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The Quest for the Natural Rate:
Evidence from a Measure
of Labor Market Turbulence
Ellen R. Rissman
Working Papers Series
Research Department
Federal Reserve Bank of Chicago
September 1998 (W P -98-10)
FEDERAL RESERVE B A N K
OF CHICAGO
T h e
Q u e s t f or t h e N a t u r a l R a t e :
Evidence from
a M e a s u r e of L a b o r M a r k e t Turbulence.*
Ellen R. Rissman
October 5, 1998
Abstract
The natural rate of unemployment changes over time as a result of
structural shifts in the distribution of employment across sectors. This
paper develops a Kalman filtering approach to measuring these structural
shiftsor turbulence. The technique isuseful in distinguishing those changes
in employment shares within an industry that are cyclical from those that
are idiosyncratic. A measure of the business cycle is generated that is
consistent with alternative measures of the cycle. A time-varying measure
of the natural rate is constructed as the rate of unemployment that is
consistent with the economy growing at its equilibrium rate, conditional on
the level of labor market turbulence. The natural rate of unemployment
increased over the seventies in response to increased dispersion in the labor
market and has declined steadily since the late eighties. Results indicate
that the gap between the actual and natural rates of unemployment enters
negatively and significantlyin inflation regressions. This result holds across
a variety of specifications.
Introduction
Friedman (1968) and Phelps (1968) originated the concept of a “natural rate”
of unemployment almost three decades ago and sparked a debate that contin*The author would like to thank Ken Kuttner for providing his invaluable Kalman filtering
programs, Ken Housinger for his excellent and timely research assistance, and Dan Sullivan and
Charlie Evans for their helpful comments and assitance. The views expressed here are not
necessarily those of the Federal Reserve Bank of Chicago’s or the Federal Reserve System’s.
Any errors ofomission or commission are purely my own.
ues today.1 The basic premise is that when unemployment falls below its nat
ural rate, excess demand for labor produces upward pressure on real wage rates.
Conversely, for unemployment above the natural rate, excess labor supply puts
downward pressure on real wage rates. The twin benefits of low inflation and
low unemployment that has occurred in the late nineties would appear to be the
result of a natural rate either around or below the current rate of unemployment.
To much discomfort, point estimates of the natural rate are rarely found below
5.5%.2
Since Friedman’s original insight, the quest for the natural rate has proved to
be chimerical. There is little consensus as to how to measure the natural rate
nor agreement on the mechanisms linking the natural rate to inflation. After
examining the evidence, Blanchard and Katz (1997) concluded that “Economists
are a long way from having a good quantitative understanding of the determinants
of the natural rate.” 3
Despite such pessimism, King and Watson (1994) document that “there is
a pronounced, negative correlation of inflation and unemployment at business
cycle frequencies, which is remarkably stable over the post-war period.4 This
result is repeated in Staiger, Stock and Watson (1997), prompting the authors
to state that “there is a downward-sloping Phillips curve; it simply is difficult to
estimate the level of unemployment at which the curve predicts a constant rate
of inflation.”5After reviewing the evidence, Stiglitz (1997) is convinced that the
natural rate concept “is useful as a theory to understand the causes of inflation.
It is useful as a general guideline for thinking about macroeconomic policy. And,
it is useful as an empirical basis for predicting changes in the inflation rate.”
Friedman (1968) was careful to note that the natural rate is not “immutable
and unchangeable.”6 Nonetheless, most empirical work assumes either a constant
natural rate or one that changes only in a very simple way. For example, Gordon
(1982), Weiner (1994), and Tootell (1994) model the natural rate of unemployment
as having discrete jump points. More recently, Gordon (1997) acknowledges the
1The terms “natural rate ofunemployment” and “Non-Accelerating Inflation Rate ofUnem
ployment,” or NAIRU, are used interchangeably.
2The Congressional Budget Office estimates the NAIRU to be 5.8% in 1996. The councilof
Economic Advisors estimates the natural rate to be 5.7% in 1996 and Gordon (1997) finds itto
be 5.6%.
3Page 52.
4Page 29.
5Page 47.
®Page 9.
2
limitations of this approach in explaining the recent economic environment and
proposes an alternative method that models the natural rate as following a random
walk. He uses a time-varying parameter regression model to obtain estimates.
Staiger, Stock, and Watson (1997) also estimate a time-varying natural rate of
unemployment. They model the natural rate as a cubic spline with two knot
points at evenly spaced intervals in their sample.
Although these attempts to make the empirical specification more flexible have
succeeded in lowering current estimates of the natural rate, they suffer from three
serious problems. First, as emphasized in Staiger, et. al (1997), confidence
intervals are uncomfortably large. Although Gordon (1997) does not report
confidence intervals for his estimates ofthe natural rate, the problem would appear
to exist in his formulation as well. Second, the assumptions of the natural rate
following a random walk or the alternative of it being modeled as a cubic spline
do not offer any insight into the nature of the change in the NAIRXJ. The reasons
behind the perceived changes in the natural rate are not elucidated. Third, the
estimation of a time-varying natural rate in these formulations requires leads in
the data to correctly evaluate the natural rate at any given time. For policy
makers who must make decisions in real-time, waiting for additional data to
pinpoint the current N A I R U is not a meaningful way to formulate policy.
Friedman (1968) recognized that certain market characteristics such as mini
m u m wage requirements and labor union strength would tend to make the natural
rate somewhat higher than itwould otherwise be. Furthermore, any activity that
would facilitate the flow of workers to employers, such as improved quality and
availability of job information, would tend to reduce the natural rate of unem
ployment. Lilien (1982) attempted to quantify this last point by emphasizing
the effect of structural change on the natural rate. According to Lilien, when
a sectoral shift occurs in the structure of the economy, whether it be across in
dustries, occupations, or geographical regions, the natural rate of unemployment
rises. In the declining sector, workers lose their jobs but cannot find reemploy
ment instantaneously. The displaced worker may lack the necessary skills to be
hired in the expanding sector or he may be in the wrong location. Information
flows may be imperfect and he may not accurately gauge the distribution of op
portunities available to him. In any event, a mismatch arises and a period of
increased unemployment ensues even though there is no change in the overall level
of economic activity. This increased temporary unemployment can be thought of
as structural unemployment and should be reflected in a rise in the natural rate.
In order to test his basic insight that sectoral shifts lead to increases in the
3
natural rate of unemployment, Lilien (1982) first had to measure sectoral shifts.
He measures structural change as the dispersion in employment growth across
industries and found that dispersion increases markedly during economic down
turns. When Lilien included cyclical variables along with this dispersion measure
in a regression with the unemployment rate as the dependent variable, he found
the coefficient on the dispersion measure to be large and statistically significant.
He interpreted this as support for his thesis that increased sectoral shifts raise the
natural rate of unemployment, given the level of economic activity.
Abraham and Katz (1986) argued that Lilien’s dispersion measure was simply
another proxy for the business cycle. They showed that dispersion can increase
both because of shifts in the distribution of employment brought about by struc
tural change and because of shifts that occur as a result of normal business cycle
activity having a differential impact across industries. This differential impact of
the business cycle on industry employment has been well-documented.7 For ex
ample, economic downturns typically coincide with manufacturing’s employment
share declining and services’share increasing. The increased employment growth
dispersion that Lilien documented during cyclical downturns may merely reflect
the fact that economic activity was low rather than some fundamental shift in the
composition of employment across industries. As a result of the Abraham and
Katz critique, Lilien’s results were suspect.
Several economists have attempted to refine Lilien’s measurement of structural
change. Loungani, Rush, and Tave (1990), Brainard and Cutler (1993), and
Genay and Loungani (1997) construct a dispersion measure that is based upon
stock returns rather than industry employment. The common theme is that by
focusing on stock returns which are forward looking over a long time horizon, the
effects of temporary cyclical movements are minimized. Taking a different tack,
Neumann and Topel (1991) and Rissman (1993) attempt to eliminate short term
cyclical fluctuations in employment shares by decomposing changes in employment
shares into permanent and temporary movements. Permanent movements are
assumed to be attributable to sectoral shifts while temporary movements are
believed to be due to the cycle. The dispersion measure then generated from
the permanent movements is quite similar to that originally constructed in Lilien
(1982) in that permanent dispersion appears to rise when the overall level of
economic activity is weak. The main difficulty with this approach is that it relies
on future information to determine whether a current movement in an industry’s
7Mitchell (1927) and Burns and Mitchell (1946) are early examples of this literature. More
recent work includes Stock and Watson (1992) and Rissman (1993).
4
employment share is “permanent” and, therefore, is of little practical use in the
policy debate.
The remainder of this paper explores the effect of an alternative measure of
labor market dispersion on unemployment and inflation. In Section 1, a Kalman
filtering model is presented in which changes in industry employment shares de
pend upon an unobserved common component thought of as the business cycle,
and an idiosyncratic shock to the industry. A turbulence measure is constructed
from the estimated industry shocks that is independent of the business cycle. In
Section 2 estimates of the natural rate of unemployment are presented. These
estimates are constructed directly from unemployment rate regressions as in Lilien
(1982), rather than the standard construction of inferring the natural rate from
inflation regressions. Finally, the relevance of these constructed natural rate se
ries for the inflation process is assessed in Section 3. Conclusions are contained
in Section 4.
1. Industry E m p l o y m e n t G r o w t h a n d the Business C y c l e
Employment growth in a given industry reflects both cyclical and industry-specific
factors. In obtaining a measure of sectoral shifts, it is important to distinguish
between the two. The approach taken here is in the same spirit as Neumann and
Topel (1990) and Rissman (1993). However, the procedure does not rely upon ad
hoc definitions of permanence to separate shifts in the distribution of employment
across industries into those that are structural in origin from those that are cycli
cal. A model of net employment growth is proposed which explicitly incorporates
cyclical movements as well as an idiosyncratic or structural shift. A measure of
labor market turbulence is computed as the dispersion in the portion of industry
employment growth that is independent of the business cycle, in a way that is
similar to Lilien’s (1982) original construct. The turbulence measure constructed
here is more intuitive and does not suffer from the difficulty in applying it to a
policy context in a timely manner.
