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However, it is also found that the estimation of the output gap on the basis of an
estimated VAR is imprecise, which is consistent with results obtained by Staiger,
Stock and
 Exact

Watson (1996)
 Suffix

with a different methodology. The spectra of the transitory
components (output gaps) resulting from the empirical applications of the CO, MBN
and LRRO methodologies differ from one another. Indeed, only the LRRO transitory
component has a peak at businesscycle frequencies, i.e., cycles lasting between 6 and
32 quarters.
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Les auteurs
constatent cependant que l'écart de production calculé à l'aide d'un vecteur
autorégressif estimé manque de précision, ce qui est conforme aux résultats obtenus
par Staiger, Stock et
 Exact

Watson (1996)
 Suffix

au moyen d'une autre méthode. Les spectres des
composantes transitoires (écarts de production) qui résultent de l'application
empirique des méthodes CO, MBN et LRRO diffèrent les uns des autres.
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As a result, various methods have been
proposed to uncover the permanent and transitory components of output.
One of these methods consists of using mechanical filters such as the
HodrickPrescott (HP) filter or the bandpass filter (BK) proposed by Baxter and
 Exact

King (1995).
 Suffix

However, mechanical filters have been criticized. For example,
Harvey and Jaeger (1993) and Cogley and Nason (1995) show that spurious
cyclicality can be induced by the HP filter when it is used with integrated or nearly
integrated data.
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Guay and StAmant (1996) reach the more general conclusion
that the HP and BK filters perform poorly in identifying the cyclical component of
time series that have a spectrum or pseudospectrum with Granger’s typical
shape, i.e., the shape characteristic of most macroeconomic time series.
Baxter and
 Exact

King (1995) and
 Suffix

others note that twosided filters such as
the HP and BK filters become illdefined at the beginning and the end of samples.
For this reason, they recommend discarding three years of quarterly data at both
ends of the sample when using the HP filter.
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Baxter and King (1995) and others note that twosided filters such as
the HP and BK filters become illdefined at the beginning and the end of samples.
For this reason, they recommend discarding three years of quarterly data at both
ends of the sample when using the HP filter. Van
 Exact

Norden (1995)
 Suffix

stresses the fact
that this is a very significant limitation for policymakers interested in estimating
1. For a discussion of how the estimation of potential output can affect the formulation of
monetary policy, see Boschen and Mills (1990) or Laxton and Tetlow (1992).
the current level of the output gap.
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of how the estimation of potential output can affect the formulation of
monetary policy, see Boschen and Mills (1990) or Laxton and Tetlow (1992).
the current level of the output gap.
Another strategy for identifying the permanent and transitory
components of output involves the use of univariate techniques such as the
unobserved components approach suggested by
 Exact

Watson (1986) and
 Suffix

the BeveridgeNelson (1981) method. However, Quah (1992) has shown that “without additional
ad hoc restrictions those [univariate] characterizations are completely
uninformative for the relative importance of the underlying permanent and
transitory components.
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Another strategy for identifying the permanent and transitory
components of output involves the use of univariate techniques such as the
unobserved components approach suggested by Watson (1986) and the Beveridge
 Exact

Nelson (1981)
 Suffix

method. However, Quah (1992) has shown that “without additional
ad hoc restrictions those [univariate] characterizations are completely
uninformative for the relative importance of the underlying permanent and
transitory components.
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 Prefix

Another strategy for identifying the permanent and transitory
components of output involves the use of univariate techniques such as the
unobserved components approach suggested by Watson (1986) and the BeveridgeNelson (1981) method. However,
 Exact

Quah (1992)
 Suffix

has shown that “without additional
ad hoc restrictions those [univariate] characterizations are completely
uninformative for the relative importance of the underlying permanent and
transitory components.
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In this paper, we compare some of the techniques briefly introduced
above with the structural vector autoregression methodology based on longrun
restrictions imposed on output (LRRO) proposed by Blanchard and
 Exact

