2.1 percent. The trajectories of the natural
rate of interest and the output gap are heavily driven by the model’s view that capital stocks are
currently well below those that would have prevailed in the absence of nominal rigidities, and that
the investment-related shocks responsible for this condition are likely to dissipate slowly.
Consistent with the gradual return of inflation and the output gap to their long-run values, the
federal funds rate is projected to increase gradually over the forecast horizon, reaching 3 34 percent
by the end of 2020. At the end of the projection horizon, the federal funds rate is still below its
∗ The author is affiliated with the Division of Research and Statistics of the Federal Reserve Board. Sections 2 and
3 contain background material on the EDO model, as in previous rounds. These sections were co-written with Hess
Chung and Jean-Philippe Laforte.
1
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Figure 1: Recent History and Forecasts
long-run value of 4.1 percent, reflecting the inertia in the policy rule and the persistently negative
output gap.
Compared with the March 2018 projection, the EDO model’s forecast of real GDP growth in this
round is stronger until the first half of 2019 because of various transitory factors; thereafter, real
GDP growth is only a touch weaker relative to March. Core PCE inflation is, on average, unchanged
over the forecast horizon. The output gap has been revised up, on average, 15 basis points over
the projection period since March, while the forecast of the real natural rate of interest has been
revised up 17 basis points, on average, since March, with a noticeable upward revision in 2017 due
to positive revisions to the contribution of TFP growth shocks. The path of the federal funds is
higher this round than in March, following the upward revision of the output gap.
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Figure 2: Recent History and Forecasts: Latent Variables
2
An Overview of Key Model Features
Figure 3 provides a graphical overview of the model. While similar to most related models, EDO
has a more detailed description of production and expenditure than most other models.1
Specifically, the model possesses two final good sectors in order to capture key long-run growth
facts and to differentiate between the cyclical properties of different categories of durable expenditure
(for example, housing, consumer durables, and nonresidential investment). For example, technological progress has been faster in the production of business capital and consumer durables (such as
computers and electronics).
The disaggregation of production (aggregate supply) leads naturally to some disaggregation of
expenditures (aggregate demand). We move beyond the typical model with just two categories of
(private domestic) demand (consumption and investment) and distinguish between four categories
of private demand: consumer nondurable goods and nonhousing services, consumer durable goods,
residential investment, and nonresidential investment. The boxes surrounding the producers in the
figure illustrate how we structure the sources of each demand category. Consumer nondurable goods
and services are sold directly to households; consumer durable goods, residential capital goods, and
nonresidential capital goods are intermediated through capital-goods intermediaries (owned by the
households), who then rent these capital stocks to households. Consumer nondurable goods and
services and residential capital goods are purchased (by households and residential capital goods
owners, respectively) from the first of economy’s two final goods-producing sectors, while consumer
durable goods and nonresidential capital goods are purchased (by consumer durable and residential
capital goods owners, respectively) from the second sector. In addition to consuming the nondurable
goods and services that they purchase, households supply labor to the intermediate goods-producing
firms in both sectors of the economy.
The remainder of this section provides an overview of the main properties of the model. In
1 Chung,
Kiley, and Laforte (2010) provide much more detail regarding the model specification, estimated parameters, and model properties.
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Figure 3: Model Overview
particular, the model has five key features:
• A New-Keynesian structure for price and wage dynamics. Unemployment measures the difference between the amount workers are willing to be employed and firms’ employment demand.
As a result, unemployment is an indicator of wage and, hence, price pressures as in Gali (2011).
• Production of goods and services occurs in two sectors, with differential rates of technological
progress across sectors. In particular, productivity growth in the investment and consumer
durable goods sector exceeds that in the production of other goods and services, helping the
model match facts regarding long-run growth and relative price movements.
• A disaggregated specification of household preferences and firm production processes that
leads to separate modeling of nondurables and services consumption, durables consumption,
residential investment, and business investment.
• Risk premiums associated with different investment decisions play a central role in the model.
These include, first, an aggregate risk premium, or natural rate of interest, shock driving a
wedge between the short-term policy rate and the interest rate faced by private decisionmakers
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(as in Smets and Wouters (2007)) and, second, fluctuations in the discount factor/risk premiums faced by the intermediaries financing household (residential and consumer durable) and
business investment.
2.1
Two-sector production structure
It is well known (for example, Edge, Kiley, and Laforte (2008)) that real outlays for business investment and consumer durables have substantially outpaced those on other goods and services,
while the prices of these goods (relative to others) has fallen. For example, real outlays on consumer
durables have far outpaced those on other consumption while prices for consumer durables have been
flat and those for other consumption have risen substantially; as a result, the ratio of nominal outlays
in the two categories has been much more stable, although consumer durable outlays plummeted in
the Great Recession. Many models fail to account for this fact.
EDO accounts for this development by assuming that business investment and consumer durables
are produced in one sector and other goods and services in another sector. Specifically, production by
firm j in each sector s (where s equals kb for the sector producing business investment and consumer
durables and cbi for the sector producing other goods and services) is governed by a Cobb-Douglas
production function with sector-specific technologies:
1−α
Xts (j) = (Ztm Zts Lst (j))
α
(Ktu,nr,s (j)) , for s = cbi, kb.
(1)
In equation (1), Z m represents (labor-augmenting) aggregate technology, while Z s represents (laboraugmenting) sector-specific technology; we assume that sector-specific technological change affects
the business investment and consumer durables sector only. Ls is labor input and K u,nr,s is capital input (that is, utilized nonresidential business capital (and hence the nr and u terms in the
superscript). Growth in this sector-specific technology accounts for the long-run trends, while highfrequency fluctuations allow for the possibility that investment-specific technological change is a
source of business cycle fluctuations, as in Fisher (2006).
2.2
The structure of demand
EDO differentiates between several categories of expenditure. Specifically, business investment
spending determines nonresidential capital used in production, and households value consumer nondurables goods and services, consumer durable goods, and residential capital (for example, housing).
Differentiation across these categories is important, as fluctuations in these categories of expenditure
can differ notably, with the cycles in housing and business investment, for example, occurring at
different points over the last three decades.
Valuations of these goods and services, in terms of household utility, is given by the following
utility function:
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E0
∞
X
cnn
β t ς cnn ln(Etcnn (i)−hEt−1
(i))+ς cd ln(Ktcd (i))
t=0
+ς r ln(Ktr (i)) −ΛLpref
ΘH
t
t
X Z
s=cbi,kb
1
ς l,s Lst (i)
1+σN
σN
1+
1+σh
0
di ,
(2)
where E cnn represents expenditures on consumption of nondurable goods and services, K cd and
K r represent the stocks of consumer durables and residential capital (housing), ΛLpref
represents a
t
labor supply shock, Θt is an endogenous preference shifter whose role is to reconcile the existence of
a long-run balance growth path with a small short-term wealth effect2 , Lcbi and Lkb represent the
labor supplied to each productive sector (with hours worked causing disutility), and the remaining
terms represent parameters (such as the discount factor, relative value in utility of each service flow,
and the elasticity of labor supply). Gali, Smets, and Wouters (2011) state that the introduction
of the endogenous preference shifter is key in order to match the joint behavior of the labor force,
consumption, and wages over the business cycle.
By modeling preferences over these disaggregated categories of expenditure, EDO attempts to
account for the disparate forces driving consumption of nondurables and durables, residential investment, and business investment —thereby speaking to issues such as the surge in business investment
in the second half of the 1990s or the housing cycle in the early 2000s recession and the most recent
downturn. Many other models do not distinguish between developments across these categories of
spending.
2.3
Risk premiums, financial shocks, and economic fluctuations
The structure of the EDO model implies that households value durable stocks according to their
expected returns, including any expected service flows, and according to their risk characteristics,
with a premium on assets that have high expected returns in adverse states of the world. However,
the behavior of models such as EDO is conventionally characterized under the assumption that this
second component is negligible. In the absence of risk adjustment, the model would then imply that
households adjust their portfolios until expected returns on all assets are equal.
Empirically, however, this risk adjustment may not be negligible and, moreover, there may be a
variety of factors, not explicitly modeled in EDO, that limit the ability of households to arbitrage
away expected return differentials across different assets. To account for this possibility, EDO
features several exogenous shocks to the rates of return required by the household to hold the assets
in question. Following such a shock —an increase in the premium on a given asset, for example
—households will wish to alter their portfolio composition to favor the affected asset, leading to
changes in the prices of all assets and, ultimately, to changes in the expected path of production
underlying these claims.
2 The
cnn , where Z =
endogenous preference shifter is defined as ΘH
t
t = Zt Λt
1−ν
Zt−1
Λcnn
t
and Λcnn
is the shadow price of
t
nondurable consumption. The importance of the short-term wealth effect is determined by the parameter ν ∈ (0, 1].
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The “sector specific” risk shocks affect the composition of spending more than the path of
GDP itself. This occurs because a shock to these premiums leads to sizable substitution across
residential, consumer durable, and business investment; for example, an increase in the risk premiums
on residential investment leads households to shift away from residential investment and toward
other types of productive investment. Consequently, it is intuitive that a large fraction of the noncyclical, or idiosyncratic, component of investment flows to physical stocks will be accounted for by
movements in the associated premiums.
Shocks to the required rate of return on the nominal risk-free asset play an especially large role
in EDO. Following an increase in the premium, in the absence of nominal rigidities, the households’
desire for higher real holdings of the risk-free asset would be satisfied entirely by a fall in prices,
that is, the premium is a shock to the natural rate of interest. Given nominal rigidities, however,
the desire for higher risk-free savings must be offset, in part, through a fall in real income, a decline
which is distributed across all spending components. Because this response is capable of generating
co-movement across spending categories, the model naturally exploits such shocks to explain the
business cycle. Reflecting this role, we denote this shock as the “aggregate risk-premium.”
Movements in financial markets and economic activity in recent years have made clear the role
that frictions in financial markets play in economic fluctuations. This role was apparent much earlier,
motivating a large body of research (for example, Bernanke, Gertler, and Gilchrist (1999)). While
the range of frameworks used to incorporate such frictions has varied across researchers studying
different questions, a common theme is that imperfections in financial markets —for example, related
to imperfect information on the outlook for investment projects or earnings of borrowers —drives a
wedge between the cost of riskless funds and the cost of funds facing households and firms. Much
of the literature on financial frictions has worked to develop frameworks in which risk premiums
fluctuate for endogenous reasons (for example, because of movements in the net worth of borrowers).
Because the risk-premium shocks induces a wedge between the short-term nominal risk-free rate and
the rate of return on the affected risky rates, these shocks may thus also be interpreted as a reflection
of financial frictions not explicitly modeled in EDO. The sector-specific risk premiums in EDO enter
the model in much the same way as does the exogenous component of risk premiums in models with
some endogenous mechanism (such as the financial accelerator framework used Boivin, Kiley, and
Mishkin (2010)), and the exogenous component is quantitatively the most significant one in that
research.3
2.4
Labor market dynamics in the EDO model
This version of the EDO model assumes that labor input consists of both employment and hours per
worker. Workers differ in the disutility they associate with employment. Moreover, the labor market
is characterized by monopolistic competition. As a result, unemployment arises in equilibrium – some
workers are willing to be employed at the prevailing wage rate, but cannot find employment because
firms are unwilling to hire additional workers at the prevailing wage.
3 Specifically, the risk premiums enter EDO to a first-order (log)linear approximation in the same way as in the
cited research if the parameter on net worth in the equation determining the borrowers cost of funds is set to zero; in
practice, this parameter is often fairly small in financial accelerator models.
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As emphasized by Gali (2011), this framework for unemployment is simple and implies that the
unemployment rate reflects wage pressures: When the unemployment rate is unusually high, the
prevailing wage rate exceeds the marginal rate of substitution between leisure and consumption,
implying that workers would prefer to work more.
The new preference specification and the incorporation of labor force participation in the information set impose discipline in the overall labor market dynamics of the EDO model. The estimated
short-run wealth effect on labor supply is relatively attenuated with respect to previous versions of
the EDO model. Therefore, the dynamics of both labor force participation and employment are
more aligned with the empirical evidence.
In addition, in our environment, nominal wage adjustment is sticky, and this slow adjustment
of wages implies that the economy can experience sizable swings in unemployment with only slow
wage adjustment. Our specific implementation of the wage adjustment process yields a relatively
standard New Keynesian wage Phillips curve. The presence of both price and wage rigidities implies
that stabilization of inflation is not, in general, the best possible policy objective (although a primary
role for price stability in policy objectives remains).
While the specific model on the labor market is suitable for discussion of the links between
employment and wage/price inflation, it leaves out many features of labor market dynamics. Most
notably, it does not consider separations, hires, and vacancies, and is hence not amenable to analysis
of issues related to the Beveridge curve.
The decline in employment during the Great Recession primarily reflected, according to the
EDO model, the weak demand that arose from elevated risk premiums that depressed spending,
as illustrated by the light blue and red bars in figure 1. The role played by these demand factors
in explaining the cyclical movements in employment is only determinant during the 1980s and
during the Great Recession. As apparent in figure 1, the most relevant drivers of employment in the
remaining of the sample are labor supply (preference) and markup shocks as shown by the blue bars.
Specifically, favorable supply developments in the labor market are estimated to have placed upward
pressure on employment until 2010; these developments have reversed, and some of the currently
low level for employment growth is, according to EDO, attributable to adverse labor market supply
developments. As discussed previously, these developments are simply exogenous within EDO and
are not informed by data on a range of labor market developments (such as gross worker flows and
vacancies).
2.5
New Keynesian price and wage Phillips curves
As in most of the related literature, nominal prices and wages are both “sticky” in EDO. This
friction implies that nominal disturbances —that is, changes in monetary policy —have effects on
real economic activity. In addition, the presence of both price and wage rigidities implies that
stabilization of inflation is not, in general, the best possible policy objective (although a primary
role for price stability in policy objectives remains).
Given the widespread use of the New Keynesian Phillips curve, it is perhaps easiest to consider
the form of the price and wage Phillips curves in EDO at the estimated parameters. The price
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Phillips curve (governing price adjustment in both productive sectors) has the form
p,s
p,s
πtp,s = 0.22πt−1
+ 0.76Et πt+1
+ .017mcst + θts
(3)
where mc is marginal cost and θ is a markup shock. As the parameters indicate, inflation is
primarily forward looking in EDO.
The wage (w) Phillips curve for each sector has the form
s
s
s
w
4wts = 0.014wt−1
+ 0.95Et 4wt+1
+ .012 mrsc,l
−
w
t
t + θt + adj. costs.
(4)
where mrs represents the marginal rate of substitution between consumption and leisure. Wages
are primarily forward looking and relatively insensitive to the gap between households’ valuation of
time spent working and the wage.
The top right panel of figure 1 presents the decomposition of inflation fluctuations into the
exogenous disturbances that enter the EDO model. As can be seen, aggregate demand fluctuations,
including aggregate risk premiums and monetary policy surprises, contribute little to the fluctuations
in inflation according to the model. This is not surprising: In modern DSGE models, transitory
demand disturbances do not lead to an unmooring of inflation (so long as monetary policy responds
systematically to inflation and remains committed to price stability). In the short run, inflation
fluctuations primarily reflect transitory price and wage shocks, or markup shocks in the language of
EDO. Technological developments can also exert persistent pressure on costs, most notably during
and following the strong productivity performance of the second half of the 1990s, which is estimated
to have lowered marginal costs and inflation through the early 2000s. More recently, disappointing
labor productivity readings over the course of 2011 have led the model to infer sizable negative
technology shocks in both sectors, contributing noticeably to inflationary pressure over that period
(as illustrated by the blue bars in figure 1).
2.6
Monetary authority and a long-term interest rate
We now turn to the last agent in our model, the monetary authority. It sets monetary policy in
accordance with an Taylor-type interest rate feedback rule. Policymakers smoothly adjust the actual
interest rate Rt to its target level R̄t
ρr
Rt = (Rt−1 )
R̄t
1−ρr
exp [rt ] ,
(5)
where the parameter ρr reflects the degree of interest rate smoothing, while rt represents a monetary
policy shock. The central bank’s target nominal interest rate, R̄t depends on the deviation of output
from the level consistent with current technologies and “normal” (steady-state) utilization of capital
and labor (X̃ pf , the “production function” output gap). Also, the change in the output gap and
consumer price inflation enter the target. The target equation is
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R̄t = X̃t
pf
r y
dX̃t
pf
rdy Πc rπ
t
Πc∗
R∗ .
(6)
pf
In equation (6), R∗ denotes the economy’s steady-state nominal interest rate, dX̃t denotes the
change in the output gap and ry , rdy and rπ denote the weights in the feedback rule. Consumer
price inflation, Πct , is the weighted average of inflation in the nominal prices of the goods produced
in each sector, Πp,cbi
and Πp,kb
:
t
t
Πct = (Πp,cbi
)1−wcd (Πp,kb
)wcd .
t
t
(7)
The parameter wcd is the share of the durable goods in nominal consumption expenditures.
The model also includes a long-term interest rate (RLt ), which is governed by the expectations
hypothesis subject to an exogenous term premiums shock:
RLt = Et ΠN
τ =0 Rτ · Υt .
(8)
where Υ is the exogenous term premium, governed by
Ln (Υt ) = 1 − ρΥ Ln (Υ∗ ) + ρΥ Ln (Υt−1 ) + Υ
t .
(9)
In this version of EDO, the long-term interest rate plays no allocative role; nonetheless, the term
structure contains information on economic developments useful for forecasting (for example, Edge,
Kiley, and Laforte (2010)), and hence RL is included in the model and its estimation.
2.7
Summary of model specification
Our brief presentation of the model highlights several points. First, although our model considers
production and expenditure decisions in a bit more detail, it shares many similar features with other
DSGE models in the literature, such as imperfect competition, nominal price and wage rigidities, and
real frictions like adjustment costs and habit-persistence. The rich specification of structural shocks
(to aggregate and investment-specific productivity, aggregate and sector-specific risk premiums, and
markups) and adjustment costs allows our model to be brought to the data with some chance of
finding empirical validation.
Within EDO, fluctuations in all economic variables are driven by 13 structural shocks. It is most
convenient to summarize these shocks into five broad categories:
• Permanent technology shocks: This category consists of shocks to aggregate and investmentspecific (or fast-growing sector) technology.
• A labor supply shock: This shock affects the willingness to supply labor. As was apparent in our
earlier description of labor market dynamics and in the presentation of the structural drivers
below, this shock captures the dynamics of the labor force participation rate in the sample and
those of employment. While EDO labels such movements labor supply shocks, an alternative
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interpretation would describe these as movements in the labor force and employment that
reflect structural features not otherwise captured by the model.
• Financial, or intertemporal, shocks: This category consists of shocks to risk premiums. In
EDO, variation in risk premiums —both the premium households receive relative to the federal
funds rate on nominal bond holdings and the additional variation in discount rates applied
to the investment decisions of capital intermediaries —are purely exogenous. Nonetheless,
the specification captures aspects of related models with more explicit financial sectors (for
example, Bernanke, Gertler, and Gilchrist (1999)), as we discuss in our presentation of the
model’s properties below.