Let S u be the share of total employment in industry i . Define y it = A i n S u =
9 it — 9 u where g it and g t are employment growth in industry i and aggregate em
ployment growth respectively. Figure 1 shows net annualized growth rates ( y it )
using quarterly data for the following industries: Construction (CON), Mining
(MIN), Durable Manufacturing (MFGD), Nondurable Manufacturing (MFGN),
Finance, Insurance and Real Estate (FIRE), Government (GOV), Services (SRV),
Wholesale Tirade (TW), Retail Trade (TR), and Transportation and Public Util
5
ities (TPU) from 1955 through 1997Q2. In looking at these graphs, there are
several important points to note. First, the scale differs markedly from industry
to industry with, for example, Mining exhibiting relatively stable employment
growth punctuated by a few large swings. Other industries, such as Nondurable
Manufacturing, and Transportation and Public Utilities show a similar pattern
but with more moderate swings. Second, some industries clearly show a trend in
employment share, these include a shrinking Durable and Nondurable Manufac
turing sector and an expanding Services industry. Third, there is a pronounced
cyclical pattern in some industries, most notably in Durable Manufacturing, Con
struction, Services, Retail Trade, Wholesale Trade, and Finance, Insurance, and
Real Estate.
The data shown in Figure 1 suggest three reasons why industry employment
growth can differ from the aggregate. First, an industry’s employment share may
be trending upwards or downwards over long periods oftime. Second, the business
cycle may cause employment shares to fluctuate over time. Finally, there is an
idiosyncratic portion that is industry-specific, an example of which occurred in
the mid-seventies in Nondurable Manufacturing.
Let y it have the following specification:
Vit — <H
+
b i( L ) C t
+
un
(1-1)
where a* is a constant varying across industries. It is interpreted as the mean net
employment growth in industry i . From Figure 1, we expect, for example, this
term to be negative in Nondurable Manufacturing and positive in Services. C t is
some common factor that affects the growth in an industry’s employment share.
This common factor is interpreted here as the business cycle and its construction
will be discussed more fully below.8 u u isthe idiosyncratic shock affecting industry
net employment growth at time t; and bt { L ) is a polynomial in the lag operator,
also referred to as the factor loadings.9 Specifically,
b i( L )
= 6? +
b\L
+
b \ l} .
(1.2)
8Rather than modeling C t as an unobserved component, an alternative specification would
include detrended GDP as an explanatory variable instead. The approach taken here remains
agnostic about the proper interpretation of C t and estimates it from the co-movements in
employment growth acrossindustries. To the extent that C t appears correlatedwith thebusiness
cycle, then itcan be interpreted as a proxy for the cycle.
9In the text the terms net employment growth in industry i and growth ofemployment share
in industry t are used interchangeably.
6
The polynomial b i ( L ) is a flexible but parsimonious way to allow for the business
cycle to have a differential impact across industries on net employment growth.
It permits the cycle to lead in one industry and lag in another. It also permits
the cycle to have a greater impact in one industry than in another.
Since Mitchell (1927) and later Burns and Mitchell (1946) the concept of a
business cycle has been defined as “expansions occurring at about the same time
in many economic activities, followed by similarly general recessions, contractions,
and revivals which merge into the expansion phase of the next cycle.” Thus, the
business cycle is essentially unobservable but can be inferred only through its
effects on many dimensions simultaneously.10 In developing a measure of the
business cycle, it is assumed that the business cycle component, C t, is not observ
able directly. However, its time series properties are restricted to follow an AR(2)
specification so that:
Ct
= fcCt-i +
4> iC t-i
+ £t-
(1-3)
The imposition of an AR(2) process generating the business cycle allows for a
succinct characterization of Mitchell’s description of recessions being followed by
expansions.
To completely specify the model it is necessary to give structure to the two
types of shocks, u n and et,where u it can be thought of as a sectoral disturbance
and e t is a business cycle shock. Specifically, it is assumed that the two types
of shocks are mean zero, E ( u it ) , E { e t ) = 0, for all t and i . Furthermore, the
shocks are serially uncorrelated, E ( u i tU i t-a) = E f a e t - s ) = 0 for all i , t , and
5 ^ 0 . 11Nor are the shocks correlated with one another, E { e tU it- a) = 0 for all
i , i,and s . 12The shock in one industry is uncorrelated with the shock in another
industry, E { u itU j t - a) = 0 for all s , i ^ j . 13Finally, each shock has a finite variance,
E{ u % ) = o f and E ( e t) = o * .
There are many ways in which this model can be expanded. For example,
an additional unobserved component could be introduced that would explain the
co-movements across subsets of the I industries. Additionally, the variance of the
tttf’s could have a cyclical component. These extensions are useful in obtaining a
10In fact, this isthe concept behind the business cycle expansion and contraction dates pub
lished by the National Bureau of Economic Research. Another example is found in Stock and
Watson (1992).;
11This assumption about the serial correlation properties of the u h 's could be relaxed fairly
easily.
12This assumption isimportant for purposes of estimation.
13This assumption could be relaxed but with some care so that there are not too .many
additional parameters to be identified.
7
richer econometric specification. However, the purpose here is to focus on shifts
that are not the result of a common cycle. This objective is met in the current
specification. The additional complexity introduced by these extensions does
not appear warranted in light of the difficulties with estimating the relatively few
parameters in this model.
As currently specified, the model is not identified without additional restric
tions.14 Neither the scale or sign of the business cycle is defined. To see this,
suppose that the common cycle C t is rescaled by multiplying it by some constant
(3 and define C t
* = (3 C t . Then C * = <j>\Ct_x +
+ £*, where £* = /3et and
var(ej) =
. Restricting crf=1 is not sufficient to identify the parameters
of the model since setting f3 = — 1 results in the same likelihood. To address
this problem of indeterminate sign, 61 is set equal to 1 in Durable Manufacturing.
This restriction ensures that the estimate of the cycle is positively correlated with
other more traditional measures of the business cycle such as real G D P growth.15
It also allows for a straightforward comparison of the relative responsiveness of
a particular industry, where the comparison is made implicitly with the Durable
Manufacturing sector.
The model is very much in the spirit of Bums and Mitchell’s (1946) idea of
comovement. However, here its interpretation is that the cycle represents that
portion of net employment growth that is common across the various industries,
while permitting the cycle to differ in its impact on industry employment growth
in terms oftiming, magnitude, and sign through the &’•parameters.16 The Kalman
filter can be used to estimate the following: C t , u i t , e t , a i ,
and cr* for all
i and j = 1,2,3.17
Although the data introduced in Figure 1 cover 10 industry categories, in
practice the estimation was implemented using only 8 sectors. Both Services and
Mining have been omitted, but for different reasons. Mining is a small sector in
terms of its share of total employment. However, its employment is also quite
volatile over the time period considered due to strikes. Because of its volatility
and relatively small magnitude in the total, it was omitted. There is a potential
multicolinearity problem that occurs because the sum of the yit’s is approximately
14See Harvey’s (1989) discussion of common trends.
15The estimation also restricts o* = 1.
16Ideally, each sector should somehow be weighted according to some scheme. The estimation
does not currently take this point into consideration. Rather, it treats each sector as being
equally important indetermining themeasure ofthe business cycle, Ct- Clearly, an improvement
would be to treat larger industries differently from smaller sectors.
17See Quah and Sargent (1993) for an example.
8
0. By omitting a second industry, in this case Services, from the estimation, the
problem is avoided.
Preliminary results that are not reported here indicate that further restrictions
are needed to guarantee that the Hessian is well-behaved. These restrictions
include setting 61 = 0 for both Construction and Nondurable Manufacturing,
and setting 62 = 0 in Transportation and Public Utilities. These restrictions are
the result of the data imprecisely estimating these coefficients rather than any
identification issue. Since the unrestricted coefficients were close to 0, the results
reported in the text set these parameters equal to 0. Estimation was carried out for
the period from 1954:2 to 1997:2. The B F G S algorithm was used in maximizing
the likelihood function. After having obtained estimates of the business cycle, C t ,
conditional on information prior to time t , a Kalman smoothing technique was
applied that uses all available information through 1997:2 to generate smoothed
estimates of C t -18
Estimates of the industry parameters ai,6°,6j,6f, and cr* are found below
in Table 1. Note that the estimation results also include parameter estimates
for both Services and Mining even though these two industries were not directly
included in the Kalman filtering exercise. These estimates were derived from a
secondary procedure. After having estimated the common cycle, C t , two addi
tional regressions were run essentially treating C t as a known exogenous variable.
Each regression is of the form found in Equation (1.1). The standard errors re
ported in Table 1 for both Services and Mining are too small in that they do not
take into account the uncertainty in the estimates of C t..
There are several interesting points to note. First, the constant term is sig
nificant in all but Construction and Wholesale Tirade, indicating that in these
two industries there is no discernible long term trend in employment share. The
remaining industries exhibit the familiar story of declining employment share in
goods-producing industries and the mirrored increasing employment shares in
service-producing industries.
Focusing only on the b° parameters, service-producing industries tend to be
countercyclical while goods-producing industries are procyclical. Within these
broad categories, there is a great deal of variation in the magnitude of each in
dustry’s responsiveness to changes in economic activity. Relative to Durable
Manufacturing, for example Nondurable Manufacturing is not very cyclical, as
expected from a glance at Figure 1. Timing also seems to matter. Some indus
tries receive only a small contemporaneous effect from cyclical movements and
18The interested reader can find more detail in Chapter 4 ofHarvey (1989).
9
respond to changes in the business cycle only with a lag.
Equation (1.3) is estimated as:
C t =
1.3995 * C
(0.0890)
t- i
-
0.5519 * C t_2
(0.0838)
+ et
The actual smoothed estimates for C t are shown in Figure 2 along with its accom
panying standard errors.19 The Kalman filter technique relied only on informa
tion about the employment shares in the 8 industries examined. Gross Domestic
Product, for example, did not enter into the estimation. Yet the estimates of the
cycle, C t , axe remarkably similar in timing to the N B E R Expansions and Con
tractions.20 C t typically declines prior to the peak of an N B E R expansion and has
a turning point c o n s is t e n t ly within one quarter of the dated N B E R trough. The
‘mini-recession’ of the mid ’80s shows up clearly. According to this estimation
technique, the ’90-’91 recession was only a minor event in comparison to prior
recessions. In fact this ‘recession’ does not even qualify as a significant event
when compared to the standard error bands. Confirming the results of other eco
nomic data, the ‘recovery’for this recession was slow to take hold relative to other
recoveries.
