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Finance and Economics Discussion Series
Divisions of Research & Statistics and Monetary Affairs
Federal Reserve Board, Washington, D.C.
The Reliability of Inflation Forecasts Based on Output
Gap Estimates in Real Time
Athanasios Orphanides and Simon van Norden
2004-68
NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS)
are preliminary materials circulated to stimulate discussion and critical comment. The
analysis and conclusions set forth are those of the authors and do not indicate
concurrence by other members of the research staff or the Board of Governors.
References in publications to the Finance and Economics Discussion Series (other than
acknowledgement) should be cleared with the author(s) to protect the tentative character
of these papers.
The Reliability of Inflation Forecasts Based on
Output Gap Estimates in Real Time
Athanasios Orphanides and Simon van Norden∗
November 2004
Abstract
A stable predictive relationship between inflation and the output gap, often referred to as
a Phillips curve, provides the basis for countercyclical monetary policy in many models. In
this paper, we evaluate the usefulness of alternative univariate and multivariate estimates
of the output gap for predicting inflation. Many of the ex post output gap measures we
examine appear to be quite useful for predicting inflation. However, forecasts using realtime estimates of the same measures do not perform nearly as well. The relative usefulness
of real-time output gap estimates diminishes further when compared to simple bivariate
forecasting models which use past inflation and output growth. Forecast performance also
appears to be unstable over time, with models often performing differently over periods of
high and low inflation. These results call into question the practical usefulness of the output
gap concept for forecasting inflation.
Keywords: Phillips curve, output gap, inflation forecasts, real-time data.
JEL Classification System: E37, C53.
Athanasios Orphanides is an adviser in the Division of Monetary Affairs at the Board of Governors
of the Federal Reserve System, a research fellow of the Centre for Economic Policy Research, and
a fellow of the Center for Financial Studies. E-mail: Athanasios.Orphanides@frb.gov. Simon van
Norden is a Professeur Agrégé at the HEC Montréal and a CIRANO fellow. E-mail: simon.vannorden@hec.ca.
∗
We benefited from presentations of earlier drafts at the European Central Bank, CIRANO, the Federal Reserve Bank of Philadelphia Conference on Real Time Data Analysis, the Centre for Growth
and Business Cycle Research, as well as at the annual meetings of the American Economics Association, the European Economics Association and the Canadian Economics Association. We would
also like to thank Sharon Kozicki, Tim Cogley, Jeremy Piger, Todd Clark, Desire Vencatachellom,
Ken West and two anonymous referees for useful comments and discussions. Athanasios Orphanides
wishes to thank the Sveriges Riksbank and European Central Bank for their hospitality during
September 2001 when part of this work was completed. Simon van Norden wishes to thank the
SSHRC and the HEC Montréal for their financial support. The opinions expressed are those of the
authors and do not necessarily reflect the views of the Board of Governors of the Federal Reserve
System.
1
Introduction
A stable predictive relationship between inflation and a measure of deviations of aggregate
demand from the economy’s potential supply—the “output gap”—provides the basis for
many formulations of activist countercyclical stabilization policy. Such a relationship, referred to as a Phillips curve, is often seen as a helpful guide for policymakers aiming to
maintain low inflation and stable economic growth. According to this paradigm, when aggregate demand exceeds potential output, the economy is subject to inflationary pressures
and inflation should be expected to rise. Under these circumstances, policymakers aiming
to contain the acceleration in prices might wish to adopt policies restricting aggregate demand. Similarly, when aggregate demand falls short of potential supply, inflation should
be expected to fall, prompting policymakers to consider the adoption of expansionary policies.1 Even assuming that the theoretical motivation for a relationship between the output
gap and inflation is fundamentally correct, a number of issues may complicate its use for
forecasting in practice. First, the definition of “potential output”—and the accompanying
“output gap”—that might be useful in practice is far from clear. Given a definition of
the output gap, its exact empirical relationship with inflation is not known a priori and
would need to be determined from the data. Second, even if the proper conceptual and
empirical relationships were identified, the operational usefulness of the output gap will be
limited by the availability of timely and reliable estimates of the identified concept. As is
well known, empirical estimates of the output gap are generally subject to significant and
highly persistent revisions. (For example, see Orphanides and van Norden (2002).) The
subsequent evolution of the economy leads to improved historical estimates of the gap by
providing useful information about the state of the business cycle. As a result, considerable
uncertainty regarding the value of the gap remains even long after it would be needed for
1
The widespread use of models featuring estimated “Phillips curves” of various forms for monetary policy
analysis at numerous central banks and other institutions is evidence of the appeal of this paradigm. See
Bryant, Hooper and Mann (1993) and Taylor (1999) for collections of monetary policy evaluations that
feature such estimated models.
1
forecasting inflation. This suggests that although the output gap may be quite useful for
historical analysis, its practical usefulness for forecasting inflation in real time may be quite
limited.
In this paper we assess the usefulness of alternative estimation methods of the output gap for predicting inflation, paying particular attention to the distinction between
suggested usefulness—based on ex post analysis using revised output gaps, and operational
usefulness—based on simulated real-time out-of-sample analysis.2 First, using out-of-sample
analysis based on ex post estimates of the output gap, we confirm that many concepts appear
to be useful for predicting inflation. This is as would be expected since the implicit Phillips
curve relationships recovered in this manner are similar to the relationships commonly found
in empirical macroeconometric models. To assess their operational usefulness, we generate
out-of-sample forecasts based on real-time output gap measures; those constructed using
only data (and parameter estimates) available at the time forecasts are generated.3 We
compare the resulting forecasts to both autoregressive forecasts of inflation and bivariate
forecasts that employ information from output growth as well as past inflation.
Our findings show that forecasts using ex post estimates of the output gap severely overstate the gap’s usefulness for predicting inflation. Real-time forecasts using the output gap
are often less accurate than forecasts that abstract from the output gap concept altogether.
And the relative usefulness of real-time output gap estimates diminishes further when compared to simple bivariate forecasting models which use past inflation and output growth.
