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This paper evaluates some of the
competing methodologies based on their ability to accurately measure the output gap in a model
economy.
Because the output gap is unobservable, competing methodologies for estimating it are difficult to
assess, and evaluation techniques have varied.
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Canova (1994),
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for example, uses the NBER
definition of business cycle turning points as a metric for evaluating a battery of detrending
methods. He finds that the HodrickPrescott (HP) (1997) filter does a good job, relative to other
measures, of identifying turning points in U.
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Also, importantly, the core of
NAOMI is consistent with the central paradigm of models used in forecasting and projection at
the Bank; monetary conditions affect the output gap via an IS curve, which in turn affects inflation
via a Phillips curve. Potential output is defined in NAOMI as the level of output consistent with
nonaccelerating inflation.
1.
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Murchison (2001)
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provides a detailed description of NAOMI’s properties.
Some changes to NAOMI were necessary for this study. First, an equation for potential output was
added to its specification. In most of the experiments conducted in this paper, the growth rate of
potential output is determined by an AR(1) process:
,(1)
where is potential output and is an identically, independently distributed (i.i.d.) shock.2
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.
2.This is equivalent to the form;, where.
3.The estimated (1981 to 2001) historical standard deviation and AR(1) for firstdifferenced log GDP
are 0.79 and 0.53, respectively.
∆yt∗μδ∆y∗t1–εt++=
yt∗εt
y∗tαy∗t1–νt++=νtδνt1–εt+=αμδ
i
=∑
δ
yt∗αβtρy∗t1–εt+++=
Another popular method of estimating the output gap is the unobservedcomponent method,
which includes the statespace model of, for example,
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Kuttner (1994).
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Unfortunately, owing to the
typical instability of the maximumlikelihood estimates of the statespace model’s parameters, we
were unable to incorporate it into this study. In preliminary attempts to incorporate the model, the
parameter estimates, in particular either the variance of potential output or outputgap shocks,
tended towards zero in most samples.
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Alternatively, by varying the ratio of demandtosupply shocks in our
DGP, we can examine the costs of assuming the “wrong” value for.
3.2The multivariate HP filter
The MVF combines the HP filter with at least one additional source of information (see
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Laxton and Tetlow 1992).
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Two versions of the MVF are examined in this study. The first (MVF1) adds
the error from a prespecified Phillips curve equation to the set of information used by the HP
filter. Specifically, it chooses a profile for that minimizes the following function:
,(4)
λ
λ
yt∗
T
∑
2
W1εt
π
()
2
T
∑λ∆
2
()yt∗
t1=
T
+∑
2
=+
Θtytyt∗–()
t0=
t4=
εt
π
W1
λβi
where is the error from a reducedform Phillips curve
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Unfortunately, within this
W1λβi
framework there is no formal way to choose values for these parameters in an optimal fashion.4
Although the Phillips curve parameters can be estimated a priori, the choice of the weighting
coefficient and the smoothing prior is more ad hoc. We use values for the weighting coefficients
consistent with those used in the Bank’s extended multivariate filter (see
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Butler 1996);
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equals
1600, the weighting coefficients and are set to one, and is set to 64 for the last 16
quarters of estimation and zero elsewhere.5 is estimated simply as the mean growth rate of
real output in the period of estimation.
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used in the Bank’s extended multivariate filter (see Butler 1996); equals
1600, the weighting coefficients and are set to one, and is set to 64 for the last 16
quarters of estimation and zero elsewhere.5 is estimated simply as the mean growth rate of
real output in the period of estimation.
An estimate of the Phillips curve parameters requires an initial estimate of the output gap.
Following
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Conway and Hunt (1997),
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we use an HPfiltered output gap, with, as the
initial estimate of the gap. Given that this experiment is being conducted from the standpoint of an
economist who does not know with certainty the true structure of the economy, we impose the
following general form for the Phillips curve:
,(6)
where is inflation and is the HPfiltered output gap.
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The procedure continues until the change in the outputgap estimate from one step to the next
falls below a prespecified convergence criterion.
λ
W1W2W3
gss
λ1600=
4
4
πtαiπti–βigti–εt+
+∑
=∑
i0=
i1=
πtgt
βi
4.Alternatively, this problem could be mapped into an unobservedcomponents model, in which the
parameters are estimated by maximum likelihood.