The downturn in C t that shows up for the last couple years does not bode
well. Although C t is currently above the expected long term average of 0, its re
cent decline has been quite sharp. If history sets any precedent, it would indicate
that declines of this magnitude are usually followed by contractions. However, it
should be noted that there is substantial uncertainty associated with these mea
sures, both because of normal parameter uncertainty and model uncertainty. In
addition, the estimation technique implicitly assumes that the nature of the busi
ness cycle remains stable over the estimation period. If, for example, Durable
Manufacturing has become a less important determinant of the underlying busi
ness cycle, the technique will tend to overstate Durable Manufacturing’s impor
tance in the more recent past and understate its importance in the earlier part of
the sample.
In addition to the estimates of the business cycle generated by the Kalman
filter, the idiosyncratic shocks, u i t , are also of interest. The estimated w# ’s are
shown in Figure 3 for the 10 industry categories. Note that the scale varies widely
19Standard errorsaxe calculated as discussed inHamilton (1994), page 398. The Monte Carlo
method provides estimates of the mean squared error of the state vector taking into account
both the filter uncertainty and the parameter uncertainty.
20NBER contractions have been shaded.
10
with Construction exhibiting the largest shocks and Retail Trade the smallest
on average. Transportation and Public Utilities exhibited relatively small distur
bances with the exception of a large shock in the early ’80s that is coincident with
the timing of the P A T C O strike. Similarly, Mining has experienced only small
disturbances with the exception of a few large deviations. The large swings in the
late ’70s are related to the strike by the Bituminous Coal Operators Association
affecting approximately 160,000 workers. In addition, the relatively large distur
bances in Nondurable Manufacturing occurring in 1975 are likely due to the oil
price shock’s affect on the Petroleum and Chemicals industries.
1.1. M easu rin g E conom ic T urbulence
The Kalman filterexercise provides an intuitive technique for measuring structural
shifts in the composition of employment where these shifts are constructed so as
to be independent of the business cycle. The measure proposed here isin the same
spirit as Lilien (1982) in that it focuses on the dispersion in employment growth
across broad industry categories. However, Lilien’s measure failed to recognize the
effects of the business cycle on dispersion. Thus, the measure of sectoral shifts he
proposed does not clearly distinguish between movements in employment share
that are the result of the business cycle and the sectoral shifts it purports to
measure. For purposes of measuring sectoral shifts, it is the portion of dispersion
in employment growth that is unrelated to the business cycle that isof importance.
In other words, it is the idiosyncratic shock, u ^ , that reflects the shifts in an
industry’s employment growth that are orthogonal to the business cycle.
Let a t be the acyclic measure of dispersion. It is defined as:
at
(1.4)
=
where S# is a measure of employment shares in industry i that is constructed to
be independent of the cycle, and the u * ’s are the estimates of the idiosyncratic
shock to net employment growth in industry i that are obtained directly from
the Kalman filter. The acyclic employment share,
is constructed from some
initial starting condition, 5*0> and imposing C t = 0 for all t = 0, ...,T. In other
words, it is what the employment share would have been ifthe economy had not
experienced any cyclical variation but responded only to idiosyncratic shocks and
long term trends. Thus, <rt is a measure of the standard deviation of changes in
11
employment shares across industries where these changes axe orthogonal to the
business cycle.
This turbulence measure, oV,is shown in Figure 4 along with Lilien’s original
dispersion measure. Note that all 10 industries have been included in the mea
sure with Services and Mining disturbances estimated by auxiliary regressions of
the form described above. Compared to Lilien’s measure of sectoral shifts, this
acyclic measure fluctuates quite a bit more. However, the timing of the peaks is
similar to that found in Lilien. It clearly indicates that the early ’70s to early
’80s was a period of structural change, so that one would expect to see a rise
in the natural rate of unemployment over this time period. Here, as well as in
Lilien’s original measure, increases in turbulence axe coincidental with economic
contractions. For Lilien, this increase in dispersion during economic downturns
is fundamentally linked to his failure to realize that dispersion is itself cyclical.
Such is not the case here. To the extent that the business cycle is reflected in co
movements in employment shares, the measure constructed here is independent
of these co-movements. Nontheless, the turbulence measure still appears to be
highly cyclical. This fact is not necessarily troubling. For example, suppose that
the fundamental distrubance is an oil price shock. Such an event has broad impli
cations for the overall economy in which most sectors are affected simultaneously.
However, the shock itselfwill result in some labor reallocation across industries as
those that are more oil-price sensitve shrink. The increase in turbulence during
this time reflects the structural reallocative effect of this disturbance rather than
the cyclical effect.
Since the early ’80s, the economy has been remarkably stable in that there have
been no large sectoral shifts between and among industries that impinge upon the
labor market. The brief increase in dispersion recorded in the early ’90s was much
smaller in magnitude than that observed in the previous two decades. In fact, the
turbulence measure shows that the current state of labor demand is very stable
by historical standards with Uttle deviation in employment distribution. Thus,
we would expect to see a lower natural rate than would be found in the ’70s and
’80s and, indeed, even in the earlier part of the decade.
Closer examination of the relative importance of the various sectors in total
dispersion provides an interesting insight. Most of the increase in employment
dispersion in the early ’70s was directly attributable to shocks in Durable Manu
facturing, combined with those in Transportation and Public Utilities. In contrast,
the increase in turbulence that occurred about the time of the ’74-’75 recession
was due to industry-specific shocks in Construction and Nondurable Manufactur
12
ing, and to a lesser extent disturbances in Durable Manufacturing and Finance,
Insurance, and Real Estate. Surprisingly, the most recent increase in turbulence
in the early ’90s was due primarily to a shift in the Government sector.
Figure 5 shows a scatter plot ofthe dispersion measure developed above against
Lilien’s measure of sectoral shifts. Although there does appear to be a positive
correlation between the two measures. This correlation is not perfect. In fact
there are several periods in which a t is high while Lilien’s corresponding measure,
ox* is low. Conversely, there are many quarters when Lilien’s measure is high
while a t is low. Such a period appears to have occurred in the mid to late
seventies.
2. Estimates of t he N a t u r a l R a t e
The natural rate is typically estimated from a regression of inflation on various
explanatory variables, including the unemployment rate and past inflation. The
natural rate of unemployment isthen calculated as the rate of unemployment that
is consistent with stable long run inflation.21 As noted by Staiger, Stock, and
Watson (1997), this technique of constructing estimates of the natural rate from
inflation regressions results in imprecise estimates that are not too informative for
predicting inflation.
Rather than start with a definition of the natural rate as being defined by the
long term inflation-unemployment trade-off, Lilien (1982) instead modeled the
underlying determinants of the natural rate. In Lilien’s framework the natural
rate of unemployment is defined as the rate of unemployment that would occur if
there were no cyclical variation affecting the unemployment rate in the long run
In this methodology, it is the unemployment rate that would be ground out by
the Walrasian auctioneer noted in Friedman (1968).
Lilien generated alternative estimates of the natural rate of unemployment
based upon an unemployment rather than an inflation regression. He modeled
the unemployment rate as depending upon cyclical factors and structural shifts
that affect the natural rate. The natural rate is then constructed as the rate of
unemployment consistent with stable growth in the economy.
Using a different measure of structural shifts, Rissman (1993) showed that
such natural rate estimates explained the apparent shifts in the Phillips curve that
occurred in the 1970s and 1980s. The methodology is attractive in that a time21Gordon (1997 and 1982) provide examples of thismethod.
13
varying natural rate is a natural outcome of the estimation. Furthermore, unlike
other attempts to estimate a time-varying natural rate directly from inflation
regressions, the underlying determinants of the natural rate are dealt with more
explicitly here.22 In this formulation the movements in the natural rate are
the result of structural shifts, but the underlying reasons for these shifts are not
elucidated.
2.1. M odeling th e U nem ploym ent Rate:
Assume that the unemployment rate isdescribed by the following regression equa
tion:
U t = a + 0 ( L ) U t -! + 7 ( L ) Z t + 6 ( L ) X t + v t .
(2.1)
is the unemployment rate at time t; / 3 (L) is a polynomial in the lag operator;
Z t reflects current and lagged cyclical factors; X t is a vector of variables thought
to affect the unemployment rate but that are independent of the cycle. Finally,
v t is an iid error with finite variance.
The analysis presented here uses the Perry-weighted unemployment rate as
the dependent variable, where the unemployment rate has been adjusted to re
flect shifts in the age, sex, and racial composition of the labor force. Augmented
Dickey-Fuller tests were performed with the result that the null hypothesis of a
unit root in the unemployment rate could not be rejected at conventional signif
icance levels. This result held for lag lengths up to and including 12 quarters.23
Consequently, in the following unemployment rate regressions the dependent vari
able is entered as a first difference.
There are potentially several different measures of the cycle that could be used.
The results discussed here include the measure of the business cycle that the
Kalman filter exercise generates, C t . In addition shocks to nonborrowed reserves
(N B R D ) and a quarterly variable constructed similar to Barro’s unanticipated
money growth (U A G M ) are also included in the vector Z t .24 Included in X t
is a measure of sectoral shifts. Two different measures were evaluated. These
include Lilien’s original measure (07), updated through 1997Q2, and the measure
developed in Section 1.1,a .
Ut
22See Staiger, Stock, and Watson (1997), Gordon (1997), and Weiner (1994) for examples of
the former approach.
23
For example, the Dickey-Fuller test statistic (7 7 ,) with a lag length of 8 is estimated to be
-1.8664. This is well above the 10% critical value of -2.58 so that the hypothesis of a unit root
cannot be rejected.
24The Fed Funds rate shock is constructed as in Christiano, Eichenbaum, and Evans (1996).
14
Table 2 describes the results for equations estimated without any of the cyclical
explanatory variables. Table 3 describes selected results for equations estimated
with cyclical explanatory variables. The variable S M P L is a d u m m y variable
taking on the value of 1 for the post 1970 period.