In some cases, we find certain measures of the output gap produce superior forecasts of
inflation. However, relative performance seems to vary considerably over time, with models
which perform relatively well in some periods performing relatively poorly in others. Thus,
2
Our analysis is related to investigations of the usefulness of the unemployment gap for forecasting
inflation, such as Stock and Watson (1999), Atkeson and Ohanian (2001), and Fisher, Liu and Zhou (2002).
In some macroeconometric models, unemployment gaps and output gaps are related through Okun’s law.
3
For this exercise, we rely on the real-time dataset for macroeconomists which was created and is maintained by the Federal Reserve Bank of Philadelphia. See Croushore and Stark (2001) for background
information regarding this database.
2
past forecast performance may provide little guidance in selecting an operationally useful
definition of the output gap going forward.
The remainder of this paper is organized as follows. In sections 2 and 3 we define the
output gap concepts used and detail the methodology of our forecasting exercise. The main
results are presented in section 4 and section 5 concludes.
2
Trends and Cycles Ex Post and in Real Time
One way to define the output gap is as the difference between actual output and an underlying unobserved trend towards which output would revert in the absence of business cycle
fluctuations. Let qt denote the (natural logarithm of) actual output during quarter t, and
µt its trend. Then, the output gap, yt can be defined as the cyclic component resulting
from the decomposition of output into a trend and cycle component:
qt = µt + yt
Since the underlying trend is unobserved, its measurement, and the resulting measurement
of the output gap, very much depends on the choice of estimation method, underlying
assumptions and available data that are brought to bear on the measurement problem. For
any given method, simple changes in historical data and the availability of additional data
can change, sometimes drastically, the resulting estimates of the cycle for a given quarter.
Evidence of the difference between historical and real-time estimates of output gaps
has been presented by Orphanides and van Norden (2002). In Table 1, we present some
of the summary reliability indicators they examine for twelve alternative measures of the
output gap which we employ in our analysis.4 These results mirror those of Orphanides
and van Norden (2002). We find that revisions in real-time estimates are often of the same
magnitude as the historical estimates themselves and that, for many of the alternative
4
Brief descriptions of the various measures appear in Appendix A. Further details, including the output
gaps used in this study, as well as the programs and data used to create them, are freely available from the
authors at http://www.hec.ca/pages/simon.van-norden.
3
methods, historical and real-time estimates frequently have opposite signs.
The importance of ex post revisions to output gap estimates suggests that the presence
of a predictive relationship between inflation and ex post estimated output gap measures
does not guarantee that the output gap will be useful for forecasting inflation in practice.
Simply, the ex post estimates of output gaps at a point in time may differ substantially
from estimates which could be made without the benefit of hindsight. As well, these differences may hinder the real-time estimation of the presumed predictive relationship, further
complicating the real-time forecasting problem.
2.1
Data Sources and Vintages
We use the term vintage to describe the values for data series as published at a particular point in time. Most of our data is taken from the real-time data set compiled by
Croushore and Stark (2001); we use the quarterly vintages from 1965Q1 to 2003Q3 for real
output. Construction of the output series and its revision over time is further described in
Orphanides and van Norden (1999, 2002). We use 2003Q3 data as “final data” recognizing,
of course, that “final” is very much an ephemeral concept in the measurement of output.
To measure inflation, we use the change in the log of the consumer price index (CPI). We
use this both for our forecasting experiments and also to estimate measures of the output
gap in multivariate models that include inflation. CPI data are revised much less than
output data, with changes in seasonal factors causing most of the revisions. We therefore
use the 2003Q3 vintage of CPI data for all of our analysis. This allows us to focus on the
effects of revisions in the output data and the estimated output gap in our analysis. One
of our models (Structural VAR) also uses data on interest rates, which are never revised.
2.2
Measuring Output Gaps
We construct output gap estimates using a variety of different models, as listed in Table 1.
Each of the output gap models is used to produce gap estimates of varying vintages. Each
4
output gap vintage uses precisely one vintage of the output data. An estimated output
gap is called a final estimate if it uses the final data vintage. Note that all the output
gap estimation techniques (aside from the Hodrick-Prescott filter) require that one or more
parameters be estimated to fit the data. Such estimation was repeated for every combination
of technique and vintage. This means, for example, that in constructing output gap vintages
from an unobserved components (UC) model spanning the period 1969Q1-2003Q3 (139
quarters), we reestimate the model’s parameters 139 times, and then store 139 series of
smoothed estimates.
3
A Forecasting Experiment
We are interested in quantifying the extent to which the output gap concept provides a
practical means of improving forecasts of inflation. The answer will clearly depend on a
large number of factors, such as the time period of interest, the way in which forecasts
are constructed, the benchmark against which such forecasts are compared, and the loss
function used to evaluate the quality of different forecasts. We restrict our attention to
US CPI inflation since 1969 and use the mean-squared forecast error (MSFE) to compare
forecast quality.
3.1
Forecasting Inflation and Benchmarks
Let πth = log(Pt ) − log(Pt−h ) denote inflation over h quarters ending in quarter t. We
examined forecasts of inflation at various horizons but use one year (h=4) as our baseline.
Note that because of reporting lags, data for quarter t first become available in quarter t+1.
Thus, a four-quarter ahead forecast is a forecast five quarters ahead of the last quarter for
4
which actual data are available.5 Our objective, therefore, is to forecast πt+4
with data for
quarter t − 1 and earlier periods.
5
Since the last datapoint in our sample is for the 2003Q2 quarter, this implies that 2002Q1 is the last
datapoint available for forming a forecast we can use in our evaluation experiment.
5
We examine simple linear forecasting models of the form:
h
πt+h
=α+
n
X
1
βi · πt−i
+
i=1
m
X
γi · yt−i + et+h
(1)
i=1
where n and m denote the number of lags of inflation and the output gap in the equation.
We estimate the unknown coefficients {α, βi , γi } by ordinary least squares. We set n and
m using a variety of different methods; in the results presented here we use the Bayes
Information Criterion (BIC). Results with other lag selection methods were found to give
similar conclusions.
To provide a benchmark for comparison, we estimate a univariate forecasting model of
inflation based on equation (1) but omitting the output gaps. We refer to this model as the
autoregressive (AR) benchmark. Of course, the problem faced by forecasters in practice is
more complex than the one we consider. One obvious and important difference is that the
information set available to policymakers is much richer. It is therefore possible that output
gaps might improve on simple univariate forecasts of inflation but not on forecasts using a
broader range of inputs. For this reason, tests against an autoregressive forecast benchmark
should be considered to be weak tests of the utility of empirical output gap models.
