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USING NOISY INDICATORS TO
M EASURE PO T EN TIA L O U TPUT
Kenneth N. Kuttner
Working Paper Series
Macro Economic Issues
Research Department
Federal Reserve Bank of Chicago
June, 1991 (WP-91-14)
Using Noisy Indicators to
Measure Potential Output
Kenneth N. Kuttner*
Introduction
Potential output, like the real interest rate and the natural rate of unemployment,
is one of the key unobservable economic variables of macroeconomics and mon
etary policy. While there is little agreement on which observable quantity best
substitutes for potential output, most economists would endow it with a variety of
distinct, but not mutually exclusive, attributes. Arthur Okun’s original definition
emphasizes inflation. According to this interpretation, potential output is the level
of production that does not add to inflationary pressures. A second characteriza
tion is as the sustainable level of output — that is, a level consistent with stable
future rates of output growth. A third corresponds to the full-employment level of
production, where unemployment is equal to the natural rate.1
This paper synthesizes these definitions by modeling real GNP growth, inflation,
and the unemployment rate as three distinct indicators of a single underlying unob
servable variable, which we call potential output or potential GNP. The resulting
model is an example of a dynamic multiple indicator specification, which opti
mally combines data from these imperfect sources in an estimate of the underlying
level of potential output. 2
This multiple-indicator approach to measuring potential output has two chief ad
vantages relative to more traditional measures. The first is that by incorporating
a number of alternative definitions of potential, it constructs a best-fit measure
based on a variety of indicators. Such a composite promises to be more precise
and reliable than measures based on only a single proxy variable.
A second advantage is that this technique delivers natural measures of the standard
error associated with the estimates of potential output. Although Okun himself
acknowledged that it was “at best an uncertain estimate, and not a firm, precise
measure,” the estimates in this paper are the first to include a set of confidence
* Senior economist, Economic Research Department, Federal Reserve Bank o f Chicago, 230 South
LaSalle Street, Chicago, 1L 60604. A previous version of this paper bore the title “A N ew Approach
to Non-Inflationary Potential GNP.” This version has benefitted from discussions with Marty Eichenbaum, Steve Strongin, and Mark Watson, and from seminars at Northwestern University and the Federal
Reserve Bank of Chicago.
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1
bounds. Some gauge of the precision of the estimates are essential to policymakers
acting on the basis of a measure of the economy’s potential.
A b rief history o f potential output
The origins of the potential output idea can be traced to Arthur Okun’s work from
the 1960s. He viewed it as essentially an aggregate supply measure, or an index
of productive capacity, describing it as “the maximum production without infla
tionary pressure, ...or more precisely... the point of balance between more output
and greater stability.”3 Deviations of output from potential were attributed to an
inappropriate level of aggregate demand. While his definition of potential output
rests on its relationship to inflation, as an empirical matter, Okun used the unem
ployment rate as the sole indicator of the gap between output X and potential X*,
according to the equation that became known as “Okun’s Law”:
X* = X[1 + 0.032 (t/-t/* )],
where the natural rate of unemployment, U*, was assumed to be a constant 4 per
cent
Until the 1970s, a simple log-linear trend served as an excellent approximation to
the potential output series derived from Okun’s Law with a time-invariant natural
rate. (In fact, Okun himself recommended the trend method, preferring to ignore
the “wiggles and jiggles” in the measure introduced by random fluctuations in the
unemployment rate.) After the supply shocks of the 1970s, however, economists
interested in the relationship between aggregate demand and inflation recognized
the importance of changes in the growth rate of potential output, and replaced the
linear trend with some time-varying measure of potential, as in Gordon (1975,
1977).
Most traditional measures of potential output have employed one of three alter
native techniques in their construction: segmented trends, adjustments to Okun’s
Law, or an aggregate production function. The segmented trend group includes
some of the more familiar measures: the high-employment and mid-expansion se
ries of the Bureau of Economic Analysis, for example, which use business cycle
dates as the basis for the segments’ endpoints. Other examples are the Federal Re
serve Board’s new potential GNP series,4 and the “trend output” series in Gordon
(1990a, 1990b).
