Abstract
This paper examines economic developments and policies in Canada during 1990–95. Spurred by the robust growth in the United States and the easing of monetary conditions between 1991 and 1993, economic growth in Canada continued to strengthen during 1994. Real GDP grew by 4.5 percent in 1994 after growing by 2.2 percent in 1993 and 0.6 percent in 1992. Economic growth in 1994 was led by exports and investment in machinery and equipment. However, growth was more broadly based in 1994; private consumption strengthened, and there was a rebound in residential and nonresidential construction.
IV. A Survey of Alternative Methodologies for Estimating Potential Output 1/
1. Introduction
Estimates of potential output often play an important role in assessing the stance of monetary and fiscal policy over the course of the business cycle. However, such estimates are subject to considerable uncertainty, and questions regarding the size of the output gap in Canada and the reliability of alternative estimates of potential output have been subject to extensive debate. In the discussion below, the strengths and weaknesses of alternative measures of potential output are reviewed, and estimates of the Canadian output gap are provided.
2. Atheoretical approaches
The two most common techniques used to calculate potential output are the segmented-trend and univariate Hodrick-Prescott (HP) filter methods. The segmented-trend method assumes that the rate of growth of potential output changes at specific structural break points, but is constant between those points. Although the break points are usually imposed exogenously, they can be chosen on the basis of recursive residual tests for structural breaks. The HP filter allows a curve to be fitted through the real output series, trading off the smoothness of the trend against deviations of the actual series from trend. The HP filter is obtained as the solution to the following problem:
where yt is the original series, qt is the estimate of the trend component and yt - qt is the residual. 2/ The λ coefficient is the so-called “smoothness parameter;” as λ rises the HP filter tends toward a straight line.
An estimate of potential output was computed using a segmented trend, assuming a structural break in the fourth quarter of 1978 to coincide with the onset of the second OPEC oil shock. Prior to this date annual potential output growth is estimated to be 5 percent, whereas after this date the estimate declines to 2.6 percent. An estimate based on the HP filter sets the smoothness parameter λ at 1600, a value that is commonly used for quarterly data.
Estimates of output gaps and potential output levels using both methods are presented in Charts IV-1 and IV-2 respectively. Although both output gaps move in the same direction over most of the period, the estimate of the output gap based on the HP filter is considerably more stable than the segmented-trend estimate of the output gap. Moreover, the estimates have diverged in the most recent period. Specifically, the HP-filter method would suggest a relatively small output gap since early 1991, whereas the segmented-trend method would suggest the opening of a large output gap.
Both methods have shortcomings. The choice of a break point for the segmented trend is often arbitrary, as is the choice of the weighting factor in the smoothness criterion of the HP filter (it is usually set equal to the value chosen in Hodrick and Prescott’s seminal paper). Moreover, the HP-filter method places a large weight on the first and last observations, causing the estimate of potential output to be pulled toward the path of actual output at the end of the series. This problem is clearly illustrated in Chart IV-1 where, despite the slow Canadian recovery since 1991 and the evidence of a large gap, 1/ the HP-filtered estimate of the output gap over this period has been virtually zero. An additional drawback to both the HP-filter and the segmented-trend approaches is the fact that they explain potential output growth simply in terms of the trend growth in output and do not relate potential output to other macroeconomic variables.
3. The production-function approach
The production-function approach to gauging potential output assumes that output is a function of factor inputs (labor and capital) and a measure of technological change--total factor productivity. Total factor productivity often is assumed to evolve according to a trend. 2/ However, recent developments in endogenous growth theory suggest that total factor productivity may depend on structural variables such as research and development and the degree of openness in the economy. 3/ Moreover, underlying the commitment of many central banks to price stability is the premise that low inflation yields improvements in potential output. In the econometric analysis below both the inflation rate and a measure of openness are assumed to be determinants of total factor productivity.