To provide a slightly stronger test, we also consider benchmark forecasts which replace
the output gap in (1) with the first difference of the log of real output. As St-Amant and
van Norden (1998) argue, using output growth in this way can be interpreted as implicitly
defining an estimated output gap as a one-sided filter of output growth with weights based
on the estimated coefficients of equation (1). van Norden (1995) refers to such estimates as
TOFU gaps (Trivial Optimal Filter–Unrestricted). We refer to this as the TF benchmark
forecast and interpret it as a simple reduced-form inflation forecast that uses a slightly
larger information set than the AR benchmark, one which contains historical information
on both prices and output growth. Comparing forecasts based on output gaps to the TF
benchmark aids in isolating the usefulness (or lack thereof) of the economic structure and
other restrictions embedded in the construction of the output gaps.
6
3.2
Forecasting and Output Gap Revisions
Several practical issues complicate the use of (1) for inflation forecasting. Since the suitable
number of lags of inflation and the output gap n and m, and the coefficients of the equation
are not known a priori, these need to be estimated with available data. As our sample
increases and additional data become available, these estimates change. In addition, output
gap estimates (like output data) are revised over time. This in turn, can influence the
selected number of lags and the coefficients of equation (1) estimated in any given sample.
In addition, given the parameters of the equation, revisions in the output gap will directly
change the forecast value of inflation.
We therefore use (1) to construct 3 to 4 different kinds of forecasts for each output gap
model. These forecasts differ in the way lag lengths are determined and in the way the
output gap model is used.
Let yti,j be an estimate of the output gap at time t formed using data of vintage i, where
i > t and j = t or i − 1. For non-UC models (i.e. all except the Watson, Harvey-Clark,
Harvey-Jaeger, Kuttner and Gerlach-Smets models) the index j is irrelevant; yti,t = yti,i−1 .
For UC models, j = t denotes a filtered output gap estimate; although the model parameters
are estimated from using data up to i − 1, the Kalman filter recursions to estimate the gap
do not use data beyond t. For these same models, j = i − 1 denotes a smoothed estimate;
although yti,t and yti,i−1 use the same parameter estimates to calculate the output gap, the
latter also uses the data after t to recursively update its estimate of yt . When T = 2003Q3,
the terminology of Orphanides and van Norden (2002) refers to the time series {ytT,T −1 } as
Final estimates of the gap and to {ytT,t } as Quasi-Final estimates. We will commonly refer
to these as FL and QF estimates.
These different kinds of output gap estimates are used to construct different kinds of
forecasts. The first of these uses fixed lag lengths with final estimates of the output gap to
7
recursively estimate the forecasting equation
h
πt+h
= α̂t−1 +
n̂
X
1
β̂it−1 · πt−i
+
i=1
m̂
X
T,T −1
γ̂it−1 · yt−i
+ et+h
(2)
i=1
where T refers to 2003Q3. This replicates the kind of recursively-estimated, out-of-sample
forecasting experiments which are commonly performed but which ignore output gap revision. These forecasts are infeasible because they require information (Final estimates of
output gaps) which is not available at the time the forecast is made. They also estimate
the optimal lag lengths m̂, n̂ ex post. We refer to this Fixed-Lag Final-estimate forecast as
FL-FL.
In the case of UC models, we can construct similar forecasts using Quasi-Final rather
than Final estimates of the output gap
h
πt+h
= α̂t−1 +
n̂
X
1
β̂it−1 · πt−i
+
i=1
m̂
X
T,t
γ̂it−1 · yt−i
+ et+h
(3)
i=1
Orphanides and van Norden (2002) note that the difference between the Final and QuasiFinal estimates of the output accounts for the bulk of the revisions in the output gaps
they examine. The difference between the accuracy of these and the Final gap forecasts
above helps us to understand the relative importance of errors in gap estimation for forecast
accuracy. Like the Final gap forecasts, these forecasts are infeasible. We refer to these as
FL-QF forecasts.
We also construct feasible forecasts which attempt to mirror closely the forecasts which
practitioners would construct using such output gap models. Specifically, in these forecasts
the lag lengths for both explanatory variables vary over time and are estimated recursively.
The output gap series is also updated with its latest available vintage every time the parameters of the forecasting equation are re-estimated. The resulting Variable-Lag Real-Time
8
output gap (VL-RT) forecasting equation takes the form6
h
πt+h
= α̂
t−1
+
t−1
n̂X
β̂it−1
·
1
πt−i
i=1
+
t−1
m̂
X
t,t−1
γ̂it−1 · yt−i
+ et+h
(4)
i=1
where the superscripts on (m̂, n̂) indicate the information set used to estimate the lag
lengths. While these are the most realistic forecasts we examine, they are also the most
difficult to compute. Among other things, they require more than just the real-time gap
estimates presented in Orphanides and van Norden (2002); they require all vintages of the
complete estimated output gap series.
To summarize, we can construct two or three series of forecasts for each output gap model
we analyze: (1) using recursive estimation, fixed lag lengths and final output gap estimates,
(2) using recursive estimation, fixed lag lengths and quasi-final output gap estimates (which
are only available for the 5 UC models we examine), and (3) using recursive estimation,
variable lag lengths and all vintages of smoothed output gap estimates. We also examine
one other type of forecast, one which uses variable lag lengths and final output gaps and
which we refer to as VL-FL. Like the FL-QF forecast, this helps to isolate the contribution
of output gap revision to forecast accuracy. As we will see below, however, these methods
differ in the appropriate ways one should conduct inference.
3.3
Forecast Evaluation
We wish to evaluate the quality of the resulting forecasts by testing the null hypothesis that
a given pair of models have equal MSFEs. Various tests of equal forecast accuracy have been
proposed in recent years, notably by Diebold and Mariano (1995) for forecasting models
without estimated parameters and by West (1996) for models with estimated parameters.
