Draft Draft Searching for the Output Gap: Economic Variable or Statistical Illusion? Mark W. Longbrake* The Ohio State University J. Huston McCulloch The Ohio State University August, 2007 Abstract This paper estimates the long term trend in US real GDP using a new econometric technique, Adaptive Least Squares (ALS). ALS is a special case of the Kalman Filter that allows for a time varying parameter model to be estimated relatively easily. The estimated trend is then used to estimate the output gap. We estimate the gap using both a one-sided and a two-sided filter. The results of our estimation suggest that GDP does not follow even a time-varying long term trend, so the output gap is illusory. We also compare our result with the Congressional Budget Office estimates of the output gap. *Corresponding author, email: longbrake.13@osu.edu 1
I. Introduction There is currently a large body of research that examines the question of finding optimal monetary policy. This research has progressed down several different paths that incorporate different underlying assumptions about the economy. Yet all of these different methods require the central bank to know the current output gap. Surprisingly there has not been a lot of focus put on estimation of the output gap, researchers simply use a simple H-P filter, or the Congressional Budget Office estimates. This paper takes advantage of a new econometric technique developed by McCulloch (2005) called Adaptive Least Squares (ALS) to objectively estimate the output gap. ALS is a modified Kalman filter that allows for time varying parameters. This allows parsimonious estimation of the trend in US real GDP, by assuming a linear trend, but allowing the trend to change over time. The change in the trend is determined, not by the econometrician, but by the underlying model. After estimating the output gap using the two-sided ALS smoother, we find that our estimates are similar to the Congressional Budget Office's (CBO) estimates. However, when we estimate the output gap using the one-sided ALS filter, our estimates while still similar to the CBO estimates, show several significant differences. This leads to the conclusion that the CBO s estimate of the output gap may rely on future information. If this is the case any projections made using CBO estimates will be subject to revision after the fact, and hence will be inaccurate. Section II describes the background of the output gap along with a discussion of its relevance. Section III outlines the issues of estimating a long run trend for GDP. Section IV gives an overview of the estimation technique used in this paper, Adaptive Least Squares. Section V gives the results of the estimation, Section VI examines the unit root question, Section VII conducts CBO and real-time comparisons, and Section VIII concludes. 2
II. The Output Gap 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 in 1993, who also used the output gap in determining optimal policy. Ball, Mankiw, and Reis in their 2003 paper 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 (1997) goes so far as to suggest the use of nominal income in monetary policy rules because potential GDP estimates are so uncertain. This paper takes advantage of a new estimation technique, ALS, to estimate potential GDP and the output gap. We explain in the section on ALS below why we believe that our method and estimate of the output gap to be a useful addition to the literature. 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 3
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 technique. The CBO uses 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 business cycle is not relevant at all to ALS, so our estimates do not have this problem. The detrending methods used in other papers are outlined in the next section. III. Long Run Trend Estimation For this paper we will follow Taylor (1993) and others by specifying the output gap as the deviation of GDP (y) from a trend. The implicit assumption in this specification is that GDP has an underlying trend. The natural place to start with the analysis is by examining a linear trend: y = a + bt +, (1) t g t Δy = b + Δg, where y t is ln real GDP at time t, gt is the output gap at time t, and a and b are trend parameters. In order for g to be meaningful it must be stationary. If g is not stationary, then y is like a random walk with updrift. If y is indeed an integrated process of order one as suggested by Nelson and Plosser (1982) then the output gap as specified is illusory. It is impossible to measure the deviation from a trend, if there is no trend. 4
Quarterly ln GDP 1947 2005 with OLS Trend above below ln GDP Figure 1 OLS Trend 1 One specification for the output gap that makes g stationary with zero mean is : g t = γ1 g t-1 + γ 2 g t-2 + ε t, (2) with γ + γ 1 2 < 1 a necessary condition for stationarity. A stationary process for g implies a trend stationary process for y. The specific example above combined with equation (1) yields the following process for GDP: Where u is a function of lags of ε and with, t y t = β 1 + β2t + β3yt-1 + β 4 y t-2 + u t (3) β 2 b = [1 ( β + β )] t β1 β 2 ( β 3 + 2β 4 ) a = [1 ( β + β )] [1 ( β + β )] γ 1 = β 3 γ 2 = β. 4 3 3 4 4 3 4 2 (4) 1 The AR(2) specification shown is found to fit well later in the paper, but the output gap could in principle follow any stationary process. A higher order AR specification simply adds more lag terms to equation (3). 5
