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- Butler, L. 1996.A Semi-Structural Method to Estimate Potential Output: Combining Economic Theory with a Time-Series Filter. Technical Report No. 77. Ottawa: Bank of Canada.

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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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- The procedure continues until the change in the output-gap estimate from one step to the next falls below a pre-specified convergence criterion. λ W1W2W3 gss λ1600= 4 4 πtαiπti–βigti–εt+ +∑ =∑ i0= i1= πtgt βi 4.Alternatively, this problem could be mapped into an unobserved-components 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 nonaccelerating-inflation 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 nonaccelerating-inflation 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 Blanchard-Quah SVAR The Blanchard-Quah (1989) SVAR methodology uses limited long-run restrictions to separate the temporary and permanent components of output.

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- Canova, F. 1994. “Detrending and Turning Points.”European Economic Review 38: 614–23. ———. 1998. “Detrending and Business Cycle Facts.”Journal of Monetary Economics 41: 475– 512.

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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 Hodrick-Prescott (HP) (1997) filter does a good job, relative to other measures, of identifying turning points in U.

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- Conway, P. and B. Hunt. 1997. “Estimating Potential Output: A Semi-Structural Approach.”

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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 HP-filtered 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 HP-filtered output gap.

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- Cooley, T.F. and M. Dwyer. 1998. “Business Cycle Analysis Without Much Theory: A Look at

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- It would be unsuitable, however, to compare the accuracy of a three-variable SVAR with a method such as the MVF, which uses data only on output and inflation. 7.Indeed, this long-run 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 non-stationarity 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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- 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 post-war 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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- Ehlgen, J. 1998. “Distortionary Effects of the Optimal Hodrick-Prescott Filter.”Economics Letters 61: 345–49.

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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 non-stationarity 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 mean-squared-error 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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- King, R.G. and S.T. Rebello. 1993. “Low-Frequency Filtering and Real Business Cycles.”Journal of Economic Dynamics and Control 17: 207–31.

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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 non-stationarity 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 mean-squared-error 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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- Indeed, as expected, the performance of the SVAR deteriorates under these conditions. There is, however, a noticeable deterioration in the performance of the HP-based approaches, particularly at the end-of-sample. This phenomenon is not unexpected;
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- King and Rebello (1993)
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- discuss conditions under which the HP-filter 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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- Kuttner, K.N. 1994. “Estimating Potential Output as a Latent Variable.”Journal of Business and

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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 first-differenced 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 unobserved-component method, which includes the state-space model of, for example,
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- Kuttner (1994).
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- Unfortunately, owing to the typical instability of the maximum-likelihood estimates of the state-space 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 output-gap shocks, tended towards zero in most samples.

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- Laxton, D. and R. Tetlow. 1992.A Simple Multivariate Filter for the Measurement of Potential Output. Technical Report No. 59. Ottawa: Bank of Canada.

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- Alternatively, by varying the ratio of demand-to-supply 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 pre-specified 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 reduced-form Phillips curve

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- Lutkepohl, H. and D.S. Poskitt. 1996. “Testing for Causation Using Infinite Order Vector Autoregressive Processes.”Econometric Theory 7: 61–87.

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- Cooley and Dwyer (1998) show that a violation of the assumptions about the non-stationarity 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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- Murchison, S. 2001. “NAOMI: A New Quarterly Forecasting Model Part II: A Guide to Canadian

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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 non-accelerating inflation. 1.
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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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- 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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- 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 demand-to-supply 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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