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both univariate mechanical filters, as well as their
hybrid extensions, may have problems in adequately separating output into trend and meanreverting components.3 These filters perform poorly in extracting business cycle frequencies of 6
to 32 quarters from series such as real GDP which have an important permanent component. In
addition, they are subject to severe endofsample problems.
1.See
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Butler (1996).
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2.An example of the application of these methods to U.S. data can be found in Dupasquier, Guay and StAmant
(1999).
3.See also Harvey and Jaeger (1993) and King and Rebelo (1993).
As one alternative to the above class of models, StAmant and van Norden suggest the use
of structural VAR (SVAR) methods for measuring trend output.4 These models are favoured
because they are not confronted with endof
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output into trend and meanreverting components.3 These filters perform poorly in extracting business cycle frequencies of 6
to 32 quarters from series such as real GDP which have an important permanent component. In
addition, they are subject to severe endofsample problems.
1.See Butler (1996). 2.An example of the application of these methods to U.S. data can be found in Dupasquier, Guay and
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StAmant (1999).
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3.See also Harvey and Jaeger (1993) and King and Rebelo (1993).
As one alternative to the above class of models, StAmant and van Norden suggest the use
of structural VAR (SVAR) methods for measuring trend output.4 These models are favoured
because they are not confronted with endofsample issues.
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the equations in the statespace system on the other, we can still analyze a welldefined macroeconomic system and
interpret the parameters and residuals in a straightforward fashion.6(ii) The capacity of the model
to provide us directly with confidence intervals around the measured gap or potential.7 (iii) The
4.Another is the TOFU method proposed in van Norden (1995).
5.For more details, see
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Phillips (1989).
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6.Ideally, it would be desirable to make use of a fully specified structural and stochastic model. Unfortunately,
this also is problematic as it necessitates, among other things, making assumptions on the equilibrium level of
the various components used in the production function, as well as its aggregated form.
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Studies employing
these types of models frequently encounter difficulties in finding suitable data for their longrun values.
7.This is the mean square error associated with the estimated state vector and is obtained as a byproduct of
applying the Kalman filter. It is also known as the filter uncertainty of the model. However, as pointed out by
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Hamilton (1986),
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two types of uncertainties are associated with these models: filter uncertainty and parameter
uncertainty. If one adopts the Bayesian perspective that the true value of the state vector is random, then our
knowledge of it, based on observable variables, is reflected in a probability distribution.
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uncertainty, must be added to the above.
fact that we can also directly obtain outofsample forecasts on the observable variables in the
model, which provide an easy way of assessing the goodness of fit of the adopted specification
and of ensuring that our estimates are useful in the formulation of policy.
The statespace framework was first used to estimate trend output on U.S. data by
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Kuttner (1994).
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It was subsequently extended by Gerlach and Smets (1997) and applied to the G7
countries. For our part, we examine the implications of using different versions and extensions of
the Gerlach and Smets (1997) model (hereafter GS) for Canada.
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Notice that this is a very general statespace framework so that when no ARCH errors are
present in the system, the above filter reduces to the wellknown Kalman filter.
3.The Gerlach and Smets model
We now move to the presentation of the Gerlach and Smets (1997) model which is a special case
of the general model presented above. As mentioned, it is a modification of the
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Kuttner (1994)
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model which was originally applied to the United States and where potential output and the slope
of the Phillips curve were simultaneously estimated in an unobserved components framework.
The log of quarterly real output,, is assumed to be the sum of log real potential output,
, and a log cyclical component,.
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These variables are included to capture the effects of temporary relative price shocks on inflation.
Finally, the error terms are assumed to be mean zero and normally distributed. They enter the
11.This is the assumption made by
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Watson (1986).
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It has since been used in a standard fashion in various
macroeconomic models.
inflation equation as a moving average process to capture any remaining inertia in supply shock
variables. This also allows the model to be identified.
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However, it remains an important question and
is left for future research.
15.In fact, if the true process is I(1), but instead an I(2) is assumed, the model may not be wellidentified. This
adversely affects any inference results from the model.
16.See, for instance,
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Duguay (1994).
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17.One such study is by Laxton, Rose and Tetlow (1993).
18.An example is the study by Ricketts and Rose (1995) for Canadian inflation data. See also Fillion and Léonard
(1997) for an estimated Phillips curve with various regimes.
4.
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This supposes that the observed
20.Since these models are nonnested, we do not use likelihood ratio tests to assess the overall goodness of fit.
Instead, we rely on indirect criteria to select the best specification.
21.For more detail, see
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Stuber (1986).
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slowdown took place very gradually. Yet, an explanation based on a one or twotime break in the
drift term is equally likely.
Opinions diverge as to the right explanation, but, at this stage, both appear equally valid.
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This debate has
been ongoing for a long time now and has spawned numerous studies on methods of testing for
break points when the break time is difficult to pin down.
The latest in this category is the extensive work by Bai and
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Perron (1998).
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These authors
discuss the properties of least squares estimators in a wide class of linear regression models in the
presence of multiple structural breaks with unknown break points. These models include cases
where the residuals are autocorrelated and heteroskedastic and where there are lagged dependent
variables in the regressors.
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These modifications were
motivated partly in response to existing empirical findings on Canadian data, and partly because
of the results of diagnostics checking on the basic model.
First we focus on the assumed specification for the output equation of the model and apply
the Bai and
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Perron (1998)
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structural change test with unknown change point. Since we find that
we cannot reject a onetime break in the drift of output growth, we estimate versions of the model
with a breaking drift. We also estimate a version of the GS model with a modified error structure
after detecting the presence of ARCH effects in this equation.
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