OLS Estimate of the Output Gap Figure 2 It can easily be seen that if g is not stationary, i.e. γ 1 + γ2 = 1, then both a and b become undefined. It is instructive to examine the simple OLS estimate of the trend outlined in equation (3). Figure 1 shows the OLS estimate of equation (3) superimposed over quarterly GDP from 1947 2005. Figure 2 shows the resulting estimate of the output gap, which is simply the difference between the trend line and actual ln GDP in Figure 1. 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 -2.83 and the 10% critical value is -3.14. 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 a unit root. One of the first to address this issue was Pierre Perron in 1989 and again in 1997. Perron 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, 6
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. Also in 1989, Hamilton developed a two-state regime switching model of GNP growth. This model uses a Markov process to transition between the 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 found in their 1995 paper 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 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 avoid this problem by proposing a model that only allows the intercept term to be time-varying (1973). Stock and Watson 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 (1998). 7
IV. Adaptive Least Squares Adaptive least squares is an adaptation of the Kalman filter developed by McCulloch to estimate time varying relationships (2005). ALS is an estimation technique that has several advantages over the alternatives offered in the previous section. One reason for this is that although ALS allows for all of the coefficients in the model to change over time there is only a single parameter, ρ, to estimate. This means ALS 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 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 derives the trend from an actual process, as opposed to the HP filter where there is an arbitrary smoothing parameter 2. 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 any 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. Even though both estimates are of interest, unless otherwise indicated, this paper we will use smoother estimates since they are more accurate. 2 Even if the smoothing parameter is optimally estimated, the HP filter models GDP based on a model that does not match the data. 8
ALS Coefficient Point Estimates (filter) ß 3 ß 1 ß 2 *100 ß 4 Figure 3 For a full explanation and derivation of ALS please 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. V. Model and Results The model of GDP we estimate is an AR(2) process with a time-varying linear trend. The model is similar to equation (3) above, with the slight change of allowing the coefficients to vary over time. y t = β 1,t + β 2,t t + β 3,t y t-1 + β 4,t y t-2 + u t. (5) The trend can then be extracted using equation (4), and the output gap can be estimated as the deviation from the trend. Figure 3 shows the coefficient estimates for (5) using the ALS filter, with β 2 scaled up for presentation purposes. The first few periods of the ALS filter estimates are very noisy, because we assume a diffuse prior and it takes the model a few periods to learn. Figure 4 shows the ALS Smoother 9
ALS Coefficient Point Estimates (smoother) ß 3 ß 1 ß 2 *100 ß 4 Figure 4 coefficient estimates. Since the filter is two sided, using both past and future data, there is no noise in the early periods. Since we need the output gap to be mean reverting, an AR process is a natural choice. As shown above, an AR process for the output gap implies an AR process for GDP. Because US real GDP data is quarterly, an AR(4) process was our first choice for the output gap. This would allow information from up to one year ago to be explanatory. In general we would favor a shorter AR process over a longer one. Thus when the coefficient on the AR(4) term in the regression was not significantly different from zero in either the ALS filter or ALS smoother, it was dropped. The same happened with the AR(3) coefficient and as a result we chose to use an AR(2) specification. Both the AR(1) and AR(2) coefficients were significant in both the ALS filter and the ALS smoother. Using seasonally adjusted quarterly US real GDP (chained 2000 $'s) for 1947 2005, equation (5) was estimated utilizing ALS. The estimate of the signal/noise 10