• Markup shocks: This category includes the price and wage markup shocks.
• Other demand shocks: This category includes the shock to autonomous demand and a monetary policy shock.
3
Estimation: Data and Properties
3.1
Data
The empirical implementation of the model takes a log-linear approximation to the first-order conditions and constraints that describe the economy’s equilibrium, casts this resulting system in its
state-space representation for the set of (in our case, 13) observable variables, uses the Kalman
filter to evaluate the likelihood of the observed variables, and forms the posterior distribution of the
parameters of interest by combining the likelihood function with a joint density characterizing some
prior beliefs. Since we do not have a closed-form solution of the posterior, we rely on Markov-Chain
Monte Carlo (MCMC) methods.
The model is estimated using 13 data series over the sample period from 1984:Q4 to 2015:Q3.
The series are the following:
1. The growth rate of real gross domestic product (∆GDP );
2. The growth rate of real consumption expenditure on nondurables and services (∆C);
3. The growth rate of real consumption expenditure on durables (∆CD);
4. The growth rate of real residential investment expenditure (∆Res);
5. The growth rate of real business investment expenditure (∆I);
6. Consumer price inflation, as measured by the growth rate of the Personal Consumption Expenditure (PCE) price index (∆PC,total );
7. Consumer price inflation, as measured by the growth rate of the PCE price index excluding
food and energy prices (∆PC,core );
8. Inflation for consumer durable goods, as measured by the growth rate of the PCE price index
for durable goods (∆Pcd );
9. Hours, which equals hours of all persons in the nonfarm business sector from the Bureau of
Labor Statistics (H);
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10. Civilian employment-population ratio, defined as civilian employment from the Current Population Survey (household survey) divided by the noninstitutional population, age 16 and over
(N );
11. Labor force participation rate;
12. The growth rate of real wages, as given by compensation per hour in the non-farm business
sector from the Bureau of Labor Statistics divided by the GDP price index (∆RW ); and
13. The federal funds rate (R).
Our implementation adds measurement error processes to the likelihood implied by the model
for all of the observed series used in estimation except the short-term nominal interest rate series.
3.2
Estimates of latent variable paths
Figures 4, 5, and 6 report estimates of the model’s persistent exogenous fundamentals (for example,
risk premiums and autonomous demand). These series have recognizable patterns for those familiar
with U.S. economic fluctuations. For example, the risk premiums jump at the end of 2008, reflecting
the financial crisis and the model’s identification of risk premiums, both economy-wide and for
housing, as key drivers.
Of course, these stories from a glance at the exogenous drivers, yield applications for alternative
versions of the EDO model and future model enhancements. For example, the exogenous risk
premiums can easily be made to have an endogenous component, following the approach of Bernanke,
Gertler, and Gilchrist (1999) (and, indeed, we have considered models of that type). At this point,
we view incorporation of such mechanisms in our baseline approach as premature, pending ongoing
research on financial frictions, banking, and intermediation in dynamic general equilibrium models.
Nonetheless, the EDO model captured the key financial disturbances during the last several years
in its current specification, and examining the endogenous factors that explain these developments
will be a topic of further study.
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Figure 4: Model Estimates of Risk Premiums
Aggregate Risk
Capital Risk
2.5
6
2
4
1.5
2
1
0
0.5
-2
0
-4
-0.5
-6
-1
1990
2000
2010
2020
1990
Housing Risk
2000
2010
2020
Durables Risk
6
40
4
20
2
0
0
-20
-2
-40
-4
-60
-6
1990
2000
2010
2020
1990
2000
2010
2020
Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible
set. Light blue intervals: 95 percent credible set.
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Figure 5: Model Estimates of Key Supply-side Variables
Aggregate Tech
Capital Tech
3
2
2
1
1
0
0
-1
-1
-2
-2
1990
2000
2010
2020
2010
2020
1990
2000
2010
2020
Labor Pref
60
40
20
0
-20
1990
2000
Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible
set. Light blue intervals: 95 percent credible set.
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Figure 6: Model Estimates of Selected Other Exogenous Drivers
Wage Markup
Exog Spending
50
10
5
0
0
-5
-10
-50
1990
2000
2010
2020
1990
2000
2010
2020
Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible
set. Light blue intervals: 95 percent credible set.
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Class II FOMC – Restricted (FR)
References
[Bernanke, Gertler, and Gilchrist (1999)] Bernanke, B., M. Gertler, and S. Gilchrist. 1999. The financial accelerator in a quantitative business cycle framework, in: John B. Taylor and Michael
Woodford, Editor(s), Handbook of Macroeconomics, Elsevier, 1999, volume 1, part 3, pages 13411393.
[Boivin, Kiley, and Mishkin (2010)] Boivin, J., M. Kiley, and F.S. Mishkin. 2010. How Has the Monetary Transmission Mechanism Evolved Over Time? In B. Friedman and M. Woodford, eds., The
Handbook of Monetary Economics, Elsevier.
[Chung, Kiley, and Laforte (2010)] Chung, H., M. Kiley, and J.P. Laforte. 2010. Documentation of
the Estimated, Dynamic, Optimization-based (EDO) model of the U.S. economy: 2010 version.
Finance and Economics Discussion Series, 2010-29. Board of Governors of the Federal Reserve
System (U.S.).
[Edge, Kiley, and Laforte (2008)] Edge, R., M. Kiley, and J.P. Laforte. 2008. Natural rate measures
in an estimated DSGE model of the U.S. economy. Journal of Economic Dynamics and Control,
vol. 32(8), pages 2512-2535.
[Edge, Kiley, and Laforte (2010)] Edge, R., M. Kiley, and J.P. Laforte. 2010. A comparison of forecast performance between Federal Reserve staff forecasts, simple reduced-form models, and a
DSGE model. Journal of Applied Econometrics vol. 25(4), pages 720-754.
[Fisher (2006)] Fisher, Jonas D. M. 2006. The Dynamic Effects of Neutral and Investment-Specific
Technology Shocks. Journal of Political Economy, University of Chicago Press, vol. 114(3), pages
413-451.
[Gali (2011)] Gali, J. 2011. The Return Of The Wage Phillips Curve. Journal of the European
Economic Association, vol. 9(3), pages 436-461.
[Gali, Smets, and Wouters (2011)] Gali, J., F. Smets, and R. Wouters. 2011. Unemployment in an
Estimated New Keynesian Model. NBER Macroeconomics Annual vol. 26(1), pages 329-360.
[Smets and Wouters (2007)] Smets, F., and R. Wouters. 2007. Shocks and Frictions in the US
Busines Cycles: A Bayesian DSGE Approach. American Economic Review, American Economic
Association, vol. 97(3), pages 586-606.
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New York Fed DSGE Model:
Research Directors Draft
May 31, 2018
Introduction
This document describes the New York Fed DSGE model, which we use both for internal
forecasting and for creating our contributions to the System DSGE memo distributed quarterly to the FOMC. The document is structured as follows. First, we provide a description
and interpretation of the forecast for the current forecast horizon. Next, we describe the
structure of the DSGE model followed by the impulse response functions to various shocks.
Model Forecast
The New York Fed model forecasts are obtained using data released through 2018Q1, augmented for 2018Q2 with the New York Fed staff forecasts (as of May 31) for real GDP growth
and core PCE inflation, and with values of the federal funds rate, the 10-year Treasury yield
and the spread between Baa corporate bonds and 10-year Treasury yields based on 2018Q2
averages up to May 31.
Table 1 shows both the conditional and unconditional forecasts of real GDP growth,
core PCE inflation, federal funds rate, real natural rate of interest and the output gap.
Unconditional forecasts are obtained using data up to the quarter for which we have the
most recent GDP release, as well as the federal funds rate, 10-year Treasury yield, and
spreads data for the following (“current”) quarter. Conditional forecasts further include the
current-quarter New York Fed staff projections for GDP growth and core PCE inflation as
additional data points.
Figure 1 plots the conditional and unconditional forecasts of real GDP growth, core
PCE inflation and the federal funds rate. Figure 2 provides a comparison of current and
previous quarterly forecasts while Figure 3 depicts the shock decomposition of the conditional
forecasts, where different colored bars indicate the contribution of different shocks to the
conditional forecast of GDP growth, inflation and federal funds rate. Finally, Figure 4 plots
the historical estimates and forecast of the output gap in the top panel, and the real natural
rate in the bottom panel.
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The output gap is defined as the difference between actual output and potential output.
Potential output is defined as the level that output would take in a world where capital and
labor are fully utilized, i.e., where there are no nominal rigidities or shocks to markups.1
A positive (negative) output gap indicates that output is above (below) its potential. The
natural rate of interest is a concept analogous to potential output: it represents the rate of
interest that would prevail in the economy absent nominal rigidities and markup shocks.
Current Forecast
We project real GDP growth of 2.3 percent in 2018 on a Q4/Q4 basis, in line with the March
forecast. GDP growth is anticipated to decline to 1.9 percent in 2019 and 2020, slightly below
the March forecast of 2.0 percent and 2.1 percent in 2019 and 2020 respectively. Inflation
is forecast to be slightly higher in the short term, at 1.9 percent relative to 1.8 percent in
the March projection. While inflation in 2018 is close to the FOMC’s longer run goal of
2 percent, the model projects that inflation will decline to 1.5 percent in 2019 and 2020
(unchanged from the March projection).
The output gap is projected to be -0.4 percent in 2018, -0.3 percent in 2019 and -0.3
percent in 2020. This represents an improvement relative to the March forecast of the
output gap (-0.7 percent, -0.6 percent and -0.4 percent in 2018, 2019 and 2020 respectively),
especially in the near term. The forecast for the natural rate of interest is unchanged from
March, rising from 1.0 percent in 2018 to 1.4 percent by the end of 2020.
The federal funds rate is forecast to be 0.1 percentage points lower than the March
projections throughout the forecast horizon: 2.2, 2.6 and 2.8 percent in 2018, 2019 and 2020
respectively. This shallower path translates into approximately two more hikes in 2018, one
more in 2019 and another in 2020.
The projections for all the variables are surrounded by significant uncertainty. For instance, the 68 percent posterior probability interval for GDP growth includes negative readings for the years between 2019 and 2020. In comparison, the posterior probability intervals
for inflation are tighter, with their upper bound never exceeding 3 percent throughout the
forecast horizon.
As in March, the model explains the above long-run average real GDP growth rate in 2018
with continued improvement in financial conditions, as captured by positive contributions of
1
Markup shocks represent exogenous fluctuations in price and wage inflation arising from various sources,
such as variations in the degree of market power, or in the price of commodities.
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both the financial and marginal efficiency of investment shocks. Over the medium term however, these factors are offset by lower TFP growth and the gradual withdrawal of monetary
accommodation (see the top panel of Figure 3). Sluggish TFP growth also explains, at least
in part, the negative output gap over the forecast horizon. The model projects near-target
inflation for 2018 (see middle panel of Figure 3), driven by a temporary increase in price
markups. However, as in March, the model projects below-target inflation in 2019 and 2020,
driven by a confluence of several factors. These factors include the lingering effects of the
financial headwinds that have hampered the recovery, as well as negative shocks to wage and
price markups. Finally, the federal funds rate is projected to remain below its long-run level
of 4 percent throughout the forecast horizon owing to persistence in the interest rate rule,
weak projected inflation and a persistently negative output gap (see lower panel of Figure
3).
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Table 1: Forecasts
2018
Real GDP
Growth (Q4/Q4)
Core PCE
Inflation (Q4/Q4)
Federal Funds
Rate (Q4)
Real Natural
Rate (Q4)
Output
Gap (Q4)
Jun.
1.9
(−0.2,3.8)
1.8
(1.2,2.3)
2.1
(0.7,3.6)
1.0
(−0.6,2.6)
−0.6
(−2.5,1.1)
Mar.
2.6
(0.1,5.0)
1.4
(0.7,2.1)
2.2
(0.8,3.6)
1.0
(−0.6,2.7)
−0.5
(−2.5,1.3)
Unconditional Forecast
2019
Jun.
Mar.
1.8
2.1
(−1.1,4.4) (−0.7,4.7)
1.4
1.4
(0.5,2.4)
(0.4,2.3)
2.6
2.6
(1.0,4.4)
(1.0,4.4)
1.3
1.3
(−0.5,3.0) (−0.5,3.1)
−0.6
−0.4
(−3.5,1.9) (−3.2,2.3)
2020
Jun.
Mar.
1.9
2.1
(−0.9,4.6) (−0.7,4.7)
1.5
1.5
(0.4,2.6)
(0.3,2.6)
2.8
2.8
(1.0,4.7)
(1.0,4.8)
1.3
1.4
(−0.5,3.2) (−0.5,3.3)
−0.6
−0.2
(−4.0,2.5) (−3.7,2.9)
Conditional Forecast
2019
2020
Mar.
Jun.
Mar.
Jun.
Mar.
2.3
1.9
2.0
1.9
2.1
(0.2,4.2)
(−1.0,4.5) (−0.8,4.7) (−0.9,4.7) (−0.8,4.8)
1.8
1.5
1.5
1.5
1.5
(1.3,2.4)
(0.5,2.4)
(0.5,2.5)
(0.4,2.6)
(0.4,2.7)
2.3
2.6
2.7
2.8
2.9
(0.9,3.7)
(1.0,4.4)
(1.0,4.4)
(1.1,4.7)
(1.1,4.8)
1.0
1.3
1.3
1.4
1.4
(−0.7,2.6) (−0.5,3.1) (−0.6,3.1) (−0.5,3.2) (−0.6,3.3)
−0.7
−0.3
−0.6
−0.3
−0.4
(−2.6,1.0) (−3.1,2.1) (−3.3,1.9) (−3.6,2.6) (−3.8,2.6)
2018
Real GDP
Growth (Q4/Q4)
Core PCE
Inflation (Q4/Q4)
Federal Funds
Rate (Q4)
Real Natural
Rate (Q4)
Output
Gap (Q4)
Jun.
2.3
(0.8,3.8)
1.9
(1.5,2.3)
2.2
(0.7,3.7)
1.0
(−0.6,2.6)
−0.4
(−2.1,1.2)
The unconditional forecasts use data up to the quarter for which we have the most recent GDP release, as well as the federal
funds rate, 10-year Treasury yield, and spreads data for the following (“current”) quarter. In the conditional forecasts, we
further include the current-quarter New York Fed staff projections for GDP growth and core PCE inflation as additional data
points. Numbers in parentheses indicate 68 percent probability intervals.
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Figure 1: Forecasts
Unconditional
Conditional
Real GDP Growth
7.5
Percent Q/Q Annualized
Percent Q/Q Annualized
Real GDP Growth
5.0
2.5
0.0
-2.5
2012
2014
2016
2018
7.5
5.0
2.5
0.0
-2.5
2020
2012
2014
3
2
1
0
2012
2014
2016
2018
2020
2
1
0
2012
2014
2016
2018
2020
Nominal FFR
6
6
5
5
Percent Annualized
Percent Annualized
2020
3
Nominal FFR
4
3
2
4
3
2
1
1
2012
2018
Core PCE Inflation
Percent Q/Q Annualized
Percent Q/Q Annualized
Core PCE Inflation
2016
2014
2016
2018
2020
2012
2014
2016
2018
2020
Quarterly forecasts, both unconditional (left panels) and conditional (right panels). The black line represents data, the red line
indicates the mean forecast, and the shaded areas mark the 50, 60, 70, 80 and 90 percent probability intervals for the forecasts,
reflecting both parameter and shock uncertainty.
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Figure 2: Change in Forecasts
Unconditional
Conditional
Real GDP Growth
Percent Q/Q Annualized
Percent Q/Q Annualized
Real GDP Growth
7.5
5.0
2.5
0.0
-2.5
2012
2014
2016
2018
7.5
5.0
2.5
0.0
-2.5
2020
2012
2014
3
2
1
0
2012
2014
2016
2018
2020
2
1
0
2012
2014
2016
2018
2020
Nominal FFR
6
6
5
5
Percent Annualized
Percent Annualized
2020
3
Nominal FFR
4
3
2
1
2012
2018
Core PCE Inflation
Percent Q/Q Annualized
Percent Q/Q Annualized
Core PCE Inflation
2016
4
3
2
1
2014
2016
2018
2020
2012
2014
2016
2018
2020
Comparison of current and previous quarterly forecasts. Solid (dashed) red and blue lines represent the mean and the 90 percent
probability intervals, respectively, of the current (previous) forecast.
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Figure 3: Shock Decomposition
Real GDP Growth
Percent Q/Q Annualized
(deviations from mean)
3
2
1
0
-1
-2
-3
-4
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2018
2019
2020
2021
2018
2019
2020
2021
Core PCE Inflation
Percent Q/Q Annualized
(deviations from mean)
0.5
0.0
-0.5
-1.0
-1.5
2012
2013
2014
2015
2016
2017
Nominal FFR
Percent Annualized
(deviations from mean)
1
0
-1
-2
-3
-4
-5
2012
Gov't
Financial
TFP
Mark-Up
Policy
MEI
2013
2014
2015
2016
2017
Shock decomposition of the conditional forecast. The solid lines (black for realized data, red for mean forecast) show each
variable in deviation from its steady state. The bars represent the shock contributions; specifically, the bars for each shock
represent the counterfactual values for the observables (in deviations from the mean) obtained by setting all other shocks to
zero.
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Figure 4: Output Gap and Natural Interest Rate
Output Gap
2.5
0.0
-2.5
-5.0
2012
2014
2016
2018
2020
Natural Rate & Ex-Ante Real Rate
4
Percent Annualized
2
0
-2
Ex-Ante Real Rate
Real Natural Rate
-4
2012
2014
2016
2018
2020
Historical estimates and forecasts of the output gap (upper panel) and the real natural rate of interest and the ex-ante real
interest rate (lower panel). In the upper panel, the black line represents the mean historical estimate, the red line the mean
forecast. In the lower panel, the solid lines represent historical estimates and the dashed lines represent forecasts of the natural
rate (red) and ex-ante rate (black). In both panels, the shaded areas mark the 50, 60, 70, 80, and 90 percent probability
intervals for the historical estimates and forecasts, reflecting both parameter and shock uncertainty.
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The Model
The following section contains a description of the New York Fed DSGE model and plots of
impulse response functions.
General structure
The New York Fed DSGE model is a medium scale, one-sector dynamic stochastic general
equilibrium model which is based on the New Keynesian model with financial frictions used
in Del Negro et al. (2015). The core of the model is based on the work of Smets and
Wouters (2007) (henceforth SW) and Christiano et al. (2005): It builds on the neo-classical
growth model by adding nominal wage and price rigidities, variable capital utilization, costs
of adjusting investment, habit formation in consumption. The model also includes credit
frictions as in the financial accelerator model developed by Bernanke et al. (1999b) where
the actual implementation of credit frictions follows closely Christiano et al. (2014), and
accounts for forward guidance in monetary policy by including anticipated policy shocks as
in Laseen and Svensson (2011).
