The Factor Utilization Gap. Mark Longbrake*

Transcription

1 Draft Draft The Factor Utilization Gap Mark Longbrake* The Ohio State University May, 2008 Abstract For the amount that the output gap shows up in the monetary policy literature there is a surprisingly small amount of time spent on its actual estimation. Due to an increasing amount of uncertainty surrounding the quality of the estimation of the output gap, due both to potential unit root problems and large ex post revisions, a number of authors have been using the unemployment gap as a proxy. However, the use of the unemployment gap implies that capital is always used to its full capacity. This paper derives both an unemployment gap and a capacity utilization gap, using Adaptive Least Squares (ALS), and combines them to estimate the output gap. Adaptive Least Squares is a special case of the Kalman Filter that allows for a time varying parameter model to be estimated relatively easily. The use of both unemployment and capacity utilization allows us to consider the effects of both labor and capital under or over utilization, thus eliminating a potential substitution bias from the unemployment gap, and avoiding unit root problems from a univariate estimation of the output gap. * longbrake.13@osu.edu 1

2 I. Introduction One result of using micro founded macroeconomic models in the monetary policy arena is the need to estimate the output gap. Starting with Taylor in 1993 and continuing until the present, almost every optimal monetary policy model contains the output gap as an explanatory variable. While the output gap is a straightforward theoretical concept, it is impossible to observe directly. This has made for a series of interesting papers from Orphanides (2001, 2003) and Orphanides and van Norden (2002) calling into question the accuracy of output gap estimates, specifically noting that there are usually large ex post revisions to the initial estimate. In addition to inaccuracy in estimation, there is also a question about output gap specification. The question of whether or not U.S. GDP follows a unit root process goes back at least as far as Nelson and Plosser in In an attempt to estimate the output gap as the deviation from trend GDP, Longbrake and McCulloch (2007) found support for the unit root hypothesis. The combination of inaccurate estimates, large revisions and unit root questions has pushed researchers to find an alternative to the output gap to use in monetary policy formation. The most commonly used alternative is the unemployment gap. The unemployment gap is usually specified as the deviation of unemployment from its natural rate. Since unemployment is estimated relatively accurately and is easily assumed to be a stationary process 1, using the unemployment gap to proxy for the output gap avoids the pitfalls of estimating the output gap directly. It is also theoretically attractive, because when output is very high, we would expect unemployment to be low, and vice versa. Thus these two gaps should be highly negatively correlated. However every growth model in economics assumes that growth is a function of at least labor and capital, and quite often a number of other parameters as well. Simply using the unemployment gap as a proxy ignores the capital side of the market. The implicit assumption of using the unemployment gap to proxy for the output gap 1 Unemployment is bounded between zero and one and thus cannot be truly non-stationary. 2

3 is that capital is always being used to its full capacity 2. In 1979, Perloff and Wachter made this assumption explicitly based on uncertainty about the capacity utilization data. Still in 2001, the Congressional Budget Office (CBO) makes this assumption in their output gap estimation method despite the fact that capacity utilization data has improved 3. One look at capacity utilization data (Figure 8) suggests that this is an incorrect assumption. Since 1948, the capacity utilization rate has varied between 92% and 68%. This large amount of variation implies that there have been periods of time when capacity that could be used has sat idle. Since the data is available, and the capital side of the economy is important, assuming capital is fully utilized is an unnecessary assumption. Thus, this paper will remove this assumption and include the production side of the market in our estimate of the output gap. Section II of this paper gives background information on the importance of the output gap. The section also discusses the potential problems with output gap estimation and suggests a method for avoiding these problems. Section III details a number of potential detrending methods that have been used in the literature to estimate the output gap. The overall outcome of these univariate detrending methods will be shown to be lacking, which will lead us to suggest the multivariate approach that this paper takes. Then we explain the method that is used in this paper, and why we believe this method to be a good choice. Section IV derives the unemployment gap using adaptive least squares (ALS) as the estimation method. This section also contains a brief summary of the ALS procedure. Section V addresses the theory behind why the capacity utilization data is useful in accurately estimating the output gap. It also uses ALS to estimate the capacity utilization gap. Section VI argues that an estimate of the output gap that only relies on unemployment data is incomplete. As a result we develop a model that allows estimation of the output gap using both the unemployment gap and the capacity utilization gap. Section VII concludes. II. The Output Gap in Monetary Policy Literature 2 There is another possible implicit assumption, that the capacity utilization gap is the same as the unemployment gap. However this assumption is never made explicitly. 3 The OECD makes a similar assumption for its output gap estimations. 3