The best example of the modem approach to Okun’s Law is Clark (1983), who
suggests using the smoothed difference between the observed unemployment and
a time-varying natural rate as the basis for potential output Those who have pro
posed aggregate production function measures include Perry (1977), Clark (1979),
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2
Perloff and Wachter (1979), and (indirectly) Boschen and Mills (1990). Among
these studies, the Perloff and Wachter paper deserves note; while based on an ag
gregate production function, their measure is unique for its use of inflation data as
an indicator of an unobserved time-varying production function parameter.
The technique described in this paper bears a resemblance to that of Perloff and
Wachter, in that it uses the inflation rate to extract information on an unobserv
able. However, here potential output is itself the unobservable of interest, and the
latent-variable econometric framework explicitly addresses the statistical noise as
sociated with its indicators. As decomposition of aggregate output fluctuations
into persistent and transitory components, it is also related to the unobserved com
ponent model of Watson (1986), and the structural VARs of Blanchard and Quah
(1989) and Galf (1989).
Potential output as an unobserved stochastic trend
Constructing a meaningful potential output series first requires specifying the form
of its time variation. An attractive alternative to the traditional segmented-trend
method is to model potential as a stochastic trend, along the lines suggested by
Stock and Watson (1988). Specifically, (log) potential output, x*, is assumed to
follow an integrated stochastic process — a random walk with drift:
Ax* = p* + £,.
(1)
This specification allows its growth rate to depart from its long-run growth rate
p* over the short run, generating persistent deviations from a deterministic linear
trend.
Measured real GNP can now be modeled as an error-correction equation in which
(log) real output, x, reverts to potential over time:
8x(L)Axt = p* + 0(x* - xt) + ^(L)^*.
(2)
The S*(L) and XX(L) lag polynomials and the coefficient 0 are to be estimated;
the constant term p* is set equal to 8*(l)p* to ensure that potential and measured
output grow at the same rate on average. The innovation is uncorrelated with
e( by assumption.
In the context of the unit roots literature, (1) and (2) imply that potential and mea
sured real output are 1(1) processes, while their difference — the “output gap” —
is stationary. Despite the fact that x* is unobservable, the interpretation of errorcorrection mechanism is straightforward: when output exceeds potential, real GNP
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growth is expected to fall below its mean. Conversely, output below potential im
plies higher-than-average future growth rates. The disturbance term rtf induces
purely transitory deviations in output from potential.5
Viewed as a latent variable model, (2) says that Ax, (real GNP growth) may be used
as an indicator of the unobserved x*. Thus, with the appropriate restrictions on the
dynamics of the error-correction equation, the two equations may be estimated,
and an estimate of potential output may be extracted. This is particularly easy to
verify in the one-indicator case, as the two equations may be rearranged to express
Ax, in canonical form6 as an ARMA process with a composite error term:
[5X(L)A + 0]Ax, = 6e, + AX*(Z,)itf.
Therefore, while the model’s spectral density is that of an ARMA, estimating (1)
and (2) parameterizes the spectrum of Ax in terms of two distinct shocks; inno
vations to the extracted potential output “signal” correspond to the low-frequency
potion of the spectrum of real GNP growth. Further discussion of the models’
spectral properties appears in a subsequent section of the paper.
The real payoff to the latent variable methodology comes from the use of addi
tional variables as indicators of the level of potential output. This improves the
precision of the estimates, and it adds to the economic interpretation of the model,
associating the unobserved x* with alternative descriptions of potential output
We begin by explicitly linking potential output to the rate of inflation by way of a
simple dynamic aggregate supply relation:
8*(L)Ajt, = a (L)(x, - x*) + X”(L)q? + 7 v„
(3)
where An, is the change in the inflation rate, the vector v, includes exogenous de
terminants of inflation not related to the current state of aggregate demand, and
q* is a disturbance term. Specification in terms of the change in inflation is ap
propriate given the apparent non-stadonarity of the inflation rate over the 1960-90
sample period. It also means that an output gap equal to zero is consistent with
any constant rate of inflation, rather than zero inflation; in other words, potential
GNP corresponds to the “non-accelerating-inflation” (or, more precisely, the nonincrearing-inflation) level of output Excluding Ax, from direct inclusion in the
equation for Art, imposes a factor structure on the co-movements of output and
inflation growth; all covariation between the two is assumed to come from the
common factor, (x* -x ,).
With the addition of inflation as a second indicator of the unobservable level of
potential output, equations 2-4 form what is referred to as a dynamic multiple in
dicator model.7 The specifications discussed in the following section of the paper
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are one-indicator output-only and two-indicator output-inflation versions. A sub
sequent section of the paper introduces the unemployment rate as a third indicator
of potential.