The current version of the model has several features that improve upon the version
presented in the New York Fed Staff Report no. 647. It features both a deterministic
and a stochastic trend in productivity and allows for exogenous movements in risk premia;
the inflation target is time-varying, following Del Negro and Schorfheide (2012); households
preferences are non-separable in consumption and leisure; the Dixit-Stiglitz aggregator of
intermediate goods has been replaced by the more flexible Kimball aggregator; we include
indexation in the price and wage adjustment processes.
Here is a brief overview. The model economy is populated by eight classes of agents: 1) a
continuum of households, who consume and supply differentiated labor; 2) competitive labor
aggregators that combine labor supplied by individual households; 3) competitive final goodproducing firms that aggregate the intermediate goods into a final product; 4) a continuum
of monopolistically competitive intermediate good producing firms; 5) competitive capital
producers that convert final goods into capital; 6) a continuum of entrepreneurs who purchase
capital using both internal and borrowed funds and rent it to intermediate good producing
firms; 7) a representative bank collecting deposits from the households and lending funds to
the entrepreneurs; and finally 8) a government, composed of a monetary authority that sets
short-term interest rates and a fiscal authority that sets public spending and collects taxes.
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Growth in the economy is driven by technological progress. We specify a process for
technology Zt∗ which includes both a deterministic and a stochastic trend, and a stationary
component:
1
Zt∗ = e 1−α z̃t Ztp eγt ,
(1)
where γ is the steady state growth rate of the economy, Ztp is a stochastic trend and z̃t is
the stationary component.
The production function is
Yt (i) = max{ez̃t Kt (i)α Lt (i)eγt Ztp
1−α
− ΦZt∗ , 0},
(2)
where ΦZt∗ is a fixed cost.
Trending variables are divided by Zt∗ to express the model’s equilibrium conditions in
terms of the stationary variables. In what follows we present a summary of the log-linearized
equilibrium conditions, where all variables are expressed in log deviations from their nonstochastic steady state.
Log-linear equilibrium conditions
The stationary component of productivity z̃t evolves as:
z̃t = ρz z̃t−1 + σz εz,t .
(3)
p
Since Ztp is a non stationary process, we define its growth rate as ztp = log(Ztp /Zt−1
) and
assume that it follows an AR(1) process:
p
ztp = ρzp zt−1
+ σzp zp ,t , zp ,t ∼ N (0, 1).
(4)
It follows that
∗
zt ≡ log(Zt∗ /Zt−1
)−γ =
1
1
(ρz − 1)z̃t−1 +
σz z,t + ztp ,
1−α
1−α
(5)
where γ is the steady-state growth rate of the economy. Steady-state values are denoted by
∗-subscripts, and steady-state formulas are provided in the technical appendix of Del Negro
and Schorfheide (2012), which is available online.
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The optimal allocation of consumption satisfies the following consumption Euler equation:
ct = −
he−γ
(1 − he−γ )
(R
−
I
E
[π
]
+
b
)
+
(ct−1 − zt )
t
t t+1
t
σc (1 + he−γ )
(1 + he−γ )
1
(σc − 1) w∗ L∗
+
I
E
[c
+
z
]
+
(Lt − IE t [Lt+1 ]) , (6)
t
t+1
t+1
(1 + he−γ )
σc (1 + he−γ ) c∗
where ct is consumption, Lt is labor supply, Rt is the nominal interest rate, and πt is inflation. The exogenous process bt drives a wedge between the intertemporal marginal utility of
consumption and the riskless real return Rt −IE t [πt+1 ], and is meant to capture risk-premium
shocks.2 This shock follows an AR(1) process with parameters ρb and σb . The parameters
σc and h capture the degree of relative risk aversion and the degree of habit persistence in
the utility function, respectively.
The optimal investment decision satisfies the following relationship between the level of
investment it , measured in terms of consumption goods, and the value of capital in terms of
consumption qtk :
it =
1
β̄
qtk
+
(i
−
z
)
+
IE t [it+1 + zt+1 ] + µt .
t−1
t
S 00 e2γ (1 + β̄) 1 + β̄
1 + β̄
(7)
This relationship shows that investment is affected by investment adjustment costs (S 00 is
the second derivative of the adjustment cost function) and by an exogenous process µt , which
we call “marginal efficiency of investment”, that alters the rate of transformation between
consumption and installed capital (see Greenwood et al. (1998)). The shock µt follows an
AR(1) process with parameters ρµ and σµ . The parameter β̄ depends on the intertemporal
discount rate in the household utility function, β, on the degree of relative risk aversion σc ,
and on the steady-state growth rate γ: β̄ = βe(1−σc )γ .
The capital stock, k̄t , which we refer to as “installed capital”, evolves as
k̄t =
i∗
1−
k̄∗
i∗
i∗ 00
k̄t−1 − zt + it + S e2γ (1 + β̄)µt ,
k̄∗
k̄∗
(8)
where i∗ /k̄∗ is the steady state investment to capital ratio.
Capital is subject to variable capacity utilization ut ; effective capital rented out to firms,
2
In the code, the bt shock is normalized to be in the same units as consumption, i.e., we estimate the
−γ
)
shock b̃t = − σ(1−he
−γ ) bt .
c (1+he
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kt , is related to k̄t by:
kt = ut − zt + k̄t−1 .
(9)
The optimality condition determining the rate of capital utilization is given by
1−ψ k
r = ut ,
ψ t
(10)
where rtk is the rental rate of capital and ψ captures the utilization costs in terms of foregone
consumption.
Real marginal costs for firms are given by
mct = wt + αLt − αkt ,
(11)
where wt is the real wage and α is the income share of capital (after paying mark-ups and
fixed costs) in the production function.
From the optimality conditions of goods producers it follows that all firms have the same
capital-labor ratio:
kt = wt − rtk + Lt .
(12)
We include financial frictions in the model, building on the work of Bernanke et al.
(1999a), Christiano et al. (2003), De Graeve (2008), and Christiano et al. (2014). We assume
that banks collect deposits from households and lend to entrepreneurs who use these funds
as well as their own wealth to acquire physical capital, which is rented to intermediate goods
producers. Entrepreneurs are subject to idiosyncratic disturbances that affect their ability
to manage capital. Their revenue may thus turn out to be too low to pay back the loans
received by the banks. The banks therefore protect themselves against default risk by pooling
all loans and charging a spread over the deposit rate. This spread may vary as a function of
entrepreneurs’ leverage and riskiness.
The realized return on capital is given by:
R̃tk − πt =
(1 − δ)
r∗k
k
rtk + k
qtk − qt−1
,
k
r∗ + (1 − δ)
r∗ + (1 − δ)
(13)
where R̃tk is the gross nominal return on capital for entrepreneurs, r∗k is the steady state
value of the rental rate of capital rtk , and δ is the depreciation rate.
The excess return on capital (the spread between the expected return on capital and the
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riskless rate) can be expressed as a function of the entrepreneurs’ leverage (i.e. the ratio of
the value of capital to nominal net worth) and exogenous fluctuations in the volatility of
entrepreneurs’ idiosyncratic productivity:
Et
h
k
R̃t+1
i
− Rt = bt + ζsp,b qtk + k̄t − nt + σ̃ω,t ,
(14)
where nt is entrepreneurs’ net worth, ζsp,b is the elasticity of the credit spread to the entrepreneurs’ leverage (qtk + k̄t − nt ), and σ̃ω,t captures mean-preserving changes in the crosssectional dispersion of ability across entrepreneurs (see Christiano et al. (2014)). σ̃ω,t follows
an AR(1) process with parameters ρσω and σσω .
Entrepreneurs’ net worth nt evolves according to:
k
+ k̄t−1 + ζn,n nt−1
nt = ζn,R̃k R̃tk − πt − ζn,R (Rt−1 − πt + bt−1 ) + ζn,qK qt−1
−γ∗ nv∗∗ zt −
ζn,σω
ζsp,σω
(15)
σ̃ω,t−1 ,
where the ζ’s denote elasticities, that depend among others on the entrepreneurs’ steadystate default probability F (ω̄), where γ∗ is the fraction of entrepreneurs that survive and
continue operating for another period, and where v∗ is the entrepreneurs’ real equity divided
by Zt∗ , in steady state.
The production function is
yt = Φp (αkt + (1 − α) Lt ) ,
where Φp =
y∗ +Φ
,
y∗
(16)
and the resource constraint is:
yt = g∗ gt +
c∗
i∗
rk k∗
c t + it + ∗ u t .
y∗
y∗
y∗
(17)
∗
where gt = log( Z ∗Gy∗t g∗ ) and g∗ = 1 − c∗y+i
.
∗
t
Government spending gt is assumed to follow the exogenous process:
gt = ρg gt−1 + σg εg,t + ηgz σz εz,t .
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The price and wage Phillips curves are, respectively:
πt = κ mct +
ιp
β̄
πt−1 +
IE t [πt+1 ] + λf,t ,
1 + ιp β̄
1 + ιp β̄
(18)
and
wt =
1 + ιw β̄
(1 − ζw β̄)(1 − ζw )
1
wth − wt −
πt +
(wt−1 − zt + ιw πt−1 )
(1 + β̄)ζw ((λw − 1)w + 1)
1 + β̄
1 + β̄
β̄
IE t [wt+1 + zt+1 + πt+1 ] + λw,t , (19)
+
1 + β̄
p β̄)(1−ζp )
, the parameters ζp , ιp , and p are the Calvo parameter, the
where κ = (1+ιp(1−ζ
β̄)ζp ((Φp −1)p +1)
degree of indexation, and the curvature parameter in the Kimball aggregator for prices, and
ζw , ιw , and w are the corresponding parameters for wages. wth measures the household’s
marginal rate of substitution between consumption and labor, and is given by:
wth =
1
ct − he−γ ct−1 + he−γ zt + νl Lt ,
−γ
1 − he
(20)
where νl characterizes the curvature of the disutility of labor (and would equal the inverse
of the Frisch elasticity in the absence of wage rigidities). The mark-ups λf,t and λw,t follow
exogenous ARMA(1,1) processes:
λf,t = ρλf λf,t−1 + σλf ελf ,t − ηλf σλf ελf ,t−1 ,
and
λw,t = ρλw λw,t−1 + σλw ελw ,t − ηλw σλw ελw ,t−1 ,
respectively.
Finally, the monetary authority follows a generalized policy feedback rule:
πt∗ )
Rt = ρR Rt−1 + (1 − ρR ) ψ1 (πt −
+ ψ2 (yt −
f
+ψ3 (yt − ytf ) − (yt−1 − yt−1
) + rtm .
ytf )
(21)
where ytf is the flexible price/wage output, obtained from solving the version of the model
without nominal rigidities and markup shocks (that is, Equations (6) through (20) with
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ζp = ζw = 0, and λf,t = λw,t = 0), and the residual rtm follows an AR(1) process with
parameters ρrm and σrm .
In this version of the model we have replaced a constant inflation target with a timevarying inflation target πt∗ , to capture the rise and fall of inflation and interest rates in
the estimation sample. Although time-varying target rates have been frequently used for
the specification of monetary policy rules in DSGE model (e.g., Erceg and Levin (2003) and
Smets and Wouters (2003), among others), we follow the approach of Aruoba and Schorfheide
(2008) and Del Negro and Eusepi (2011) and include data on long-run inflation expectations
as an observable for the estimation of the model. At each point in time, long-run inflation
expectations essentially determine the level of the target inflation rate. To the extent that
long-run inflation expectations at the forecast origin contain information about the central
bank’s objective function, e.g. the desire to stabilize inflation at 2%, this information is
automatically included in the forecast.
The time-varying inflation target evolves according to:
∗
πt∗ = ρπ∗ πt−1
+ σπ∗ π∗ ,t ,
(22)
where 0 < ρπ∗ < 1 and π∗ ,t is an iid shock. We model πt∗ as a stationary process, although
our prior for ρπ∗ will force this process to be highly persistent. The assumption that the
changes in the target inflation rate are exogenous is, to some extent, a short-cut. For instance,
the learning models of Sargent (1999) or Primiceri (2006) imply that the rise in the target
inflation rate in the 1970’s and the subsequent drop is due to policy makers learning about
the output-inflation trade-off and trying to set inflation optimally. We are abstracting from
such a mechanism in our specification.
Anticipated policy shocks
This section describes the introduction of anticipated policy shocks in the model, which
follows Laseen and Svensson (2011). We modify the exogenous component of the policy
rule (21) as follows:
K
X
m
m
R
rt = ρrm rt−1 + t +
R
(23)
k,t−k ,
k=1
R
where R
t is the usual contemporaneous policy shock, and k,t−k is a policy shock that is
known to agents at time t − k, but affects the policy rule k periods later, that is, at time t.
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2
We assume that R
k,t−k ∼ N (0, σk,r ), i.i.d.
In order to solve the model we need to express the anticipated shocks in recursive form.
For this purpose, we augment the state vector st (described below) with K additional states
R
whose law of motion is as follows:
νtR ,. . . ,νt−K
R
ν1,t
R
ν2,t
R
= ν2,t−1
+ R
1,t
R
= ν3,t−1 + R
2,t
..
.
R
= R
νK,t
K,t
and rewrite the exogenous component of the policy rule (23) as3
m
R
rtm = ρrm rt−1
+ R
t + ν1,t−1 .
Parameters
The following tables describe the parameters used in the New York Fed DSGE model. Table 2
gives the prior distributions for each parameter. Table 3 gives the posterior mean, 5th
percentile, and 95th percentile for each parameter.
PK
R
R
It is easy to verify that ν1,t−1
= k=1 R
k,t−k , that is, ν1,t−1 is a “bin” that collects all anticipated shocks
that affect the policy rule in period t.
3
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Table 2: Priors
Dist
Policy Parameters
ψ1
ψ2
ψ3
ρR
Normal
Normal
Normal
Beta
Mean
Std Dev
Dist
Mean
Std Dev
1.50
0.12
0.12
0.75
0.25
0.05
0.05
0.10
ρr m
σr m
σant1
Beta
InvG
InvG
0.50
0.10
0.20
0.20
2.00
4.00
0.10
0.15
ζw
ιw
w
Beta
Beta
-
0.50
0.50
10.00
0.10
0.15
S 00
ψ
π∗
Normal
Beta
Normal
Normal
Normal
-
4.00
0.50
0.50
1.00
0.00
-45.00
1.50
0.18
1.50
0.15
Nominal Rigidities Parameters
ζp
ιp
p
Beta
Beta
-
0.50
0.50
10.00
fixed
fixed
Other Endogenous Propagation and Steady State Parameters
100γ
α
100(β −1 − 1)
σc
h
νl
δ
Φp
Normal
Normal
Gamma
Normal
Beta
Normal
Normal
0.40
0.30
0.25
1.50
0.70
2.00
0.03
1.25
0.10
0.05
0.10
0.37
0.10
0.75
fixed
0.12
γgdpdef
δgdpdef
L̄
λw
g∗
fixed
2.00
2.00
5.00
fixed
fixed
Financial Frictions Parameters
F (ω̄)
SP∗
Gamma
0.03
2.00
fixed
0.10
ζsp,b
γ∗
Beta
-
0.05
0.99
0.00
σg
σb
σµ
σz
σσω
σπ∗
σz p
σλf
σ λw
ηgz
InvG
InvG
InvG
InvG
InvG
InvG
InvG
InvG
InvG
Beta
0.10
0.10
0.10
0.10
0.05
0.03
0.10
0.10
0.10
0.50
2.00
2.00
2.00
2.00
4.00
6.00
2.00
2.00
2.00
0.20
fixed
Exogenous Process Parameters
ρg
ρb
ρµ
ρz
ρσ ω
ρπ ∗
ρz p
ρλ f
ρλ w
ηλf
η λw
Beta
Beta
Beta
Beta
Beta
Beta
Beta
Beta
Beta
Beta
0.50
0.50
0.50
0.50
0.75
0.99
0.50
0.50
0.50
0.50
0.50
0.20
0.20
0.20
0.20
0.15
fixed
0.20
0.20
0.20
0.20
0.20
Measurement Error Parameters
Note: For Inverse Gamma prior mean and SD, τ and ν reported.
σant1 through σant12 all have the same distribution.
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Cme
ρgdp
ρgdi
ρ10y
ρtf p
ρgdpdef
ρpce
Dist
Normal
Normal
Beta
Beta
Beta
Beta
Mean
1.00
0.00
0.00
0.50
0.50
0.50
0.50
May 31, 2018
Std Dev
fixed
0.20
0.20
0.20
0.20
0.20
0.20
%gdp
σgdp
σgdi
σ10y
σtf p
σgdpdef
σpce
Dist
Normal
InvG
InvG
InvG
InvG
InvG
InvG
Mean
0.00
0.10
0.10
0.75
0.10
0.10
0.10
Std Dev
0.40
2.00
2.00
2.00
2.00
2.00
2.00
Note: For Inverse Gamma prior mean and SD, τ and ν reported.
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Table 3: Posteriors
Mean
Policy Parameters
ψ1
ψ2
ψ3
ρR
ρr m
σr m
1.45
0.05
0.25
0.66
0.31
0.23
(p5, p95)
(1.21,
(0.03,
(0.20,
(0.59,
(0.20,
(0.21,
Mean
1.68)
0.08)
0.31)
0.73)
0.42)
0.25)
(p5, p95)
σant1
σant2
σant3
σant4
σant5
σant6
0.09
0.09
0.09
0.08
0.09
0.11
(0.07,
(0.07,
(0.07,
(0.06,
(0.07,
(0.08,
0.11)
0.10)
0.10)
0.10)
0.11)
0.13)
ζw
ιw
w
0.92
0.55
10.00
(0.90, 0.94)
(0.34, 0.76)
(2.11, 4.12)
(0.42, 0.69)
γgdpdef
δgdpdef
L̄
λw
g∗
3.15
0.56
0.50
1.03
0.00
-48.03
1.50
0.18
ζsp,b
γ∗
0.05
0.99
(0.05, 0.06)
σg
σb
σµ
σz
σσω
σπ∗
σz p
σλf
σ λw
ηgz
2.26
0.03
0.44
0.57
0.04
0.03
0.15
0.08
0.40
0.38
(2.07,
(0.02,
(0.38,
(0.52,
(0.03,
(0.02,
(0.10,
(0.06,
(0.35,
(0.10,
%gdp
-0.15
Nominal Rigidities Parameters
ζp
ιp
p
0.93
0.22
10.00
(0.91, 0.95)
(0.10, 0.35)
fixed
fixed
Other Endogenous Propagation and Steady State Parameters
100γ
α
100(β −1 − 1)
σc
h
νl
δ
Φp
0.35
0.18
0.15
1.00
0.48
2.30
0.03
1.10
(0.28,
(0.15,
(0.07,
(0.77,
(0.38,
(1.50,
0.43)
0.20)
0.23)
1.25)
0.58)
3.08)
fixed
(1.04, 1.16)
S 00
ψ
π∗
fixed
(0.95, 1.10)
(-0.05, 0.05)
(-50.06, -46.03)
fixed
fixed
Financial Frictions Parameters
F (ω̄)
SP∗
0.03
1.81
fixed
(1.67, 1.94)
fixed
Exogenous Process Parameters
ρg
ρb
ρµ
ρz
ρσω
ρπ ∗
ρz p
ρλ f
ρλ w
ηλf
η λw
0.99
0.95
0.81
0.96
0.98
0.99
0.89
0.80
0.36
0.65
0.40
(0.98,
(0.94,
(0.75,
(0.94,
(0.97,
1.00)
0.96)
0.87)
0.98)
1.00)
fixed
(0.84,
(0.69,
(0.09,
(0.46,
(0.18,
0.95)
0.91)
0.61)
0.85)
0.62)
2.45)
0.03)
0.50)
0.63)
0.05)
0.04)
0.21)
0.10)
0.44)
0.64)
Measurement Error Parameters
Cme
1.00
fixed
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ρgdp
ρgdi
ρ10y
ρtf p
ρgdpdef
ρpce
Mean
0.04
0.93
0.96
0.21
0.51
0.30
(p5, p95)
(-0.17, 0.25)
(0.89, 0.97)
(0.94, 0.99)
(0.10, 0.32)
(0.38, 0.64)
(0.08, 0.51)
May 31, 2018
σgdp
σgdi
σ10y
σtf p
σgdpdef
σpce
Mean
0.24
0.31
0.12
0.77
0.16
0.10
(p5, p95)
(0.20, 0.28)
(0.28, 0.35)
(0.11, 0.13)
(0.70, 0.84)
(0.15, 0.18)
(0.08, 0.12)
Impulse Responses
The following figures depict impulse response functions to various shocks. Figure 5 depicts
the response of the economy to a discount factor shock, Figure 6 to a spread shock, Figure 7
to a shock to the marginal efficiency of investment (MEI), Figure 8 to a TFP shock, Figure
9 to a price markup shock, and Figure 10 to a monetary policy shock.