4 The output gap has been widely used in recent monetary policy literature. In their paper, "The Science of Monetary Policy" Clarida, Gali and Gertler use a New Keynesian Phillips curve to analyze optimal monetary policy (1999). The key components of the New Keynesian Phillips curve are the expectations of future inflation and the output gap. Their work builds on the work of Taylor (1993) who used the output gap as an input in the Fed s policy reaction function. Ball, Mankiw, and Reis (2003) derive a sticky information Phillips curve that also incorporates the output gap. This body of literature suggests a number of different optimal monetary policy rules. There is a distinction between the rigidities in the models, whether to use past, current or future inflation expectations, as well as the ability of the central bank to commit; however amidst all of their differences every one of the suggested optimal policy rules require that the central bank know the output gap. All of this literature takes for granted that the central bank knows the size of the output gap at any point in time. The output gap is specified as the deviation of output from its potential, and as such is easily calculated given current and potential output levels. However the estimation of potential output is crucial. A sizeable fraction or the current literature uses the CBO s estimates for potential output. The remaining fraction uses a simple detrending of GDP. The difficulties in accurately measuring the output gap are well documented. In Rudebusch s 2002 paper he acknowledges that output gap estimates are uncertain. In order to model the uncertainty of the output gap estimation, Rudebusch assumes the output gap is estimated with an error. Orphanides (2001) goes so far as to suggest the use of nominal income in monetary policy rules because potential GDP estimates are so uncertain. More recently authors have taken to simply using the unemployment gap as a proxy for the output gap. This removes the need to estimate the output gap, but adds the need to estimate the unemployment gap. Generally speaking the estimation methods are the same for estimating the output gap and the unemployment gap. Section IV will outline why the unemployment gap is often preferred. This paper takes advantage of a new estimation technique, Adaptive Least Squares (ALS), to estimate potential GDP and the output gap. The methodology of ALS will be explained in Section III below. 4

5 The CBO s methods of estimating the output gap are explained by the CBO in their 2001 paper CBO s Method for Estimating Potential Output: An Update and in their 2004 paper A Summary of Alternative Methods for Estimating Potential GDP. The CBO uses a Solow growth model for GDP, and estimates potential levels for labor force and unemployment using a piecewise linear interpolation technique. The CBO uses NBER business cycle peaks as the break points for its piecewise linear trend. Thus it is easy to see why there is substantial revision error in the CBO estimates, because business cycle peaks are not known until well after they occur. The CBO s method differs from simple detrending because it utilizes a model of the economy. However, it can not get completely away from detrending, because even in their model it is necessary to detrend multiple variables. The detrending methods used by the CBO and in other papers are outlined in the next section. This paper combines a parsimonious detrending method, ALS, with a simple Solow growth model in order to incorporate both unemployment and capacity utilization data into an estimation of the output gap. By not using real GDP data, this method avoids all potential unit root problems, and by utilizing ALS methodology we avoid ex post revision problems. III. Long Run Trend Estimation The simplest form of long run trend estimation is OLS. However in the case of GDP OLS estimates are clearly lacking. It is instructive to examine the simple OLS estimate of the trend in quarterly ln real GDP. Figure 1 shows the OLS estimate of the long term trend in GDP superimposed over quarterly ln GDP from For the first third of the sample the output gap is always negative, and for the second third the gap is always positive. Only for the data from the mid 80's to the present does the OLS trend seem to fit the data. The Augmented Dickey-Fuller test statistic for the OLS estimate is and the 10% critical value is Thus the null hypothesis of a unit root can not be rejected even at the 10% level, lending support to Nelson and Plosser s (1982) suggestion that GDP is an I(1) process. However the idea that GDP follows a unit root process is theoretically unattractive, and this led to a body of literature that tried to show GDP did not follow 5

6 a unit root. From Figure 1 we can see that the long run trend in ln real GDP appears to have changed over time. In response to this, Pierre Perron in 1989 and again in 1997 allowed the trend in GDP to have a structural break in 1973 which he attributed to an oil price shock. By allowing the slope of the trend to change in 1973, Perron is able to reject the unit root hypothesis at the 5% level. One problem with this result is that the timing of the structural break was imposed ex post by Perron. In his 1997 paper, Perron, does not have the date of the structural break fixed. Instead the data is allowed to choose the time of the break. While this is a good modification to the initial model, even under the modified procedure the econometrician is still required to choose the number and type of structural breaks. Another drawback to this method is that it lacks any predictive power since there is no way to predict the timing or magnitude of future structural breaks. The CBO also uses a piecewise linear trend, but as mentioned above they use business cycle peaks as the break points. This is far less attractive than Perron s methods and leads to the need for, sometimes large, ex post revisions. Quarterly ln GDP with OLS Trend above below D-F Stat = % critical value: ln GDP OLS Trend Figure 1 Attempting to go in another direction Hamilton, in 1989, developed a two-state regime switching model of GNP growth. This model uses a Markov process to 6