Estimating the model
For the purposes of estimation, the indicator equations (2) and (3) need to be re
stricted; 8X(L) is taken to be a second-order lag polynomial, while the \ X(L) and
XK(L) moving-average polynomials are both assumed to be first-order. The §”(L)
polynomial is set to unity, so that inflation changes are entirely a function of the
output gap, plus a moving-average error term. These restrictions are consistent
with (unreported) Wald tests of more profligate specifications.
Next, setting a (L) equal to aiL + a 2D 2 specifies inflation rate changes as a func
tion of two lags of the output gap. Finally, one exogenous variable is chosen for
inclusion in v(: the lagged difference between the growth rates of M2 and nominal
income, or the opposite of the differenced velocity of M2. The idea, which is con
sistent with an error-correction mechanism between the growth rates of M2 and
nominal income, is that this difference represents the inflationary pressure from
any monetary expansion not yet incorporated into nominal income.
One convenient way to estimate multiple indicator models like the one presented
here is through the application of the Kalman Filter to its state-space representa
tion,
z, = Az,_i + B e, + Gw,
y, = C z, + D u, + Hwt,
where z, y and w are a vectors of state, observable and exogenous variables, re
spectively. The model in equations 1-3 can be expressed by including current and
lagged values of x*, qx and q* in the state vector. The observable y, obviously
includes An, and Ax,, while w, includes a vector of ones, lagged observables, and
the monetary term discussed above.
As described in Harvey (1981, 1989), the Kalman filter computes the minimum
mean square error forecast of the period t state vector and its error variance, con
ditional on period t - 1 information, according to the prediction equations,
z<|»-1 = Azm m + Gwm
fltt'-i =
+ B B '.
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Conditional on period t data, the filter extracts the optimal estimate of the state vec
tor, and computes the period t signal-extraction error,
by way of the updating
equations,
z<|t = ztp—
1 + Kt £y<—( C z ^ i +
Qn, =
-K tC O m -u where
K t^ a ^ iC 'ic a ^ iC '+ D D '}.
If the elements of the matrices A, B, C, D , G, and H were known, extracting esti
mates of the unobserved z( and its variance would be trivial. As they are not known,
the series of uncorrelated prediction errors generated by the Kalman filter can be
used to evaluate a likelihood function under the assumption of normally distributed
disturbances. Standard optimization routines8 can compute maximum-likelihood
estimates of the parameters.
The one additional piece of information required for estimation is the initial value
of the state vector, zo, and the variance associated with that value. The Bayesian
interpretation is that zo embodies prior information on the value of the state vector
at the beginning of the sample. The estimates are not generally sensitive to the
choice of z0: as of 1959:4, log potential GNP is assumed to equal the Federal
Reserve Board’s estimate of7.428, with a standard error of 5%.9 The initial values
of the q disturbances are set to zero, with standard errors of 3%.
The lack of consistent pre-1959 money stock data requires beginning the estima
tion in 1960; the sample ends in 1991:1, for a total of 125 observations. While
the obvious measure of real output is standard NIPA real GNP series, the choice
between alternative price series is less clear. Because it is less volatile and less
subject to fluctuations in oil prices, the fixed-weight GNP deflator is used as the
basis for the inflation rate.10 Relative to the implicit deflator, the fixed-weight
version is preferable as a better measure of pure price changes.11
Param eter estim ates
Maximum-likelihood estimates of the one- and two- indicates' models appear in Ta
ble 1. The estimated 6 and oe (respectively, the coefficient on the error-correction
term and the standard deviation of the potential output innovation) generally suppot the idea of time-varying potential GNP. Although these two parameters are
not individually significant in the single-indicator model (specification I), they are
much more precisely estimated when the inflation rate is included as a second in
dicator in specification II; their asymptotic standard errors fall by one-half. As oe
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is expressed in percentage terms, its specification n estimate of 0.73 corresponds
to a quarterly potential GNP shock with a variance of roughly half a percent
Both models yield estimates of p*. the mean growth rate of potential output close
to 0.74% on a quarterly basis (2.96% annualized). The estimates of the standard
deviation of the transitory qx shocks are both roughly 0.9S, again in quarterly per
centage terms. In contrast the estimated lag polynomials 8* and Xx change some
what from specification I to n. However, these changes are small relative to the
coefficients standard errors, and have little impact on the models’ spectral density.