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Figure 5: Responses to a Discount Factor Shock bt
Hours Per Capita
Real GDP Growth
0.0
-0.1
-0.2
-0.3
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
10
20
30
40
10
Percent Change in Wages
0.00
-0.01
-0.02
-0.03
10
20
20
30
40
30
40
30
40
GDP Deflator
30
40
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
10
Core PCE Inflation
20
Nominal FFR
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.025
-0.050
-0.075
-0.100
10
20
30
40
10
Consumption growth per capita
20
Real Investment per capita
0.0
-0.1
-0.2
-0.3
0.0
-0.1
-0.2
-0.3
10
20
30
40
10
20
30
40
Long term inflation expectations
BAA - 10yr Treasury Spread
0.08
0.06
0.04
0.02
-0.005
-0.010
-0.015
-0.020
10
20
30
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.00
-0.01
-0.02
-0.03
-0.01
-0.02
-0.03
-0.04
-0.05
10
20
30
40
30
40
10
20
30
40
Real GDI Growth
0.0
-0.1
-0.2
-0.3
10
20
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Figure 6: Responses to a Spread Shock σ̃ω,t
Real GDP Growth
Hours Per Capita
0.01
0.00
-0.01
-0.02
-0.03
-0.04
0.00
-0.05
-0.10
-0.15
10
20
30
40
10
Percent Change in Wages
30
40
30
40
30
40
GDP Deflator
0.002
0.000
-0.002
-0.004
-0.006
0.001
0.000
-0.001
-0.002
10
20
30
40
10
Core PCE Inflation
20
Nominal FFR
0.005
0.000
-0.005
-0.010
-0.015
0.001
0.000
-0.001
-0.002
10
20
30
40
10
0.04
0.03
0.02
0.01
0.00
-0.01
10
20
20
Real Investment per capita
Consumption growth per capita
30
40
0.0
-0.1
-0.2
-0.3
-0.4
BAA - 10yr Treasury Spread
0.05
0.04
0.03
0.02
0.01
0.00
20
10
20
30
40
Long term inflation expectations
0.0010
0.0005
0.0000
-0.0005
10
20
30
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.002
0.000
-0.002
-0.004
0.002
0.000
-0.002
-0.004
10
20
30
40
30
40
10
20
30
40
Real GDI Growth
0.01
0.00
-0.01
-0.02
-0.03
-0.04
10
20
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Figure 7: Responses to an MEI Shock µt
Real GDP Growth
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
Hours Per Capita
0.2
0.0
-0.2
-0.4
10
20
30
40
10
20
30
40
30
40
30
40
GDP Deflator
Percent Change in Wages
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0.005
0.000
-0.005
-0.010
-0.015
10
20
30
40
10
Core PCE Inflation
20
Nominal FFR
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0.02
0.00
-0.02
-0.04
-0.06
10
20
30
40
10
Real Investment per capita
Consumption growth per capita
0.025
0.000
-0.025
-0.050
10
20
20
30
40
0.0
-0.5
-1.0
-1.5
BAA - 10yr Treasury Spread
10
20
30
40
Long term inflation expectations
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
0.004
0.003
0.002
0.001
10
20
30
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.012
0.009
0.006
0.003
0.000
10
20
30
40
30
40
0.01
0.00
-0.01
-0.02
-0.03
10
20
30
40
Real GDI Growth
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
10
20
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Figure 8: Responses to a TFP Shock z̃t
Real GDP Growth
Hours Per Capita
0.4
0.3
0.2
0.1
0.0
0.1
0.0
-0.1
-0.2
10
20
30
40
10
20
30
40
30
40
30
40
GDP Deflator
Percent Change in Wages
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
0.03
0.02
0.01
0.00
10
20
30
40
10
20
Nominal FFR
Core PCE Inflation
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
10
20
30
40
10
Consumption growth per capita
20
Real Investment per capita
0.3
0.2
0.1
0.0
0.20
0.15
0.10
0.05
0.00
10
20
30
40
10
BAA - 10yr Treasury Spread
20
30
40
Long term inflation expectations
0.010
0.008
0.006
0.004
0.002
0.000
-0.002
0.000
-0.005
-0.010
-0.015
10
20
30
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.000
-0.005
-0.010
-0.015
-0.020
0.6
0.4
0.2
0.0
10
20
30
40
30
40
10
20
30
40
Real GDI Growth
0.4
0.3
0.2
0.1
0.0
10
20
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Figure 9: Responses to a Price Markup Shock λf,t
Real GDP Growth
Hours Per Capita
0.00
-0.03
-0.06
-0.09
-0.12
0.0
-0.1
-0.2
-0.3
10
20
30
40
10
Percent Change in Wages
20
30
40
0.15
0.10
0.05
0.00
10
Core PCE Inflation
0.15
0.10
0.05
0.00
30
40
30
40
30
40
GDP Deflator
0.00
-0.05
-0.10
-0.15
10
20
20
Nominal FFR
0.06
0.05
0.04
0.03
0.02
0.01
0.00
10
20
30
40
10
Consumption growth per capita
20
Real Investment per capita
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
10
20
30
40
10
BAA - 10yr Treasury Spread
20
30
40
Long term inflation expectations
0.010
0.008
0.006
0.004
0.002
0.000
-0.002
0.0000
-0.0025
-0.0050
-0.0075
-0.0100
10
20
30
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.015
0.010
0.005
0.000
0.00
-0.01
-0.02
-0.03
10
20
30
40
30
40
10
20
30
40
Real GDI Growth
0.00
-0.03
-0.06
-0.09
-0.12
10
20
New York Fed DSGE Team, Research and Statistics
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New York Fed DSGE Model: Research Directors Draft
May 31, 2018
Figure 10: Responses to a Monetary Policy Shock rtm
Real GDP Growth
Hours Per Capita
0.5
0.4
0.3
0.2
0.1
0.0
0.3
0.2
0.1
0.0
10
20
30
40
10
20
30
40
GDP Deflator
Percent Change in Wages
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
0.015
0.010
0.005
0.000
10
20
30
40
10
Core PCE Inflation
20
30
40
30
40
Nominal FFR
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
-0.025
-0.050
-0.075
-0.100
-0.125
10
20
30
40
10
Consumption growth per capita
20
Real Investment per capita
0.4
0.3
0.2
0.1
0.0
0.6
0.4
0.2
0.0
10
20
30
40
10
BAA - 10yr Treasury Spread
20
30
40
Long term inflation expectations
0.005
0.000
-0.005
-0.010
10
20
30
40
0.0010
0.0005
0.0000
-0.0005
-0.0010
Long term interest rate expectations
10
20
30
40
Total Factor Productivity
0.000
-0.005
-0.010
-0.015
0.04
0.03
0.02
0.01
0.00
10
20
30
40
30
40
10
20
30
40
Real GDI Growth
0.3
0.2
0.1
0.0
10
20
New York Fed DSGE Team, Research and Statistics
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References
Aruoba, S. B. and F. Schorfheide (2008): “Insights from an Estimated Search-Based
Monetary Model with Nominal Rigidities,” Working Paper.
Bernanke, B., M. Gertler, and S. Gilchrist (1999a): “The Financial Accelerator in
a Quantitative Business Cycle Framework,” in Handbook of Macroeconomics, ed. by J. B.
Taylor and M. Woodford, North Holland, Amsterdam, vol. 1C.
Bernanke, B. S., M. Gertler, and S. Gilchrist (1999b): “The Financial Accelerator
in a Quantitative Business Cycle Framework,” in Handbook of Macroeconomics, ed. by
J. B. Taylor and M. Woodford, Amsterdam: North-Holland, vol. 1C, chap. 21, 1341–93.
Christiano, L., R. Motto, and M. Rostagno (2003): “The Great Depression and the
Friedman-Schwartz Hypothesis,” Journal of Money, Credit and Banking, 35, 1119–1197.
Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): “Nominal Rigidities and
the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113,
1–45.
Christiano, L. J., R. Motto, and M. Rostagno (2014): “Risk Shocks,” American
Economic Review, 104, 27–65.
De Graeve, F. (2008): “The External Finance Premium and the Macroeconomy: US
Post-WWII Evidence,” Journal of Economic Dynamics and Control, 32, 3415 – 3440.
Del Negro, M. and S. Eusepi (2011): “Fitting Observed Ination Expectations,” Journal
of Economic Dynamics and Control, 35, 2105–2131.
Del Negro, M., M. P. Giannoni, and F. Schorfheide (2015): “Inflation in the Great
Recession and New Keynesian Models,” American Economic Journal: Macroeconomics,
7, 168–196.
Del Negro, M. and F. Schorfheide (2012): “DSGE Model-Based Forecasting,” Federal
Reserve Bank of New York Working Paper.
Erceg, C. J. and A. T. Levin (2003): “Imperfect Credibility and Inflation Persistence,”
Journal of Monetary Economics, 50, 915–944.
New York Fed DSGE Team, Research and Statistics
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Greenwood, J., Z. Hercovitz, and P. Krusell (1998): “Long-Run Implications of
Investment-Specific Technological Change,” American Economic Review, 87, 342–36.
Laseen, S. and L. E. Svensson (2011): “Anticipated Alternative Policy-Rate Paths in
Policy Simulations,” International Journal of Central Banking, 7, 1–35.
Primiceri, G. (2006): “Why Inflation Rose and Fell: Policymakers Beliefs and US Postwar
Stabilization Policy,” Quarterly Journal of Economics, 121, 867–901.
Sargent, T. J. (1999): The Conquest of American Inflation, Princeton University Press,
Princeton.
Smets, F. and R. Wouters (2003): “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1, 1123
– 1175.
——— (2007): “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach,”
American Economic Review, 97, 586 – 606.
New York Fed DSGE Team, Research and Statistics
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Detailed Philadelphia (PRISM) Forecast Overview
June 2018
Keith Sill
Forecast Summary
The FRB Philadelphia DSGE model denoted PRISM, projects that real GDP growth will
be slightly above its trend pace over the next three years with real output growth running at about
a 3 percent pace. Core PCE inflation is only marginally below the FOMC target of 2 percent over
most of the next three years. The funds rate rises to 2.7 percent in 2018Q4 and increases steadily
to reach 3.7 percent at the end of 2020. The current gap between the level of output and its trend
level remains significant in the estimated model and, absent any shocks, the model continues to
predict a recovery to the trend level. The relatively slow pace of growth and low inflation that
have tended to characterize U.S. economic performance over the past few years require the
presence of shocks to offset the strength of the model’s internal propagation channels. The
PRISM model does not take into account the recent tax reform except to the extent that it is
represented in current data observations.
The Current Forecast and Shock Identification
The PRISM model is an estimated New Keynesian DSGE model with sticky wages,
sticky prices, investment adjustment costs, and habit persistence. The model is similar to the
Smets & Wouters 2007 model and is described more fully in Schorfheide, Sill, and Kryshko
2010. Unlike in that paper though, we estimate PRISM directly on core PCE inflation rather
than projecting core inflation as a non-modeled variable. Details on the model and its estimation
are available in a Technical Appendix that was distributed for the June 2011 FOMC meeting or
is available on request.
The current forecasts for real GDP growth, core PCE inflation, and the federal funds rate
are shown in Figures 1a-1c along with 68 percent probability coverage intervals. The forecast
uses data through 2018Q1 supplemented by a 2018Q2 nowcast. The model takes the 2018Q2
nowcast for output growth of 3.2 percent as given and the projection begins with 2018Q3.
PRISM anticipates that output growth edges down to 3 percent in 2018Q3, with growth then
running at about that pace through the end of 2019. Growth tapers off slightly in 2020 to run at
2.8 percent by the end of the year. Overall, the growth forecast for this round is very similar to
that from the March projection. While output growth is fairly robust going forward, core PCE
inflation stays contained and runs at a pace slightly below the 2 percent target over the forecast
horizon. Based on the 68 percent coverage interval, the model sees a minimal chance of deflation
or recession (measured as negative quarters of real GDP growth) over the next 3 years. The
federal funds rate is determined by an estimated policy rule and the funds rate rises from 1.7
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percent in 2018Q2, to 2.7 percent in 2018Q4, 3.4 percent in 2019Q4, and 3.7 percent in 2020Q4.
This path for the funds rate is similar to that in the March projection.
Figures 1d and 1e plot the model’s estimates of the output gap and the natural real rate of
interest. The output gap is defined as the log deviation of the level of output from the level that
obtains under flexible wages and prices. The real natural rate of interest is the short term real
interest rate that obtains in the model when prices and wages are flexible. The model currently
estimates the output gap at -1.6 percent, a more or less continued improvement from the -3
percent reached during the recession. The output gap is expected to improve slowly, reaching
about -0.9 percent at the end of 2020. The estimated real natural rate of interest is low at 0.13
percent in 2018Q2. The natural rate then rises over the next three years to reach 1.4 percent at
the end of 2020. Note though that the natural rate estimate is extremely volatile and the 68
percent confidence bands are quite large.
The key factors driving the projection are shown in the forecast shock decompositions
(Figures 2a-2e) and the smoothed estimates of the model’s primary shocks (shown in Figure 3,
where they are normalized by standard deviation). GDP growth is at an above trend pace in
2018Q2, boosted by shocks to labor supply, investment, and government spending (which
includes net exports). Thereafter, growth is at a close to a trend pace over the forecast horizon.
TFP, financial, and monetary policy shocks act as a drag on growth, but are largely offset by
rebounds in government spending, investment, and labor supply. Over the course of the recession
and recovery PRISM estimated a series of large positive shocks to leisure (negative shocks to
labor supply) that have a persistent effect on hours worked and so pushed hours well below
steady state. As these shocks unwind hours worked continue to rebound over the forecast horizon
and so support higher output growth. Similarly, the unwinding of investment shocks contribute
to output growth over the forecast horizon.
Consumption is expected to grow at a slightly below-trend pace over the forecast horizon
(Figure 2d). Consumption growth is pulled down by shocks to technology, investment, monetary
policy, and government spending. Consumption gets a boost from the unwinding of financial
and labor shocks, but not enough to offset the downward pull from the other shocks in the model.
Financial shocks exert a considerable drag on investment growth in early 2018 – these same
shocks make a positive contribution to consumption growth (Figure 2d-e). However, strong and
offsetting investment shocks are enough to keep investment growth near its trend over the
forecast horizon. All told, the model now forecasts trend growth in investment (gross private
domestic + durable goods consumption) in 2018 as the gradual unwinding of MEI shocks (see
Figures 2e and 3) are offset by the effects of financial shocks.
The forecast for core PCE inflation continues to be a story of upward pressure from the
unwinding of negative labor supply shocks and MEI shocks being offset by downward pressure
from the waning of discount factor shocks. Negative discount factor shocks have a strong and
persistent negative effect on marginal cost and inflation in the estimated model. But labor supply
shocks that push down aggregate hours also serve to put upward pressure on the real wage and
hence marginal cost. The effect is persistent -- as the labor supply shocks unwind over the
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forecast horizon they exert a waning upward push to inflation. On balance the effect of these
opposing forces keep inflation slightly below target over the next 3 years.
The federal funds rate is projected to rise fairly quickly over the forecast horizon. The
model attributes the current level of the funds rate primarily to a combination of monetary
policy, discount factor and labor supply dynamics. Looking ahead, the positive contribution from
labor supply shocks is more than offset by discount factor shock dynamics over the medium
term, but as these shocks wane the funds rate gradually rises to 3.7 percent by the end of 2020.
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References
Schorfheide, Frank, Keith Sill, and Maxym Kryshko. 2010. “DSGE model-based forecasting of
non-modelled variables.” International Journal of Forecasting, 26(2): 348-373.
Smets, Frank, and Rafael Wouters. 2007. “Shocks and Frictions in U.S. Business Cycles: A
Bayesian DSGE Approach.” American Economic Review, 97(3): 586-606.