Quarterly ln GDP 1947 2005 with ALS Smoother Trend ln GDP Figure 5 ALS Trend variance ratio, ρ, was 0.000296. The likelihood ratio of ρ = 0 is 2.66. The 5% critical value for the likelihood ratio is approximately 2.33, meaning that we can reject the hypothesis that ρ = 0 at the 5% level. The estimate of ρ implies a limiting effective sample size of 58.59 quarters, or 14.65 years, in turn giving the model an asymptotic gain of 1.7%. The effective sample size says that the model parameters change approximately every 14.65 years, or that there are, roughly speaking, four different episodes in the US real GDP time series from 1947-2004. Figure 5 shows the filter estimate of the trend, a t + b t t (from equation (4)), superimposed over the US real GDP time series, y t. The filter estimate of the long term trend in GDP has a couple of interesting features. The first is the fact that the trend actually looks negative for the period between 1981 and 1983, and a negative trend is unrealistic for the real GDP time series. However, what is actually showing up here is a function of the way the overall trend is built. ALS allows the trend coefficients, a and b, to change every period. This gives a different linear trend for each period. Figure 6 shows the two trend lines that coincide with the top and the 3 The critical values for the likelihood ratio were determined by McCulloch using Monte Carlo simulations. 11
ALS Filter Trend with Sample Trend Lines ALS Filter Trend Sample Trend Lines Sample Projections Figure 6 bottom of the decrease in the early 1980 s along with US real GDP. The dotted portion of these lines is the projected trend as of the period where the solid line ends. It is clear from this figure that the linear trend is always positive, but the intercept and slope are changing over time. Second, the ALS filter estimate is very responsive to changes in real GDP. This is because the ALS filter is using only information known up to time t. By not knowing what is to come, the trend adjusts more rapidly to the current realization. We will see below that this also makes the ALS filter estimates of the output gap smaller in relative magnitude. Figure 7 shows the ALS smoother estimate of the long-term trend in real GDP. At first glance this estimated trend appears to be linear, but when it is compared with the OLS trend in Figure 1, it is easy to see that the ALS estimate fits more tightly. The ALS smoother estimate is, well, smooth because it is using two-sided estimation. By using both past and future data, the ALS smoother is able to gradually adjust to changes that are in the process of happening. Now that we have estimated the long-term trend using both the ALS filter and smoother, it is easy to calculate the ALS estimates of the output gap. Figure 8 12
Quarterly ln ALS GDP Filter 1947 Estimate 2005 of with the ALS Output Smoother Gap Trend ln GDP Figure 7 ALS Trend ALS Filter Estimate of the Output Gap Figure 8 13
ALS Smoother Estimate of the Output Gap Figure 9 ALS Smoother and Filter Estimates of the Output Gap ALS Smoother ALS Filter Figure 10 14
shows the ALS filter estimate of the output gap which is simply, y t (a t + btt) and Figure 9 shows the ALS smoother estimate. Figure 10 shows both trends on the same graph. In this Figure you can see the magnitude of the output gap is smaller for the filter estimate. However, the two estimates are still related, with a correlation coefficient of 0.59. The significant difference of the two estimation techniques is highlighted by a six year period between 1984 and 1990 where the ALS filter and smoother give different signs for the output gap. This implies that the decision makers at the time thought the output gap was negative, when in hind sight it was actually positive. Having the incorrect sign on the output gap would have negative consequences for monetary policy. ALS Filter Estimate of Time Trend Slope (b) Figure 11 We now move to a discussion of the trend itself. It has become accepted that the growth rate of GDP has fallen in the last 30 years. This stylized fact is what led Perron to suggest a structural break in the GDP time series. If GDP growth has been falling, then the slope of the time trend of GDP should also have been falling. Figure 11 shows the ALS filter estimate of the time trend slope, b t t. This graph shows that the estimated time trend slope has indeed been decreasing over time. This decrease is 15
ALS Smoother Estimate of Time Trend Slope (b) Figure 12 even clearer in Figure 12, which is the smoother estimate of the slope. Due to the adaptive learning nature of the model, it does not show that there was a one-time shift to a different level. ALS estimation, does document the decrease in the growth rate of GDP. But it suggests that the change was gradual, in line with the expectations of the method. It has recently been suggested that there is a second structural break in the GDP time series in 1995. As can be seen in Figure 12, our estimates do not show any change in the 1990 s. 16