Because specification I (except for the effect of the initial conditions) is essentially
a reparameterized ARMA(3,2), its maximized likelihood is virtually equal to that
of the estimated ARMA.
The estimate of aggregate supply relation in specification II produces no surprises.
The a coefficients indicate a strong effect of aggregate demand on inflation. The
estimates of cti and ot2 suggest a specification in toms of the first difference of
the output gap. However, including the level of the output gap allows aggregate
demand to contribute to some of the non-stationarity in the inflation rate, which the
first-difference specification would not allow. Finally, y, the coefficient on lagged
adjusted M2 growth, appears as a significant additional determinant of inflation.
Extensions to the basic m odel
The models discussed above used output growth and the inflation rate as indicators
of the latent potential GNP. This section presents two extensions to the basic model,
both of which include the unemployment rate as a third indicator variable.
Including unemployment data
The foundation for the unemployment indicator is a modified version of Okun’s
Law:
8«(L)i/r=t/ ; + -*)+%,
(4)
where <t>(L) and 8U(L) are lag polynomials specific to the unemployment equation,
and is the disturbance term. The XJ*t term represents the natural rate of unem
ployment.
There are two problems with using this equation as an indicator of the output gap.
The first is the problem of time variation in the natural rate. While it was thought
to be roughly constant at 4% in the 1960s, its apparent upward trend in the 1970s
led to considerable research on time-variation in its structural component.12 A
related problem is the non-stationarity of the unemployment rate. Perhaps due to
highly persistent changes in its structural component, the usual statistical tests are
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Table 1
Basic Estimation Results
Specification I
Estimate
Std Error
Specification n
Estimate
Std Error
n*
Oe
0.730
0.553
0.090
0.618
0.740
0.728
0.075
0.278
6
0.064
0.276
0.215
-0.001
0.940
0.070
0.442
0.158
0.456
0.069
0.106
0.435
0.200
-0.165
0.948
0.054
0.314
0.148
0.318
0.074
0.117
-0.092
-0.680
0.058
0.256
0.030
0.028
0.224
0.028
0.038
8!
85
M
ox
ai
0-2
*?
Y
0*
LLF
SEax
SEA*
.,
.
.
.
•
4.2103
0.900
*••
9.562
0.875
0.296
Specification I: single indicator
Ax* = p* + e,
A x , = (1 - 8* - SDn* + 6(x* - x ,) + 8*
+ SjAxrf + rtf + Xi Um
Specification II: adds inflation indicator
An,
= aiCtM -ac^1) + a2(X rf-x^ 2) + YACln(M2)-ln(GNP))M +n? + ^fn*i
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June 1991, WP-1991-14
unable to rejectthe null hypothesis of a unitroot in the unemployment rate over
the postwar sample.
One possible solution isto model the naturalrate as an additional non-stationary
latentvariable, and specify the measured unemployment rateas a function of the
output gap and the time-varying natural rate. However, because the focus of this
paper ispotentialGNP ratherthan thenaturalrateofunemployment, an appealing
alternative approach issimply toremove the unitrootby firstdifferencing.
This would be a perfect solution to the time-varying natural rateproblem only if
U* were a pure random walk, U*t= lft_x+v,, whose errorswere uncoirelatedwith
the stochastic components of the output gap. In this case, firstdifferencing (4)
yields:
8“(L)At/r= <KL)A(jc;- xt)+ V,+ A$f;
or,combining thev, and
8
intoa single MA(1) composite error term X “ ( L ) t i “ ,
“(L)Af/, = «
- x,)+ X“(L)nr.
(5)
whereT}*isuncorrelatedwithqxand eatalllags. Inpractice,theseassumptionsare
ratherdubious. While serialcorrelation inv(could be taken care ofby expanding
the XU(L)polynomial, the literatureon sectoral shifts suggests sources ofnonzero
correlationbetween thestructuralunemployment innovations,v, and dieotherdis
turbances,Ty*and e,violatingthedynamic factorassumptions. At best,one would
hope that first-differencing would remove most low-frequency structural unem
ployment fluctuations. Adding the unemployment rate as a third indicator of the
outputgap, therefore, may introducea tradeoffbetween bias and precision. While
the additional indicator reduces the signal extraction error associated with esti
mated potential GNP, italso introduces the possibility thatestimated fluctuations
in the output gap may be contaminated by changes in structuralunemployment.