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Figure 1a
Real GDP Growth
10
8
6
4
2
0
-2
-4
-6
-8
-10
2014
2015
2016
2017
2018
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Figure 1b
Core PCE Inflation
6
5
4
3
2
1
0
-1
2014
2015
2016
2017
2018
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Figure 1c
Fed Funds Rate
8
6
4
2
0
-2
-4
2014
2015
2016
2017
2018
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Figure 1d
Output Gap
4
3
2
1
0
-1
-2
-3
-4
-5
1980
1985
1990
1995
2000
2005
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Figure 1e
Natural Rate of Interest
12
10
8
6
4
2
0
-2
-4
-6
-8
1980
1985
1990
1995
2000
2005
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Figure 2a
Shock Decompositions
percent
Real GDP Growth
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 2b
Shock Decompositions
percent
Core PCE Inflation
4
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 2c
Shock Decompositions
percent
Fed Funds Rate
6
6
4
4
2
2
0
0
-2
-2
-4
-4
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 2d
Shock Decompositions
percent
Real Consumption Growth
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 2e
Shock Decompositions
Real Investment Growth
30
20
20
10
10
percent
30
0
0
-10
-10
-20
-20
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 3
Smoothed Shock Estimates for Conditional Forecast Model
(normalized by standard deviation)
labor shock
discount factor shock
4
5
2
0
0
-2
-4
2010
-5
2012
2014
2016
2018
2010
2012
TFP shock
2014
2016
2018
2016
2018
mei shock
2
2
0
0
-2
-2
-4
2010
2012
2014
2016
2018
2010
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Impulse Responses to TFP shock
output growth
consumption growth
1
1
0.5
0.5
0
0
5
10
15
0
0
investment growth
0.5
0
0
0
5
10
15
-0.5
0
inflation
0.05
0
0
0
5
15
5
10
15
nominal rate
0.05
-0.05
10
aggregate hours
2
-2
5
10
15
-0.05
0
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Impulse Response to Leisure Shock
output growth
consumption growth
2
2
0
0
-2
0
5
10
15
-2
0
investment growth
5
0
0
-1
-5
0
5
10
15
-2
0
inflation
0.4
0.2
0.2
0
5
10
15
5
10
15
nominal rate
0.4
0
5
aggregate hours
10
15
0
0
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Impulse Responses to MEI Shock
output growth
consumption growth
2
0.2
0
0
-2
0
5
10
15
-0.2
0
investment growth
1
0
0.5
0
5
10
15
0
0
inflation
0.4
0
0.2
0
5
15
5
10
15
nominal rate
0.1
-0.1
10
aggregate hours
10
-10
5
10
15
0
0
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Impulse Responses to Financial Shock
output growth
consumption growth
1
2
0
0
-1
0
5
10
15
-2
0
investment growth
0.5
0
0
0
5
10
15
-0.5
0
inflation
1
0.2
0.5
0
5
15
5
10
15
nominal rate
0.4
0
10
aggregate hours
5
-5
5
10
15
0
0
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Impulse Responses to Price Markup Shock
output growth
consumption growth
0.5
0.5
0
0
-0.5
0
5
10
15
-0.5
0
investment growth
0
0
-0.1
0
5
10
15
-0.2
0
inflation
0.5
0
0
0
5
15
5
10
15
nominal rate
1
-1
10
aggregate hours
1
-1
5
10
15
-0.5
0
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Impulse Responses to Unanticipated Monetary Policy Shock
output growth
consumption growth
0.5
0.5
0
0
-0.5
0
5
10
15
-0.5
0
investment growth
0.2
0
0
0
5
10
15
-0.2
0
inflation
1
0
0
0
5
15
5
10
15
nominal rate
0.1
-0.1
10
aggregate hours
1
-1
5
10
15
-1
0
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Impulse Responses to Govt Spending Shock
output growth
consumption growth
2
0.5
0
0
-2
0
5
10
15
-0.5
0
investment growth
0.4
0
0.2
0
5
10
15
0
0
inflation
0.04
0.01
0.02
0
5
15
5
10
15
nominal rate
0.02
0
10
aggregate hours
0.2
-0.2
5
10
15
0
0
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Research Directors’ Guide to
the Chicago Fed DSGE Model∗
Jeffrey R. Campbell
Filippo Ferroni
Jonas D. M. Fisher
Leonardo Melosi
November 30, 2017
This guide describes the construction and estimation of the Chicago Fed’s DSGE
model, which we use both for internal forecasting and for creating our contributions
to the System DSGE memo distributed quarterly to the FOMC. The model has been
in use and under ongoing development since 2010. Originally, it was largely based
on Justiniano, Primiceri, and Tambalotti (2010). We published results based on
simulations from the estimated model in Campbell, Evans, Fisher, and Justiniano
(2012) and in Campbell, Fisher, Justiniano, and Melosi (2016).
The model contains many features familiar from other DSGE analyses of
monetary policy and bussiness cycles. External habit in preferences, i-dot costs of
adjusting investment, price and wage stickiness based on Calvo’s (1983) adjustment
probabilities, and partial indexation of unadjusted prices and wages using recently
observed price and wage inflation. The features which distinguish our analysis from
many otherwise similar undertakings are
• Forward Guidance Shocks: An interest-rate rule which depends on recent
(and expected future) inflation and output and is subject to stochastic
disturbances governs our model economy’s monetary policy rate. Standard
analysis prior to the great recession restricted the stochastic disturbances to be
unforecastable. Our model deviates from this historical standard by including
forward guidance shocks, as in Laséen and Svensson (2011). A j-quarter ahead
forward guidance shock revealed to the public at time t influences the interestrate rule’s stochastic intercept only at time t + j. Each period, the model’s
monetary authority reveals a vector of these shocks with one element for each
quarter from the present until the end of the forward guidance horizon. The
∗
The views expressed herein are the authors’. They do not necessarily represent those of the
Federal Reserve Bank of Chicago, the Federal Reserve System, or its Board of Governors.
1
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vector’s elements may be correlated with each other, so the monetary authority
could routinely reveal persistent shifts in the interest-rate rule’s stochastic
intercept. However, the forward guidance shocks are serially uncorrelated
over time, as is required for them to match the definition of “news.”
• Investment-Specific Technological Change: As in the Real Business
Cycle models from which modern DSGE models decend (King, Plosser, and
Rebelo, 1988a), stochastic trend productivity growth both short-run and longrun fluctuations. Our model features two such stochastic trends, one to
Hicks-neutral productivity (King, Plosser, and Rebelo, 1988b) and one to
the technology for converting consumption goods into investment goods (as
in Fisher (2006)). This investment-specific technological change allows our
model to reproduce the dynamics of the relative price of investment goods
to consumption goods, which is a necessary input into the formula we use to
create Fisher-ideal chain-weighted index of real GDP.
• A Mixed Calibration-Bayesian Estimation Empirical Strategy:
Bayesian estimation of structural business cycle models attempts to match all
features of the data’s probability distribution using the model’s parameters.
Since no structural model embodies Platonic “truth,” this exercise inevitably
requires trading off between the model’s ability to replicate first moments
with its fidelty to the business cycles in second moments. Since the criteria for
this tradeoff are not always clear, we adopt an alternative “first-moments-first”
strategy. This selects the values of model parameters which govern the model’s
steady-state growth path, such as the growth rates of Hicks-neutral and
investment-specific technology, to match estimates of selected first moments.
These parameter choices are then fixed for Bayesian estimation, which chooses
values for model parameters which only influence second moments, such as
technology innovation variances. (Since we employ a log linear solution of our
model and all shocks to its primitives have Gaussian distributions, our analysis
has no non-trivial implications for third and higher moments of the data.)
The guide proceeds as follows. The next section presents the model economy’s
primitives, while Section 2 presents the agents’ first-order conditions. Section 3
gives the formulas used to remove nominal and technological trends from model
variables and thereby induce model stationarity, and Sections 4 and 5 discuss
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the stationary economy’s steady state and the log linearization of its equilibrium
necessary conditions around it. Section 6 discusses measurement issues which arise
when comparing model-generated data with data measured by the BEA and BLS.
Section 7 describes our mixed Calibration-Bayesian Estimation empirical strategy
and presents the resulting parameter values we use for model simulations and
forecasting.
1
The Model’s Primitives
Eight kinds of agents populate the model economy:
• Households,
• Investment producers,
• Competitive final goods producers,
• Monopolistically-competitive differentiated goods producers,
• Labor Packers,
• Monopolistically-competitive guilds,
• a Fiscal Authority and
• a Monetary Authority.
These agents interact with each other in markets for
• final goods used for consumption
• investment goods used to augment the stock of productive capital
• differentiated intermediate goods
• capital services
• raw labor
• differentiated labor
• composite labor
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• government bonds
• privately-issued bonds, and
• state-contingent claims.
The households have preferences over streams of an aggregate consumption good,
leisure, and the real value of the fiscal authority’s bonds in their portfolios. Our
specification for preferences displays balanced growth. They also feature external
habit in consumption; which creates a channel for the endogenous propagation of
shocks. Our bonds-in-the-utility-function preferences follow those of Fisher (2015),
and they allow us to incorporate a persistent spread between the monetary policy
rate and the return on productive capital. The aggregate consumption good has a
single alternative use, as the only input into the linear production function operated
by investment producers. These firms sell their output to the households. In turn,
households produce capital services from their capital stocks, which they then sell
to differentiated goods producers. Producers of final goods operate a constantreturns-to-scale technology with a constant elasticity of substitution between its
inputs, which are differentiated goods produced by the monopolistically-competitive
firms. These firms operate technologies with affine cost curves (a constant fixed cost
and linear marginal cost), which employs capital services and composite labor as
inputs. The labor packers produce composite labor using a constant-returns-toscale technology with a constant elasticity of substitution between its inputs, the
differentiated labor sold by guilds. Each of these produces differentiated labor from
the raw labor provided by the households with a linear technology, and they sell their
outputs to the labor packers. There is a nominal unit of account, called the “dollar.”
The fiscal authority issues one-period nominally risk-free bonds, provides public
goods through government spending, and assesses lump-sum taxes on households.
The monetary authority sets the interest rate on the fiscal authority’s one-period
bond according to an interest-rate rule.
All non-financial trade is denominated in dollars, and all private agents take
prices as given with two exceptions: the monopolistically-competitive differentiatedgoods producers and guilds. These choose output prices to maximize the current
value of expected future profits taking as given their demand curves and all relevant
input prices. Financial markets are complete, but all securities excepting equities
in differentiated-goods producers are in zero net supply. These producers’ profits
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and losses are rebated to the households (who own the firms’ equities) lump-sum
period-by-period, as are the profits and losses of the guilds. Given both a process
for government spending and taxes and a rule for the monetary authority’s interest
rate choice, a competitive equilibrium consists of allocations and prices that are
consistent with households’ utility maximization, firms’ profit maximization, guilds’
profit maximization, and market clearing.
The economy is subject to stochastic disturbances in technology, preferences,
and government policy. Without nominal rigidities, the economy’s real allocations
in competitive equilibrium can be separated from inflation and other dollardenominated variables. Specifically, monetary policy only influences inflation. To
connect real and nominal variables in the model and thereby consider the impact
of monetary policy on the business cycle, we introduce Calvo-style wage and price
setting. That is, nature endows both differentiated goods producers and guilds with
stochastic opportunities to incorporate all available information into their nominal
price choices. Those producers and guilds without such a opportunity must set their
prices according to simple indexing formulas. These two pricing frictions create two
forward-looking Phillips curves, one for prices and another for wages, which form
the core of the new Keynesian approach to monetary policy analysis.
The model economy is stochastic and features complete markets in statecontingent claims. To place these features on a sound footing, we base all shocks on
a general Markovian stochastic process st . Denote the history of this vector from
an initial period 0 through τ with sτ ≡ (s0 , s1 , . . . , sτ ). The support of sτ is Στ , and
the probability density of sτ given st for some t < τ is ( סsτ st ). (The Hebrew letter
ס, pronounced “samekh,” corresponds to the Greek letter σ.) All model shocks are
implicit functions of st , and all endogenous variables are implicit functions of st . We
refer to all such implicit functions as “state-contingent sequences.” We use braces to
denote such a sequence. For example, {Xt } represents the state-contingent sequence
for a generic variable Xt .
1.1
Households
Our model’s households are the ultimate owners of all assets in positive net supply
(the capital stock, differentiated goods producers, and guilds). They provide labor
and divide their current after-tax income (from wages and assets) between current
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consumption, investment in productive capital, and purchases of financial assets,
both those issued by the government and those issued by other households. The
individual household divides its current resources between consumption and the
available vehicles for intertemporal substitution (capital and financial assets) to
maximize a discounted sum of current and expected future felicity.
∞
Et [ ∑ β τ εbt+τ (Ut+τ + εst+τ L (
τ =0
Bt+τ
))]
Pt+τ Rt+τ
with
Ut =
1
(1−γc )
((Ct − %C̄t−1 )(1 − Ht1+γh ))
1 − γc
(1)
The function L(⋅) is strictly increasing, concave, and differentiable everywhere
on [0, ∞). In particular, L′ (0) exists and is finite. Without loss of generality, we
set L′ (0) to one. The argument of L(⋅) equals the real value of government bonds
in the household’s portfolio: their period t + 1 redemption value Bt divided by their
nominal yield Rt expressed in units of the consumption good with the nominal price
index Pt . The time-varying coefficient multiplying this felicity from bond holdings,
εst , is the liquidity preference shock introduced by Fisher (2015). A separate shock
influences the household’s discounting of future utility to the present, εbt . Specifically,
the household discounts a certain utility in t + τ back to t with β τ Et [εbt+τ /εbt ]. In
logarithms, these two preference shocks follow independent autoregressive processes.
ln εbt = (1 − ρb ) ln εb∗ + ρb ln εbt−1 + ηtb , ηtb ∼ N(0, σb )
(2)
ln εst = (1 − ρs ) ln εs∗ + ρs ln εst−1 + ηts , ηts ∼ N(0, σs ).
(3)
A household’s wealth at the beginning of period t consists of its nominal
government bond holdings, Bt , its net holdings of privately-issued financial assets,
and its capital stock Kt−1 . The household chooses a rate of capital utilization ut , and
the capital services resulting from this choice equal ut Kt−1 . The cost of increasing
utilization is higher depreciation. An increasing, convex and differentiable function
δ(U ) gives the capital depreciation rate. We specify this as
δ(u) = δ0 + δ1 (u − u⋆ ) +
δ2
2
(u − u⋆ ) .
2
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A household can augment its capital stock with investment, It . Investment
requires paying adjustment costs of the “i-dot” form introduced by Christiano,
Eichenbaum, and Evans (2005). Also, an investment demand shock alters the
efficiency of investment in augmenting the capital stock. Altogether, if the
household’s investment in the previous period was It−1 , and it purchases It units
of the investment good today, then the stock of capital available in the next period
is
Kt = (1 − δ(ut )) Kt−1 + εit (1 − S (
AK
t−1 It
)) It .
K
At It−1
(4)
In (4), AK
t equals the productivity level of capital goods production, described in
more detail below, and εit is the investment demand shock. In logarithms, this
follows a first-order autoregression with a normally-distributed innovation.
ln εit = (1 − ρi ) ln εi∗ + ρi ln εit−1 + ηti , ηti ∼ N(0, σi )
1.2
(5)
Production
The producers of investment goods use a linear technology to transform the final
good into investment goods. The technological rate of exchange from the final good
to the investment good in period t is AIt . We denote ∆ ln AIt with ωt , which we call the
investment-specific technology shock and which follows first-order autogregression
with normally distributed innovations.
ωt = (1 − ρω )ω⋆ + ρω ωt−1 + ηtω , ηtω ∼ N(0, σω2 )
(6)
Investment goods producers are perfectly competitive.
Final good producers also operate a constant-returns-to-scale technology; which
takes as inputs the products of the differentiated goods producers. To specify this,
let Yit denote the quantity of good i purchased by the representative final good
producer in period t, for i ∈ [0, 1]. The representative final good producer’s output
then equals
Yt ≡ (∫
1
0
1
p
1+λ
t
Yit
1+λpt
di)
.
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With this technology, the elasticity of substitution between any two differentiated
products equals 1 + 1/λpt in period t. Although this is constant across products
within a time period, it varies stochastically over time according to an ARMA(1, 1)
in logarithms.
p
ln λpt = (1 − ρp ) ln λp⋆ + ρp ln λpt−1 − θp ηt−1
+ ηtp , ηtp ∼ N(0, σp )
(7)
Given nominal prices for the intermediate goods Pit , it is a standard exercise to
show that the final goods producers’ marginal cost equals
Pt = (∫
1
0
−
Pit
1
p
λ
t
−λpt
di)
(8)
Just like investment goods firms, the final goods’ producers are perfectly
competitive. Therefore, profit maximization and positive final goods output together
require the competitive output price to equal Pt . Therefore, we can define inflation
of the nominal final good price as πt ≡ ln(Pt /Pt−1 ).
The intermediate goods producers each use the technology
1−α
α
Yit = (Kite ) (AYt Hitd )
− At Φ
(9)
Here, Kite and Hitd are the capital services and labor services used by firm i, and
AYt is the level of neutral technology. Its growth rate, νt ≡ ln(AYt /AYt−1 ), follows a
first-order autogregression.
νt = (1 − ρν ) ν∗ + ρv νt−1 + ηtν , ηtν ∼ N(0, σν ),
(10)
The final term in (9) represents the fixed costs of production. These grow with
α
At ≡ AYt (AIt ) 1−α .
(11)
We demonstrate below that At is the stochastic trend in equilibrium output and
consumption, measured in units of the final good. We denote its growth rate with
zt = νt +
α
ωt
1−α
(12)
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Similarly, define
I
AK
t ≡ At A t
(13)
In the specification of the capital accumulation technology, we labelled AK
t the
“productivity level of capital goods production.” We demonstrate below that this
is indeed the case with the definition in (13).
Each intermediate goods producer chooses prices subject to a Calvo (1983)
pricing scheme. With probability ζp ∈ [0, 1], producer i has the opportunity to
set Pit without constraints. With the complementary probability, Pit is set with the
indexing rule
p
Pit = Pit−1 πt−1
π⋆
ι
1−ιp
.
(14)
In (14), π⋆ is the gross rate of price growth along the steady-state growth path, and
ιp ∈ [0, 1].1
1.3
Labor Markets
Households’ hours worked pass through two intermediaries, guilds and labor packers,
in their transformation into labor services used by the intermediate goods producers.
The guilds take the households’ homogeneous hours as their only input and produce
differentiated labor services. These are then sold to the labor packers, who assemble
the guilds’ services into composite labor services.
The labor packers operate a constant-returns-to-scale technology with a constant
elasticity of substitution between the guilds’ differentiated labor services. For its
specification, let Hit denote the hours of differenziated labor purchased from guild
i at time t by the representative labor packer. Then that packer’s production of
composite labor services, Hts are given by
Hts
1
= (∫ (Hit )
1
1+λw
t
1+λw
t
di)
.
0
As with the final good producer’s technology, an ARMA(1, 1) in logarithms governs
To model firms’ price-setting opportunities as functions of st , define a random variable upt
which is independent over time and uniformly distributed on [0, 1]. Then, firm i gets a pricesetting opportunity if either upt ≥ ζp and i ∈ [upt − ζp , upt ] or if upt < ζp and i ∈ [0, upt ] ∪ [1 + upt − ζp , 1].
1
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the constant elasticity of substitution between any two guilds’ labor services.
w
w
w
w w
2
ln λw
t = (1 − ρw ) ln λ⋆ + ρw ln λt−1 − θw ηt−1 + ηt , ηt ∼ N(0, σw )
(15)
Just as with the final goods producers, we can easily show that the labor packers’
marginal cost equals
1
− λ1w
Wt = (∫ (Wit )
t
−λw
t
di)
.
(16)
0
Here, Wit is the nominal price charged by guild i per hour of differentiated labor.
Since labor packers are perfectly competitive, their profit maximization and positive
output together require that the price of composite labor services equals their
marginal cost.
Each guild produces it’s differentiated labor service using a linear technology
with the household’s hours worked as its only input. A Calvo (1983) pricing
scheme similar to that of the differentiated goods producers constrains their nominal
prices. Guild i has an unconstrained opportunity to choose its nominal price with
probability ζw ∈ [0, 1]. With the complementary probability, Wit is set with an
indexing rule based on πt−1 and last period’s trend growth rate, zt−1 .
Wit = Wit−1 (πt−1 ezt−1 ) w (π⋆ ez⋆ )
ι
1−ιw
.
(17)
α
ω⋆ is the unconditional mean of zt and ιw ∈ [0, 1].