In the past, openness has been proxied by the ratio of total exports and imports to GDP (Quah and Rauch (1990)) and by the real growth in traded goods (see Michaely (1979) and Balassa (1982)). Both approaches have drawbacks. The use of the trade ratio can be misleading in that it can be affected by the size of the national market. For example-, while the trade ratio for the United States is considerably lower than for many other countries, this is more likely to be the result of the size of its own national market rather than a lack of openness. 1/ Similarly, the growth of traded goods may be higher in countries that have only recently lowered tariffs and opened up trade barriers, than in countries that have set low tariffs for many years (and conventionally are considered to be more open). Both estimates of openness are considered in the estimates described below.
Two methods are commonly used to derive potential output using a production function. The two-stage approach derives total factor productivity as a residual from a production function with labor and capital as inputs, and then regresses the residual on a set of variables assumed to be structural determinants of total factor productivity. The cointegrating vector approach simply includes the structural determinants of total factor productivity with labor and capital in the equation determining output. Both approaches are illustrated below with reference to Canadian data over the period 1971 to 1993.
a. The two -stage method
Assume that the aggregate production function exhibits constant returns to scale and there are two inputs, labor and capital
where Yt is the log of output, At is the log of total factor productivity, Lt is the log of total employment and Kt is the log of the real capital stock. 2/
In the first stage, the change in total factor productivity is estimated as a residual from equation (1), assuming that α equals the labor share of GDP at factor cost (0.627 over the historical period 1971-1993):
CANADA MEASURES OF THE OUTPUT GAP
Citation: IMF Staff Country Reports 1995, 046; 10.5089/9781451806823.002.A004
Source: Statistics Canada (supplied by DRI); and Fund staff estimates.CANADA MEASURES OF POTENTIAL OUTPUT
Citation: IMF Staff Country Reports 1995, 046; 10.5089/9781451806823.002.A004
Source: Statistics Canada (supplied by DRI); and Fund staff estimates.In the second stage, the estimate of total factor productivity from equation (2) is regressed on inflation, alternative openness measures, and on the change in the capacity utilization rate, which is used as the measure of the cycle. 1/ We include a measure of the cycle because total factor productivity is assumed to be procyclical--firms are sluggish in laying-off and rehiring workers in cyclical downturns and upturns, respectively, preferring to adjust hours rather than employment. 2/
Stationarity tests indicated that each variable except for the GDP deflator is integrated of order 1. 3/ However, Chadha and Prasad (1994) document that Canada’s GDP deflator inflation rate is stationary using a longer time period, so it was assumed that all variables were stationary in first differences. Therefore, an equation explaining total factor productivity was estimated in first differences so that correct statistical inferences could be made. 4/
As noted above, the growth in total factor productivity is assumed to be positively related to the change in the capacity utilization rate (ΔCAP), and the growth in the volume of tradeables (ΔTV), and negatively related to the rate of increase of the GDP deflator (ΔGDPD). 5/
Since the inflation rate, the capacity utilization rate, and total factor productivity are likely to be simultaneously determined, instrumental variables were used to correct for simultaneity bias. The ratio of defense purchases to GDP was used to instrument for capacity utilization and the ratio of the average wage to the capital price deflator was used as an instrument for inflation. 6/
The tabulation below presents instrumental variable estimates of equation (3). These indicate that the growth in the volume of tradeables is a significant factor explaining productivity growth (at the 10 percent significance level) but that the inflation rate is not a significant factor. 1/ The insignificance of inflation is consistent with the work of Rudebusch and Wilcox (1994) and Fortin (1993) who find very little effect of inflation on productivity in the United States and Canada, respectively. As can be seen in the second column of the tabulation, excluding the inflation rate has very little effect on the estimates of the remaining coefficients.