7 transition between two states, expansion and contraction. Hamilton's model was extended by Lam in 1990 by allowing the transition probabilities to vary over time. Yet even in this modified version, there is still the limitation that the world is only allowed to have two states. A popular method of finding the trend in a given time series is to use a Hodrick- Prescott (HP) filter. The HP filter has two significant drawbacks. The first is that it requires the use of an arbitrary smoothing parameter. The econometrician in effect must choose the length of the cycles to be removed from the trend, and therefore introduced into the gap. Cogley and Nason (1995) found that cycles found in HP filtered data might be due to the filter, not the underlying data. The other drawback is the fact that the HP filter has no predictive power. In order to avoid the criticisms of the structural break/piecewise linear literature, some have proposed using a time-varying parameter model. This would allow any structural break to occur naturally simply by letting the parameter values change over time. In addition this method would allow the observation of the time paths of the parameters, which could then be used for predictive purposes. The major drawback to this approach is that it requires the estimation of a large number of parameters. And as the GDP series is not incredibly long, this quickly becomes a problem. Cooley and Prescott (1973) avoid this problem by proposing a model that only allows the trend intercept term to be time-varying. Thus the slope of the trend and the AR parameters are not allowed to change. Stock and Watson (1998) find no time variation in output growth rate using a model similar to Cooley and Prescott's, but again only the intercept parameter is allowed to change over time. However this model has nothing to say about the slope of the trend because, similar to OLS, it is not allowed to change. Since the structural break that Perron imposes is a change in the slope of the trend, the Cooley and Prescott and Stock and Watson models are instructive, but do not completely address the question. Our method of choice for this paper is called Adaptive Least Squares (ALS). ALS is a variant of the Kalman filter developed by McCulloch to estimate time varying relationships (2005). ALS is an estimation technique superior to the alternatives offered earlier in this section. One reason for this is that although ALS allows for all 7

8 of the coefficients in the model to change over time there is only a single parameter, ρ, that governs these changes. ρ is a measure of the signal/noise variance ratio in the model, which also determines the asymptotic gain of the model, or how quickly the model learns. The learning aspect of ALS means that its estimation has all the benefits of a time varying parameter model, without having a large number of time varying parameters to estimate. In addition although the entire system is dependent on only one parameter all of the coefficients, including both the intercept and slope parameters, are allowed to change over time. This makes ALS superior to models that only allow the intercept to be time-varying. Also by allowing time varying coefficients, ALS allows for a different trend in each period. This means that the series can have an infinite number of states, as opposed to the limit of two states imposed by Markov switching models. ALS also estimates ρ from the actual data, as opposed to the HP filter where there is an arbitrary smoothing parameter 4. This keeps the econometrician from imposing cycles upon the data. Another benefit of the adaptive least squares technique is that it allows the information to be analyzed in two ways. The ALS filter estimates the model using only information from the past. This gives the perspective of the best estimate that could have been made at each point in time. These estimates are very instructive because they show the best estimates policy makers could have when making policy decisions. The ALS smoother is both forward and backward looking, giving the advantage of being a more precise estimate. However, this estimate is only useful ex post. Both estimates are of interest since they address different questions. For a full explanation and derivation of ALS see McCulloch (2005). This paper follows the estimation technique described therein, using a GAUSS procedure also developed by McCulloch. A brief summary of the ALS procedure is shown in Appendix 1. Longbrake and McCulloch (2007) estimated the output gap using only GDP data. The outcome was unsatisfying because, like Nelson and Plosser, they were unable to reject that GDP follows a unit root process even locally. If the GDP time series truly 4 The HP filter implicitly models GDP based on a model that does not match the data, so there is no point in trying to estimate the smoothing parameter optimally. 8

