In this paper, we discuss some methodologies for estimating potential output and the output gap that have recently been studied at the Bank of Canada. The assumptions and econometric techniques used by the different methodologies are discussed in turn, and applications to Canadian data are presented.

The use of the Hodrick-Prescott (HP) filter to measure the output gap has been justified on the basis that this filter extracts business-cycle frequencies from the data and that it can estimate an unobserved cyclical component. We note that the HP filter is unlikely to do well in achieving these objectives for series whose spectra have the typical Granger shape, such as real output, and that it will often fail to measure cyclical components adequately. The problems of the HP filter are accentuated at the end of samples, which is the place most relevant for policymakers. Finally, we note that univariate filters will only be able to give us information about the current output gap if the gap is Granger-caused by output growth; this is not the case if we believe that potential output is exogenous.

Extensions to the HP filter, such as those proposed by Laxton and Tetlow (1992) and Butler (1996), have focussed on incorporating additional information derived from assumed or estimated economic relationships. The motivation behind these "hybrid" methods is a desire to obtain estimates of the output gap that are conditioned by structural information but that remain "smooth." However, existing hybrid methods have proved hard to estimate. In addition, they may not be robust to alternative reasonable calibrations, and they do not allow for easy calculation of confidence intervals. We also find that Butler's method does not perform as well as the simple HP filter in terms of isolating fluctuations of output originating from business-cycle frequencies. We also discuss the "TOFU" approach (a Trivial Optimal Filter that may be Useful), which replaces the smoothness assumptions of the hybrid methods with an unrestricted but linear filter.

We then turn to multivariate filtering methods based on VARs that incorporate long-run restrictions. Unlike univariate filters, VAR-based methods do not suffer from obvious end-of-sample problems, and they can provide projected values for the output gap. Relative to other multivariate methods (such as the multivariate Beveridge-Nelson method), one advantage of the VAR method using long-run restrictions is that it does not restrict the dynamics of potential output a priori. We investigate the implications of long-run restrictions on real output only and on real output and inflation. We argue that the latter approach should be of interest for policy-makers focussing on movements of real output associated with movements in the trend of inflation. Unfortunately, the VAR applications that we consider display wide confidence intervals, similar to those reported on the basis of other methods. Using VARMAs or constrained VARs instead of unconstrained VARs may reduce that uncertainty.