Including population growth
A second extension involves including population as a determinant of potential
output. The simplestway todo thisistosubtractthe logarithm ofpopulation from
thelog ofpotentialoutput,x*-nt= x*,and expressper capitapotentialoutput,X*
as a stochastic trend:
Ax(
*= p* + e„
(6)
asbeforeinequation 1. The threeindicatorequations 2,3, and 5,remain thesame,
exceptthesum ofthelatentvariableX* and the (exogenous) log ofpopulation now
appear in place ofx*.
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Table 2
Extended Estimation Results
SpecificationIII
Estimate
StdError
Specification TV
Estimate
Std Error
n*
Oe
0.733
0.505
0.052
0.134
0.349
0.545
0.060
0.164
e
0.084
0.411
0.209
-0.132
0.948
0.048
0.356
0.160
0.364
0.078
0.076
0.494
0.038
0.273
0.162
0.285
0.076
0 .0 2 1
1
0%
0.096
-0.075
-0.549
0.051
0.273
<>0
2 1 .1 1
♦l
8 ?
X?
6.62
0.338
-0.283
0.170
%
«5
Ox
(Xl
a2
Ou
LLF
SEa,
SEa*
SEaii
0 .2 2 2
-0.188
0.954
0 .0 2 1
0.089
-0.072
-0.505
0.044
0.279
0 .0 2 1
2.17
3.66
0.106
0.198
0.026
21.27
7.71
0.300
-0.351
0.157
2.13
3.49
0.103
0.293
0.039
0.019
0.095
0.026
10.648
0.846
0.291
0.207
0 .0 2 2
0 .0 2 1
0.082
0.025
10.637
0.894
0.296
0.204
Specification III:adds unemployment indicator
A U, - 4>oA(j:*- x ,) + <>!A(jc(
*_!- xM )+ 6?AC/M + rtf+ X^JLi
Specification IV: potentialoutput inper capitaterms
x* = n,+5* where Ax* = p* + e,
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The results for these two models, designated specifications m and IV, appear in
table2. Adding the unemployment rateindicatorhas a relatively small impact on
most of the parameter estimates. The most noticeable change isin the size of the
shocks topotentialoutput; specificationm delivers an estimatedaeofonly 0.S0S,
compared with 0.728 in specification II.
The specificationIV resultsaresimilartothosefrom III.The main differenceisthat
p* now represents thegrowth rate ofper captiapotentialoutput, whose estimated
annual growth rate isroughly 1.4%. Surprisingly, as measured by the standard er
rorof theone-step-aheadforecasts, theoutput and inflationequations’fitdeclines
slightlyrelativetospecification in.
Estim ates o f potential GNP
Perhaps themost usefulresultsofthesemodels aretheestimatedpathsofpotential
GNP and theirstandard errors. Extracting an estimate of the latentx* isstraight
forward; conditionalon themaximum-likelihoodparameterestimates,theKalman
filtercan compute one-sided estimates ofpotentialoutputand itssignalextraction
error,denoted% and
Applying the Kalman smoothing algorithm yieldsanal
ogous two-sided estimates,x^t and Q 4 7 .
However, theseestimatesofthesignalextractionerrorneglecttheuncertaintyfrom
the parameter estimates’variance. To compute thisvariance, we adopt theproce
dure proposed by Hamilton (1986). This involves decomposing the totalvariance
of state vector element of interest, say xjjr,into the signal extraction or “filter”
uncertainty,
£ {tf-*3r.fc)2IZr.po},
and theparameter uncertainty,
{ (xqr.fL ~ x3r,Po)21Zr} ,
where Po istheestimatedparametervector,andZj denotes thedatathrough period
T. The Q ( | 7 from the Kalman smoother approximates the filteruncertainty. The
parameter uncertainty isestimated via Monte-Carlo: an artificialsample of Ps is
drawn from a multivariate normal population, generating a sample of xjjr series,
which is then used to compute the sample variance of each observation in the
series. The same procedure applies to the one-sided estimates,
Figure 1 shows thetwo-sided estimateof thepotentialGNP seriesfrom specifica
tion H (the two-indicator model with inflation), along with the 5th and 95th per
centiles of itsempirical distribution (analogous to the 90% confidence interval).