Determinants of Total Factor Productivity
Determinants of Total Factor Productivity
| Coefficient estimates 2/ | |||
|---|---|---|---|
| Change in the Capacity | 0.0030 | 0.0030 | |
| Utilization Rate | (4.00) | (4.06) | |
| Inflation Rate | 0.0041 | -- | |
| (0.05) | |||
| Tradeables Growth | 0.0470 | 0.0469 | |
| (1.77) | (1.78) | ||
| R2 | 0.47 | 0.47 | |
| DW statistic | 2.09 | 2.09 | |
Determinants of Total Factor Productivity
| Coefficient estimates 2/ | |||
|---|---|---|---|
| Change in the Capacity | 0.0030 | 0.0030 | |
| Utilization Rate | (4.00) | (4.06) | |
| Inflation Rate | 0.0041 | -- | |
| (0.05) | |||
| Tradeables Growth | 0.0470 | 0.0469 | |
| (1.77) | (1.78) | ||
| R2 | 0.47 | 0.47 | |
| DW statistic | 2.09 | 2.09 | |
The estimate of potential output is calculated by aggregating the weighted sum of the factor inputs and adding to it the fitted value of long-run total factor productivity with the change in capacity set to zero. 3/ Charts IV-1 and IV-2 illustrate the output gap and potential output level calculated using this approach.
As can be seen from Chart IV-1, the gap moves from a small positive position in the mid-seventies to minus 6 percent in 1982. The gap then narrows, turning positive in 1987, and reaches a peak of 4 percent in 1988. Most recently the gap has turned negative, reaching a trough of 5 percent in 1993. The estimated gap rises to 1 1/2 percent in the fourth quarter of 1994. This estimate of the output gap is very similar to the estimate obtained using the segmented-trend method. The principal difference is that the estimate of potential growth using the two-stage estimate is 3.8 percent over the 1974-1978 period, whereas the estimate using the segmented-trend method is 5 percent.
The basic limitation of the two-stage approach is the need to control for cyclical effects in determining long-run total factor productivity. The capacity utilization variable that is used as the measure of the cycle already incorporates an implicit estimate of the output gap and therefore the estimate is subject to some degree of circularity. Other authors avoid this problem by detrending the total factor productivity series using the HP filter or a linear trend. However, these alternative methods cannot provide a consistent estimate of total factor productivity in a forecast.
b. The cointegrating vector approach
The second method of estimating potential output using the production function approach tries to find a long-run cointegrating vector between output and its long run-determinants. Coe and Moghadam (1993) provide an example of this technique, whereby estimates of potential output for France are obtained by the Johansen multivariate maximum-likelihood method. 1/ The basic economic assumption underlying this approach is that long-run output can be expressed as a log linear function of factor inputs (e.g., labor and capital) and structural determinants of total factor productivity (e.g., inflation and the trade ratio). The advantage of this approach is that there is no need to control for cyclical effects because, by assumption, they are not present in a long-run relationship. The estimated equation is as follows:
The tabulation below presents test statistics and the estimated cointegrating vector from the Johansen procedure assuming increasing returns to scale (no significant cointegrating vector could be found when constant returns to scale was imposed). Panel A reports the maximal eigenvalue test. For the null hypothesis that there are no cointegrating vectors (τ=0) the test statistic (49.4) is greater than the 95 percent critical value, rejecting the null hypothesis. The null hypothesis of τ≤1 against τ=2 however, cannot be rejected, indicating that there is a unique cointegrating vector. Panel B reports the trace test, indicating that the null of τ=0 against τ≥1 is rejected but that the null of τ≤1 against τ≥2 cannot be rejected. Both tests therefore indicate that there is a unique cointegrating vector.