9 does not follow a trend-stationary process then any attempt to detrend the series will inevitably meet with failure, because there is no underlying trend. This suggests that in order to find the output gap we need to look outside of the GDP data. As mentioned earlier this has led to suggestions of using everything from nominal income to unemployment in place of GDP. In the next section we explain why unemployment is a possible proxy for GDP and estimate an unemployment gap using ALS. IV. The Unemployment Gap The relationship between the output gap and the unemployment gap was suggested by Arthur Okun (1962). Based mainly on empirical observation, Okun stated that the amount that unemployment was above or below the natural rate of unemployment was inversely related to the amount that output was above or below the natural rate of output. This relationship came to be known as Okun s Law 5. It is intuitive that the unemployment rate and output should be related, because labor is one of the inputs into production. Thus if there is high unemployment and as a result productive labor resources are being under-utilized, naturally we would expect output to fall. Since the ultimate goal is to find the output gap, the deviation of output from its long run level, the amount that unemployment is above or below its natural rate should be a good proxy for the amount output is above or below its natural rate. The natural rate of unemployment, UN, is specified as the rate of unemployment associated with output being at its natural rate. This is exactly the rate that we are trying to measure here. In order to estimate the unemployment gap, we first need to measure the natural rate of unemployment. 6 For this paper we will specify the unemployment gap as the deviation of actual unemployment from a long-term possibly time varying natural rate, specifically, 5 Okun s Law is described as a linear relationship between output growth and the change in unemployment. This assumes that neither the natural rate of unemployment, nor the natural rate of output change over time. By estimating a time-varying parameter model, we use the intuition of Okun s Law but allow the relationship to change over time. 6 For simplicity, we will use the natural rate of unemployment rather than the NAIRU. 9

10 u t = UN t + g u,t, (1) where u t is the unemployment rate in period t, UN t is the value of the natural rate of unemployment in period t, and g u,t is the unemployment gap in period t. With this specification, we can solve for the unemployment gap, g u, by taking the difference between actual unemployment, u, and its long-term trend or natural rate, UN t. This specification assumes that unemployment follows a possibly time-varying constant trend. In order for this specification to be meaningful, we need the unemployment gap to be stationary. We have chosen to specify the gap as an autoregressive process, g u,t = γ u,1 g u,t-1 + γ u,2 g u,t-2 + γ u,3 g u,t-3 + γ u,4 g u,t-4 + e u,t. 7 (2) This implies that the unemployment time series is stationary and while this is may not be the best assumption for GDP data, it is a reasonable assumption for the unemployment data. In fact the unemployment rate was the only series, of the fourteen examined, that Nelson and Plosser (1982) were able to reject their unit root hypothesis for. Figure 2 shows a graph of Bureau of Labor Statistics monthly seasonally adjusted unemployment data from 1948 to From looking at the unemployment data it seems like the natural rate of unemployment may have increased in the late 1970 s and fallen again in the 1990 s. If indeed the natural rate has changed over time then we will need to estimate a model that can take the changes into account. A simple OLS estimate of the natural rate of unemployment is included in Figure 2, holding the γ s and hence UN constant. This estimate implies that the natural rate of unemployment has been 5.63% over the last 60 years. 7 Adding additional lags to the specification of the output gap simply adds additional γ s. 10

11 OLS trend with Unemployment Unemployment OLS Trend Figure 2 However, there is a period of about twenty years where unemployment is almost always below its natural rate, followed by a period of thirteen years where unemployment is always above its natural rate. This suggests rather long unemployment cycles. It seems as if there may have been a fundamental change in the period In order to take this into account we need to allow for the natural rate of unemployment to adjust over time. There are several options of models that will accomplish this goal. The first option, and the one used by the Congressional Budget Office in their output gap estimate, is to use a piecewise linear trend. The CBO uses the NAIRU in their estimate of the unemployment gap, but then converts the unemployment gap into a labor force growth rate using a piecewise linear trend. This type of model has a major drawback in that the econometrician needs to choose when the breaks in the trend occur. 8 Even if a Perron (1997) type model is used, which allows the data to determine the location of the breaks, the econometrician still needs to choose how many breaks there are and what types of breaks occur. This method also requires the 8 The CBO uses business cycle peaks as the break points in its piecewise linear trend. This is doubly problematic because business cycle peaks are not identifiable until multiple periods after they occur. Thus large ex post revisions are required any time a new business cycle peak is identified. 11