While such tests have been popular, the assumptions they require are unfortunately violated
t,t−1
Note that in equation (4) we use smoothed estimates of the output gap (yt−i
) rather than filtered
t,t−i
estimates (yt−i ). This reflects the common practice of practitioners, which is to use the most accurate
possible estimate of the gap in estimating their forecast equations. Limited experiments which replaced
these smoothed estimates with filtered estimates suggest that this does not have a major impact on forecast
performance. Koenig, Dolmas and Piger (2003) discuss how the use of data of varying vintage affects forecast
accuracy.
6
9
for some of the hypotheses of interest here.
First, the use of Diebold-Mariano statistics with standard normal critical values for
asymptotic inference is justified only if the two models being compared are not nested.
However, when using suitable lag lengths, the output gap models nest the AR benchmark
model. Clark and McCracken (2001) suggest alternative tests for the case of nested models,
while Clark and McCracken (2002) find that the limiting distribution of these statistics is
non-pivotal for forecast horizons greater than one period. To compare these models, we
therefore use the MSE-F statistic proposed by McCracken (2000), which takes the form
MSE-F = P ·
(M SF E1 − M SF E2 )
M SF E2
(5)
where P is the number of forecasts, M SF E1 is the MSFE of the restricted model and
M SF E2 is the MSFE of the unrestricted model. The distribution of the statistic under
the null hypothesis of equal MSFE is estimated via a bootstrap experiment with 2000
replications, as detailed in Appendix B. Because these distributions are non-pivotal, the
test statistics are bootstrapped anew for every different choice of (P, h, y, m, n). This means
that every p-value we report for the AR benchmark is based on its own set of 2000 bootstrap
experiments.
Second, while the available asymptotic theory underlying all such tests allows for the
coefficients in an equation like (1) to be re-estimated over time, it assumes that lag lengths
are fixed during the recursive estimation, that the data remain fixed during the recursive
estimation, and that the data are not estimated.
All these assumptions are violated for the VL-RT forecasts we construct, so no p-values
are presented for this case.
Inference in the case of the TF benchmark is more straightforward as the models of
interest are no longer nested. Accordingly, we base our inference on the test statistics
proposed by Diebold and Mariano (1995) and West (1996). Specifically, letting dt ≡ e2it − e2jt
be the difference in squared forecast errors between model i and model j at time t, d ≡
10
T −1 ·
PT
t=1 (dt )
the mean difference, and ρτ ≡ T −1 ·
PT
t=τ +1 (dt − d) · (dt−τ
− d) the estimated
autocovariance of dt at lag τ , we compute the test statictic:
d
Ω/T
z=p
where Ω ≡
P6
l=−6 (1 − |l|/7) · ρl
(6)
is the Newey-West (1986) Heteroscedasticity and Autocor-
relation (HAC) robust estimator of the long-run variance of dt . West (1996) shows that
under conventional assumptions this statistic is asymptotically normally distributed under
the null hypothesis of equal forecast accuracy when the parameters of the forecast model are
estimated by ordinary least squares. We therefore calculate and report 2-sided p-values for
the TF benchmark using the standard normal distribution. Again, this asymptotic theory
is not applicable to the VL-RT forecasts, so no p-values are reported in this case.
4
4.1
Does the Output Gap Improve Forecasts of Inflation?
Are Improvements in Forecast Accuracy Significant?
Our next step is to examine the results of the forecasting experiments described above. Table 2 shows the results of formal tests for differences in MSFE between the two benchmark
models and the twelve output gap models. The upper panel of the table compares forecasts
constructed using final output data, final estimates of the output gap, and constant lag
lengths in the forecasting equation (FL-FL). The middle panel of the table shows the comparable results when using quasi-final rather than final (i.e. filtered rather than smoothed)
estimates of the output gap (FL-QF). Since such estimates can only be constructed from
UC models of the output gap, only results for the five UC models are presented. In both
cases, we see the MSFE of the benchmark models, the fractional improvement in MSFE
relative to the benchmark models ((M SF EBenchmark − M SF EGap )/M SF EGap ) and the
p-value for the test of the null hypothesis that the MSFEs of the benchmark and the gap
model are equal. Differences between these two panels are entirely due to the effects of ex
post revisions of output gaps.
11
The first thing apparent from the top panel of the table is that all the gap models
forecast better than the autoregressive benchmark model when using final output gaps. In
all but one case the differences in MSFE are greater than 10 per cent, and in four of the
twelve cases they are greater than 30 per cent. The suggested improvement is statistically
significant at the 5 per cent level for all but the SVAR model and at the one per cent level
for nine of the twelve models. These results confirm the conventional wisdom that ex post
output gaps appear to help forecast inflation. They also show that out-of-sample tests have
sufficient power to detect relevant differences in MSFE.
The evidence supporting the usefulness of output gaps is weakened when the benchmark
model is changed by adding real output growth to the forecasting equation (the TF model).
As can be seen on the right side of the top panel, three of the twelve gap models now have
larger MSFEs than the benchmark, and only five of the twelve show an improvement of
more than 10 per cent. The differences in MSFE are significant at the 10 per cent level in
only three cases and are never significant at the 5 per cent level. However, comparison of
the significance of the differences in MSFE across the two benchmarks is complicated by
differences in the tests used for nested and non-nested models, as explained in section 3.3.
Note, in particular, that the reported p-values for nested models (the AR benchmark) are
based on one-sided tests, while those for non-nested models (the TF benchmark) are based
on two-sided tests. In addition, Clark and McCracken (2001, 2002) suggest that the MSE-F
statistic, which is used for the AR benchmark, is more powerful than the z statistic used
for the TF benchmark.
The apparent superiority of output-gap based forecasts is also weakened by the use of
quasi-final rather than final estimates of the gap, shown in the middle panel. Improvements
over the AR benchmark are now lower in every case, falling 10 to 20 per cent, and in one case
output-gap-based forecasts are less accurate than the benchmark. However, improvements
in forecast accuracy are still significant at or near the 5 per cent significance level in the four
12
remaining cases. The situation changes further if we instead use the TF benchmark. Four
of the five models now forecast less accurately than the benchmark model. Ignoring the
effects of output gap revisions evidently tends to overstate the importance and significance
of output gaps for forecasting inflation.