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Figure 1
Figure 2
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Figure 3
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Table 3
Sources of uncertainty in potential GNP
Specification
I
II
III
IV
Parameter variance
7.68
9.54
2.07
1.74
2.08
1.02
0.64
1.17
Overall standard error
4.07
1.75
1.54
1.80
One-sided
Filter variance
T\vo-sided
Filter variance
4.44
1.03
0.95
1.12
Parameter variance
6.80
0.65
0.40
0.87
Overall standard error
3.30
1.29
1.16
1.41
The estimated parameter variances are based on 200 draws.
The reported averages exclude the first 8 quarters o f the sample.
The solid line plots log real GNP. The local trend in potential output varies con
siderably over the sample; from a relatively rapid growth rate in the early 1960s,
it begins to slow late in the decade, a tendency which continues throughout the
1970s. In the early 1980s, however, its growth rate picks up once again.
As measured by the two-indicator specification n , the output gap is statistically
significant during only four episodes: the 19 6 5 -9 ,1 9 7 3 -4 and 1978-9 expansions
and the 1981-2 recession. The 1974-5 recession and the late-80s expansion are
marginal episodes. Here, the deviations barely exceed the two-sided 90% confi
dence bounds, but not the one-sided bounds shown in figure 2. Figure 3 shows
the one-sided estimate from specification III. Including the unemployment rate as
a third indicator makes it possible to distinguish the 7 4-75 recession, although the
late 80s remain within the 90% bounds.
Except for the fact that the parameters are estimated over the entire sample, the
one-sided estimates correspond to the estimates that would have been available
to policymakers in “real tim e.” The fact that som e o f the deviations o f output
from potential are discernible in the two-sided but not in the one-sided estim ates
illustrates the point that discerning the appropriate course o f monetary policy is
frequently easier with the benefit o f hindsight.
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Table 3 reports the average uncertainty associated with the estimates o f potential
GNP, including the separate contributions o f filter and parameter uncertainty. The
table demonstrates the significant reduction in the standard error obtained in going
from a one- to a two-sided estimate; in the case o f specification U, it shrinks by
30%. The table also shows the beneficial effect o f adding additional indicators o f
the output gap. The reduction in uncertainty is particularly spectacular in going
from specification I to II, in which the overall standard errors fall by over 60%.
Including the unemployment rate as an additional indicator yields an additional
improvement in the standard errors. M oving to the per capita specification, how
ever, increases the standard errors, perhaps due to the deterioration in the m odel’s
fit in the output equation. Except in the case o f the (somewhat overparameterized)
specification I, signal extraction contributes more uncertainty to the estim ates than
the sampling variance o f the parameter estimates.
Spectral properties o f the estim ated m odels
Another interesting way to view the results is from the frequency domain, exam
ining the spectral density function o f Ax im plicit in the specification. The contri
butions o f the two shocks to the spectrum o f the change in log real GNP can be
computed from their transfer functions:
/ at(co) = oj\
6
8*(e’ to) ( l - t r to) + e
+ o?
(l-e-^GT*0)
because the e and t |x shocks are uncorrelated, the spectral density o f Ax is just
the sum o f the two shocks’ individual contributions. The spectral density from
the two-indicator specification II appears in figure 4. Except for sm all differences
in the size o f the frequency-zero variance, the other specifications’ spectra are
virtually identical.
The shapes o f the shocks’ individual contributions reflect the error-correction
structure o f equation 2. The transitory q* shocks contribute nothing to the spectrum
at frequency zero, while the persistent e shocks contribute spectral mass only near
the zero frequency; by co = 0.S (12.5 quarters), its contribution is virtually z o o .
The decomposition o f the k-step-ahead forecast error variance, shown in table 4 , is
an alternative way to view the information in the spectral density function. W hile
almost none o f the one-quarter-ahead variance comes from the potential output
shocks, that proportion rises sharply at horizons longer than 12 quarters. A t 16
quarters, the e shock accounts for more than three-quarters o f the variance.
As mentioned earlier, the univariate representation o f Ax, im plied by these poten
tial output specifications is equivalent to an ARM A(3,2). An interesting restriction
on these specifications involves imposing 0 = 0 (which also renders o e irrelevant).