12 econometrician to have the entire time series with which to figure out potential trend breaks. In order to pick up the potential change in the natural rate of unemployment, we choose to estimate the natural rate using a time-varying parameter model. We are using ALS to estimate the trend. The advantages of this method were outlined in section III, and an overview of the methodology is shown in Appendix I. ALS has the added advantage of nesting OLS, so if the parameters do not change over time the model will not force them to. We estimate the unemployment rate as an AR(4) process with a time-varying level. The length of the AR process was initially estimated at 12, which with monthly data would give us one year of lags, but the coefficient on the AR(12) parameter was not significantly different from zero. We assumed that a simpler process trumps a more complicated one and thus dropped the AR(12) term. We repeated this process until the final AR coefficient was significant, leaving us with an AR(4) process. The data we used is the Bureau of Labor Statistics seasonally adjusted monthly Current Population Survey unemployment series. 9 The final model is as follows: u t = β 0,t + β 1,t u t-1 + β 2,t u t-2 + β 3,t u t-3 + β 4,t u t-4 + e t. (3) The natural rate of unemployment can be extracted from the results with the following equation: β 0, t UN t =, for all t. (4) 4 1 βi, t i= 1 This makes the unemployment gap easy to find since it is specified as the difference between actual unemployment (u) and its natural rate (UN). The model in equation (3) was estimated using both the ALS filter and smoother. The estimate of the signal/noise variance ratio, ρ, was The likelihood ratio statistic for ρ = 0 was 3.03 and since the 5% critical value for the likelihood ratio is 9 Using the unadjusted series and including seasonal dummies does not significantly affect our results. 12

13 approximately 2.3 we can reject the hypothesis that ρ = 0 with 95% confidence 10. This estimate of ρ implies a limiting effective sample size of months, or years. This means that the model parameters change substantially every years, implying that in our sample, the natural rate of unemployment goes through a bit more than 4 episodes. The estimates of ρ, the likelihood ratio and the effective sample size are the same for both the ALS filter and smoother. Sum of AR coefficients ALS Filter Estimates Figure 3 According to our specification the unemployment gap must be stationary, and a necessary condition for stationarity is that the sum of the AR coefficients is less than one. The ALS filter estimates of the sum of the AR coefficients are shown in Figure 3. It is interesting to note that from 1980 to 1983 there is a substantial increase in this sum, culminating in a value that is basically one. This near one value for the sum of the AR coefficients creates an anomaly in the ALS filter estimate of UN (see below). The ALS smoother estimates do not show this sharp run up, and hence are always safely below one. In fact there is virtually no change in the value of the smoother estimate of the sum of the AR coefficients in the early 1980s (Figure 4). 10 The critical values for the likelihood ratio were derived by McCulloch using Monte Carlo methods, since the null is on the boundary of the parameter space. 13

14 The ALS filter estimate of the natural rate of unemployment is shown in Figure 5. Recall that the ALS filter only uses information up to and including time t. This shows us the best estimate that we could have made at any given point in time. Note that in 1983, when the sum of the AR coefficients is very close to one, the ALS filter estimate of the natural rate of unemployment shoots up to almost 14%. This is clearly incorrect, but it stems from the fact that in the period leading up to 1983 it appeared that the unemployment rate was exploding. Only in hindsight did we know that 1983 was a peak. The ALS filter estimate of the unemployment gap, u UN, is shown as the dashed red line in Figure 7. There is an anomalous sharp dip in the ALS filter estimate of the unemployment gap in This matches the spike in the natural rate that was due to near unity in the sum of the AR coefficients. The early points in the ALS filter graphs are very volatile because the system is initialized with a diffuse prior and it takes a few periods for the model to learn the coefficients. This problem does not occur with the smoother because it is looking both forward and backwards. Thus at t 1 the smoother already has 13.9 years of information, while the filter has only one quarters worth of information. Sum of AR coefficients ALS Smoother Estimates Figure 4 14

15 Figure 6 shows the ALS smoother estimate of the natural rate of output on top of the actual unemployment rate. It may be seen that the natural rate of unemployment starts under 5% in 1948, and climbs to over 6% in the early 1980s before falling in recent years. The smoother estimate of the natural rate of unemployment does not show a spike in This is because by using both past and future data, the smoother is able to see that the unemployment rate subsequently fell from the high rates in the early 1980s. The solid blue line in Figure 7 shows the ALS smoother estimate of the unemployment gap, u UN. When we convert this estimate of the unemployment gap to a quarterly frequency 11 and compare it with the ALS smoother estimate of the output gap from Longbrake and McCulloch 2007, the correlation coefficient is This relatively high negative correlation and the fact that there is no debate over whether the unemployment rate is stationary are the reasons that many economists use the negative of the unemployment gap, the employment gap, to proxy for the output gap. The high correlation also vindicates the univariate measure of the output gap from Longbrake and McCulloch even though they were not able to reject that GDP follows a unit root process. 11 We simply used the end of quarter value, but using a quarterly average, or the middle of the quarter value does not significantly change our results. 12 If we were to compare to the CBO output gap rather than the ALS smoother, our results would be similar since the correlation between these two measures is