Quah (1989),
 Suffix

Shapiro and Watson (1988), and King et al. (1991) in theory (Section 2) and in
applications (Section 3). In Section 2, we note that one characteristic of the LRRO
method is that it does not impose restrictions on the shortrun dynamics of the
permanent component of output.
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In this paper, we compare some of the techniques briefly introduced
above with the structural vector autoregression methodology based on longrun
restrictions imposed on output (LRRO) proposed by Blanchard and Quah (1989), Shapiro and
 Exact

Watson (1988), and
 Suffix

King et al. (1991) in theory (Section 2) and in
applications (Section 3). In Section 2, we note that one characteristic of the LRRO
method is that it does not impose restrictions on the shortrun dynamics of the
permanent component of output.
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We find
that the answer is “yes” when the entire output gap series is considered, but “not
really” when one is interested in estimating the output gap at a specific point in
time. In the latter case, the estimation of potential output and the output gap is
indeed imprecise. This is consistent with recent results reported by Staiger, Stock
and
 Exact

Watson (1996)
 Suffix

for the estimation of the NAIRU. Another interesting result is
that, of the methods we consider, only the LRRObased one generates an output
gap with a peak at business cycle frequencies as defined by Burns and Mitchell
(1946), i.e., cycles lasting between 6 and 32 quarters.
2 Methodologies used for estimating the trend in output
2.1 The approach based on the LRRO methodology
In this section,
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vector including a n1vector of I(1)
variables and a n2vector of I(0) variables such that.3 By the
Wold decomposition theorem, can be expressed as the following reduced form:
(1)
Zt
Zt∆X1t′X2t′,()′=
Zt
Ztδt()CL()εt+=
δt()CL() Σi0=
∞
CiL
i
=
where is deterministic, is a matrix of polynomial lags,
is the identity matrix, the vector is the onestepahead forecast errors
2. See
 Exact

Watson (1993)
 Suffix

for a more detailed presentation of the LRRO approach.
3. I(d) denotes a variable that is integrated of order d.
C0In=εt
in given information on lagged values of,, and
with positive definite. We suppose that the determinantal polynomial
has all its roots on or outside the unit circle, which rules out the nonfundamental
representations emphasized by Lippi and Reichlin (1993).
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From (2) and (3) we have:
(4)
CL() ΓL()Γ0
–1
=
C1()ΩC1()′
C1()ΩC1()′ Γ1()Γ1()′=
Γ0
C1()
C1()
This relation suggests that we can identify matrix with an appropriate
number of restrictions on the longrun covariance matrix of the structural form.
Blanchard and
 Exact

Quah (1989) and
 Suffix

Shapiro and Watson (1988) use longrun
restrictions to identify shocks with having full rank. King et al. (1991)
work in a context where the rank of is less than n1 and they use
cointegration restrictions.
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From (2) and (3) we have:
(4)
CL() ΓL()Γ0
–1
=
C1()ΩC1()′
C1()ΩC1()′ Γ1()Γ1()′=
Γ0
C1()
C1()
This relation suggests that we can identify matrix with an appropriate
number of restrictions on the longrun covariance matrix of the structural form.
Blanchard and Quah (1989) and Shapiro and
 Exact

Watson (1988)
 Suffix

use longrun
restrictions to identify shocks with having full rank. King et al. (1991)
work in a context where the rank of is less than n1 and they use
cointegration restrictions.
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Potential output
based on the LRRO method is then:
(6)
∆yt
p
μyΓ1
p
()ηLt
p
=+
Thus, “potential output” corresponds to the permanent component of output. The
part of output due to transitory shocks is defined as the “output gap.” It is
important to note that we do not talk in terms of “demand” or “supply” shocks as
in Blanchard and
 Exact