Prom Table 2, the estimated regressions suggest that the unemployment rate
is best described as a random walk with no drift. In general, the unemployment
rate regressions that use the Lilien measure of sectoral shifts <t l have higher ad
justed R 2 values than those that use the measure of turbulence a constructed
above. There is no compelling reason why the turbulence measure a should be
calculated as the square root of the sums of the weighted squared shocks. There
are innumerable alternatives. For example, ifone believed that the seventies and
early eighties were particularly volatile, an alternative measure such as the vari
ance, <72 ,would be appropriate since it has the effect of increasing the turbulence
during this time period relative to others. To test whether this variance measure
better describes the unemployment rate, both a and a 2 were included both sep
arately and together in the regressions. The results are found in regressions (1),
(3), and (5). W hen entered separately, the contemporaneous effect of an increase
in turbulence is to increase the unemployment rate. Lags of both turbulence and
its square do not follow any apparent pattern and are largely insignificant. In
regression (5), although turbulence stillhas a positive effect on the unemployment
rate, neither turbulence nor its square enters significantly.
A similar analysis is done for the Lilien measure <j l and its square, cr2 . The
results of these regressions are found in columns (2), (4), and (6) of Table 2. In
contrast to the turbulence measure described above, there does appear to be a
difference in the adjusted R 2 depending upon whether ctl or cr£ is employed, with
the latter measure giving a better fitto the data. Interestingly, the contemporane
ous effect of structural shifts on the unemployment rate is positive and significant
while the lags are negative, significant, and of the same magnitude. This same
result is documented in Rissman (1993). W h e n both crL and a \ enter the re
gression, only the a \ parameters are significant with contemporaneous structural
change increasing the unemployment rate.
Regression results when cyclical variables are included as explanatory variables
in the unemployment rate regressions are documented in Table 3. The lag of
changes in the unemployment rate ishighly significant in each of these regressions.
Equation (1) of Table 3 repeats equation (5) of Table 2 but includes the business
cycle variable C estimated from the Kalman filter exercise as an explanatory
variable. The fit of the equation increases markedly as a result of its inclusion
15
__2
with the R increasing from 0.4224 to 0.7303. Contemporaneous turbulence tends
to increase the unemployment rate significantly and has a significant nonlinear
effect. This compares with insignificant parameter estimates on turbulence in the
regression where the business cycle measure C is not included in the regression
As anticipated, the business cycle variable has a negative impact on changes in
the unemployment rate.
The same analysis is repeated using the Lilien sectoral shifts measure with the
corresponding regression results found in (3.2) and (2.6)25 As with the turbulence
_2
measure, the inclusion of the cyclical variable C makes a difference in the R of the
regression. The effect of sectoral shifts is less precisely estimated in Regression
(3.2) with the sum of lags of a \ not significantly different from 0. There also
appears to be a significant increase in the change in the unemployment rate in
the latter part of the sample period once C is included in the analysis.
Columns (3)-(6) examine the separate effect of alternative cyclical measures
on changes in the unemployment rate. These measures include shocks to NonBorrowed Reserves and Unanticipated Money Growth. To summarize, when
current and lags of shocks to Non-Borrowed Reserves and Unanticipated Money
Growth are included in the regressions without inclusion of the cyclical term C,
their cumulative and individual effects on changes in the unemployment rate are
_2
insignificantly different from 0 and consequently have little impact on the R
statistics.
Columns (7) through (10) report the regression results for estimation that
includes both C and either current and lagged U A G M or N B R D . Again, the
inclusion of Unanticipated Money Growth has little impact upon the results as
shown in columns (7) and (8). Interestingly, when shocks to Non-Borrowed
Reserves are included simultaneously with the cyclical measure C , the effect on
changes in the unemployment rate is negative and significant.26 This can be seen
from columns (9) and (10).
Finally, regression results that include C , U A G M , and N B R D are found in
columns (11) and (12). When all three cyclical variables are included in the
regressions, U A G M still has little impact upon the results. Contemporaneous
turbulence enters significantly and has the theorized sign. The sum of the lags
are insignificantly different from 0. In addition the equation appears to be stable
in that the intercept term is insignificant throughout the entire period examined.
This contrasts with results for Lilien’s structural change measure. The intercept
25The notation (a.6 ) refers to Column b of Table o.
26This corresponds with the results found in Christiano, Eichenbaum, and Evans (1996)
16
term is significant and increases in later periods, as shown in column (12).
2.2.
C onstructing th e N atural R ate o f U nem ploym ent:
The results of the previous section can now be used to generate estimates of the
natural rate of unemployment, U * . Recall that the Perry-weighted unemployment
rate appears to follow a random walk with the following form:
2
A
Ut = a + P A U t-i
+ X
t=0
q
p
+
X
S iN B R D t - i
i=0
+ £
1=0
+
vt.
(2.2)
The natmral rate is the rate of unemployment that would have occurred had the
economy been growing at its equilibrium rate, which by definition implies that
the business cycle is neutral in its effects on the natural rate. This definition
does not imply that the natural rate is itself constant, since other factors such as
turbulence cause the amount of search unemployment to vary over time. In the
context of Equation (2.2), when the economy is in equilibrium, cyclical variables
are set to their long-run values. This implies that the change in the natural rate
is constructed as:
Au
+
; = a + 0 A U U
J 2 W
t-i
(2.3)
t=0
where the notation ‘x’is the estimated value of x. The change in the natural rate
between period t and period 0, f7t* — U q , can be constructed as:
t
v ; - v s
= Y
. A % -
t=l
<2-4>
Figure 6 shows estimates of the deviation of the natural rate from its starting
value, U ? — U q , for three different unemployment regressions such as those found
in equation (2.2). The starting period is 1956:4 and q = 8 for all three of the
estimated models. The solid line reflects natural rate changes for a model in which
the effect of labor market turbulence is described by the turbulence measure a .
The short dashed line shows natural rate changes for a model in which the effect
of turbulence is captured by <r2. Finally, the last set of estimates shown in the
graph is for a model in which both a and a 2 are included. The time series pattern
is quite similar for all three of the models. The sixties were a relatively stable
17
period in which the natural rate remained fairly constant, although it increased
relative to its level in the reference period. In the early seventies the natural
rate increased rapidly as a result of the economic turbulence that is captured
by the acyclic shocks to changes in industry employment share. By the early
eighties, the natural rate had risen to over 4% higher than its level in 1956:4.
The subsequent period from the early eighties to the present has been remarkably
devoid of structural change. This is reflected in the steady decline in the natural
rate over this time period. By 1997:2 the natural rate had declined over 2.5
percentage points from its high in the early eighties, although it was still in excess
of its reference level in 1956:4.
___
Calculating confidence intervals for the natural rate estimates, C/t* — U q , is
somewhat complicated due to the cumulative nature of the computation. A boot
strapping methodology provides a way to construct these confidence intervals in
a relatively straightforward fashion. A two-step process is employed. First,
equation (2.2) is estimated using the available data and estimates of the natural
rate £/t* — U q are constructed. The estimated errors from the regression are used
to construct an artificial dataset where shocks are drawn from the distribution of
estimated errors with replacement. In the second step, this newly constructed
shock series is used to generate artificial data. This data assumes the parameters
of equation (2.2) are known constants equal to the estimated coefficients in the
first step of the process. Initial values are given by the initial values of the origi
nal data. Equation (2.2) is then reestimated using the new artificial data and an
alternative estimate of the natural rate is constructed. This process is repeated
2000 times. The 95% confidence interval is constructed as the range of values for
U t — U q that occurred in 95% of the estimates.
Figure 7 shows bootstrapped confidence intervals for the natural rate derived
from estimates of equation (2.2) in which a and its square are included as ex
planatory variables. The natural rate is constructed over the period from 1957:1
through 1997:2 with t = 0 corresponding to 1956:4. The confidence interval for
Uj< — U $ is so large that one cannot say with any degree of certainty whether the
natural rate has increased or decreased relative to its value in 1956:4. The unem
ployment rate is estimated in first differences rather than in levels. To obtain
natural rate estimates relative to its starting value at time 0, changes in the nat
ural rate are cumulated. This cumulation implies that, roughly speaking, past
errors are added to current errors, giving rise to expanding confidence bands.
The expanding confidence bands does not imply that short term changes in
the natural cannot be determined with accuracy. Indeed, changes in the natural
18
rate can be estimated reasonably well over short time horizons. Only when these
changes are accumulated over long periods of time is the range uncomfortably
large. For example, to assess whether the natural rate has declined relative to its
maximum in 1985:2, the same bootstrapping method is used. Estimates of the
natural rate relative to its value in 1985:2 are constructed. This estimate along
with the relevant 95% confidence intervals are shown in Figure 8.27 In 1997:2,
only 6 % of the 2000 bootstrapped estimates showed an increase in the natural
rate of unemployment for this model relative to its level in 1985:2; 81% exhibited
a greater than 1% decline and 57% exhibited at least a 2 % decline. Put another
way, the likelihood that the natural rate was lower in 1997:2 than it was in 1985:2
was 94%.
3. T h e N a t u r a l R a t e a n d Inflation
In contrast to the methodology employed here, the N A I R U has most often been
inferred by a process which takes as a starting point an expression relating inflation
or the change in inflation to the gap between the unemployment rate and the
NAIRU. For example, Staiger, Stock, and Watson (1997) begin their analysis of
the N A I R U with a regression model such as the following:
Airt =
b (L ){ U R t -
U ;)
+ 7X
t + vt
(3.1)
where b ( L ) is a polynomial in the lag operator, U * is the natural rate at time r,
X* is a vector of additional variables that includes lags of A7rt as well as supply
shock variables. Assuming a constant natural rate of U , the above equation can
be rearranged and estimated by OLS. Specifically,
A7rt =
where /x= — ^
b iU .
n + b (L )U R t
+
iX t
+
vt
The natural rate is simply estimated as:
&=-£/£?<
i
where the ’a;’signifies the OLS estimate of x. Various procedures have been pro
posed to obtain time-varying estimates of the natural rate, again taking as the
27The natural rate series is obtained from unemployment regressions employing a as the only
explanatory variable describing turbulence. Note that this estimate of the natural rate is in
general higher than the other two series constructed and shown in Figure 6 .
19
foundation the relation between inflation and the unemployment gap that is assinned in equation (3.1). Essentially, these methods rely upon a specification for
how the natural rate evolves over time. This specification is then incorporated
into the estimation of equation (3.1). For example, Staiger, Stock, and Watson
(1997) model the natural rate as a cubic spline with two knot points. Gordon
(1982), Weiner (1993), and Tootell (1994) model the natural rate as having dis
crete changes while remaining constant over long periods.