Johansen Maximum Likelihood Tests and Parameter Estimates
Johansen Maximum Likelihood Tests and Parameter Estimates
| A. | Cointegration likelihood ratio test based on maximal eigenvalue of the stochastic matrix | ||
|---|---|---|---|
| Hypothesis | 95 Percent | ||
| Null | Alternative | Test Statistic | Critical Value |
| τ=0 | τ=1 | 49.4 | 33.32 |
| τ≤1 | τ=2 | 18.7 | 27.14 |
| τ≤2 | τ=3 | 11.6 | 21.07 |
| τ≤3 | τ=4 | 4.5 | 14.90 |
| τ≤4 | τ=5 | 0.3 | 8.18 |
| B. | Cointegration likelihood ratio test based on trace of the stochastic matrix | ||
| Null | Alternative | Test Statistic | 95 Percent Critical Value |
| τ=0 | τ≥1 | 84.3 | 70.60 |
| τ≤1 | τ≥2 | 34.8 | 48.30 |
| τ≤2 | τ≥3 | 16.1 | 31.50 |
| τ≤3 | τ≥4 | 4.5 | 17.95 |
| τ≤4 | τ≥5 | 0.3 | 8.18 |
Johansen Maximum Likelihood Tests and Parameter Estimates
| A. | Cointegration likelihood ratio test based on maximal eigenvalue of the stochastic matrix | ||
|---|---|---|---|
| Hypothesis | 95 Percent | ||
| Null | Alternative | Test Statistic | Critical Value |
| τ=0 | τ=1 | 49.4 | 33.32 |
| τ≤1 | τ=2 | 18.7 | 27.14 |
| τ≤2 | τ=3 | 11.6 | 21.07 |
| τ≤3 | τ=4 | 4.5 | 14.90 |
| τ≤4 | τ=5 | 0.3 | 8.18 |
| B. | Cointegration likelihood ratio test based on trace of the stochastic matrix | ||
| Null | Alternative | Test Statistic | 95 Percent Critical Value |
| τ=0 | τ≥1 | 84.3 | 70.60 |
| τ≤1 | τ≥2 | 34.8 | 48.30 |
| τ≤2 | τ≥3 | 16.1 | 31.50 |
| τ≤3 | τ≥4 | 4.5 | 17.95 |
| τ≤4 | τ≥5 | 0.3 | 8.18 |
The assumption of this approach is that in the long run actual output equals potential output so that the parameters of the cointegrating vector provide an estimate of potential output. 1/
Except for the most recent period when the series have diverged considerably, the output gap calculated from the cointegrating relationship follows a similar profile to the output gap calculated using the two-stage method (see Chart IV-1). The reason for the recent divergence is that the potential output estimate from the cointegration method is penalized less severely in a climate of low inflation, whereas the estimate from the two-stage method is not affected by inflation. The inflation rate has averaged less than 2 percent over the past few years and this low inflationary climate generates an estimate of potential output growth of over 4 percent in the 1990s using the cointegration method. The corresponding estimate of potential output growth using the two step method is 2.6 percent.
Although the cointegrating vector approach to estimating potential output is the most attractive approach on methodological grounds, its coefficient estimates are not particularly robust. A cointegrating vector could not be found when the assumption of constant returns to scale was imposed. Moreover, the size of the coefficient on the inflation rate appears implausibly high; a cointegrating vector was found excluding inflation but the signs of the remaining coefficient estimates were not plausible.
4. Evaluation of alternative approaches
This paper has considered a number of ways of estimating potential output and finds that each method has its shortcomings. However, two criteria can be used to differentiate between the estimates. The preferred estimate should be consistent with other economic data and should generate an output gap estimate that is consistent with a measure of excess demand in the labor market. On this basis the preferred approach is the two-stage method, since it is based on the production function and produces estimates of potential output that are robust and consistent with measures of the unemployment gap.
The Johansen cointegrating technique would be preferred on methodological grounds because it eliminates the need to control for cyclical movements. However, this method produces coefficient estimates that are heavily dependent on the equation specification. In addition, it produces an estimate of the output gap of 11 percent in the fourth quarter of 1994, which is much higher than the corresponding estimate of the excess demand gap from the labor market (2 1/2 percent). 1/ In contrast, the output gap estimate for the fourth quarter of 1994 from the two-stage production function method is 1 1/2 percent.
References
Balassa, B., Development Strategies in Semi-Industrial Economies, Oxford University Press (1982).
Chadha, B. and E. Prasad, “Interpreting the Cyclical Behavior of Prices,” Staff Papers, International Monetary Fund (June 1993).
Coe, D. and R. Moghadam, “Capital and Trade as Engines of Growth in France,” Staff Papers, International Monetary Fund (September 1993).
Edwards, S., “Trade and Growth in Developing Countries,” Journal of Economic Literature (September 1993).
Fortin, P., “The Unbearable Lightness of Zero Inflation Optimism,” Canadian Business Economics (Spring 1993).