The bottom panel of Table 2 shows the results of tests for differences in MSFE between
the two benchmark models and the twelve output gap models when the forecasts are constructed with time-varying lag lengths and real-time output gap estimates (VL-RT). This
change also increases the MSFE of the benchmark AR model by a little over 10 per cent.
The relative accuracy of these real-time forecasts is almost always lower than that of
the ex post forecasts analysed in the top panel of the table. Drops in relative MSFE are
substantial for many models. As noted earlier. the normal asymptotic theory results are
not valid in this case so no p-values are reported. Crude simulations based on bootstrapped
MSE-F statistics, however, suggested that several output gap models which appeared to
forecast significantly better than the AR benchmark in the top panel no longer showed a
significant difference in accuracy.
The reversal in the performance of the output gap models relative to the output growth
(TF) benchmark, is even more striking. This can be seen by comparing the top and bottom
panels on the right-hand side of the table. In real time, none of the output gap models
examined forecasts better than the TF benchmark.
4.2
The Effect of Output Gap Revisions on Relative Forecast Accuracy
To better understand the causes for the changes in MSFE noted above, Table 3 compares the
MSFEs of three different forecasting experiments. The first is identical to that documented
in the upper panel of the previous table, using final output data and gap estimates as well
as constant lag lengths in the forecasting equation (FL-FL). The second experiment uses
the same output data and gap estimates, but now updates the lag lengths each time the
forecast coefficients are recursively re-estimated (VL-FL). The third experiment is identical
13
to that documented in the bottom panel of the previous table, using time-varying lag lengths
and real-time output gap estimates (VL-RT). Differences in outcomes between the first two
experiments isolate the effects of variations in lag length. Differences between the second
two experiments similarly isolate the effects of output gap revision.
The table shows that the introduction of time-varying lag lengths has important effects
on forecast accuracy. A priori, such time-variation may improve forecasts if the underlying
relationship is unstable over time. On the other hand, it may introduce another source of
estimation error, which could reduce forecast accuracy. The table shows that all forecasts
see a reduction in accuracy, averaging 15 per cent. The benchmarks forecasts see changes
in MSFEs which are very close to the average.
Moving from Final to real-time output gap estimates has no effect on the AR benchmark
forecast, but tends to make other forecasts less accurate. While the average effects of this
change are smaller than those of changes in lag length, the impact varies much more across
models. Four models see their accuracy improve while three see their MSFE rise by more
than 20 per cent. Note that the TF benchmark sees the greatest improvement in accuracy.
Evidently, revisions in output growth contain useful information about future inflation.
The net effect of the changes in lag length determination and data vintage worsens
forecast accuracy in all but one case. The net effect on the AR benchmark is somewhat less
than average, while the TF benchmark improves more than any other model.
The results above suggest that some output gap models forecast inflation more accurately than an autoregressive model, even when using real-time output gap estimates.
However, none of the output gap models we examine forecasts inflation as well as simple
models which use both past inflation and output growth. Further, the relative performance
of different models is greatly affected by the use of real-time rather than ex post output gap
estimates. Finally, uncertainty about the lag structure also adds considerably to MSFEs.
14
4.3
The Robustness of Changes in Forecast Accuracy
We now investigate the robustness of the results presented in Table 2. Table 4 examines
the effects of changing the period over which forecasts are evaluated. The full 1969-2002
sample is split into two roughly equal halves, with the 1969-1983 portion characterized by
relatively high and volatile inflation, whereas prices were more stable over the 1984-2002
period. The greater volatility of inflation in the former period implies that least-squares
methods applied to the full sample tend to emphasize the fit of the model over the former
period. Perhaps as a consequence, the full-sample results presented in Table 2 largely reflect
forecast performance over the first half of the sample. Results for the low-inflation period
after 1983 may be a more relevant guide for contemporary decision-making, but they differ
from the full-sample results in several ways.
First, looking at forecasts with final output gaps, we see that the AR benchmark has
become harder to beat. Nine of the 12 models see their relative MSFEs decline, and only
five can reject the null of equal forecast accuracy at the 5 per cent level (compared to 11 in
the earlier portion of the sample). This decline in the predictability of inflation has been
noted previously in other studies, for example, Atkeson and Ohanian (2001), and Fisher,
Liu and Zhou (2002). The picture for the TF benchmark is less clear; while the relative
performance of the output gap models improves somewhat in the latter sample, there is
little evidence of significantly different forecast accuracy.
Second, looking at forecasts with real-time output gaps, it appears that it has become
increasingly difficult to forecast as well as the benchmarks. Out of 12 models 11 (10) have
larger MSFEs than the AR (TF) benchmark in the post-83 period. The Band-Pass filter
is the only model to forecast inflation better than either benchmark in the recent period,
giving over a 20 per cent reduction in MSFE. It is also interesting to note that, consistent
with the reported decline in the predictability of inflation, the AR benchmark now forecasts
slightly better in real time than the TF benchmark.
15
One possible explanation for the difference in results across the two sample periods
is parameter instability, a feature which has been noted by other research on inflation
forecasts, in particular, Stock and Watson (1996, 1999), and Clark and McCracken (2003).
Indeed, examination of changes in the period over which the forecasting model is estimated
suggested some evidence of such instability for some of our output gap forecasting models.
We also considered the effects of changing the forecasting horizons, forecasting changes
rather than levels of inflation, using different lag selection criteria, and using nominal rather
than real income growth as a benchmark. (Detailed results are available from the authors
upon request.) Based on a review of these findings, it appears that the results shown in
Table 2 are among the best that can be obtained for inflation forecasts from simple linear
forecasting models using output gaps.
Having considered this evidence, one might also ask which of the output gap models
examined here a practitioner should use to forecast inflation (if forced to do so.) It would
appear that the deterministic trend models (Linear, Quadratic and Breaking) were often
among the worst-performing in real-time, and should probably be avoided for that reason.
UC models which estimated Phillips Curves (Kuttner and Gerlach-Smets) had some of the
largest differences in performance when used with real-time rather than final estimates.
The Band-Pass and the Beveridge-Nelson methods perform better in our simulated realtime experiments. However, their success appeared to be sensitive to the forecast horizon
used. Rather than rely on any of these output gap models, our analysis suggests that a
practitioner could do well by simply taking into account the information contained in real
output growth without attempting to measure the level of the output gap—the TOFU
model. This model was consistently among the best performers, particularly over the post1983 forecast sample.