The methodology proposed here is distinct from the traditional analysis in
that the natural rate of unemployment has been constructed separately with no
reference as to itseffect on inflation. In addition, the N A I R U that results from this
approach is clearly time-varying. The separation of estimation of the natural rate
of unemployment from the inflation process provides an opportunity to evaluate
whether the estimates of the N AIRU that are obtained from this approach are
consistent with the model of the inflation process proposed in equation (3.1).
Let U t = U ? —
expressed as:
Uq
as constructed above in Section (2.2). Equation (3.1) can be
A trt
= fi + b ( L ) ( U R t -
U t)
+
jX t
+
vt
(3.2)
where /j l = —
. This regression can be estimated by OLS. Furthermore, as
a by-product of this regression, a true time-varying estimate of the natural rate
can be constructed as:
where the expression in parentheses isthe estimate of U q that isobtained from the
OLS regression of equation (3.2). Note that the N A I R U at time 0 is constructed
in a way which is analogous to the estimation of a constant natural rate in the
traditional methodology discussed above.
Table 4 below shows results for inflation regressions of the form found in
equation (3.2). In addition to the three different estimates of the natural rate,
Ut.constructed above, four different measures of inflation have been examined.
These measures of inflation include the Personal Consumption Expenditures De
flator (PCE), Personal Consumption Expenditures Less Food and Energy Deflator
(PCEX), the Consumer Price Index for Urban Workers (CPI-U), and the Consinner Price Index Less Food and Energy (CPDC).
Table 4.a reports the results in which the change in inflation is the dependent
variable and inflation is calculated as a four quarter difference in the log of the
P C E Deflator (a * = In( P C E t) — ] n ( P C E t- 4) ) . Similarly, Table 4.b reports re
20
gression results for inflation as measured by the growth rate of P C E X . Table
4.c and 4.d report results for inflation measured by the growth of C P I — U and
C P I X respectively. The G A P series used in the regressions for Columns (1)
and (2) on all four tables is based upon unemployment rate regressions in which
a measures turbulence. The results reported in Columns (3) and (4) rely upon
G A P estimates for which a 2 measures turbulence in the unemployment regres
sion. Finally, the results of Columns (5) and (6) show results for which the G A P
estimates are derived from unemployment regressions in which both a and a 1 have
been introduced as separate regressors. Akaike and Schwarz tests were performed
to determine the appropriate lag length for both the dependent and independent
variables. Generally, four lags of the dependent variable and 5 lags of the G A P
tended to adequately fit the data. Odd numbered columns have been estimated
without restrictions. Even numbered columns report Shiller estimates in which
the coefficients on G A P -1 to G A P s are restricted to be smooth while G A P 0 is
left unrestricted.28
From Table 4.a results are consistent across different measures of the G A P .
This is not surprising given the high correlation between the different U series.
Coefficients on lags of A P C E greater than 2 are not significant. For the unre
stricted regressions, the G A P tends to decrease changes in inflation. This can
be seen from the statistics reported for ]T)G A P . From column (1) for example,
a sustained positive gap between the unemployment rate and the natural rate
U of 1% results in inflation declines as measured by the P C E deflator of 0.1817
quarterly. In the unrestricted regressions, the G A P tended to be significant only
for lags of 4 and 5. The Shiller specification forces the coefficient estimates to
become more negative and significant for contemporaneous G A P and generally
reduces the magnitude of the coefficients on lags of the G A P . It should be noted
that the R 2 from these regressions are not particularly high. In fact the R 2 sta
tistics from the Shiller estimates are substantially lower, leading to the possibility
that the restrictions imposed are not well-supported in the data. Serial corre
lation in the residuals does not appear to be a problem. Finally, the estimates
of the starting value of the natural rate ( U q ) are reported. Note that for G A P
28Suppose that a second degree polynomial describes the /?<so that /?»= ao + ai * i + c*2 * t2.
This specification implies that A 3Pi = 0. The Shiller method instead assumes that A 3/?* = Wi
where «;< ~ iidN(0 ,<r2) and A is the difference operator. The resulting estimator is a special
case of a Ridge estimator. An iterative procedure suggested by Lindley and Smith (1972) isused
to determine the value of the weighting parameter, k = o^/cr2 ,which isthe ratio of the variance
of the inflation error to the variance of the smoothness error. Generally, estimates converged
after 3 iterations.
21
estimates that are derived from turbulence measures a and o 2 found in columns
(5) and (6), the value of the natural rate at time 0 is roughly half a percentage
point larger than the other estimates. This is due to the fact that the U t series
constructed from unemployment rate regressions that have a and a 2 entered as
separate explanatory variables is lower than either of the other two estimates.
The higher estimate of ( U ) in this case corrects for this, leaving the levels similar
over time.
In Table 4.b the volatile food and energy components have been omitted from
the P C E deflator (PCEX). The sum of the G A P coefficients are negative for
each of the regressions reported and their magnitudes are marginally, although
not significantly larger than those reported for the P C E regressions. In contrast
to the P C E results of Table 4.a, the contemporaneous G A P is large, negative, and
significantly different from 0. There is no significant change in the R 2 statistics
between the restricted and unrestricted regressions leading to the conclusion that
the Shiller estimates explain the data reasonably well. The estimates of U q repeat
the same pattern as for the P C E regression results. Specifically, the starting value
for the NAIRU derived from the G A P estimates obtained from using both the
turbulence measure and its square in the unemployment rate regression are higher
by about half a percentage point.
Regression results for inflation as measured by the CPI-U are found in Table
4.c. As for the other inflation measures, the sum of the G A P coefficients is
negative and significant. The sum is also larger in magnitude than for the P C E
deflator measures. The R 2 is larger as well for these equations than for their
counterparts in Tables 4.a and 4.b. The result of applying the Shiller ridge
estimator to the G A P coefficients is to shift the weights of the coefficients towards
the contemporaneous G A P and away from longer lags.
Regression results for inflation as measured by core inflation (C P I X ) are found
in Table 4.d. Again the sum of the G A P coefficients is negative and significantly
different from 0. There is not a great deal of difference in magnitudes between the
unrestricted and restricted R 2s, so that the Shiller estimates appear to adequately
describe the data. The result of the Shiller procedure is to make the coefficient
on the contemporaneous G A P measure more negative and significantly different
from 0.
The results described above are robust with respect to other variables included
in the vector X t . These other variables include current and lagged values of the
Kalman filter measure of the cycle, C<,and current and lagged shocks to nonborrowed reserves, N B R D . Results are unreported here for the sake of brevity.
q
22
However, in allcases the G A P was consistently negatively and usually significantly
related to changes in inflation.
4. Conclusions
Estimates of the natural rate vary substantially. This appears to be the result of
differences in methodology as well as its being estimated only imprecisely. The
assumption of a constant natural rate appears to have little support in the data,
as many researchers have noted.29 The evidence reported in previous research is
consistent with a stable natural rate over the sixties, an increasing natural rate
during the seventies to mid- eighties, and a declining natural rate since the lateeighties. However, when confidence intervals have been reported, it is clear that
even these broad characterizations are subject to a great deal of uncertainty.
Attempts to model the natural rate of unemployment as time-varying have
focused on using statistical techniques to capture time-variation in a very simple
way. The standard approach is to infer the N A I R U from parameter estimates of
an inflation or change in inflation regression. Within this framework for example,
the natural rate has been modeled as having discrete changes where the timing
and magnitude of these changes are estimated from the data. Alternatively, the
natural rate has been modeled as a random walk, again relying on the inflation
regression for estimation of the parameters. Still another approach has used a
cubic spline to describe the natural rate time series. These techniques, while
informative, provide little explanation as to why the natural rate has changed
over time.
Prom a policy perspective many of these approaches are also lacking since the
econometrician is able to infer that the natural rate has changed only after some
period of time has passed. For example, ifthe natural rate were to move at date t ,
and it was modeled as changing only at discrete points, the econometrician would
require several more periods of data in order to say with any confidence that the
natural rate had indeed shifted. A more reliable current estimate of the natural
rate of unemployment would be a valuable input in the policy process.
The focus of the work here is to explicitly model how the natural rate changes
over time and to do so within a framework that provides a way to obtain current
estimates of the natural rate in a timely manner. The natural rate as constructed
29Examples include Lilien (1982), Loungani, Rush, and Ihve (1990), Brainard and Cutler
(1993), Rissman (1993), Tootell (1994), Gordon (1997), Blandiard and Katz (1997), and Staiger,
Stock, and Watson (1997),
23
here is calculated without explicit reference to inflation. This distinguishes it from
the traditional approach which relies on inflation to define the natural rate. In
fact, the natural rate of unemployment constructed here is determined by an
unemployment regression, rather than an inflation regression. The natural rate
is assumed to change depending upon the amount of labor market turbulence
affecting the distribution of employment across industries. Only after estimates
of the gap between the actual unemployment rate and the natural rate have been
constructed is inflation expressly considered. There is no a p r i o r i reason why the
natural rate as defined above should be consistent with the traditional view of the
natural rate as the non-accelerating inflation rate of unemployment (NAIRU).
The results are encouraging. First, estimates of the natural rate derived from
this approach appear reasonable. The natural rate seems to have been relatively
flat over the sixties, increased over the seventies to mid- eighties, and declined
continuously since then. Unfortunately, standard errors are uncomfortably large
when calculating the level of the natural rate. However, changes in the natural
rate over shorter time periods are estimated more precisely. This problem of
large confidence intervals appears in the traditional approach as well. Second,
the estimates of the gap between the actual unemployment rate and the natural
rate has the anticipated effect on inflation. Specifically, for a wide variety of
natural rate models and inflation measures, increases in the gap tend to lead
to declines in inflation changes. Whether these natural rate estimates provide
better forecasts of inflation than alternative natural rate models remains open.
It is important to note that the purpose of this research is not to obtain the best
forecasts of inflation. If this were the goal, then a V A R approach would be the
most productive. Instead, the objective is to succinctly capture economic events
that plausibly underlie a more structural understanding of the natural rate and
to ascertain ifsuch an approach offers some insight into the nature of the inflation
process.
Finally, there are presumably many different factors affecting the natural rate.