Quah (1989),
 Suffix

but simply in terms of permanent and transitory
shocks.
2.2 Comparison with other multivariate methods
In this section, we examine the features of two alternatives to the LRRO
approach: the multivariate BeveridgeNelson decomposition (MBN) and
Cochrane’s outputconsumption decomposition (CO).4
The MBN decomposition defines potential output as the level of
out
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With reference to equation (2), where output is the first element of, we write the
following decomposition:
(7)
Zt
∆ytμyC11()εtC1∗L()εt++=
Potential output is defined by the first two terms on the righthand side of (7):
(8)
∆yt
p
=μyC11()εt+
4. See
 Exact

Cogley (1995)
 Suffix

for another comparison of the MBN and CO methodologies.
Potential output is thus simply a random walk with drift.
Cochrane (1994) uses a twovariable VAR including GNP and
consumption to identify the permanent and transitory components of GNP.
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With reference to equation (2), where output is the first element of, we write the
following decomposition:
(7)
Zt
∆ytμyC11()εtC1∗L()εt++=
Potential output is defined by the first two terms on the righthand side of (7):
(8)
∆yt
p
=μyC11()εt+
4. See Cogley (1995) for another comparison of the MBN and CO methodologies.
Potential output is thus simply a random walk with drift.
 Exact

Cochrane (1994)
 Suffix

uses a twovariable VAR including GNP and
consumption to identify the permanent and transitory components of GNP. The
bivariate representation is augmented with lags of the ratio consumption to GNP.
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Indeed, the validity of the
permanentincome hypothesis would imply that the last two terms of the
consumption equation are equal to zero and that. It is not clear to
what the CO decomposition corresponds if consumption is not a random walk.5
 Exact

Cochrane (1994)
 Suffix

notes that the measure of potential output obtained on the basis
of the CO method would be equivalent to the one obtained from the LRRO
approach if the transitory effect of permanent shocks to GNP and consumption
were exactly the same, i.e., if and.
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al. (1988) or King et al. (1991) — imply that
the ratio of the log of GNP to the log of consumption is stationary but that consumption is
not a random walk because the real interest rate is not constant. In these models, the
transitory component of permanent shocks to consumption is not equal to zero. The LRRO
decomposition is compatible with the prediction of these models.
6.
 Exact

Kuttner (1994)
 Suffix

proposes a method based on the univariate unobserved stochastic trend
decomposition of Watson (1986) augmented with a Phillipscurve equation. As with the
BeveridgeNelson decomposition, Kuttner’s approach constrains potential output to follow
a randomwalk process.
μΓy
p
()η1t
p
+
Γ
p∗
()L
Γy
p
() Γ1c
p
=()1
Γy
p∗
() ΓLc
p∗
=()LΓy
p
() Γ1c
p
=()1
One implication of def
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In these models, the
transitory component of permanent shocks to consumption is not equal to zero. The LRRO
decomposition is compatible with the prediction of these models.
6. Kuttner (1994) proposes a method based on the univariate unobserved stochastic trend
decomposition of
 Exact

Watson (1986)
 Suffix

augmented with a Phillipscurve equation. As with the
BeveridgeNelson decomposition, Kuttner’s approach constrains potential output to follow
a randomwalk process.
μΓy
p
()η1t
p
+
Γ
p∗
()L
Γy
p
() Γ1c
p
=()1
Γy
p∗
() ΓLc
p∗
=()LΓy
p
() Γ1c
p
=()1
One implication of defining potential output as a random walk with
drift is that when the contemporary effect of a positive permanent sh
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The autoregressive reducedform VAR of the model is first estimated:
q
ZtΠiZti–et+
=∑
i1=
withqthe number of lags and a vector of estimated residuals with
.
It is crucial that the estimated VARs include a sufficient number of
et
Eetet()Σ=
lags. Indeed, Monte Carlo simulations by DeSerres and
 Exact