In (17), z⋆ ≡ ν⋆ + 1−α
1.4
Fiscal and Monetary Policy
The model economy hosts two policy authorities, each of which follows exogenouslyspecified rules that receive stochastic disturbances. The fiscal authority issues bonds,
Bt , collects lump-sum taxes Tt , and buys “wasteful” public goods Gt . Its periodby-period budget constraint is
Gt + Bt−1 = Tt +
Bt
.
Rt
(18)
The left-hand side gives the government’s uses of funds, public goods spending and
the retirement of existing debt. The left-hand side gives the sources of funds, taxes
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and the proceeds of new debt issuance at the interest rate Rt . We assume that the
fiscal authority keeps its budget balanced period-by-period, so Bt = 0. Furthermore,
the fiscal authority sets public goods expenditure equal to a stochastic share of
output, expressed in consumption units.
Gt = (1 − 1/gt )Yt ,
(19)
ln gt = (1 − ρg ) ln sg⋆ + ρg ln gt−1 + ηtg , ηtg ∼ N(0, σg2 ).
(20)
with
The monetary authority sets the nominal interest rate on government bonds, Rt .
For this, it employs a Taylor rule with interest-rate smoothing and forward guidance
shocks.
M
j
ln Rt = ρR ln Rt−1 + (1 − ρR ) ln Rtn + ∑ ξt−j
.
(21)
j=0
1
M
The monetary policy disturbances in (21) are ξt0 , ξt−1
, . . . , ξt−M
. The public learns
j
the value of ξt−j in period t − j. The conventional unforecastable shock to current
monetary policy is ξt0 , while for j ≥ 1, these disturbances are forward guidance
shocks. We gather all monetary shocks revealed at time t into the vector ε1t . This is
normally distributed and i.i.d. across time. However, its elements may be correlated
with each other. That is,
ε1t ≡ (ξt0 , ξt1 , . . . , ξtM ) ∼ N(0, Σ1 ).
(22)
The off-diagonal elements of Σ1 are not necessarily zero, so forward-guidance shocks
need not randomly impact expected future monetary policy at two adjacent dates
independently. Current economic circumstances influence Rt through the notional
interest rate, Rtn .
ln Rtn = ln r⋆ + ln πt⋆ +
1
1
φ1
φ2
Et ∑ (ln πt+j − ln πt⋆ ) + Et ∑ (ln Yt+j − ln y ⋆ − ln At+j ) .
4 j=−2
4 j=−2
(23)
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The constant r⋆ equals the real interest rate along a steady-state growth path, and πt⋆
is the central bank’s intermediate target for inflation. We call this the inflation-drift
shock. it follows a first-order autoregression with a normally-distributed innovation.
Its unconditional mean equals π⋆ , the inflation rate on a steady-state growth path.
⋆
ln πt⋆ = (1 − ρπ )π⋆ + ρπ ln πt−1
+ ηtπ , ηtπ ∼ N(0, σπ2 )
(24)
Allowing πt⋆ to change over time enables our model to capture the persistent
decline in inflation from the early 1990s through the early 2000s engineered by
the Greenspan FOMC.
1.5
Other Financial Markets and Equilibrium Definition
All households participate in the market for nominal risk-free government debt.
Additionally, they can buy and sell two classes of privately issued assets without
restriction. The first is one-period nominal risk-free private debt. We denote the
P
value of household’s net holdings of such debt at the beginning of period t with Bt−1
P
and the interest rate on such debt issued in period t maturing in t + 1 with Rt+1
.
The second asset class consists of a complete set of real state-contingent claims. As
of the end of period t, the household’s ownership of securities that pay off one unit
of the aggregate consumption good in period τ if history sτ occurs is Qt (sτ ), and
the nominal price of such a security in the same period is Jt (sτ ).
We define an equilibrium for our economy in the usual way: Households maximize
their utility given all prices, taxes, and dividends from both producers and guilds;
final goods producers and labor packers maximize profits taking their input and
output prices as given; differentiated goods producers and guilds maximize the
market value of their dividend streams taking as given all input and financial-market
prices; differentiated goods producers and guilds produce to satisfy demand at their
posted prices; and otherwise all product, labor, and financial markets clear.
2
First Order Conditions
In this section we present the first-order conditions associated with the optimization
problems that the agents in our model solve.
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2.1
Households
Given initial financial asset holdings holdings, a stock of productive capital,
investment in the previous period (which influences investment adjustment costs),
and the external habit stock; households’ choices of consumption, capital investment,
capital utilization, hours worked, and financial investments maximize utility subject
to the constraints of the capital accumulation and utilization technology and a
sequence of one-period budget constraints. To specify these budget constraints,
denote the nominal wage-per-hour paid by labor guilds to households with Wth , the
nominal rental rate for capital services with Rtk , the nominal price of investment
goods with PtI , and the dividends paid by labor guilds added to those paid by
differentiated good producers with Dt . With this notation, writing the period t
budget constraint with uses of funds on the left and sources of funds on the right
yields
Ct +
P
Bt
BP
Tt Bt−1 Bt−1
W h Ht Rtk ut Kt−1 Dt
PtI It
+
+ Pt +
≤
+
+ t
+
+
Pt
Rt Pt Rt Pt Pt
Pt
Pt
Pt
Pt
Pt
(25)
Denote the Lagrange multiplier on (25) with β t Λ1t , and that on the capital
accumulation constraint in (4) with β t Λ2t . With these definitions, the first-order
conditions for a household’s utility maximization problem are
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Λ1t = εbt ((Ct − %C̄t−1 )(1 − εht Ht1+γh ))
−γc
(1 − εht Ht1+γh )
Wth
−γc
= (1 + γh )εbt ((Ct − %C̄t−1 )(1 − εht Ht1+γh )) (Ct − %C̄t−1 )εht Htγh
Pt
1
Λ1
Λt
εs
Bt
− εbt+q L′ (
) t
= βEt [ t+1 ]
Rt Pt
Rt Pt Rt Pt
Pt+1
Λ1t
Λ1t+1
Λ1t
=
βE
[
]
t
Pt+1
RtP Pt
Λ2t = βE [Λ1t+1
k
Rt+1
ut+1
+ Λ2t+1 (1 − δ(ut+1 ))]
Pt+1
Λ1t Rtk
= Λ2t δ ′ (ut )
Pt
Λ1t = εit Λ2t ((1 − St (⋅)) − St′ (⋅)
it
it−1
)
′
+βEt [εit+1 e(1−γC )zt+1 λ2t+1 St+1
(⋅)
i2t+1
]
i2t
In equilibrium, C̄t = Ct always.
2.2
2.2.1
Goods Sector
Final Goods Producers
The nominal marginal cost of final goods producers equals the right-hand side of
(8). A producer of final goods maximizes profit by shutting down if Pt is less than
this marginal cost and can make an arbitrarily large profit if Pt exceeds it. When
(8) holds, an individual final goods producer’s output is indeterminate.
Final goods producers’ demand for intermediate goods takes the familiar
constant-elasticity form. If we use Yt to denote total final goods output, then the
amount of differentiated good i demanded by final goods producers is
Pit −
Yit = Yt ( )
Pt
p
1+λ
t
p
λ
t
.
Given the choice of a reset price, we wish to calculate the overall price level.
All intermediate goods producers with a price-setting opportunity choose P̃t . The
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remaining producers use the price-indexing rule in (14). The aggregate price level
is given by
1
λp,t −1
Pt = [(1 − ζp )P̃t
+ ζp ((πt−1 ) (π∗ )
1−ιp
ιp
Pt−1 )
1
λp,t −1
λp,t −1
]
where P̃t is the optimal reset price
2.2.2
Intermediate Goods Producers
Intermediate goods producers’ cost minimization reads as follows:
d
e
max Wt Ht,i
+ Rtk Ki,t
e
Ht,i ,Ki,t
1−α
e α
d
)
s.t. Yt,i = εat (Kt,i
) (Ayt Ht,i
− At Φ
We get the following optimal capital-labor ratio.
α Wt (Kite )s
=
d
1 − α Rtk
Ht,i
Notice how for each firm, the idiosyncratic capital to labor ratio is not a function of
any firm-specific component. Hence, each firm has the same capital to labor ratio.
In equilibrium,
Kte = ut Kt−1
To find the marginal cost, we differentiate the variable part of production with
respect to output, and substitute in the capital-labor ratio.
−(1−α)
M Ct,i = (εat )−1 (Ayt )
Wt1−α Rtkα α−α (1 − α)−(1−α)
Again, notice that each firm as the same marginal cost.
Given cost minimization, a differentiated goods producer with an opportunity to
adjust its nominal price does so to maximize the present-discounted value of profits
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earned until the next opportunity to adjust prices arrives. Formally,
∞
max Et ∑ ζps
P̃t,i
s=0
β s Λ1t+s Pt
y
[P̃t,i Xt,s
− M Ct+s ] Yt+s,i
1
Λt Pt+s
λp,t
1−λp,t
y P̃t,i
)
s.t. Yt (i) = (Xt,s
Yt
Pt
⎧
⎪
∶s=0
⎪ 1
y
where Xt,s = ⎨ s ιp
1−ιp
⎪
∶ s = 1, . . . , ∞
⎪
⎩ ∏l=1 πt+l−1 π∗
⎫
⎪
⎪
⎬
⎪
⎪
⎭
This problem leads to the following price-setting equation for firms that are allowed
to reoptimize their price:
0=
⎡
s 1
⎢
s β Λt+s Pt
Yit+s ⎢⎢λp,t+s M Ct+s
Et ∑ ζp 1
Λt Pt+s
⎢
s=0
∞
⎣
⎤
⎥
− Xt,s P̃it ⎥⎥
⎥
⎦
It can be shown that the producers that are allowed to reoptimize choose the
same price. So henceforth, P̃it = P̃t .
2.2.3
Investment Goods Producers
Characterizing the profit-maximizing choices of investment goods and final goods
producers is straightforward. If PtI > Pt /AIt , then each investment goods producer
can make infinite profit by choosing an arbitrarily large output. On the other
hand, if PtI < Pt /AIt , then investment goods producers maximize profits with zero
production. Finally, their profit-maximizing production is indeterminate when
PtI = Pt /AIt .
(26)
−1
The relative price of investment to consumption is equal to (AIt ) . Making
this substitution into the household f.o.c and noting that Pt YtI is an intermediate
input that will not be reflected in the aggregate resource constraint, it suffices to
−1
substitute the relative price (AIt ) in the constraint for the household.
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2.3
Labor Sector
2.3.1
Labor Packers
The labor packers choose the the labor inputs supplied by guilds, pack them into a
composite labor service to be sold to the intermediate goods producers. Formally,
labor packers’ problem reads as follows:
max
Wt Hts − ∫
s
Ht ,Hit
s.t. [∫
1
0
1
1
1+λw,t
Hit
Wit Hit di
0
1+λw,t
= Hts
di]
We obtain the following labor demand equation for guild i:
Wit −
)
Hit = (
Wt
1+λw,t
λw,t
Ht
(27)
As for the goods sector, we can show that aggregate wage is given by the following
equation:
−λ1
Wt = [(1 − ζw )W̃t
w,t
+ ζw ((ezt−1 πt−1 ) w (π∗ ez∗ )
ι
1−ιw
−λ1
Wt−1 )
−λw,t
w,t
]
where W̃ is the optimal reset wage for guilds.
2.3.2
Guilds
Each guild with an opportunity to set its nominal price does so to maximize the
current value of the stream of dividends returned to the household. Formally, their
problem reads
∞
max Et ∑ ζws (
W̃it
s=0
s.t. Hit+s =
β s Λ1t+s Pt
l
h
] Hit+s
) [Xt+s
W̃it − Wt+s
Λ1t Pt+s
l
W̃it ⎞
⎛ Xt,s
−
1+λw,t+s
λw,t+s
Ht+s
⎝ Wt+s ⎠
⎧
⎪
∶s=0
⎪ 1
l
where Xt,s = ⎨ s
1−ι
w
At+j−1
⎪
(πeγ )ιw ∶ s = 1, . . . , ∞
⎪
⎩ ∏j=1 (πt+j−1 At+j−2 )
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⎫
⎪
⎪
⎬
⎪
⎪
⎭
Authorized for public release by the FOMC Secretariat on 1/12/2024
W̃t is the optimal reset wage. This optimal wage is chosen by the guilds who are
allowed, with probability ζw , to change their prices in a given period. Also, we index
the nominal wage inflation rate with ιw .
This maximization problem gives a wage-setting equation that reads as follows:
∞
0 = Et ∑ ζws
s=0
1
β s Λ1t+s Pt
h
l
((1 + λw,t+s )Wt+s
Hit+s
− Xt,s
W̃it )
1
Λt Pt+s
λw,t+s
It can be shown that the guilds that are allowed to reoptimize choose the same wage.
So henceforth, W̃it = W̃t .
3
Detrending
To remove nominal and real trends, we deflate nominal variables by their matching
price deflators, and we detrend any resulting real variables influenced permanently
by technological change. All scaled versions of variables are the lower-case
counterparts.
ct =
kt =
wt =
p̃t =
yt =
rtk =
Ct
At
Kt
At AIt
Wt
A t Pt
P̃t
Pt
Yt
At
Rtk AIt
Pt
it =
kte =
w̃t =
πt =
mct =
wth =
λ1t = Λ1t Aγt C
It
At AIt
Kte
At AIt
W̃t
At P t
Pt
Pt−1
M Ct
Pt
Wth
At P t
λ2t = Λ2t Aγt C AIt
εst = Aγt C εst
3.1
Detrended Equations
The detrended equations describing our model are listed in the following sections.
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Households’ FOC
−γc
ct−1
h 1+γh
h
(1
)]
(1 − εht h1+γ
)
)
−
ε
h
t t
t
ezt
−γc
ct−1
ct−1
(1+σ )
λ1t wth = (1 + γh )εbt [(ct − % zt ) (1 − εht ht h )] (ct − % zt ) εht hγt h
e
e
1
1
−γ
z
t+1
C
λ e
λt
]
= βEt [ t+1
P
πt+1
Rt
λ1
λ1t
εb εs
− L′ (0) t t = βEt t+1 e−zt+1 γC
Rt
Rt
πt+1
i2
it
′
) + βEt [εit+1 e(1−γC )zt+1 λ2t+1 St+1
(⋅) t+1
]
λ1t = εit λ2t ((1 − St (⋅)) − St′ (⋅)
it−1
i2t
λ1t = εbt [(ct − %
k
λ2t = βEt [e−γC zt+1 −ωt+1 (λ1t+1 rt+1
ut+1 + λ2t+1 (1 − δ(ut+1 )))]
λ1t rtk = λ2t δ ′ (ut )
kt = (1 − δ(ut )) kt−1 e−zt −ωt + εit (1 − S(⋅)) it
kte = ut kt−1 e−zt −ωt
Final Goods Price Index
1
1−λp,t
1 = [(1 − ζp )p̃t
1−λp,t
1
ιp
∗(1−ιp ) −1 1−λp,t
+ ζp (πt−1 π
πt )
]
Intermediate Goods Firms: Capital-Labor Ratio
kte
α wt
=
d
ht 1 − α rtk
Intermediate Goods Firms: Real Marginal Costs
α
mct =
wt1−α (rtk )
εat αα (1 − α)1−α
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Intermediate Goods Firms: Price-Setting Equation
∞
0 =Et ∑ ζps β s λ1t+s
s=0
ỹt,t+s
At+s 1−γC
p
[λp,t+s mct+s − X̃t,s
p̃t ]
(
)
λp,t+s − 1 At
where
p
X̃t,s
⎧
⎪
⎪
⎪ 1
1−ιp
ι
= ⎨ ∏sj=1 πt+j−1
π∗p
⎪
⎪
s
⎪
⎩ ∏j=1 πt+j
⎫
⎪
⎪
⎪
⎬
⎪
∶ s = 1, . . . , ∞ ⎪
⎪
⎭
∶s=0
ỹt,t+s denotes the time t + j output sold by the producers that have optimized at
time t the last time they have reoptimized. Since it can be shown that optimizing
producers all choose the same price, then we do not have to carry the i-subscript.
Labor Packers: Aggregate Wage Index
−λ1
wt = [(1 − ζw )w̃t
w,t
ιw zt−1 −zt (1−ιw )z∗
+ ζw (e
e
−λw,t
−λ1
−1 1−ιw
ι
w,t
πt−1 πt π∗ wt−1 )
]
Guilds: Wage-Setting Equation
∞
0 =Et ∑ ζws βλ1t+s (
s=0
At+s 1−γC h̃t,t+s
l
h
((1 + λw,t+s )wt+s
− X̃t,s
w̃t )
)
At
λw,t+s
where
l
X̃t,s
⎧
⎪
⎪
⎪ 1
= ⎨ ∏sj=1 (πt+j−1 ezt+j−1 )1−ιw (πγ)ιw
⎪
z
⎪
⎪
∏sj=1 πt+j e t+j
⎩
⎫
⎪
⎪
⎪
⎬
∶ s = 1, . . . , ∞ ⎪
⎪
⎪
⎭
∶s=0
h̃t,t+s denotes the time t + j labor supplied by the guild that have optimized at time
t the last time they have reoptimized. Since it can be shown that optimizing guilds
all choose the same wage, then we do not have to carry the i-subscript.
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Monetary Authority
ψ1
ψ2 ⎤1−ρR
⎡
⎢
1
M
1
4
4 ⎥
y
π
t+j
t+j
⎥
ρR ⎢
⎢r∗ πt∗ ( ∏ ∗ ) ( ∏ ∗ ) ⎥
Rt = Rt−1
ξt−j,j
∏
⎥
⎢
j=−2 y
j=0
j=−2 πt
⎥
⎢
⎦
⎣
The Aggregate Resource Constraint
yt
=ct + it
gt
Production Function
yt =εat (kte ) (hdt )1−α − Φ
α
Labor Market Clearing Condition
ht = hdt
4
Steady State
We normalize most shocks and the utilization rate:
u⋆ =1
εi =1
εa =1
εb =1
Next, we set the following restriction on adjustment costs:
S(⋅∗ ) ≡ 0
S ′ (⋅∗ ) ≡ 0
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4.1
Prices and Interest Rates
Given β, z∗ , γC , and π∗ , we can solve for the steady-state nominal interest rate on
private bonds R∗P by using the FOC on private bonds:
R∗P =
π∗
(βe−γC z∗ )
(28)
From the definition of δ(u), we have
δ(1) =δ0
δ ′ (1) =δ1 .