The work here has focused on just one, namely, the effect of sectoral shocks on
the distribution of employment across industries. While the way in which tur
bulence is constructed here does not suffer from some of the criticisms leveled at
earlier measures, turbulence may arise from other sectors as well. For example,
the natural rate may respond to geographical shocks as well as international re
structuring. Incorporating such concepts into models of the natural rate remains
to be done.
24
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Lilien, David, “Sectoral Shifts and
Cyclical Unemployment,” J o u r n a l
o f P o lit ic a l E c o n o m y , Vol. 90, No.
4, August 1982, pp. 777-793.
[Loungani, Rush, and Tave (1990)]
Loungani,
Prakash,
Mark
Rush,
and William Have,
“Stock Market Dispersion and
Unemployment,” J o u r n a l o f M o n
e t a r y E c o n o m ic s , Vol. 24, No. 3,
June 1990, pp. 367-388.
[Mitchell (1927)]
Mitchell, Wesley C.,
C y c le s :
T h e P r o b le m
B u s in e s s
a n d I ts S e t
N ew York: National Bureau
of Economic Research, Inc., 1927.
tin g ,
[Neumann and Topel (1991)]
Neumann, George R., and
Robert H. Topel, “Employment
Risk, Diversification, and Unem
ployment,” Q u a r t e r ly J o u r n a l o f
E c o n o m ic s , Vol. 106, 1991, pp.
1341-1366.
[Phelps (1968)]
Phelps, Edmund, ’’Money-Wage
Dynamics and Labor-Market Equi
librium,” J o u r n a l o f P o lit ic a l E c o n
o m y , Vol. 76, Part 2, July/August
1968, pp.678-711.
[Quah and Sargent (1993)]
Quah, Danny, and T h o m a s J.
Sargent, “A Dynamic Index Model
for Large Cross Sections,” in N B E R
S t u d ie s i n B u s in e s s C y c le s , vol. 28,
University of Chicago Press, 1993,
pages 285-306.
27
[Rissman (1993)]
Rissman,
Ellen R., “Wage
Growth
and
Sectoral shifts:
Phillips Curve Redux,” J o u r n a l o f
M o n e t a r y E c o n o m ic s , Vol. 31, No.
3, June 1993, pp. 396-416.
[Rissman (1997)]
Rissman, Ellen R., “Measuring
Labor Market Turbulence,” E c o
n o m ic P e r s p e c t iv e s ,Federal Reserve
Bank of Chicago, Vol. 21 No, 3,
May/June 1997, pp. 2-14.
[Staiger, Stock, and Watson (1997)]
Staiger, Douglas, James H.
Stock, and M a r k W . Watson,
‘The NAIRU, Unemployment and
Monetary Policy,” J o u r n a l o f E c o
n o m ic P e r s p e c t iv e s , Vol. 11, No. 1,
Winter 1997, pp. 51-72.(1997)
[Stiglitz (1997)]
Stiglitz, Joseph, “Reflections
on the Natural Rate Hypothe
sis,” J o u r n a l o f E c o n o m ic P e r s p e c
tiv e s , Vol. 11, No. 1, Winter 1997,
pp. 33-52.
[Stock and Watson (1992)]
Stock, James H., and M a r k W .
Watson, “A procedure for Predict
ing Recessions with Leading Indica
tors: Econometric Issues and Re
cent Experience,” National Bureau
of Economic Research, working pa
per, No. 4, March 1992.
[Tootell (1994)]
Tootell, Geoffrey M . B., “Re
structuring the NAIRU, and the
Phillips Curve, ” N e w E n g la n d
E c o n o m ic
R e v ie w
of
th e
F e d e ra l
September/October 1994, pp. 31-44. '
R e s e rv e
28
Bank
o f B o s to n ,
[Weiner (1994)]
Weiner, Stuart E., “The Natural
Rate and Inflationary Pressures,”
E c o n o m ic R e v ie w o f th e F e d e r a l R e
s e r v e B a n k o f K a n s a s C it y ,
Summer 1994, pp. 5-9.
29
Vol. 79,
Figure 1: Industry E m p l o y m e n t G r o w t h Less Aggregate E m p l o y m e n t G r o w t h
1955:1 - 1997:2
Government
is
10
s
0
-5
i
/
\
N
(
j
W
.
/V
j_i
J
....
•10
•IS
RetailTrade
Service
10.0
7.5
5.0
l ivn I 1r
J\J r
^1
y
•2.5
V
-5.0
•7.5
•10.0
1055
1061
I06>
*1073
*1070*
imS
iMl
F i g u r e 2: K a l m a n
Filter E s t i m a t e s o f t h e B u s i n e s s C y c l e
2 x S ta n d a rd E rro r B a n d s ,
1 9 5 5 :1 -1 9 9 7 :2
Figure 3: Estimated Industry S h o c k s
1955:1 - 1 9 9 7 : 2
RetailTrade
Figure 4: Turbulence Measures
Rissman
F i g u r e 5: R i s s m a n
Lilien
v s . Lilien
F i g u r e 6:
1957
1963
1969
Natural R a t e Estimates
1975
1981
1987
1993
F i g u r e 7: N a t u r a l R a t e E s t i m a t e s w i t h C o n f i d e n c e i n t e r v a l s
N u m b e r o f d ra w s = 2 0 0 0
F i g u r e 8: N a t u r a l R a t e E s t i m a t e s w i t h C o n f i d e n c e I n t e r v a l s
N u m b e r o f d ra w s = 2 0 0 0 ,
1 9 8 5 :2 = 0
Table 1: Industry parameter estimates from Kalman filtering exercise*
-------- —0-------b,2
ai
D|
b,'
CON
-0.9229***
-0.4146
1.0341*
***
0
(0.3851)
(0.2277)
(0.2339)
FIR
0.7171***
0.8812***
-0.6422***
-0.7054***
(0.1563)
(0.2539)
(0.1391)
(0.2218)
MFGD
-2.3775***
-1.9048***
1.0000
1.9422***
(0.3073)
(0.3150)
(0.3326)
MFGN
0.1266**
-0.3320***
-1.9971***
0
(0.0748)
(0.0791)
(0.1608)
TPU
-0.3862**
0
-1.0798***
0.6323***
(0.2383)
(0.2000)
(0.1989)
GOV
0.4854***
-0.5071***
-1.4068***
-1.2682***
(0.2024)
(0.1918)
(0.2999)
(0.1888)
-0.1807*
-0.2427
0.2446**
TR
0.4617***
(0.1312)
(0.2034)
(0.1256)
(0.1217)
TW
-0.3676*
0.5393***
-0.1037
-0.1711*
(0.2247)
(0.1404)
(0.1049)
(0.1309)
SRV
-0.1260
-1.0167***
1.0150***
2.1009***
(0.1708)
(0.1031)
(0.1003)
(0.0819)
MIN
-2.8618***
-1.2902
2.6158
-0.9995
(1.9276)
(0.9237)
(1.1628)
(1.1321)
t*
♦*
***
Significant at the 10% level.
Significant at the 5% level.
Significant at the 1% level.
CT|
5.1308***
(0.2808)
1.3843***
(0.1334)
2.2735***
(0.2025)
1.6477***
(0.0918)
2.7029***
(0.1472)
1.8845***
(0.1262)
1.2772***
(0.0714)
1.3726***
(0.0778)
1.0710
12.0898
T ab le 2: U n e m p lo y m e n t R a te R e g r e s s io n s , D e p e n d e n t v a ria b le is AW UR?
1
-0.0619
(0.0921)
0.0190
(0.0531)
0.6357***
(0.0694)
0.0440**
(0.0206)
-0.0233
(0.0367)
—
2
0.0228
(0.0774)
0.0501
(0.0502)
0.5309***
(0.0788)
—
Z'crtS
—
—
OL,t
—
OL
0.0294***
(0.0048)
-0.0323***
(0.0049)
—
E W
—
—
0.5800
55.7376***
0.0190
0.4286
45.7257
0.1285
c
SMPL
AWURU1
Oi
Eton
of
E*OL.U
Ft
Q(36)
p-value
0.4224
44.6190
0.1535
3
-0.0391
(0.0545)
0.0150
(0.0535)
0.0649***
(0.0680)
4
-0.0048
(0.0434)
0.0475
(0.0469)
0.5631***
(0.0789)
—
—
0.0046*
**
(0.0024)
-0.0007
(0.0052)
—
“
—
—
—
—
0.0005***
(0.0001)
-0.0005***
(0.0001)
0.6111
54.2921**
0.0258
5
0.0167
(0.2038)
0.0152
(0.0572)
0.6177***
(0.0709)
0.0658
(0.0629)
-0.1049
(0.1371)
-0.0029
(0.0074)
0.0119
(0.0200)
—
—
0.4388
40.6760
0.2720
6
-0.0181
(0.1561)
0.0336
(0.0496)
0.5668***
(0.0796)
—
""
—
—
-0.0055
(0.0132)
0.0067
(0.0149)
0.0005***
(0.0002)
-0.0006***
(0.0002)
0.6127
65.6680***
0.0018
+Eight lags ofdie sectoral shiftsmeasures are used. Standard errors appear in parentheses below estimates
ofthe parameter. Statisticsreported forjc(1 to 8 )= Si* Piwhere Pi isthe estimated coefficienton x,^. The
statisticQ(36) isthe Ljung— Box Q statisticusing 36 lags ofthe residual. The marginal significance level
ofthe 0 -statisticisreported below the 0 -statistic.
*
**
***
Significant atthe 10% level.
Significant atthe 5% level.
Significant atthe 1% level.