16 ALS Filter trend with Unemployment Unemployment ALS Filter Trend Figure 5 ALS smoother trend with Unemployment Unemployment ALS Trend Figure 6 The correlation coefficient between the ALS smoother estimate of the unemployment gap and the ALS filter estimate is This indicates that only a 13 The first 24 months worth of data was omitted in order to allow the ALS filter to build up a memory. 16

17 little is gained by looking at the forward information. Previously we saw that the correlation between the ALS smoother estimate of the unemployment gap and the ALS smoother estimate of the output gap was The correlation between the filter estimates of the two gaps is This again shows that the univariate estimates have value despite their inability to reject a unit root. A complete discussion of the differences between the various estimates, while important, is beyond the scope of the current paper. ALS Smoother and ALS Filter Unemployment Gap Correlation Coefficient: 0.81 ALS Smoother ALS Filter Figure 7 V. The Capacity Utilization Gap The Capacity Utilization rate, as measured by the Federal Reserve Board, is an attempt to determine what percentage of the highest sustainable level of output is being achieved. The highest sustainable level of output would allow for standard maintenance downtime and a normal work schedule. Thus a 100% capacity utilization measure does not mean that every factory in the country is operating 24 hours a day seven days a week. Since standard downtime is already built into capacity utilization, any reading below 100% suggests that potentially productive capital is sitting idle. Figure 8 shows the Federal Reserve s monthly estimates of 17

18 capacity utilization (p) for the period While the capacity utilization data is volatile and appears to be drifting downward, we know that it cannot be truly nonstationary since it is bounded between zero and one. Also looking over time it seems that the long run average has been falling since the mid-1960s. OLS trend with Capacity Utilization Capacity Utilization OLS Trend Figure 8 Our method of estimation for the capacity utilization gap follows the method used for the unemployment gap closely. In the case of capacity utilization it is less clear whether the natural rate has been changing over time. The OLS estimate of the natural rate of capacity utilization is shown as the horizontal line in Figure 8 and is estimated to be 81.2%. In this case the fact that adaptive least squares nests OLS is helpful. If the natural rate of capacity utilization ( p ) has not changed over time ALS will not force it to. On the other hand if the natural rate does change ALS will measure this where as OLS will not. As with the unemployment rate, we estimate the capacity utilization rate as an AR process with a possibly time varying constant trend. We initially estimated the autoregressive process as an AR(12), but the coefficient on the AR(12) term was not significantly different from zero. Choosing a simpler process over a more complex one, we dropped the AR(12) term and re-estimated. This process continued until the 18

19 final AR term was significantly different from zero, giving us an AR(4) process. The data was the monthly measure of capacity utilization as measured by the Federal Reserve Board. The final model is as follows: p t = β 0,t + β 1,t p t-1 + β 2,t p t-2 + β 3,t p t-3 + β 4,t p t-4 + e t. (5) The natural rate of capacity utilization ( p ) can be easily calculated using the following formula: p t β0, t = 4 1 βi, t i= 1, for all t. (6) The capacity utilization gap can then be calculated by finding the difference between actual capacity utilization, p, and its natural rate, p. Sum of AR coefficients ALS Smoother and Filter Estimates ALS Smoother ALS Filter Figure 9 The model in equation (5) was estimated using both the ALS filter and smoother. The estimate of the signal/noise variance ratio, ρ, was The likelihood ratio of ρ = 0 was 4.71 and since the 5% critical value for the likelihood ratio is approximately 2.3 we can reject the hypothesis that ρ = 0 with 95% confidence. This 19

20 estimate of ρ implies a limiting effective sample size of months, or 12.5 years. This means that the model parameters change significantly every 12.5 years, implying that in our sample, the natural rate of capacity utilization also changes substantially almost 5 times. Again, the estimates of ρ, the likelihood ratio and the effective sample size are the same for both the ALS filter and smoother. As before, the sum of the AR coefficients from our estimation is important to establish stationarity. Both the ALS filter and ALS smoother estimates of the sum of the AR coefficients is shown in Figure 9. While both estimates move higher over time, they are all safely below one. It is interesting to note that similar to the unemployment estimates, the ALS filter estimate does show a bit of a spike in the early 1980s. However in this case the up tick does not approach one. ALS Filter Natural Rate with Capacity Utilization Capacity Utilization ALS Filter Natural Rate Figure 10 The ALS filter estimate of the natural rate of capacity utilization starts around 83% and basically stays there until 1980 (Figure 10). Then in 1980 there is a rather abrupt drop down to 81% and then a leveling off. The natural rate then holds steady at 81% until 2000 when there is another abrupt drop off and leveling out at 79%. The ALS filter estimates the current natural rate of capacity utilization to be 79.6%, which is substantially different then the OLS estimate of 81.2%. 20

