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- Butler, L. 1996. The Bank of Canada’s New Quarterly Projection Model, Part 4. A Semi-

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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 end-of-sample 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 St-Amant (1999). 3.See also Harvey and Jaeger (1993) and King and Rebelo (1993). As one alternative to the above class of models, St-Amant 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 end-of

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- Duguay, P. 1994. “Empirical Evidence on the Strength of the Monetary Transmission Mechanism in Canada.”Journal of Monetary Economics 33(1).

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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 well-identified. 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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- Hamilton, J. 1986. “A Standard Error for the Estimated State Vector of a State-Space Model”. ournal of Econometrics 33:387-97.

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- Studies employing these types of models frequently encounter difficulties in finding suitable data for their long-run values. 7.This is the mean square error associated with the estimated state vector and is obtained as a by-product 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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- Kuttner, K.N. 1994. “Estimating Potential Output as a Latent Variable.”Journal of Business and Economic statistics 12(3):361-68.

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- uncertainty, must be added to the above. fact that we can also directly obtain out-of-sample 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 state-space 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 state-space framework so that when no ARCH errors are present in the system, the above filter reduces to the well-known 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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- Phillips, P.C.B. 1989. “Partially Identified Econometric Models.”Econometric Theory, 5:181-240.

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- the equations in the statespace system on the other, we can still analyze a well-defined 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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- Stuber, G. 1986. The Slowdown in Productivity Growth in the 1975-83 Period: A s Survey of Possible Explanations. Technical Report No. 43. Ottawa:Bank of Canada. van Norden, S. 1995. “Why Is It So Hard to Measure the Current Output Gap?” Mimeograph. Bank of Canada.

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- This supposes that the observed 20.Since these models are non-nested, 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 two-time 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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- Watson, M.W. 1986. “Univariate Detrending Methods with Stochastic Trends.”Journal of

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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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