T ab le 3: U n e m p lo y m e n t R a te R e g re s s io n s , D e p e n d e n t v a ria b le is AW UR .f
c
SMPL
A W U R f.i
Ot
Ot
~ zT o
1
2
3
4
5
6
-0.1159
(0.1710)
0.0652
(0.0415)
0.1928***
(0.0744)
0.0884**
(0.0438)
-0.0488
(0.1091)
-0.0087*
(0.0052)
0.0070
(0.0151)
0.02310
(0.1553)
0.0903*
**
(0.0430)
0.2792***
(0.0811)
—
0.0029
(0.2054)
0.0301
(0.0595)
0.6164***
(0.0717)
0.0755
(0.0645)
-0.1075
(0.1379)
-0.0040*
(0.0076)
0.0113
(0.0201)
-0.0073
(0.1582)
0.0439
(0.0518)
0.5682***
(0.0806)
— ■
0.0493
(0.2025)
0.0122
(0.0567)
0.6243***
(0.0708)
0.0520
(0.0637)
-0.1139
(0.1362)
-0.0008
(0.0076)
0.0131
(0.0198)
-0.0066
(0.0156)
0.0357
(0.0495)
0.5791***
(0.0797)
<*L,t
£ * &L.U
°L
“
Z o 'N B R D u
-0.0215**
(0.0100)
—
Zo'UAGMu
**“
R*
Q (36)
p -v a lu e
0.7303
44.3900
0.1855
—
““
""
-0.0101
(0.0109)
-0.0082
(0.0131)
0.0003*
(0.0002)
-0.0001
(0.0002)
-0.0343**
(0.0127)
—1
0.7444
61.84**
0.0047
—
““
—
—
—
-0.0061
(0.0134)
0.0060
(0.0151)
0.0005***
(0.0002)
-0.0006**
(0.0002)
—
—
-12.7681
(14.7012)
0.4335
35.4229
1.4959
-7.8277
(12.1779)
0.6070
64.0315***
0.0027
—
—
—
-0.0056
(0.0132)
0.0057
(0.0149)
0.0005**
(0.0002)
-0.0006**
(0.0002)
—
0.2383
(2.8581)
—
-1.3595
(2.4321)
—
0.4491
45.1399
0.1413
0.6158
77.8831***
0.0001
—
—
—
* Eight lags of die sectoral shifts measures are used. Standard errors appear in parentheses below estimates
of die parameter. Statistics reported for x(l to 8) - Zi* Pi where Pj is the estimated coefficient on x„. The
statistic Q (3 6 ) is the Ljung— Box Q statistic using 36 lags of the residual. The marginal significance level
of the ^-statistic is reported below the ^-statistic.
*
**
***
Significant at the 10% level.
Significant at the 5% level.
Significant at the 1% level.
T ab le 3: U n e m p lo y m en t R ate R e g re s s io n s , D e p e n d e n t v a ria b le is AW UR.*
c
SMPL
AWURt.i
crt
ZtOti
of
OL,t
7
8
9
10
11
12
-0.0494
(0.1778)
0.0355
(0.0482)
0.1849**
(0.0750)
0.0937**
(0.0447)
-0.0854
(0.1111)
-0.0097*
(0.0053)
0.0118
(0.0154)
—
0.2629*
*
(0.1565)
0.0700
(0.0462)
0.2568***
(0.0846)
—
-0.1673
(0.1681)
0.0752*
(0.0406)
0.1708**
(0.0769)
0.0870*
(0.0450)
•0.0433
(0.1068)
-0.0073*
(0.0054)
0.0099
(0.0149)
"**
0.2277
(0.1528)
0.1005**
(0.0424)
0.2477***
(0.0836)
—
-0.0430
(0.1760)
0.0301
(0.0486)
0.1722**
(0.0760)
0.0837*
(0.0454)
-0.1020
(0.1082)
-0.0077
(0.0054)
0.0176
(0.0150)
—
0.2765*
(0.1533)
0.0817*
(0.0464)
0.2370***
(0.0855)
—
-0.0236**
(0.0102)
—
-0.0127
(0.0111)
-0.0063
(0.0130)
0.0003**
(0.0002)
-0.0001
(0.0002)
-0.0370***
(0.0129)
—
25.3528
(21.6419)
0.7407
46.3546
0.1157
26.6063
(20.6176)
0.7525
57.8124**
0.0120
Ei crL,u
—
vL,t
—*
—*
Zo*Cu
Xo'NBRDu
Zo'UAGMu
R*
Q(36)
p-value
—
—
-0.0073
(0.0111)
—
-0.0124
—
(0.0132)
—
0.0003*
—
(0.0002)
—
-0.0003**
—
(0.0002)
-0.0203**
-0.0361***
-0.0233**
(0.0099)
(0.0127)
(0.0099)
-12.1495*** -10.1473*** -12.1418***
(3.9703)
(3.7060)
(3.9719)
—
—
35.6795
(22.9826)
0.7462
0.7544
0.7622
34.8840
68.7264***
41.0968
0.5215
0.0008
0.2571
—
—
—
-0.0100
(0.0110)
•0.0119
(0.0131)
0.0003*
(0.0002)
-0.0005**
(0.0002)
-0.0372***
(0.0126)
-10.1727***
(3.7425)
26.9380
(22.4162)
0.7679
63.3247***
0.0009
+Eight lags ofthe sectoral shiftsmeasures are used. Standard errors appear in parentheses below estimates
ofthe parameter. Statisticsreported forx(l to 8 )= Zi* Piwhere Pi isdie estimated coefficient onxt-i. The
statisticQ(36) isdie Ljung— Box Q statisticusing 36 lagsoftheresidual. The marginal significance level
ofthe ^-statistic isreported below die ^-statistic.
*
**
***
Significantatthe 1 0% level.
Significant atthe 5% level.
Significant atthe 1% level.
T ab le 4.a: OLS Inflation R e g re s s io n R e su lts, D e p e n d e n t V ariable is APCE .+
0.2
a
M
APCE,.,
A P C E ,.2
APCE,*
APCEm
GAP,
GAP,.,
GAP,.2
gapm
GAP„
GAP,*
ZoGAPu
R?
Q(36)
p-value
U*0
a and a2
1
2
3
4
5
6
0.6034*
*
(0.3220)
-0.3430***
(0.0802)
-0.1981**
(0.0852)
-0.0334
(0.0850)
-0.1234
(0.0796)
-0.3837
(0.3612)
-0.8608*
(0.7257)
1.3931
(0.8095)
0.7178
(0.7932)
-2.6082***
(0.7054)
1.5600***
| (0.3528)
-0.1817**
(0.0892)
0.2836
30.5527
0.7250
3.2321
0.7501**
(0.3440)
-0.3052***
(0.0853)
-0.2964***
(0.0882)
0.0173
(0.0892)
-0.0873
(0.0827)
-1.0221***
(0.3256)
0.8654*
(0.4400)
0.0584
(0.1082)
-0.2975
(0.2077)
-0.1963
(0.1403)
0.3692*
(0.2123)
-0.2229**
(0.0953)
0.1741
36.6890
0.4367
3.3652
0.4827
(0.3113)
-0.3383***
(0.0804)
-0.1894**
(0.0848)
-0.0198
(0.0848)
-0.1139
(0.0794)
-0.3932
(0.3621)
-0.8353
(0.7304)
1.4745*
(0.8162)
0.5969
(0.8048)
-2.5460***
(0.7169)
1.5558***
(0.3564)
-0.1473*
(0.0858)
0.2778
30.8623
0.7113
3.2758
0.6186*
(0.3322)
-0.2975***
(0.0852)
-0.2834***
(0.0880)
0.0282
(0.0890)
-0.0804
(0.0829)
-1.0272***
(0.3260)
0.9155**
(0.4423)
0.0591
(0.1088)
-0.3176
(0.2090)
-0.2090
(0.1412)
0.3935*
(0.2130)
-0.1853**
(0.0915)
0.1698
35.4977
0.4923
3.3384
0.5331
(0.3776)
-0.3299***
(0.0813)
-0.1824**
(0.0855)
-0.0172
(0.0854)
-0.0857
(0.0801)
-0.2846
(0.3738)
-1.1339
(0.7498)
1.8540**
(0.8393)
0.2352
(0.8302)
-2.2451***
(0.7400)
1.4340***
(0.3688)
-0.1403
(0.0914)
0.2580
34.1835
0.5552
3.8004
0.7057*
(0.4004)
-0.2970***
(0.0853)
-0.2761***
(0.0883
0.0400
(0.0893)
-0.0764
(0.0833)
■0.9934***
(0.3354)
0.8521*
(0.4521)
0.0671
(0.1116)
-0.2811
(0.2131
-0.1861
(0.1440
0.3590
(0.2187
-0.1824*
(0.0970
0.1562
37.6199
0.3949
3.869
fThe dependent variablex has been constructed as follows: Lety be therelevant index. Then
x, =A(ln yrln
where A isthe 1period difference operator. Columns (1),(3),and (5)have
been estimated without constraints. Columns (2),(4),and (6 )show Shillerestimates ofthe GAP
parameters with p0estimated without constraint. Columns (1) and (2) reflectGAP estimates
derived from turbulence measure o. Columns (3) and (4)reflectGAP estimates derived from
turbulence measure o2. Columns (3) and (6 )reflectGAP estimates derived from turbulence
measures o and o2.The statisticQ(36) isdie Ljung— Box Q statisticusing 36 lags ofthe residual.
The marginal significance level ofthe ^-statisticisreported below die £>statistic.
*
**
***
Significantatthe 10% level.
Significantatthe 5% level.
Significantatthe 1% level.
T ab le 4 .b : O LS Inflation R e g re s s io n R e s u lts , D e p e n d e n t V ariab le is APCEX?