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Figure 4
Spectrum of output growth, specification II
Table 4
Decomposition of forecast variance
^-step-ahead variance, %
k
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T |x
shocks
e shocks
1
94
6
4
8
12
88
82
12
18
71
29
16
23
77
16
Figure 5
Spectrum of output growth, ARMA(2,1)
Doing so produces redundant unit roots in both the autoregressive and m ovingaverage polynomials; the (1 - L) factors cancel, and the model collapses to an
ARM A(2,1). Imposing this restriction and estimating the restricted ARMA yields
the follow ing model: (1 + 0.33L - 0.32L2)Axt = (1 + 0.59L)e,.
The comparison between the restricted and unrestricted models reveals a lot about
their low-frequency behavior. The restricted ARM A(2,1) spectrum, which appears
in figure 5, differs significantly from the unrestricted m odel’s — particularly near
frequency zero, where the ARM A(2,1) spectral density function attains a peak o f
roughly 2.0. By contrast, the unrestricted model with its ARM A(3,2) represen
tation allows for a much richer characterization o f the low frequencies o f Ax, an
observation consistent with the Christiano and Eichenbaum (1990) analysis o f the
low-frequency properties o f alternative ARMA specifications.
Examining the roots o f the m odels’ characteristic polynomials is another way to
understand this behavior. The effect o f moving from zero to positive values o f 6
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and o e is m ove the redundant unit MA root away from the unit circle. Likew ise, a
positive 6 moves the AR root away from unity. The result is to create two nearly
redundant roots. In specification II, for example, the largest MA root is -0 .9 0 , and
the modulus o f the largest AR root is 0.81.
Some comparisons
An U nobserved C om ponents m odel
In his discussion o f univariate detrending methods, Watson (1986) proposes de
com posing difference-stationary time series as the sum o f an 1(1) component and
a stationary autoregressive com ponent For real GNP, Watson uses the follow ing
Unobserved Components (UC) specification:
x, = x, + ct where
X, = p + t,_i + ej and
ct = a ic t- 1 + a 2c t- 2 + ect ,
a random walk stochastic trend xf, and a stationary AR(2) cyclical com ponent ct.
The estimated parameters o f this model for the 1960:1-1991:1 sample period are:
A = 0.721, <h = 1.604, &2 = -0 .6 4 7 , d t = 0.702, and d c = 0.476. The standard
error is 0.89% .
As a decom position in which the stochastic trend is modeled as a latent variable,
the UC model has much the same flavor o f the latent-variable specification pro
posed in this paper. Indeed, the spectrum o f Ax it im plies, shown in figure 6, is
alm ost identical to the that o f the potential GNP model. The difference between
them lies in the nature o f the two shocks’ contributions to the spectrum: in the
case o f the UC model, the shock to the trend x contributes to the spectrum across
all frequencies. In contrast, by virtue o f the error-correction specification o f the
potential GNP model, the shock to the potential output trend contributes only at
very low frequencies. The result is a somewhat smoother estimated trend, free o f
the higher-frequency movements that characterize the UC trend.
The H odrick-Prescott filter
Another interesting comparison is between the estimated potential output sa le s
and the trend component from the Hodrick-Prescott (HP) filter, which is essen
tially a sophisticated two-sided m oving-average.13 Figure 7 plots the two-sided
specification n estimate o f potential along with the HP trend, with X set to 1600.
For reference, figure 7 also includes a deterministic linear trend. In general ap
pearance, the two series are comparable; the smoothness imposed by the stochas
tic trend specification far x* is similar to the smoothness delivered by X. = 1600.
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Figure 6
Spectrum of output growth, unobserved components model
One obvious distinction between the two procedures is that without an underlying
statistical model, the HP filter cannot deliver the standard errors computed by the
Kalman filter.
The series them selves also reflect a number o f important discrepancies. Two o f
these occur in 1973 and 1979, when potential output growth takes a dip not re
flected in the HP trend. The estimate o f potential apparently associates the in
creasing rate o f inflation over those periods with a depressed level o f potential
output. Or, to attach a more structural interpretation, the potential GNP estim ate
successfully picks out the 1973 and 1979 OPEC-induced supply shocks.
Another important discrepancy occurs in 1982-4, where the HP trend is depressed
relative to estimated potential. Because the HP technique simply involves smooth
ing the observed output series, som e portion o f the 1981-2 recession appears in
the filtered trend component. By contrast, the potential GNP procedure attributes
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Figure 7
Figure 8
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virtually all o f the 1981-2 decline in output to a widening output gap — in other
words, a pure demand shock.