21 ALS smoother Natural Rate with Capacity Utilization Capacity Utilization ALS Natural Rate Figure 11 Figure 11 shows the ALS smoother estimate of the natural rate of capacity utilization with the actual level. The ALS smoother estimate of the natural rate of capacity utilization is flat at about 83% until It then has a somewhat steep downward trend until 1982, after which it flattens out again at about 80%. Unlike the ALS filter estimate, there are no abrupt movements to new levels in the ALS smoother estimate. This data and trend clearly show that an assumption of 100% capacity utilization is untrue; in fact the natural rate of capacity utilization is currently at almost the lowest level it has been since The correlation coefficient between the ALS smoother capacity utilization gap and the ALS output gap from Longbrake and McCulloch is This high correlation that Longbrake and McCulloch s output gap is somewhat valid, despite their inability to reject a unit root. The correlation coefficient between the ALS smoother capacity utilization gap and the ALS smoother unemployment gap is coincidently also 0.74, which means that capacity utilization would be just as good a proxy for the output gap as unemployment. A high correlation between the output gap and the capacity utilization gap is to be expected. If capacity is being under utilized, then it follows that output will be lower than expected. But most estimates 21

22 of the composition of production put more emphasis on the labor side of the market than the capital side which implies that unemployment should be a better proxy for the output gap. In the next section we will investigate combining both unemployment and capacity utilization to improve our estimate of the output gap. ALS Smoother and ALS Filter Capacity Utilization Gap Correlation Coefficient: 0.95 ALS Smoother Trend ALS Filter Trend Figure 12 Figure 12 shows the ALS filter and smoother capacity utilization gaps together. Other than the initial periods, where the filter is building up a history, the two estimates of the capacity utilization gap are very similar. In fact the correlation coefficient between the two is In the case of capacity utilization, there seems to be little difference between one sided and two sided estimation. VI. The Combined Model We have now calculated both the unemployment gap and the capacity utilization gap using ALS. The overall goal is to find a measure of the output gap based on these estimates. As stated in section IV, the correlation coefficient between the ALS 14 The first two years of data were omitted when calculating the correlation in order to allow the filter estimates to build up a history. 22

23 smoother estimates of the unemployment gap and the output gap is equal to This is high, but it is still far from one, and as such has led to the unemployment gap s wide use as a direct proxy for the output gap. As shown in section V, the correlation between the ALS smoother estimates of the capacity utilization gap and the output gap is also This measure has not been used as a proxy for the output gap since the unemployment gap performs just as well both theoretically and practically. Additionally labor s share of the economy is larger than the share of capital. Since both of these gap measures track the output gap closely and since all three gaps are essentially measuring the same phenomenon the natural question is why not just use one of them. Figure 13 shows the ALS smoother estimates of both the negative of the unemployment gap (the employment gap) and the capacity utilization gap. The correlation between these two series is 0.74, suggesting that the series are similar, but also different. In fact there are times when capacity utilization is below its natural rate, at the same time that employment is above its natural rate. An example of this occurs in late 1998 and In this circumstance using the different gaps as a proxy for the output gap would give the output gap different signs. One possibility is that these are points in time where labor has become relatively less expensive, so firms are using more labor and reducing there reliance on equipment. The reverse could also be true. If energy and fuel costs fall it could make labor relatively more expensive. This would lead to employment being below its natural rate at the same time capacity utilization is above its natural rate. Additionally, any time there was a change in relative productivity of labor and capital we could see similar adjustments. In any of these cases using the unemployment gap or the capacity utilization gap alone may lead to the overall output gap suffering from a substitution bias. The best way to eliminate this bias is to combine the two measures to estimate the output gap. 15 This assumes that the output gap is a valid estimate, which may be a problematic assumption due to unit root issues. 23

24 ALS Smoother Capacity Utilization Gap and Employment Gap Capacity Utilization Gap -(Unemployment Gap) Figure 13 The general logic for combining these measures comes from a simple Solow type growth model, Y = f(a,l,k), where Y is production, A is some measure of productivity, L is labor and K is capital. This model suggests that both labor underutilization and capital underutilization should have an impact on the output gap. If we assume a Cobb-Douglas production function 16 we can rewrite the growth model with σ equal to labor share as: σ ( 1 σ ) Y = AL K. (7) In order to get the output gap from the above equation we need to posit a natural rate of output equation. The natural rate of output occurs when all of the inputs are operating at their natural rates, hence: σ ( 1 σ ) Y = AL K, (8) where a bar over a variable indicates natural rate. Then specifying the output gap as the difference between the actual level of output and its natural rate, after taking natural logs, we derive the following equation: ( y y) = ( a a) + σ ( l l ) + (1 σ )( k k ), (9) 16 We do not need to worry about whether the A variable is labor augmenting or capital augmenting or both, because with a Cobb-Douglass production function they are equivalent. 24