16
5
Conclusion
Forecasting inflation is a difficult but essential task for the successful implementation of
monetary policy. The hypothesis that a stable predictive relationship between inflation and
the output gap—a Phillips curve—is present in the data, suggests that output gap measures
could be useful for forecasting inflation. This has served as the basis for empirical formulations of countercyclical monetary policy in many models. We find that many alternative
measures of the output gap appear to be quite useful for forecasting inflation, on the basis
of ex post analysis. That is, a historical Phillips curve is suggested by the data, and final
(constructed ex post) estimates of the output gap are useful for understanding subsequent
movements in inflation.
However, this historical usefulness does not imply a similar operational usefulness. Our
simulated real-time forecasting experiment suggests, instead, that the predictive ability of
many different output gap measures may be illusory. Output gaps typically can not forecast
inflation as well out of sample as simple linear models of inflation and output growth (although the differences are mostly not statistically significant.) This is particularly true if we
restrict our attention to the post-1983 period. These rather pessimistic findings regarding
the output gap mirror earlier investigations regarding the predictive power for forecasting
inflation of “unemployment gaps,” that is the difference between the rate of unemployment
and estimates of the NAIRU. As demonstrated by Staiger, Stock and Watson (1997a,b) and
Stock and Watson (1999), estimates of the NAIRU are inherently unreliable, and simulated
out-of-sample forecasting exercises do not indicate a robust improvement in inflation forecasts from using information about unemployment. Stock and Watson (1999) also show
that better inflation forecasts may be obtained by indicators other than the unemployment
gap. Our analysis suggests similar conclusions regarding the output gap as well. Instead
of using output gaps, forecasts of inflation which simply incorporate information from the
growth rate of output appear to forecast inflation as well or better.
17
Finally, we note that these negative findings regarding the usefulness of real-time measures of the output gap do not necessarily invalidate the potential usefulness of the theoretical Phillips curve framework per se, nor that of ex post constructed output gaps for
historical analysis. That said, the dubious contribution of real-time measures of the output
gap for forecasting inflation brings into question their role in the formulation of reliable
real-time policy analysis.
18
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for inflation: resolving in-sample and out-of-sample evidence.” Federal Reserve Bank of
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Gerlach, Stefan and Frank Smets, “Output Gaps and Inflation: Unobserable-Components
Estimates for the G-7 Countries.” Bank for International Settlements mimeo, Basel
1997.
19
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Data in Economic Forecasting,” Review of Economics and Statistics, 85(3) August 2003,
618-628.
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Business and Economic Statistics, 12(3), 1994, 361-68.
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in Real Time,” Finance and Economics Discussion Series 1999-38, August 1999.
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2002.
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20
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1996, 1067-84.
21
22
Univariate.
Univariate.
Univariate.
Univariate.
Univariate.
Trivariate.
Univariate.
Univariate.
Univariate.
Bivariate.
Bivariate.
Quadratic Trend
Breaking Trend
Hodrick-Prescott
Band Pass
Beveridge-Nelson
Structural VAR
Watson
Harvey-Clark
Harvey-Jaeger
Kuttner
Gerlach-Smets
Harvey-Clark model and inflation equation.
Watson model and inflation equation.
Local Linear Trend and Cycle.
Local Linear Trend and AR(2).
Local Level and AR(2).
Imposes long-run restrictions.
Assumes ARIMA(1,1,2).
6–32 quarters, series padded with AR forecasts.
With λ = 1600.
Trend Break in 1973Q1, starting in 1977Q1.
Method Details
0.79
0.87
0.56
0.75
0.88
0.68
0.84
0.72
0.52
0.77
0.51
COR
0.88
0.82
0.90
0.86
0.92
0.87
0.85
0.09
0.77
0.93
0.87
0.97
AR
0.90
1.05
1.54
0.92
0.91
1.50
0.95
0.63
0.77
1.05
0.81
1.06
NSR
1.63
0.40
0.61
0.50
0.39
0.55
0.41
0.30
0.36
0.45
0.28
0.42
OPSIGN
0.58
Notes: Univariate methods employ only real GNP/GDP data. Bivariate also employ CPI inflation. Trivariate also employs
treasury bill data. The last four columns present summary measures of the reliability of real-time estimates of the output gap.
All statistics are for the 1969:1–2003:1 period. COR denotes the correlation of the real-time and final estimates of the output
gap, AR the first order serial correlation of the revision (the difference between the final and real-time series), NSR indicates
the ratio of the root of the mean square revision and the standard deviation of the final estimate of the gap, and OPSIGN
indicates the frequency with which the real-time and final gap estimates have opposite signs.
Data
Univariate.