Next, given ω∗ , δ0 , and the above, we can solve for the real return on capital r∗k
using the FOC on capital:
r∗k =
4.2
eγC z∗ +ω∗
− (1 − δ0 )
β
(29)
Ratios
Moving to the production side, we can use the aggregate price equation to solve for
p̃∗ :
p̃∗ = 1
Using this result and given λp,∗ , we can use the price Phillips curve to solve for mc∗ :
mc∗ =
1
1 + λp,∗
(30)
Given values for α and εa∗ , we can use the marginal cost equation to solve for
w∗ :
1
w∗ = (mc∗ αα (1 − α)1−α (r∗k )−α ) 1−α
(31)
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The definition of effective capital gives us a value for k∗e in terms of k∗ :
k∗e = k∗ e−z∗ −ω∗
Calculating y∗ using the labor share of output 1 − α:
y∗ =
w∗ h∗
1−α
Using capital shares based off our value of α, we can calculate the output to
capital ratio as follows:
y∗ r∗k
=
k∗e α
y∗ −z∗ −ω∗ r∗k
=e
k∗
α
Using the capital accumulation equation, we can get a value for
i∗
k∗
i∗
= 1 − (1 − δ0 )e−z∗ −ω∗
k∗
Using the resource constraint, we can get
c∗
k∗ :
y∗
i∗
c∗
=
g −
k∗ k∗ s⋆ k∗
These ratios will give us the remaining steady-state levels and ratios:
k∗ =y∗ (
c∗ =
4.3
y∗ −1
)
k∗
i∗ =
c∗
k∗
k∗
i∗
k∗
k∗
g∗ =gy y∗
Liquidity Premium
Using the aggregate wage equation, we can get the following for w̃∗ :
w̃∗ = w∗
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Combining this result with the wage Phillips curve, we get the following:
w∗h =
w∗
1 + λw,∗
We can use the FOC for consumption and the labor supply to pin down εh and
λ1∗
% −γc
(1+γ )
ε [c∗ (1 − z )] (1 − εh h∗ h ) − λ1∗ = 0
e
b
(1−γ )
−(1 + γh )εb c∗ c (1 −
−γc
% (1−γc )
h (1+γh )
εh hγ∗h + λ1∗ w∗h = 0
)
(1
−
ε
h
)
∗
z
ε
Finally, the government bond rate is calculated from
λ1∗ − εb∗ εs∗ = βR∗
π∗
π∗
− εb∗ εs∗ −γ z 1 = R∗
−γ
z
C
βe
βe C λ∗
´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶
λ1∗ −γC z
e
π∗
R∗P
Noting that R∗P =
π∗
βe−γC z
we can write
R∗P − R∗ εb∗ εs∗
= 1 .
R∗P
λ∗
This is the liquidity premium in steady state.
5
Log Linearization
Hatted variables refer to log deviations from steady-state (x̂ = ln ( xx∗t )):
ln εjt = ρj ln εjt−1 + ηtj
In the cases of zt , ωt , and νt , we have that x̂ = xt − x∗ as these variables are already
in logs.
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Households’ First Order Conditions
%
1
ez
(ĉt−1 − ẑt )
% ĉt + γc
1 − ez
1 − e%z
%
1 − γc
ez
1
h
b
h
ĉt + (1 − γc )
(ĉt−1 − ẑt )
λ̂t + ŵt − ε̂t − ε̂t −
1 − e%z
1 − e%z
ε̂bt − λ̂1t − γc
h
εh h1+γ
∗
) ĥt = 0
h 2
)
(1 − εh h1+γ
∗
R P − R∗
R∗
λ̂1t = ∗ P (ε̂st + ε̂bt ) + P (R̂t + Et [(λ̂1t+1 − π̂t+1 − γC ẑt+1 ])
R∗
R∗
(32)
(33)
− (γh + γc (1 + γh )
(34)
λ̂1t = Et [λ̂1t+1 − γC ẑt+1 + R̂t − π̂t+1 ]
(35)
λ̂1t = (ln εit + λ̂2t ) − S ′′ (ı̂t − ı̂t−1 ) + βe(1−γC )γ S ′′ Et (ı̂t+1 − ı̂t )
(36)
k
+ ût+1 )] +
λ2∗ λ̂2t = βe−γC z∗ −ω∗ [λ1∗ u∗ r∗k Et (−γC ẑt+1 − ω̂t+1 + λ̂1t+1 + r̂t+1
(37)
+ βe−γC z∗ −ω∗ [(1 − δ0 )λ2∗ Et (−γC ẑt+1 − ω̂t+1 + λ̂2t+1 ) − λ2∗ δ1 u∗ Et ût+1 ]
δ2
λ̂1t = λ̂2t + u∗ ût − r̂tk
δ1
εi∗ i∗
εi i ∗
k̂t = (1 −
) (k̂t−1 − ẑt − ω̂t ) + ∗ (ε̂it + ı̂t ) − δ1 u∗ e−z∗ −ω∗ ût
k∗
k∗
k̂te = ût + k̂t−1 − ẑt − ω̂t
(38)
(39)
(40)
Capital-Labor Ratio
k̂te = ŵt − r̂tk + ĥdt
(41)
Real Marginal Costs
̂ t = (1 − α) ŵt + αr̂tk − ε̂at
mc
(42)
The New Keynesian Phillips Curve for Inflation
(1 − βζp e(1−γC )z∗ )(1 − ζp )
λp,∗
̂ t] +
[
λ̂p,t + mc
(1−γ
)z
∗
C
1 + λp,∗
(1 + βιp e
)ζp
ιp
βe(1−γC )z∗
+
π̂
+
Et π̂t+1
t−1
1 + βιp e(1−γC )z∗
1 + βιp e(1−γC )z∗
π̂t =
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Wage Mark-Up
h
µ̂w
t = ŵt − ŵt
(44)
The New Keynesian Phillips Curve for Wages
ŵt =
1
1 + βe(1−γC
ŵ +
)z∗ t−1
βe(1−γC )z∗
βe(1−γC )z∗
ŵ
+
(Et π̂t+1 + Et ẑt+1 )+
t+1
1 + βe(1−γC )z∗
1 + βe(1−γC )z∗
(45)
ιw
1 + ιw βe(1−γC )z∗
(π̂
+
ẑ
)
−
(π̂t + ẑt )+
t−1
t−1
1 + βe(1−γC )z∗
1 + βe(1−γC )z∗
λw,∗
1 − βζw e(1−γC )z∗ 1 − ζw
λ̂w,t − µ̂w
[
t ]
(1−γ
)z
ζw
1 + λw,∗
1 + βe C ∗
The Aggregate Resource Constraint
y∗
c∗
i∗
(ŷt − ĝt ) =
ĉt +
ı̂t
g∗
c ∗ + i∗
c∗ + i ∗
(46)
The Production Function
ŷt =
1
(ln εat + αk̂te + (1 − α) ĥdt )
mc∗
(47)
Labor Market Clearing Condition
ĥt = ĥdt
(48)
Monetary Authority’s Reaction Function
R̂t = ρR R̂t−1 + (1 − ρR ) [(1 − ψ1 ) π̂t∗ +
M
ψ2 1
ψ1 1
( ∑ π̂t+j ) +
( ∑ ŷt+j )] + ∑ ξˆt−j,j
4 j=−2
4 j=−2
j=0
(49)
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6
Measurement
6.1
National Income Accounts
The model economy’s basic structure, with the representative household consuming
a single good and accumulating capital using a different good, differs in some
important ways from the accounting conventions of the Bureau of Economic Analysis
(BEA) underlying the National Income and Product Accounts (NIPA). In particular
1. The BEA treats household purchases of long-lived goods inconsistently. If
classifies purchases of residential structures as investment and treats the service
flow from their stock as part of Personal Consumption Expenditures (PCE) on
services. The BEA classifies households purchases of all other durable goods
as consumption expenditures. No service flow from the stock of household
durables enters measures of current consumption. In the model, all long-lived
investments add to the productive capital stock.
2. The BEA treats all government purchases as government consumption.
However, government at all levels makes purchases of investment goods on
behalf of the populace. In the model, these should be treated as additions to
the single stock of productive capital.
3. The BEA sums PCE and private expenditures on productive capital (Business
Fixed Investment and Residential Investment), with government spending,
inventory investment, and net exports to create Gross Domestic Product. The
model features only the first three of these.
To bridge these differences, we create four model consistent NIPA measures from
the BEA NIPA data.
1. Model-consistent GDP. Since the model’s capital stock includes both the stock
of household durable goods and the stock of government-purchased capital, a
model-consistent GDP series should include the value of both stocks’ service
flows. To construct these, we followed a five-step procedure.
(a) We begin by estimating a constant (by assumption) service-flow rate by
dividing the nominal value of housing services from NIPA Table 2.4.5
by the beginning-of-year value of the residential housing stock from the
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BEA’s Fixed Asset Table 1.1. We use annual data and average from
1947 through 2014. The resulting estimate is 0.096. That is, the annual
value of housing services equals approximately 10 percent of the housing
stock’s value each year.
(b) In the second step, we estimate estimate constant (by assumption) depreciation rates for residential structures, durable goods, and government
capital. We constructed these by first dividing observations of value lost
to depreciation over a calendar year by the end-of-year stocks. Both
variables were taken from the BEA’s Fixed Asset Tables. (Table 1.1 for
the stocks and Table 1.3 for the deprecation values.) We then averaged
these ratios from 1947 through 2014. The resulting estimates are 0.021,
0.194, and 0.044 for the three durable stocks.
(c) In the third step, we calculated the average rates of real price depreciation
for the three stocks. For this, we began with the nominal values and
implicit deflators for PCE Nondurable Goods and PCE Services from
NIPA Table 1.2. We used these series and the Fisher-ideal formula to
produce a chain-weighted implicit deflator for PCE Nondurable Goods
and Services. Then, we calculated the price for each of the three
durable good’s stocks in consumption units as the ratio of the implicit
deflator taken from Fixed Asset Table 1.2 to this deflator. Finally, we
calculated average growth rates for these series from 1947 through 2014.
The resulting estimates equal 0.0029, −0.0223, and 0.0146 for residential
housing, household durable goods, and government-purchased capital.
(d) The fourth combines the previous steps’ calculations to estimate constant
(by assumption) service-flow rates for household durable goods and
government-purchased capital. To implement this, we assumed that all
three stocks yield the same financial return along a steady-state growth
path. These returns sum the per-unit service flow with the appropriately
depreciated value of the initial investment. This delivers two equations
in two unknowns, the two unknown service-flow rates. The resulting
estimates are 0.29 and 0.12 for household durable goods and governmentpurchased capital.
(e) The fifth and final step uses the annual service-flow rates to calculate real
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and nominal service flows from the real and nominal stocks of durable
goods and government-purchased capital reported in Fixed Asset Table
1.1. This delivers an annual series. Since the stocks are measured as of
the end of the calendar year, we interpret these as the service flow values
in the next year’s first quarter. We create quarterly data by linearly
interpollating between these values.
With these real and nominal service flow series in hand, we create nominal
model-consistent GDP by summing the BEA’s definition of nominal GDP
with the nominal values of the two service flows. We create the analogous
series for model-consistent real GDP by applying the Fisher ideal formula to
the nominal values and price indices for these three components.
2. Model-consistent Investment. The nominal version of this series sums nominal
Business Fixed Investment, Residential Investment, PCE Durable Goods, and
government investment expenditures. The first three of these come from NIPA
Table 1.1.5, while government investment expenditures sums Federal Defense,
Federal Nondefense, and State and Local expenditures from NIPA Table 1.5.5.
We construct the analogous series for real Model-consistent Investment by
combining these series with their real chain-weighted counterparts found in
NIPA Tables 1.1.3 and 1.5.3 using the Fisher ideal formula. By construction,
this produces an implicit deflator for Model-consistent investment as well.
3. Model-consistent Consumption. The nominal version of this series sums
nominal PCE Nondurable Goods, PCE Services, and the series for nominal
services from the durable goods stock. The first two of thse come from
NIPA Table 1.1.5. We construct the analogous series for real Modelconsistent consumption by combining these series with their real chainweighted counterparts using the Fisher ideal formula. The two real PCE series
come from NIPA Table 1.1.3. Again, this produces an implicit deflator for
Model-consistent consumption as a by-product.
4. Model-consistent Government Purchases. Conceptually, the model’s measure
of Government Purchases includes all expenditures not otherwise classified as
Investment or Consumption: Inventory Investment, Net Exports, and actual
Government Purchases. We construct the nominal version of this series simply
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by subtracting nominal Model-consistent Investment and Consumption from
nominal Model-consistent GDP. We calculate the analogous real series using
“chain subtraction.” This applies the Fisher ideal formula to Model-consistent
GDP and the negatives of Model-consistent Consumption and Investment.
Our empirical analysis requires us to compare model-consistent series measured
from the NIPA data with their counterparts from the model’s solution. To do this,
we begin by solving the log-linearized system above, and then we feed the model
specific paths for all exogenous shocks starting from a particular initial condition.
for a given such simulation, the growth rates of Model-consistent Consumption and
Investment equal
∆ ln Ct = z∗ + ∆ĉt + zt and
∆ ln It = z∗ + ω∗ + ∆ît + zt + ωt
The measurement of GDP growth in the model is substantially more complicated,
because the variables Yt and yt denote model output in consumption units. In
contrast, we mimic the BEA by using a chain-weighted Fisher ideal index to measure
model-consistent GDP. Therefore, we construct an analogus chain-weighted GDP
index from model data. Since such an ideal index is invariant to the units with
which nominal prices are measured, we can normalize the price of consumption to
equal one and employ the prices of investment goods and government purchases
relative to current consumption. Our model identifies the first of these relative
prices as with investment-specific technology. However, the model characterizes
only government purchases in consumption units, because private agents do not
care about their division into “real” purchases and their relative price. For this
reason, we use a simple autoregression to characterize the evolution of the price of
government services in consumption units. Denote this price in quarter t with Ptg .
We construct this for the US economy by dividing the Fisher-ideal price index for
model-consistent government purchases by that for model-consistent consumption.
Then, our model for its evolution is
g
g
g
g
t
ln(Ptg /Pt−1
) = µg + θgg1 ln(Pt−1
/Pt−2
) + θgg2 ln(Pt−2
/Pt−3
) + εgg
t .
(50)
g
2
Here, εgg
t ∼ N(0, σgg ). Given an arbitrary normalization of Pt to one for some time
period, simulations from (??) can be used to construct simulated values of Ptg for
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all other time periods. With these and a simulation from the model of all other
variables in hand, we can calculate the simulation’s values for Fisher ideal GDP
growth using
Qt
≡
Qt−1
√
Q̇Pt Q̇Lt ,
(51)
where the Paasche and Laspeyres indices of quantity growth are
Ct + PtI It + PtG (Gt /PtG )
and
g
)
Ct−1 + PtI It−1 + PtG (Gt−1 /Pt−1
I
G
Ct + Pt−1
It + Pt−1
(Gt /PtG )
≡
.
I
G
G
Ct−1 + Pt−1
It−1 + Pt−1
(Gt−1 /Pt−1
)
Q̇Pt ≡
(52)
Q̇Lt
(53)
In both (52) and (53), PtI is the relative price of investment to consumption. In
equilibrium, this always equals AIt .
The above gives a complete recipe for simulating the growth of model-consistent
real GDP growth. However, we also embody its insights into our estimation with a
log-linear approximation. For this, we start by removing stochastic trends from all
variables in (52) and (53), and we proceed by taking a log-linear approximation of
the resulting expression. Details are available from the authors upon request.
6.2
Hours Worked Measurement
Empirical work using DSGE models like our own typically measure labor input with
hours worked per capita, constructed directly from BLS measures of hours worked
and the civilian non-institutional population over age 16. However, this measure
corresponds poorly with business cycle models because it contains underlying low
frequency variation. This fact led us to construct a new measure of hours for the
model using labor market trends produced for the FRB/US model and for the
Chicago Fed’s in-house labor market analysis.
We begin with a multiplicative decomposition of hours worked per capita into
hours per worker, the employment rate of those in the labor force, and the laborforce participation rate. The BLS provides CPS-based measures of the last two rates
for the US as a whole. However, its measure of hours per worker comes from the
Establishment Survey and covers only the private business sector. If we use hours
per worker in the business sector to approximate hours per worker in the economy
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as a whole, then we can measure hours per capita as
Ht HtE EtC LC
t
=
.
C
Pt EtE LC
P
t
t
Here, Ht and Pt equal total hours worked and the total population, HtE /EtE equals
hours per worker measured with the Establishment survey, EtC /LC
t equals one
C
C
minus the CPS based unemployment rate, and Lt /Pt equals the CPS based laborforce participation rate. Our measure of model-relevant hours worked deflates each
component on the right-hand side by an exogenously measured trend. The trend for
the unemployment rate comes from the Chicago Fed’s Microeconomics team, while
those for hours per worker and labor-force participation come from the FRB/US
model files.
6.3
Inflation
Our empirical analysis compares model predictions of price inflation, wage inflation,
inflation in the price of investment goods relative to consumption goods, and
inflation expectations with their observed values from the U.S. economy. We
describe our implementations of these comparisons sequentially below.
6.3.1
Price Inflation
Our model directly characterizes the inflation rate for Model-consistent Consumption. In principle, this is close to the FOMC’s preferred inflation rate, that for
the implicit deflator of PCE. However, in practice the match between the two
inflation rates is poor. In the data, short-run movements in food and energy prices
substantially influences the short-run evolution of PCE inflation. Our model lacks
such a volatile sector, so if we ask it to match observed short-run inflation dynamics,
it will attribute those to transitory shocks to intermediate goods’ producers’ desired
markups driven by λpt .
To avoid this outcome, we adopt a different strategy for matching model and data
inflation rates, which follows that of Justiniano, Primiceri, and Tambalotti (2013).
This relates three observable inflation rates – core CPI inflation, core PCE inflation,
and market-based PCE inflation – to Model-consistent consumption inflation using
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auxiliary observation equations. For core PCE inflation, this equation is
πtp1 = π∗ + π∗p1 + β1p1 π̂t + β2p1 πtD + εp1
t
(54)
In (54) as elsewhere, π∗ equals the long-run inflation rate. The constant π∗1 is
an adjustment to this long-run inflation rate which accounts for possible long-run
differences between realized inflation and the FOMC’s goal of π⋆ . The right-hand
side’s inflation rates, π̂t and πtD equal Model-consistent consumption inflation and
PCE Durables inflation. We refer to the coefficients multiplying them, β1p1 and
β2p1 , as the inflation loadings. We include PCE Durables inflation on the righthand side of (54) because the principle adjustment required to transform Modelconsistent inflation into core PCE inflation is the replacement of the price index
for durable goods services with that for durable goods purchases. The disturbance
term εp1
t follows a first-order autoregression with autocorrelation ϕp1 and normally
distributed innovations with mean zero and standard deviation σp1 .
The other two observed inflation measures, market-based PCE inflation and core
CPI inflation, have identically specified observation equations. We use p2 and p3 in
superscripts to denote these equations parameters and error terms, and we use the
same expressions as subscripts to denote the parameters governing the evolution of
p2
p3
their error terms. We assume that the error terms εp1
t , εt , and εt are independent
of each other at all leads and lags.
To produce forecasts of inflation with these these three observation equations, we
must forecast their right-hand side variables. The model itself gives forecasts of π̂t .
The forecasts of durable goods inflation come from a second-order autoregression.
D
πtD = θ0D + θ1D πt−1
+ εD
t
(55)
Its innovation is normally distributed and serially uncorrelated with standard
deviation σD .