M id-expansion trend GNP
A final instructive comparison is between this estimate o f potential GNP and the
more traditional mid-expansion measure constructed by the Bureau o f Economic
Analysis, which, because it uses ex post business cycle dating information, can
also be thought o f as a two-sided estimate. Figure 8 compares the “output gaps”
implied by the two measures, where the gap is defined as actual GNP less potential.
Again, the two series differ significantly during a number o f important episodes.
Like the HP trend, the mid-expansion measure m isses the 1973 and 1979 dips in
potential, implying smaller output gaps than those based on the specification II
measure. The discrepancy goes the other way in the late 60s. During this period,
the mid-expansion gap is about twice the size o f the one based on the specification
II estimate. Only during the 1981-2 recession do the two measures coincide.
Conclusions
The goal o f this paper has been to apply new econometric techniques to the estima
tion o f a key unobservable macroeconomic quantity, potential output. The paper
proposed using multiple indicators— output growth, inflation, and unemployment
— to extract information on the level o f potential real GNP, which was modeled
as a latent stochastic trend.
W hile consistent with Okun’s original conception o f potential output, these new
estimates embody two important advantages relative to traditional measures. First,
they distill disparate sources o f information into a single estimate o f potential out
put, promising to improve its reliability and precision. Second, they explicitly
recognize the uncertainty involved in tire estimation and provide a way o f gauging
that uncertainty.
Another way to interpret these results is as a decomposition o f real GNP fluctu
ations into potential output shocks, and deviations from output around potential
(traditionally, “ supply” and “demand” shocks). The results presented here are
consistent with the idea that shocks to potential are a very small source o f output
variance at horizons shorter than three years. However, over four-year and longer
horizons, shocks to potential output are responsible for m ost o f the variance in real
GNP.
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One possible extension to the method outlined above is to expand the supply side
o f the potential output specification, which was modeled as a stochatic trend tv as a
sim ple function o f population. Augmenting that specification to include measures
o f factor inputs is a promising area for future work.
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Footnotes
1. Some econom ists would dissent from this characterization. On one hand, clas
sical econom ists criticize the potential output concept for its lack o f a general
equilibrium foundation. Plosser and Schwert (1979) make this case. On the
other side, DeLong and Summers (1988) argue that potential output should
correspond to the highest feasible level o f production.
2. Since the 1970s, latent-variable models have found extensive application. A
partial list includes Aigner et al. (1986), Engle and Watson (1985), Hamilton
(1985), andNorrbinandSchlagenhauf (1988). A comprehensive guide to their
application to econometrics is Harvey (1989).
3. Okun (1970), pp. 132-3.
4. Although this series remains unpublished, it is widely used in Federal Reserve
Board research, such as Hallman Porter and Small (1989).
5. As the e shocks produce persistent changes in the sustainable level o f out
put, while the q* shocks produce only transitory fluctuations around that level,
one might interpret them as “supply” and “demand” shocks respectively, as in
Blanchard and Quah (1989). While appealing, this interpretation is not essen
tial to the model.
6. See Nerlove, Grether and Carvalho (1979).
7. A useful taxonomy o f latent variable econometric models appears in Watson
and Engle (1983).
8. These results use the BFGS and BHHH algorithms from the Gauss MAXLIK
procedure.
9. In the single-indicator model, real GNP changes alone provide very little infor
mation on the level o f potential output Therefore, when a very loose prior on
is specified, the procedure manages to fit a trend-stationary model by rais
ing the initial value o f log potential GNP upwards, to over 7.5. ly in g down a
more plausible initial condition with only a 2% standard error on xj forces the
procedure to fit a difference stationary model with nonzero e variance.10
10. In practice, results based on the CPI are very similar to those obtained with the
fixed-weight deflator when oil price changes are appropriately controlled for.
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11. Statistical releases from the Department o f Commerce caution against com
puting an inflation rate from the im plicit deflator “[Because] the prices are
weighted by the com position o f GNP in each period,.. .the Im plicit Price D e
flator reflects not only changes in prices but also changes in the com position
o f GNP, and its use as a measure o f price change should be avoided.”
12. See, for example, Lilien (1982) and Rissman (1986).
13. See King and Rebelo (1989).
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