Method
Linear Trend
Description of Alternative Output Gap Measures and Summary Reliability Statistics
Table 1
Table 2
Relative Improvement in MSFE
Method
AR
AR p-value
Fixed Lags, Final Gaps
Benchmark MSFE
0.494
Linear Trend
0.302
0.009
Quadratic Trend
0.168
0.010
Breaking Trend
0.106
0.034
Hodrick-Prescott
0.149
0.000
Band-Pass
0.134
0.000
Beveridge-Nelson
0.139
0.000
SVAR
0.047
0.121
Watson
0.319
0.001
Harvey-Clark
0.270
0.002
Harvey-Jaeger
0.109
0.001
Kuttner
0.336
0.008
Gerlach-Smets
0.362
0.001
Fixed Lags, Quasi-Final Gaps
Watson
0.132
0.043
Harvey-Clark
0.070
0.068
Harvey-Jaeger
−0.032
0.811
Kuttner
0.248
0.030
Gerlach-Smets
0.091
0.070
Variable Lags, Real-time Gaps
Benchmark MSFE
0.559
Linear Trend
0.045
Quadratic Trend
0.021
Breaking Trend
0.043
Hodrick-Prescott
0.132
Band-Pass
0.283
Beveridge-Nelson
0.211
SVAR
−0.093
Watson
0.121
Harvey-Clark
0.147
Harvey-Jaeger
0.080
Kuttner
0.107
Gerlach-Smets
0.099
TF
TF p-value
0.436
0.148
0.030
−0.024
0.013
0.000
0.004
−0.077
0.163
0.120
−0.022
0.178
0.201
0.164
0.779
0.778
0.900
0.997
0.309
0.474
0.060
0.162
0.811
0.079
0.052
−0.002
−0.056
−0.146
0.100
−0.038
0.979
0.374
0.382
0.250
0.414
0.416
−0.219
−0.237
−0.221
−0.154
−0.042
−0.095
−0.323
−0.163
−0.143
−0.193
−0.173
−0.179
Notes: The AR benchmark is a univariate autoregressive forecast of inflation; the TF benchmark
forecasts from a linear regression on lagged inflation and real output growth. Mean squared forecast
errors (MSFE) for the two benchmark models are shown multiplied by 1000. The remaining figures
in the AR and TF columns denote the relative improvements in MSFE for the output gap models,
measured as (A − B)/B where A is the MSFE of the benchmark and B is that of the output gap
model. The p-values for the AR benchmark are for the null that B ≥ A, based on the statistic in
equation (5). The p-values shown for the TF benchmark are for two-sided test of the null that A = B,
based on the statistic in equation (6). See section 3.3 and Appendix B for further discussion of the
construction and interpretation of the p-values. The forecast horizon is 4 quarters and forecast
performance is evaluated over the period from 1969Q1 to 2002Q1. Forecast equation estimation
starts in 1955Q1. Fixed lag lengths are (1,1) while varying lag lengths are reset every quarter using
BIC.
23
Table 3
The Effect of Lag Selection and Data Vintage
Method
AR benchmark
TF benchmark
Linear Trend
Quadratic Trend
Breaking Trend
Hodrick-Prescott
Band-Pass
Beveridge-Nelson
SVAR
Watson
Harvey-Clark
Harvey-Jaeger
Kuttner
Gerlach-Smets
Mean
Std Dev
FL-FL
0.494
0.436
0.380
0.423
0.447
0.430
0.436
0.434
0.472
0.375
0.389
0.446
0.370
0.363
MSFE
VL-FL
0.559
0.496
0.438
0.500
0.494
0.556
0.502
0.482
0.502
0.433
0.448
0.577
0.402
0.426
VL-RT
0.559
0.416
0.533
0.545
0.534
0.492
0.434
0.460
0.614
0.497
0.486
0.516
0.503
0.507
Change in MSFE (percent)
FL to VL FL to RT
Total
−13.0
0.0
−13.0
−13.7
16.0
4.6
−15.4
−21.7
−40.4
−18.1
−9.0
−28.8
−10.6
−8.0
−19.5
−29.2
11.5
−14.4
−15.2
13.5
0.4
−11.0
4.5
−6.0
−6.4
−22.3
−30.1
−15.4
−14.9
−32.6
−15.1
−8.4
−24.7
−29.5
10.7
−15.7
−8.5
−25.3
−36.0
−17.3
−19.0
−39.6
−15.6
−5.2
−21.1
6.7
14.5
14.4
Notes:
MSFE denotes the mean squared forecast error (shown multiplied by 1000.)
FL-FL refers to forecasts using fixed lag lengths and final output gap estimates.
VL-FL refers to forecasts using variable lag lengths and final output gap estimates.
VL-RT refers to forecasts using variable lag lengths and real-time output gap estimates.
FL to VL refers to the change from FL-FL to VL-FL.
FL to RT refers to the change from VL-FL to VL-RT.
Total refers to the change from FL-FL to VL-RT.
24
Table 4
Relative Improvement in MSFE: Sub-sample Evaluation
1969Q1–1983Q4
Method
AR
p-value
TF
Fixed Lags, Final Gaps
Benchmark MSFE
0.863
0.739
Linear Trend
0.247
0.025
0.068
Quadratic Trend
0.194
0.014
0.023
Breaking Trend
0.120
0.038
−0.041
Hodrick-Prescott
0.178
0.000
0.009
Band-Pass
0.172
0.003
0.004
Beveridge-Nelson
0.174
0.000
0.005
SVAR
0.013
0.359
−0.133
Watson
0.331
0.001
0.140
Harvey-Clark
0.320
0.002
0.131
Harvey-Jaeger
0.140
0.001
−0.024
Kuttner
0.317
0.024
0.128
Gerlach-Smets
0.432
0.001
0.226
Fixed Lags, Quasi-Final Gaps
Watson
0.091
0.117
−0.065
Harvey-Clark
0.081
0.074
−0.074
Harvey-Jaeger
0.252
0.002
0.072
Kuttner
0.194
0.088
0.023
Gerlach-Smets
0.115
0.076
−0.045
Variable Lags, Real-time Gaps
Benchmark MSFE
1.010
0.689
Linear Trend
0.225
−0.165
Quadratic Trend
0.228
−0.163
Breaking Trend
0.172
−0.201
Hodrick-Prescott
0.508
0.028
Band-Pass
0.301
−0.113
Beveridge-Nelson
0.288
−0.122
SVAR
−0.106
−0.391
Watson
0.209
−0.176
Harvey-Clark
0.205
−0.179
Harvey-Jaeger
0.445
−0.015
Kuttner
0.205
−0.179
Gerlach-Smets
0.153
−0.214
p-value
AR
1984Q1–2002Q1
p-value
TF
p-value
0.573
0.838
0.664
0.942
0.974
0.202
0.263
0.154
0.152
0.841
0.278
0.048
0.191
0.555
0.079
0.060
0.051
0.013
0.025
0.199
0.277
0.113
0.006
0.411
0.154
0.003
0.139
0.164
0.063
0.254
0.086
0.012
0.009
0.070
0.308
0.020
0.028
0.187
0.517
0.054
0.035
0.025
−0.011
0.000
0.170
0.247
0.086
−0.018
0.377
0.126
0.043
0.859
0.870
0.868
0.938
0.953
0.405
0.197
0.674
0.864
0.042
0.519
0.422
0.326
0.595
0.815
0.404
0.311
0.032
−0.474
0.494
0.010
0.010
0.267
1.000
0.019
0.418
0.280
0.007
−0.487
0.458
−0.015
0.024
0.931
0.198
0.045
0.865
0.191
−0.357
−0.405
−0.289
−0.451
0.215
−0.035
−0.018
−0.144
−0.046
−0.480
−0.177
−0.081
0.196
−0.341
−0.390
−0.272
−0.438
0.244
−0.011
0.006
−0.123
−0.023
−0.468
−0.158
−0.059
Notes: The AR benchmark is a univariate autoregressive forecast of inflation; the TF benchmark
forecasts from a linear regression on lagged inflation and real output growth. Mean squared forecast
errors (MSFE) for the two benchmark models are shown multiplied by 1000. The remaining figures
in the AR and TF columns denote the relative improvements in MSFE for the output gap models,
measured as (A − B)/B where A is the MSFE of the benchmark and B is that of the output gap
model. The p-values for the AR benchmark are for the null that B ≥ A, based on the statistic
in equation (5). The p-values shown for the TF benchmark are for two-sided test of the null that
A = B, based on the statistic in equation (6). See section 3.3 and Appendix B for further discussion
of the construction and interpretation of the p-values. The forecast horizon is 4 quarters and forecast
equation estimation starts in 1955Q1. Fixed lag lengths are (1,1) while varying lag lengths are reset
every quarter using BIC.