6.3.2
Wage Inflation
Although observed wage inflation does not feature the same short-run variability
as does price inflation, it does include the influences of persistent demographic
labor-market trends which we removed ex ante from our measure of hours worked.
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Therefore, we follow the same general strategy of relating observed measures of wage
inflation to the model’s predicted wage inflation with a error-augmented observation
equation. For this, we employ two measures of compensation per hour, Earnings per
Hour and Total Compensation per Hour. In parallel with our notation for inflation
measures, we use w1 and w2 to denote these two wage measures of wage inflation.
The observation equation for Earnings per Hour is
πtw1 = z∗ + π∗w1 + β w1 π̂tw + εw1
t
(56)
Just as with the price inflation measurement errors, εw1
follows a first-order
t
autoregression with autocorrelation ϕw1 and innovation standard deviation σw1 . The
observation equation for Total Compensation per Hour is analogous to (56).
6.3.3
Relative Price Inflation
To empirically ground investment-specific technological change in the model, we use
an error-augmented observation equation to relate the relative price of investment
to consumption, both model-consistent measures constructed from NIPA and Fixed
Asset tables as described above, with the model’s growth rate of the rate of
technological transformation between these two goods, ωt .
C/I
πt
= ωt + ε t
C/I
Here, we use the superscript C/I to indicate that the variables characterize the price
C/I
of Consumption relative to Investment. The measurement error εt follows a firstorder autoregression with autocorrelation ϕC/I and normally-distributed innovations
with standard deviation σC/I .
6.3.4
Inflation Expectations
We also discipline our model’s inferences about the state of the economy by
comparing expectations of one-yea and 10-year inflation from the Survey of
Professional Forecasters with the analogous expectations from our model. Just
as with all of the other inflation measures, we allow these two sets of expectations
to differ from each other by including serially correlated measurement errors. The
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observation equations are
πte4
=
π∗ + π∗e4
1 4
+ ∑ Et [π̂t+i ] + εe4
t
4 i=1
πte40 = π∗ + π∗e40 +
1 40
∑ Et [π̂t+i ] + εe40
t
40 i=1
The two measurement errors follow mutually-independent first-order autoregressions
with autocorrelations ϕe4 and ϕe40 and innovation standard deviations σe4 and σe40 .
6.4
Interest Rates and Monetary Policy Shocks
Since our model features forward guidance shocks, it has non-trivial implications
for the current policy rate as well as for expected future policy rates. We use two
distinct but complementary approaches to disciplining the parameters governing
their realizations, the elements of Σ1 , using data. The first method compares the
model’s monetary policy shocks to high-frequency interest-rate innovations informed
by event studies, such as that of Gürkaynak, Sack, and Swanson (2005). Those
authors applied a factor structure to innovations in implied expected interest rates
from futures prices around FOMC policy announcement dates. Specifically, they
show that the vector of M implied interest rate changes following an FOMC policy
announcement, ∆r, can be written as
∆r = Λf + η
Where f is a 2 × 1 vector of factors, Λ is a M × 2 matrix of factor loadings, and
η is an M × 1 vector of mutually independent shocks. Denoting the 2 × 2 diagonal
variance covariance matrix of f with Σf and the M ×M diagonal variance-covariance
matrix of η with Ψ, we can express the observed variance-covariance matrix of ∆r
as ΛΣf Λ′ + Ψ.
Our model has implications for this same variance covariance matrix. For this,
use the model’s solution to express the changes in current and future expected
interest rates following monetary policy shocks as ∆r = Γ1 ε1 . Here, ε1 is the vector
which collects the current monetary policy shock with M − 1 forward guidance
shocks, and Γ1 is an M × M matrix. In general, Γ1 does not simply equal the
identity matrix, because current and future inflation and output gaps respond to the
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monetary policy shocks and thereby influence future monetary policy “indirectly”
through the interest rate rule. Given this solution for ∆r, we can calculate its
variance-covariance matrix as Γ1 Σ1 Γ′1 . Equating these two expressions and solving
for Σ1 yields
′
′−1
Σ1 = Γ−1
1 (ΛΣf Λ + Ψ) Γ1 .
The second approach to disciplining Σ1 is more traditional: directly compare
quarterly observations of the current policy rate and expected future interest rates
– from market prices, surveys of market participants, or both – with their implied
values from the model given a particular realization of the vector of monetary policy
shocks. We use both methods in the estimation procedure described below.
7
Calibration and Bayesian Estimation
As we noted in the introduction, we follow a two-stage approach to the estimation
of our model’s parameters. In a calibration stage, we set the values of selected
parameters so that the model has empirically-sensible implications for long-run
averages from the U.S. economy. In this stage, we also enforce several normalizations
and a judgemental restriction on one of the measurement error variances. In
the second stage, we estimate the model’s remaining parameters using standard
Bayesian methods.
We employ standard prior distributions, but those governing monetary policy
shocks deserve further elaboration. Our estimation requires the variance-covariance
matrix of monetary policy shocks to be consistent with the factor-structure of
interest rate innovations used by Gürkaynak, Sack, and Swanson (2005), as described
above. Therefore, we parameterize Σ1 in terms of Λ, Σf , Ψ, and the model
parameters which influence Γ1 . We then center our priors for Λ, Σf , and Ψ at their
estimates from event-studies. However, we do not require our estimates to equal
their prior values. Our Bayesian estimation procedure employs quarterly data on
expected future interest rates, the posterior likelihood function includes Λ, Σf , and
Ψ as free parameters. It is well known that Λ and Σf are not separately identified,
so we impose two scale normalizations and one rotation normalization on Λ. The
rotation normalization requires that the first factor, which we label “Factor A”, is
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the only factor influence the current policy rate. That is, the second factor, “Factor
B” influences only future policy rates. Gürkaynak, Sack, and Swanson (2005) call
Factors A and B the “target” and “path” factors.
Our estimation’s sample period begins in the first quarter of 1993 and ends in the
fourth quarter of 2016. Of course, the FOMC substantially changed its operating
procedures in the aftermath of its persistent stay at the Zero Lower Bound, so it
would be unwise to imagine the data from this entire period being generated from
our model with time-invariant parameters. For this reason, we estimate the model
twice. For the first sample, which runs from 1993Q1 through 2008Q3, we estimate
all model parameters while allowing for four quarters of forward guidance. For
the second sample, we estimate the parameters governing monetary policy shocks
allowing for ten quarters of forward guidance, adjust the average rate of Hicksneutral productivity growth to bring potential GDP growth rate from its first-sample
value of 3 percent down to 2 percent, and hold all other model parameters fixed at
their first-sample posterior-mode values.
We report the results of our two-stage two-sample estimation in a series of tables.
Table 1 reports our most notable calibration targets. The long-run policy rate equals
1.1 percent on a quarterly basis. We target a two percent growth rate of per capita
GDP. Given an average population growth rate of one percent per year, this implies
that our potential GDP growth rate equals three percent. The other empirical
moments we target are a nominal investment to output ratio of 26 percent and
nominal government purchases to output ratio of 15 percent. Finally, we target a
capital to output ratio of approximately 10 on a quarterly basis.
Table 2 lists the parameters which we calibrate along with their given values.
The table includes many more parameters than there are targets in Table 1. This is
because Table 1 omitted calibration targets which map one-to-one with particular
parameter values. For example, we calibrate the steady-state capital depreciation
rate (δ0 ) using standard methods applied to data from the Fixed Asset tables.
It is also because Table 2 lists several parameters which are normalized prior to
estimation. Most notable among these are the three factor loadings listed at the
table’s bottom.
Tables 3 and 4 report prior distributions and posterior modes for the model’s
remaining paramters, for the first and second samples respectively.
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Table 1: Calibration Targets
Description
Expression
Fixed Interest Rate (quarterly, gross)
R
Per-Capita Steady-State Output Growth Rate (quarterly)
Yt+1 /Yt
Investment to Output Ratio
It /Yt
Capital to Output Ratio
Kt /Yt
Fraction of final good output spent on public goods
Gt /Yt
Value
1.011
1.005
0.260
10.763
0.153
Table 2: First Sample Calibrated Parameters
Parameter
Discount Factor
Steady-State Measured TFP Growth (quarterly)
Investment-Specific Technology Growth Rate
Elasticity of Output w.r.t Capital Services
Steady-State Wage Markup
Steady-State Price Markup
Steady-State Scale of the Economy
Steady-State Inflation Rate (quarterly)
Steady-State Depreciation Rate
Steady-State Marginal Depreciation Cost
Nominal Output over Nominal Private Purchases
Std. Dev Long-Run Inflation Expectations Measurement Error
Long-Run Inflation Expectations (Constant CPI Adjustment)
Average Earnings Constant
Average Total Compensation Constant
Loading Compensation
Loading Core PCE
Constant for Relative Price Inflation
Loading 0 Factor A
Loading 0 Factor B
Loading 4 Factor B
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Symbol
β
z∗
ω∗
α
λw
∗
λp∗
H∗
π∗
δ0
δ1
g∗
σe40
π∗e40
π∗w1
π∗w2
β1w2
β1p1
π∗G
λ0,1
λ0,2
λ4,2
Value
0.986
0.489
0.371
0.401
1.500
1.500
1.000
0.500
0.016
0.039
0.847
0.010
0.122
-0.237
-0.202
1.000
1.000
0.252
0.981
0.000
0.951
Authorized for public release by the FOMC Secretariat on 1/12/2024
Table 3: First Sample Estimated Parameters
Parameter
Depreciation Curve
Active Price Indexation Rate
Active Wage Indexation Rate
External Habit Weight
Labor Supply Elasticity
Price Stickiness Probability
Wage Stickiness Probability
Adjustment Cost of Investment
Elasticity of Intertemporal Substitution
Interest Rate Response to Inflation
Interest Rate Response to Output
Interest Rate Smoothing Coefficient
Autoregressive Coefficients of Shocks
Discount Factor
Inflation Drift
Exogenous Spending
Investment
Liquidity Preference
Price Markup
Wage Markup
Neutral Technology
Investment Specific Technology
Moving Average Coefficients of Shocks
Price Markup
Wage Markup
Standard Deviations of Innovations
Discount Factor
Inflation Drift
Exogenous Spending
Symbol
Density
Prior
Mean
Std.Dev
Posterior
Mode
δ2
δ1
ιp
ιw
λ
γH
ζp
ζw
ϕ
γc
ψ1
ψ2
ρR
G
B
B
B
N
B
B
G
N
G
G
B
1.0000
0.5000
0.5000
0.7500
0.6000
0.8000
0.7500
3.0000
1.5000
1.7000
0.2500
0.8000
0.150
0.150
0.150
0.025
0.050
0.050
0.050
0.750
0.375
0.150
0.100
0.100
0.499
0.280
0.082
0.790
0.591
0.833
0.904
4.326
1.915
1.833
0.488
0.791
ρb
ρπ
ρg
ρi
ρs
ρλp
ρλw
ρν
ρω
B
B
B
B
B
B
B
B
B
0.5000
0.9900
0.6000
0.5000
0.6000
0.6000
0.5000
0.3000
0.3500
0.250
0.010
0.100
0.100
0.200
0.200
0.150
0.150
0.100
0.850
0.998
0.920
0.759
0.841
0.687
0.668
0.496
0.407
θλp
θλw
B
B
0.4000
0.4000
0.200
0.200
0.608
0.306
σb
σπ
σg
U
I
U
0.5000
0.0150
1.0000
2.000
0.0075
2.000
1.187
0.094
2.500
Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform
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First Sample Estimated Parameters (Continued)
Parameter
Investment
Liquidity Preference
Price Markup
Wage Markup
Neutral Technology
Investment Specific Technology
Relative Price of Cons to Inv
Monetary Policy
Unanticipated
1Q Ahead
2Q Ahead
3Q Ahead
4Q Ahead
Total Earnings
Loading 1
Standard Deviation
AR(1) Coefficient
Total Compensation
Standard Deviation
AR(1) Coefficient
Core PCE
Constant
Loading 2
Standard Deviation
AR(1) Coefficient
Market-Based Core PCE
Constant
Loading 1
Loading 2
Standard Deviation
Symbol
Density
Prior
Mean
Std.Dev
Posterior
Mode
σi
σs
σλ p
σλ w
σν
σω
σ ci
I
U
I
I
U
I
I
0.2000
0.5000
0.1000
0.1000
0.5000
0.2000
0.0500
0.200
2.000
1.000
1.000
0.250
0.100
2.000
0.618
0.390
0.069
0.031
0.504
0.183
0.215
σµ0
σµ1
σµ2
σµ3
σµ4
N
N
N
N
N
0.0050
0.0050
0.0050
0.0050
0.0050
0.0025
0.0025
0.0025
0.0025
0.0025
0.012
0.012
0.007
0.009
0.010
β1w1
σw1
ϕw1
N
I
B
0.8000
0.0500
0.4000
0.100
0.100
0.100
0.824
0.147
0.624
σw2
ϕw2
I
B
0.0500
0.4000
0.100
0.100
0.169
0.343
π∗p1
β2p1
σp1
ϕp1
N
N
I
B
-0.1000
0.0000
0.0500
0.2000
0.100
1.000
0.100
0.100
-0.087
0.014
0.048
0.091
π∗p2
β1p2
β2p2
σp2
N
N
N
I
-0.1000
1.0000
0.0000
0.0500
0.100
0.100
1.000
0.100
-0.123
1.102
0.028
0.039
Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform
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First Sample Estimated Parameters (Continued)
Parameter
AR(1) Coefficient
Core CPI
Constant
Loading 1
Loading 2
Standard Deviation
AR(1) Coefficient
PCE Durable Goods Inflation
Constant
1st Lag Coefficient
2nd Lag Coefficient
Standard Deviation
Relative Price Inflation
1st Lag Coefficient
2nd Lag Coefficient
Standard Deviation
Factor A
Loading 1
Loading 2
Loading 3
Loading 4
Standard Deviation
Factor B
Loading 1
Loading 2
Loading 3
Standard Deviation
Symbol
Density
Prior
Mean
Std.Dev
Posterior
Mode
ϕp2
B
0.2000
0.100
0.128
π∗p3
β1p3
β2p3
σp3
ϕp3
N
N
N
I
B
0.0500
1.0000
0.0000
0.1000
0.4000
0.100
0.100
1.000
0.100
0.200
0.047
0.804
0.119
0.076
0.597
θ∗D
θD1
θD2
σD
N
N
N
I
-0.3500
0.4500
0.4000
0.2000
0.100
0.200
0.200
2.000
-0.356
0.430
0.362
0.287
θG1
θG2
σG
N
N
I
0.0000
-0.1000
0.5000
0.500
0.500
2.000
0.279
0.006
0.811
λ1,1
λ2,1
λ3,1
λ4,1
σF 1
N
N
N
N
N
0.6839
0.5224
0.4314
0.3243
0.1000
0.200
0.200
0.200
0.200
0.0750
1.256
0.857
0.361
0.032
0.041
λ1,2
λ2,2
λ3,2
σF 2
N
N
N
N
0.3310
0.6525
0.8059
0.1000
0.200
0.200
0.200
0.0750
0.698
1.162
1.199
0.072
Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform
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Table 4: Second Sample Estimated Parameters
Parameter
Total Earnings
Constant
Loading 1
Standard Deviation
AR(1) Coefficient
Total Compensation
Constant
Standard Deviation
AR(1) Coefficient
Core PCE
Loading 2
Standard Deviation
AR(1) Coefficient
Market PCE
Constant
Loading 1
Loading 2
Standard Deviation
AR(1) Coefficient
CPI
Constant
Loading 1
Loading 2
Standard Deviation
AR(1) Coefficient
Durable Goods Inflation
Constant
Standard Deviation
Relative Price Inflation
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Symbol
Prior
Mean
Std.Dev
Posterior
Mode
β∗w1
β1w1
σw1
ϕw1
-0.2370
0.8242
0.1468
0.6239
0.200
0.200
0.100
0.200
-0.096
0.252
0.183
0.529
β∗w2
σw2
ϕw2
-0.2023
0.1687
0.3430
0.200
0.100
0.200
-0.142
0.233
0.351
β2p1
σp1
ϕp1
0.0281
0.0481
0.0913
0.100
0.100
0.150
0.232
0.143
0.256
π∗p2
β1p2
β2p2
σp2
ϕp2
-0.1230
1.1022
0.0139
0.0755
0.5972
0.100
0.150
0.100
0.100
0.150
-0.114
0.358
0.219
0.121
0.527
π∗p3
β1p3
β2p3
σp3
ϕp3
0.0475
0.8039
0.1192
0.0388
0.1278
0.100
0.150
0.100
0.100
0.150
-0.022
0.305
0.206
0.085
0.220
π∗D
σD
-0.4500
0.5000
0.200
0.150
-0.463
0.291
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Second Sample Estimated Parameters (Continued)
Parameter
Constant
Standard Deviation
Factor A
Loading 0
Loading 1
Loading 2
Loading 3
Loading 4
Loading 6
Loading 7
Loading 8
Loading 9
Loading 10
Standard Deviation
Factor B
Loading 0
Loading 1
Loading 2
Loading 3
Loading 4
Loading 5
Loading 6
Loading 7
Loading 9
Standard Deviation
Standard Deviations of Monetary Policy Innovations
Unanticipated
1Q Ahead
2Q Ahead
3Q Ahead
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Symbol
Prior
Mean
Std.Dev
Posterior
Mode
π∗G
σG
0.8900
0.8143
0.400
0.080
-0.146
1.023
λ0,1
λ1,1
λ2,1
λ3,1
λ4,1
λ6,1
λ7,1
λ8,1
λ9,1
λ10,1
σF1
0.0180
0.0574
0.1941
0.3996
0.6520
1.2266
1.5237
1.8139
2.0914
2.3523
0.0442
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.100
0.158
0.356
0.496
0.623
0.820
0.990
1.122
1.102
1.087
2.515
0.072
λ0,2
λ1,2
λ2,2
λ3,2
λ4,2
λ5,2
λ6,2
λ7,2
λ9,2
σF2
-0.0181
0.2211
0.3679
0.4424
0.4612
0.4370
0.3817
0.3032
0.1074
0.0334
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.100
0.029
0.039
0.070
0.095
0.123
0.138
0.167
0.184
0.229
0.429
σµ0
σµ1
σµ2
σµ3
0.0061
0.0021
0.0004
0.0019
0.005
0.005
0.005
0.005
0.011
0.010
0.010
0.009
Authorized for public release by the FOMC Secretariat on 1/12/2024
Second Sample Estimated Parameters (Continued)
Parameter
4Q Ahead
5Q Ahead
6Q Ahead
7Q Ahead
8Q Ahead
9Q Ahead
10Q Ahead
44
Page 110 of 112
Symbol
Prior
Mean
Std.Dev
Posterior
Mode
σµ4
σµ5
σµ6
σµ7
σµ8
σµ9
σµ10
0.0001
0.0025
0.0019
0.0011
0.0001
0.0014
0.0028
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.010
0.010
0.010
0.010
0.009
0.010
0.010
Authorized for public release by the FOMC Secretariat on 1/12/2024
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@FedFRASER