Hodrick, R. and E. Prescott, “Postwar U.S. Business Cycles: An Empirical Investigation,” mimeo, Carnegie-Mellon University, (1980).
Laxton, D. and R. Tetlow, “A Simple Multivariate Filter for the Measurement of Potential Output,” Bank of Canada Technical Report No. 59 (June 1992).
Macklem, T., “Recent Advances in Growth Theory: Perspective and Policy Implication,” Bank of Canada Review (Winter 1993).
Michaely, M., “Exports and Growth: An Empirical Investigation,” Journal of Development Economics (March 1977).
Organization for Economic Cooperation and Development, “Estimating Potential Output, Output Gaps and Structural Budget Balances,” mimeo (1994).
Quah, D. and J. Rauch, “Openness and the Rate of Economic Growth,” Working Paper, University of California San Diego (October 1990).
Rudebusch, G. and D. Wilcox, “Productivity and Inflation: Evidence and Interpretations,” mimeo, Board of Governors of the Federal Reserve (May 1994).
Wilkins, C., F. Lee, and S. James, “Estimating Trend Total Factor Productivity Growth in Canada,” Department of Finance Working Paper No. 3 (1992).
Prepared by Alun H. Thomas.
A recent extension of the univariate filter is the multivariate HP filter. The multivariate filter minimizes the weighted average of the errors from a set of relationships rather than from a single output equation. In addition to the output equation, an inflation equation (often based on the Phillips curve), and an unemployment equation (often constructed so as to reflect Okun’s law) are usually included in the specification (see Laxton and Tetlow (1992) for example). The advantage of this approach is that--by construction--it produces an estimate of the output gap that has strong explanatory power in the inflation equation. The disadvantages include the standard end-point problem associated with the HP filter, and the need to arbitrarily choose the weights in the loss function assigned to the errors in each equation.
For example, the unemployment rate was at least 1 1/2 percent above most estimates of the natural rate of unemployment over the 1991-93 period.
See, for instance, OECD (1994) which uses the HP filter to determine the trend in total factor productivity, and Wilkins, Lee, and James (1992) who use a segmented trend.
See Macklem (1993) for an overview.
See Edwards (1993) for a discussion of this point.
The output variable is GDP in constant 1986 dollars, the employment variable is economy-wide employment from the Labor Force Survey and the capital stock series is gross fixed non-residential capital formation in constant 1986 dollars. All variables were obtained from DRI.
This is the approach used by Rudebusch and Wilcox (1994) and by Fortin (1993).
See Chapter II for more details on this issue.
A variable is said to be integrated of order 1 if first differencing renders it stationary.
In order to satisfy the basic assumptions of regression analysis variables must be transformed into stationary series or yield a linear combination that is stationary.
Substituting the CPI inflation rate for the GDP deflator inflation rate produced similar results. The capacity utilization rate is provided by Statistics Canada and is based on a trend fitted through the output-capital ratio and multiplied by an estimate of the capital stock. The estimate only covers the non farm goods-producing sector, which accounts for roughly 30 percent of GDP. The volume of tradeables is the sum of merchandise exports and imports in constant 1986 dollars and the GDP deflator is the implicit output deflator. All variables were obtained from DRI.
Rudebusch and Wilcox (1994) use similar instruments.
The coefficients on the level and the change in the ratio of total trade to GDP were insignificant.
t-statistics in parentheses.
The labor factor input differs slightly from the variable used to estimate total factor productivity because the long-run employment level is used instead of the actual level. The long-run level is defined as the labor force multiplied by one minus the natural rate of unemployment. See Chapter III for an analysis of the natural rate of unemployment.
Since, as discussed above, output, labor, and capital are integrated of order 1, it is valid to investigate the possibility of a long-run relationship between these variables in levels. See Coe and Moghadam (1993) for details of the methodology.
The preferred cointegrating vector was chosen on the basis that all of the estimated coefficients have the expected signs and are of reasonable magnitudes.
The excess demand gap in the labor market is defined as the difference between the actual rate of unemployment and the natural rate, adjusted for an Okun coefficient of 2.3. See Chapter III for an analysis of the natural rate of unemployment.