Guay (1995)
 Suffix

show that
using a lag structure that is too parsimonious can significantly bias the estimation
of the structural components. These authors also find that informationbased
criteria, such as the Akaike and Schwarz criteria, tend to select an insufficient
number of lags, while Wald or likelihoodratio (LR) tests, using a generaltospecific approach, perform much bett
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in which the permanent
component of output is a random walk imply that the economy is below (above)
potential in the transition period following a permanent positive (negative) shock
to output. To the extent that the transition primarily reflects factors associated
with an adjustment in the supply side of the economy, assuming that potential
7. Blanchard and
 Exact

Quah (1989) and Gali (1992),
 Suffix

among others, report similar results.
output follows a random walk can be misleading. It could, in particular, provide
misleading signals about the extent of inflationary pressures in the economy.
Chart 3 shows the output gaps calculated on the basis of the LRRO
methodology in the bivariate and trivariate cases.
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Nevertheless, the correlation between
the output gaps identified on the basis of the CO and MBN methodologies is rather
small, indicating that consumption may not be a random walk. This is consistent
with results reported in
 Exact

Watson (1993),
 Suffix

showing that the spectrum of the first
difference of consumption has a peak at businesscycle frequencies. In Section 2,
we noted that if consumption followed a randomwalk process, the CO and MBN
methodologies would give identical results.
8.
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CHART 5: Uncertainty surrounding the estimation of the output gap
MBN
Confidence interval
LRRO
Confidence interval
The main message of Chart 5, which would apply to other estimates
of the output gap reviewed in this paper, is that there is a substantial amount of
uncertainty surrounding the estimation of the output gap. Staiger, Stock and
 Exact

Watson (1996),
 Suffix

using a different methodology, reach a similar conclusion
concerning the estimation of the NAIRU. This uncertainty should probably be
taken into account by policymakers who use the output gap to guide their
decisions.
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in this paper is that, unlike mechanical filters, they reflect at least some
of the uncertainty.
3.2 Spectra analysis
Chart 6 shows the estimated spectra of the CO, MBN (trivariate case) and LRRO
(trivariate case) output gaps plus those resulting from the application of two
mechanical filters: the HodrickPrescott filter (HP) and the bandpass filter (BK)
proposed by Baxter and
 Exact

King (1995).
 Suffix

Loosely speaking, the spectrum of a series is
that series expressed as the integral of random periodic components that are
mutually orthogonal. The total area below the spectrum corresponds to the
variance of the series.
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The spectrum of the gap resulting from the twovariable MBN
application (not shown on the graph) has the same shape as the threevariable
9. For an introduction to spectral analysis see
 Exact

Hamilton (1994).
 Suffix

The order of these processes
was determined on the basis of the Akaike criteria.
case, although with a lower peak and a smaller total variance. The latter result is
not surprising, since it is well known that the MBN methodology gives a transitory
component whose importance increases with the number of series used to identify
it.
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It is difficult to compare the
spectra of MBN output gaps with the others.
4 Conclusions
In this paper, we compared different techniques that are used to measure
potential output. We started with a brief explanation of why we think that
mechanical filters such as the HodrickPrescott filter and the bandpass filter
proposed by Baxter and
 Exact

King (1995)
 Suffix

perform poorly in accomplishing this task.
We then compared the LRRO approach based on longrun restrictions with two
alternative multivariate approaches: the one proposed by Cochrane (1994) and
the multivariate BeveridgeNelson methodology.
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We started with a brief explanation of why we think that
mechanical filters such as the HodrickPrescott filter and the bandpass filter
proposed by Baxter and King (1995) perform poorly in accomplishing this task.
We then compared the LRRO approach based on longrun restrictions with two
alternative multivariate approaches: the one proposed by
 Exact

Cochrane (1994) and
 Suffix

the multivariate BeveridgeNelson methodology. We argued that one advantage
of the approach based on longrun restrictions is that it allows for estimated
transitional dynamics following permanent shocks.
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However,
the estimates are imprecise for specific points in time and it appears difficult to
distinguish between these methodologies in that context. This later result is
consistent with the conclusions of Staiger, Stock and
 Exact

Watson (1996).
 Suffix

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