5.See
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Butler (1996)
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for a discussion of the choice of these weights. Using estimates of the nonacceleratinginflation rate of unemployment (NAIRU) and the trend rate of capacity utilization, in
addition to a Phillips curve, as conditioning information, de Brouwer (1998) sets the weights on each
piece of conditioning information to be inversely proportional to the variance of the respective gap.
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Using estimates of the nonacceleratinginflation rate of unemployment (NAIRU) and the trend rate of capacity utilization, in
addition to a Phillips curve, as conditioning information, de Brouwer (1998) sets the weights on each
piece of conditioning information to be inversely proportional to the variance of the respective gap.
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Butler (1996),
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however, finds that such an approach does not produce estimates substantially different
from those produced using a scheme of equal weights on each piece of conditioning information.
3.3The BlanchardQuah SVAR
The BlanchardQuah (1989) SVAR methodology uses limited longrun restrictions to separate the
temporary and permanent components of output.
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It would be unsuitable, however,
to compare the accuracy of a threevariable SVAR with a method such as the MVF, which uses data
only on output and inflation.
7.Indeed, this longrun restriction holds in the DGP.
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Cooley and Dwyer (1998)
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show that a violation of
the assumptions about the nonstationarity of the data can have a significant impact on how well an
SVAR’s dynamics mimic those in the true data. Future work will examine the implications of the trend
growth rate of potential output being subject to structural breaks.
8.
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Cooley and Dwyer (1998) show that a violation of
the assumptions about the nonstationarity of the data can have a significant impact on how well an
SVAR’s dynamics mimic those in the true data. Future work will examine the implications of the trend
growth rate of potential output being subject to structural breaks.
8.This approach to lag selection was proposed by
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Lutkepohl and Poskitt (1996).
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y
∑π
a110
a21a22
This identifying assumption is at odds with QPM and NAOMI, both of which allow a channel
whereby productivity shocks result in business cycle fluctuation. We gauge the consequences of
incorrectly maintaining this assumption by comparing the cases in which there is no correlation
between the transitory and permanent components of output with three cases in which correlation
is allo
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We gauge the consequences of
incorrectly maintaining this assumption by comparing the cases in which there is no correlation
between the transitory and permanent components of output with three cases in which correlation
is allowed.
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Cooley and Dwyer (1998)
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stress that the SVAR’s “auxiliary” assumption of difference stationarity
in output is not as innocuous as it appears. They point to the difficulty in distinguishing trend
dependence from a unit root in postwar data to motivate an experiment in which output in the
DGP, rather than being difference stationary, is instead driven by a near unit root, while the
assumption of difference stationary outp
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This is particularly important if we are
to attempt to rank the various estimation methodologies, as the assumptions underlying certain
estimators may bias our results for or against that technique. For example, identification of the
SVAR requires an assumption about the type of nonstationarity in output data. On the other hand,
it has been shown
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(King and Rebello 1993; Ehlgen 1998)
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that one condition under which the HP
filter is optimal, in a meansquarederror sense, is the smoothing parameter,, being equal to the
ratio of the variances of innovations in the cyclical and trend components in the DGP.
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In this sample, the three estimates follow the actual gap reasonably
closely: the RMSEs for the MVF1, SVAR, and combined approach are 1.66, and 1.46, and 1.49,
respectively, lower than the standard deviation of the actual output gap of 2.58. The correlations
between the actual and estimated gap are 0.78, 0.86, and 0.83.
12.See
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Murchison (2001)
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for a discussion of supply shocks in NAOMI.
13.For Case 6, it is necessary to lower the coefficient on potential output growth to match the historical
autocorrelation of output growth. Also, the ratio of demandtosupply innovations of 0.6 is the lowest
ratio possible, given the correlation between the output gap and potential in this case.
14.
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Indeed, as expected, the performance of the SVAR deteriorates under these
conditions. There is, however, a noticeable deterioration in the performance of the HPbased
approaches, particularly at the endofsample. This phenomenon is not unexpected;
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King and Rebello (1993)
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discuss conditions under which the HPfilter is the optimal linear filter; they
include orthogonal transitory and permanent innovations. As the DGP becomes less consistent
with this condition, the performance of the HP filter worsens.
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