G.2
cJ
APCEt-1
APCEt-2
APCEX,*
APCEX^
GAPt
GAP„
GAPt-a
gapm
GAP„
gapm
L?GAPU
R*
Q(36)
p-value
U\
a and a2
1
2
3
4
5
6
0.7278*
(0.3882)
-0.6518***
(0.0849)
-0.3680***
(0.0994)
-0.1991**
(0.0992)
-0.1099
(0.0820)
-1.0404**
(0.4423)
1.3603
(0.8883)
-0.0992
(0.9814)
-0.1226
(0.0964)
-0.9603
(0.8742)
0.6526
(0.4435)
-0.2100**
(0.1078)
0.3029
47.2819*
0.0988
3.4734
0.7452*
(0.3871)
-0.6469***
(0.0833)
-0.3626***
(0.0989)
-0.2083**
(0.0982)
-0.1076
(0.0816)
-1.0489***
(0.3793)
1.3729***
(0.5156)
0.0351
(0.1198)
-0.5682**
(0.2456)
-0.4365***
(0.1630)
0.4313*
(0.2551)
-0.2144**
(0.1074)
0.3010
45.7535
0.1279
3.4757
0.5997
(0.3772)
-0.6467***
(0.0848)
-0.3594***
(0.0992)
•0.1948*
(0.0991)
-0.1061
(0.0820)
-1.0343**
(0.4440)
1.4260
(0.8937)
-0.1665
(0.9882)
-0.1621
(0.9720)
-0.8921
(0.8845)
0.6566
(0.4464)
-0.1724
(0.1042)
0.3029
46.7505
0.1082
3.2758
0.6126
(0.3761)
-0.6436***
(0.0832)
-0.3541***
(0.0987)
-0.2023**
(0.0981)
-0.1044
(0.0817)
-1.0224***
(0.3813)
1.3798***
(0.5204)
0.0192
(0.1201)
-0.5863**
(0.2476)
-0.4363***
(0.1647)
0.4700
(0.2553)
-0.1760*
(0.1038)
0.2957
45.3567
0.1364
3.4807
0.7729*
(0.4537)
-0.6519***
(0.0850)
-0.3553***
(0.1004)
-0.1924*
(0.1004)
-0.0986
(0.0828)
-0.9202**
(0.4552)
1.0197
(0.9087)
0.2226**
(0.9958)
-0.2966
(0.9758)
-0.7490***
(0.8913)
0.5307
(0.4558)
-0.1928*
(0.1101)
0.2896
47.0920
0.1021
4.0085
0.7954*
(0.4521)
-0.6453***
(0.0840)
-0.3538***
(0.1001)
-0.1998**
(0.0994)
-0.1016
(0.0823)
-0.9890**
(0.3939)
1.2297**
(0.5369)
0.0616*
(0.1241)
-0.4747*
(0.2555)
-0.3782**
(0.1697)
0.3522
(0.2653)
-0.1984*
(0.1096)
0.2882
45.3134
0.1374
4.0091
*The dependent variablex has been constructed as follows: Lety be the relevant index. Then
x, =A(lnyrlny,_J, where A isthe 1period difference operator. Columns (1), (3), and (5)have
been estimated without constraints. Columns (2),(4), and (6 )show Shillerestimates ofthe GAP
parameters with J30 estimated without constraint. Columns (1) and (2) reflectGAP estimates
derived from turbulence measure a. Columns (3) and (4)reflectGAP estimates derived from
turbulence measure a2. Columns (5) and (6 )reflectGAP estimates derived from turbulence
measures a and o2.The statisticQ(36) isthe Ljung— Box Q statisticusing 36 lags ofthe residual.
The marginal significance level ofthe ^-statistic isreported below the ^-statistic.
*
**
***
Significant at the 10% level.
Significant at the 5% level.
Significant at the 1% level.
T ab le 4 .c: OLS Inflation R e g re s s io n R e su lts, D e p e n d e n t V ariable is ACPI-U.r
G.2
O
p
ACPbU,.,
ACPbU,.2
ACPI-U,.*
ACPbUm
GAP,
GAP,.,
GAP,,2
GAPM
GAP,^
GAP,*
Zo'GAPu
R*
Q(36)
p-value
U*0
a and a2
1
2
3
4
5
6
1.0186**
(0.4309)
-0.4034***
(0.0834)
-0.3497***
(0.0896)
-0.0086**
(0.0897)
-0.1177
(0.0822)
-0.5643
(0.4812)
-1.8570*
(0.9572)
2.8915***
(1.0579)
-0.0775
(1.0708)
-2.0237**
(0.9779)
1.3285***
(0.4952)
1.1675**
(0.4512)
-0.3809***
(0.0866)
-0.4432***
(0.0906)
0.0294
(0.0913)
-0.0639
(0.0837)
-1.4463***
(0.4332)
0.7179
(0.5992)
0.2380
(0.1488)
-0.0145
(0.2866)
-0.0321
(0.1930)
0.1926
(0.2941)
0.8502**
(0.4154)
-0.3950***
(0.0834)
-0.3326***
(0.0890)
0.0059
(0.0873)
-0.1104
(0.0815)
-0.5663
(0.4820)
-1.8824*
(0.9647)
3.1251***
(1.0688)
-0.2952
(1.0866)
-1.9782**
(0.9912)
1.3424***
(0.4983)
0.9921**
(0.4363)
-0.3693***
(0.0866)
-0.4268***
(0.0904)
0.0415
(0.0910)
-0.0556
(0.0838)
-1.4840***
(0.4346)
0.8350
(0.6043)
0.2482*
(0.1494)
-0.0541
(0.2892)
-0.0644
(0.1950)
0.2250
(0.2957)
0.9423*
(0.5059)
-0.3815***
(0.0840)
-0.3383***
(0.0893)
0.0164
(0.0879)
-0.0902
(0.0821)
•0.5530
(0.4986)
-1.8761
(0.9933)
3.1415**
(1.1034)
-0.5281
(1.1208)
-1.5907
(1.0194)
1.1628**
(0.5145)
1.1045**
(0.5265)
-0.3635***
(0.0863)
-0.4207***
(0.0900)
0.0537
(0.0909)
•0.0535
(0.0840)
-1.4425***
(0.4454)
0.7680
(0.6132)
0.2515
(0.1523)
-0.0202
(0.2929)
-0.0398
(0.1973)
0.1995
(0.3020)
-0.3024**
(0.1193)
0.3707
38.8271
0.3435
3.3690
-0.3444***
(0.1249)
0.3042
34.3200
0.5486
3.3900
-0.2545**
(0.1144)
0.3700
39.4421
0.3187
3.3408
-0.2943**
(0.1201)
0.2993
34.3077
0.5492
3.3710
-0.2436**
(0.1224)
0.3459
39.6741
0.3096
3.8686
-0.2833**
(0.1274)
0.2853
35.4220
0.4959
3.8987
fThe dependent variablex has been constructed as follows: Let.ybe the relevant index. Then
x, =A(lnyrlny,_i), where A isthe 1 period difference operator. Columns (1),(3), and (5)have
been estimated without constraints. Columns (2),(4), and (6) show Shillerestimates ofthe GAP
parameters with fioestimated without constraint. Columns (1) and (2)reflectGAP estimates
derived from turbulence measure a. Columns (3) and (4)reflectGAP estimates derived from
turbulence measure a 2. Columns (3) and (6) reflectGAP estimates derived from turbulence
measures a and o2.The statisticQ(36) isthe Ljung— Box Q statisticusing 36 lagsofthe residual.
The marginal significance level ofthe ^-statisticisreportedbelow the ^-statistic.
*
**
***
Significant at the 10% level.
Significant at the 5% level.
Significant at the 1% level.
T ab le 4 .d: OLS Inflation R e g re s s io n R e s u lts , D e p e n d e n t V ariab le is ACPIXS
I
p
ACPfXf.1
acpix,
.2
ACPIXm
ACPIXm
GAPt
GAP„
GAPt.2
GAPt.3
GAPM
GAPM
IoaGAPu
F?
Q(36J
p-value
U*o
o.2
CJ
I
1
1.1775***
(0.4182)
-0.4035***
(0.0875)
-0.2474**
(0.0960)
0.0080
(0.0949)
-0.0990
(0.0871)
-0.4899
(0.4847)
-1.3132
(0.9402)
2.1988**
(1.0178)
-1.0122
(1.0177)
0.2159
(0.9301)
0.0573
(0.4782)
-0.3432***
(0.1157)
0.2351
39.8533
0.3027
3.4309
a and o2
2
3
4
5
6
0.2173***
(0.4227)
-0.3934***
(0.0881)
-0.2796***
(0.0957)
0.0162
(0.0954)
-0.0773
(0.0865)
-0.9496**
(0.4330)
0.1645
(0.5960)
0.2159
(0.1545)
0.1913
(0.2840)
0.0946
(0.1915)
-0.0713
(0.2871)
-0.3545***
(0.1169)
0.2128
36.5482
0.4432
3.4339
1.0286**
(0.4065)
-0.3924***
(0.0876)
-0.2367**
(0.0957)
0.0126
(0.0948)
-0.0991
(0.0872)
-0.4622
(0.4894)
-1.3150
(0.9569)
2.2930**
(1.0397)
-1.1789
(1.0394)
0.3021
(0.9477)
0.0607
(0.4838)
-0.3004***
(0.1118)
0.2258
40.7248
0.2703
3.4241
1.0631**
(0.4112)
-0.3814***
(0.0881)
-0.2687***
(0.0956)
0.0211
(0.0952)
-0.0774
(0.0866)
-0.9410**
(0.4360)
0.2267
(0.6021)
0.2045
(0.1548)
0.1523
(0.2871)
0.0743
(0.1936)
-0.0270
(0.2897)
-0.3103***
(0.1131)
0.2021
37.1808
0.4144
3.4260
1.2284**
(0.4874)
-0.3880***
(0.0878)
-0.2229**
(0.0959)
0.0373
(0.0949)
-0.0801
(0.0872)
-0.7076
(0.5003)
-0.9656
(0.9781)
2.0511*
(1.0613)
-0.8560
(1.0603)
0.0889
(0.9653)
0.0755
(0.4942)
-0.3107***
(0.1179)
0.2241
37.6271
0.3946
3.9540
1.2748**
(0.4908)
-0.3745***
(0.0880)
-0.2516***
(0.0954)
0.0413
(0.0948)
-0.0640
(0.0867)
-1.1256**
(0.4445)
0.3797
(0.6141)
0.2766**
(0.1573)
0.1654
(0.2919)
0.0499
(0.1973)
-0.0674
(0.2953)
-0.3241***
(0.1187)
0.2069
35.8142
0.4774
3.9334
TThe dependent variablex has been constructed as follows: Let.ybe the relevant index. Then
x, =A(lnyrlnyh4), where A isthe 1period difference operator. Columns (1),(3), and (5)have
been estimated without constraints. Columns (2),(4), and (6) show Shiller estimates of the GAP
parameters with p0 estimated without constraint. Columns (1) and (2) reflectGAP estimates
derived from turbulence measure o. Columns (3)and (4)reflect GAP estimates derived from
turbulence measure o2. Columns (5) and (6)reflectGAP estimates derived from turbulence
measures a and o2.The statisticQ(36) isthe Ljung— Box Q statisticusing 36 lagsofthe residual.
The marginal significance level ofthe ^-statistic isreported below die ^-statistic.
*
**
***
Significant at the 10% level.
Significant at the 5% level.
Significant at the 1% level.
@FedFRASER