25
Appendix A:
The Construction of Real Time Output Gaps
The output gaps used in this study, as well as the data and programs used to create them, are
freely available from the authors. The estimates examined here include all those examined in
Orphanides and van Norden (2002) plus the Band-Pass, Beveridge-Nelson, Harvey-Jaeger
and SVAR methods described below; this is identical to the list of models considered in
Orphanides and van Norden (2003). The range of available estimates were updated so that
the “final” data vintage now corresponds to 2003Q3 (i.e. data available as of mid-August
2003, so data series end in 2003Q2) rather than 2000Q1 as in these two earlier papers. Data
for real output were taken from the Real Time Data Archive of the Federal Reserve Bank
of Philadelphia in September 2003. Observations span the period from 1947Q1 to 2003Q2.
Vintages for output run from Nov. 1965 to August 2003. All CPI data are from the 2003Q3
vintage. The SVAR method also uses data for 3-month US treasury bills. Data for this
rate (secondary market) from January 1934 to August 2003 were obtained from the FRED
database of the Federal Reserve Bank of St Louis.
All output gap models we consider decompose the logarithm of output into trend and
cycle components. The linear trend (LT) and quadratic trend (QT) models are from OLS
regressions with linear and quadratic deterministic trends. The breaking trend model is
identical to the LT model until 1976Q4. Starting in 1977Q1, it allows for an estimated break
in the trend at the end of 1973. The Hodrick-Prescott(HP) method is based on the filter
proposed by Hodrick and Prescott (1997) with their recommended smoothing parameter of
1600 for quarterly data. The band-pass method (BP) is based on the Stock and Watson
(1998) adaptation of the Baxter and King (1999) approach. Following Stock and Watson
(1998), we use a filter 25 observations in width and pad the available observations with
forecasts from an AR(4) model. The Beveridge-Nelson follows Beveridge and Nelson (1981)
in modelling output as an ARIMA(p,1,Q) series. Based on results for the full sample, we use
an ARIMA(1,1,2), with parameters re-estimated by maximum likelihood methods before
26
each recalculation of the trend.
We examine five unobserved component (UC) models, all of which are estimated by
maximum likelihood. Three of the five are univariate models. The Watson (WT) model is
based on Watson (1986) and models the output trend as a random walk with drift while
the cycle is assumed to follow a stationary AR(2) process. The Harvey-Clark (CL) model
follows Harvey (1985) and Clark (1987), replacing the constant drift in the trend of the
WT model with a random walk. The Harvey-Jaeger (HJ) model has the same trend as the
CL model but replaces the AR(2) component with a stochastic cycle. All three of these
univariate models require estimation of five parameters, including variances for the assumed
Gaussian shocks. The Kuttner (KT) model appends a Phillips curve, as specified in Kuttner
(1994), to the WT model, giving a bivariate model with eight more estimated parameters
than its univariate counterpart. The Gerlach-Smets (GS) model similarly adds the Phillips
curve specified in Gerlach and Smets (1997) to the CL model, yielding a bivariate model
with six more estimated parameters than its univariate counterpart.
The Structural VAR measure of the output gap (BQ) is based on a VAR identified via
restrictions on the long-run effects of the structural shocks, as proposed by Blanchard and
Quah (1989). Our implementation is identical to that of Cayen and van Norden (2002),
who use a trivariate system including output, CPI and yields on 3-month treasury bills.
Lag lengths for the VAR are selected using finite-sample corrected LR tests and a generalto-specific testing approach.
27
Appendix B:
Evaluation of Forecast Performance
As noted in section 3.3, our statistical inference for the forecast performance of the output
gap models relative to the AR benchmark model is based on the MSE-F statistic proposed
by McCracken (2000). This takes the form
MSE-F = P ·
(M SF E1 − M SF E2 )
M SF E2
(B.1)
where P is the number of forecasts, M SF E1 is the mean squared forecast error (MSFE) of
the restricted model and M SF E2 is the MSFE of the unrestricted model.
The distribution of the MSE-F statistic under the null hypothesis of equal MSFE is
estimated via a bootstrap experiment. The bootstrap begins by estimating a constrained
VAR(12) in πt1 , ytT,T −1 in which we impose the restriction that y does not Granger-cause
π. 2000 simulated realizations of this DGP are created by simulating the estimated model
h
with shocks randomly drawn with replacement from the estimated residuals. πt+h
is then
constructed as the sum of h consecutive observations of π11 . For each simulation, the dyh , π 1 , y T,T −1 for
namic model is initialized with historical observations starting with πk+h
k−i k−i
an independently drawn value of k. MSE-F statistics are then calculated for each simulated
series and their empirical distribution is used to estimate p-values for the true data’s MSE-F
statistics. Because these distributions are non-pivotal, the distribution of the test statistics
is bootstrapped anew for every different choice of (P, h, y, m, n). The p-values for every
reported MSE-F are therefore based on independent bootstrap experiments.
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@FedFRASER