25 where lowercase letters denote natural logs. From this equation we can see that the output gap, y y, is a function of the technology/productivity gap, the labor gap and the capital gap. Equation (9) is similar to the equations used by both the CBO and the OECD in their output gap estimations. The CBO and OECD then both make the assumption that k = k, thus assuming that capital is always used at its full capacity. This allows them to estimate the output gap using only a labor gap, which is a function of unemployment, and a technology gap, which is simply taken to be the residuals from there estimations. The more natural assumption seems to be that any technological/productivity innovations would be fully utilized, implying that a = a. When using this assumption, the output gap is a function of only the labor gap and the capital gap. The labor gap can then be estimated from unemployment data as done in section IV, and the capital gap can be estimated using capacity utilization data as done in section V. Univariate GDP and Combined Model Output Gaps - Smoother Correlation Coefficient: 0.79 Univariate GDP Gap Combined Model Gap Figure 14 25

26 This logic allows us to derive the following model, with g y denoting the output gap, g y = σ(ū u) + (1 σ)(p p ) 17. Having already calculated the unemployment gap and the capacity utilization gap the only thing left to do is determine the value of σ. Estimates or the labor share in the literature are usually around 2/3. The CBO in their potential output model uses a labor share of 70%. In our model we follow the CBO and set σ = Because this combined model is based on the under or over utilization of the factors of production, hereafter it will be referred to as the Factor Utilization Model. Univariate GDP and Combined Model Output Gaps - Filter Correlation Coefficient: 0.77 Univariate GDP Gap Combined Model Gap Figure 15 Figure 14 shows the estimate of the output gap from the Factor Utilization model outlined above: g t = 0.7(ū t u t ) + 0.3(p t p t ). For this figure we used the ALS smoother estimates of the unemployment and capacity utilization gaps. Also included in the figure is the ALS smoother estimate of the output gap derived from using only GDP data. The graph shows an interesting effect. In almost every period, with the exception of 1975 and 1982, the magnitude of the directly estimated output gap is 17 Because we need the employment gap, which is the negative of the unemployment gap, u and ū have been reversed. 18 There is no reason why the labor share needs to be fixed for the entire sample. Indeed it would capture the substitution effect more accurately if the labor share was allowed to change every year. 26

27 smaller under the Factor Utilization model. However the correlation between the two series is very high, with a coefficient equal to Figure 15 shows the ALS filter Factor Utilization Gap along with the univariate GDP ALS filter estimate of the output gap. In this figure the magnitude effect seen in the smoother models is gone. If anything the reverse is true, the magnitude is higher for the Factor Utilization model. Again the correlation coefficient is a very high VII. Conclusion By combining ALS detrending of unemployment and capacity utilization with a Solow type growth model this paper develops a proxy for the output gap that avoids the flaws in direct estimates. The first problem with previous output gap estimates is that the estimates that rely only on GDP data have to deal with unit root issues. Our model avoids this problem by using two data series, unemployment and capacity utilization that do not suffer from unit root questions. The second problem with current estimates is that they have been subject to substantial ex post revisions. By using ALS as our econometric method the revisions problem is avoided and the trend is allowed to vary over time. The learning aspect of ALS removes the onus of trend break analysis from the econometrician by allowing the trend to adjust however it wants to the data. Also by combining both the capacity utilization rate and the unemployment rate into our estimates gives us a more accurate output gap estimate than using unemployment alone as a proxy. Estimates that use only unemployment may suffer from a substitution bias. This bias may cause the output gap to be overstated when the relative cost of labor is low, or the relative productivity of labor is high. Indeed when our combined model estimate of the output gap is compared with the unemployment only model, the magnitude of the output gap is smaller, suggesting that there is a substitution bias. Further work needs to be done in the area of output gap estimation. The differences between the various estimates need to be explored. Additionally there are other measures of capacity utilization that could be incorporated into our estimation. 27

28 One of the limitations of our technique is that while it gives a good estimate of the output gap, there is no way to get the level of potential output from our model. This is because the technology/productivity variable, a, falls out of the model. Thus or model does not work for any application where potential output is necessary. Overall this paper has attempted to examine current estimates of the output gap and offer potential improvements. We believe that our Factor Utilization model estimate of the output gap, while still imperfect, offers a significant improvement over direct estimates. 28

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