Sum of the AR Coefficients ˆ β + ˆ β 3 4 (Filter Estimates) +2 s.e. ˆ β + ˆ β 3 4-2 s.e. Figure 13 VI. Unit Root The next step in the analysis is to test whether the output gap is stationary, or if it follows a unit root process. Remember from equations (2) and (3) that in order for the output gap to be stationary, γ + γ 1 2 < 1, which is equivalent to β 3 + β 4 < 1. Figure 13 shows the sum of the ALS filter estimates of the AR coefficients along with standard error bands. Figure 14 is the equivalent picture using ALS smoother estimates. In both cases, because of the times series nature of the estimates, the coefficients do not have a t-distribution. However, whenever the standard error bands do incorporate the number one we know that we can not reject a unit root. The sum of the AR coefficients is uncomfortably close to one for much of the sample. As a result we need to perform a unit root test. The standard test of the unit root hypothesis for time-series data is the augmented Dickey-Fuller test. However we have to modify this test because all of the ALS model parameters vary over time. With different estimates of the coefficients in every period, the Dickey-Fuller statistic also has a different value in every period. 17
Sum of the AR Coefficients ˆ β + ˆ β 3 4 (Smoother Estimates) +2 s.e. ˆ β + ˆ β 3 4-2 s.e. Figure 14 Figure 15 shows the Dickey-Fuller statistic plotted over time for the ALS filter estimates. The Dickey-Fuller statistic is only significant at the 10% level in a few places early in the sample, and is never significant at the 5% level. Figure 16 shows the Dickey-Fuller statistics for the ALS smoother estimates. At no time do these statistics become significant at even the 10% level 4. The null hypothesis for the Dickey-Fuller test is that β 3 + β 4 = 1, and the alternative is β 3 + β 4 < 1. Thus, if the null hypothesis is true it implies that US real GDP follows a unit root process, the alternative suggests that GDP is locally trend stationary. As a result we can never reject the hypothesis that y t follows a unit root process at the 5% level, and only very early in the sample using ALS filter estimates can we reject at the 10% level. So, even allowing the trend to change at every point in time we can not reject that US real GDP follows a unit root process. 4 The sample size used for determining the critical values for the Dickey-Fuller test was 2*T -1. Where T is the effective sample size: 58.59 quarters. 18
Dickey-Fuller Test Statistic (filter estimate) T = 50 10% 5% Figure 15 Dickey-Fuller Test Statistic (smoother estimate) T = 100 10% 5% Figure 16 19
VII. Comparison of Estimates The main source for the output gap in the literature is the Congressional Budget Office's estimate. Figure 17 shows real GDP overlaid with the CBO s estimate of potential GDP. The CBO gets an interesting wavy shape due to their assumption of piecewise linear trends. Because of the wide use of CBO estimates we compare our estimates to theirs. Figure 18 shows the CBO s estimate along with the ALS filter estimate of the output gap. There is a surprisingly high correlation coefficient between the CBO estimate and the ALS filter estimate, 0.75. Figure 19 shows the CBO s estimates along with the ALS smoother estimates of the output gap. The correlation coefficient for the CBO and the ALS smoother is even higher, 0.81. Quarterly ln GDP 1947 2005 with CBO Trend ln GDP CBO Trend Figure 17 20
. ALS Filter and CBO Estimates of the Output Gap ALS Filter CBO Figure 18 ALS Smoother and CBO Estimates of the Output Gap ALS Smoother CBO Figure 19 Unsurprisingly given the high correlation coefficients, the CBO output gap and the ALS filter have almost identical sign throughout. There is only one three year period 21
in the 1950 s where the two estimates are different for any length of time. There is even less sign difference between the CBO output gap and the ALS smoother estimate, only one short period in the early 1980 s. These results accurately suggest that the CBO has both forward and backward looking components in their estimates. Notice that the ALS smoother fits the older CBO estimates, while the ALS filter fits the newer estimates. These findings make sense because the CBO makes initial estimates and then goes back and revises them when new data is available. However, comparing the ALS filter with the CBO ex post revised data is not a correct comparison. Thus we obtained government real-time estimates of the output gap. This is the series that Orphanidies used in his 2003 paper updated to contain information that was not available in 2003 5. Comparing these two series shows us estimates that do not use any future data in their construction. Figure 20 shows these two series on the same graph. This figure quickly confirms the questionability of real-time output gap estimates. VIII. Conclusion After estimating the US output gap using ALS techniques we have learned several things. The first is that there is a meaningful difference between the ALS filter and the ALS smoother estimates. This means that any decision maker may get monetary policy incorrect because they have to rely only on past data. Based on this fact, we are unsurprised by the large revisions to the output gap found by Orphanidies (1997). Also, while ALS does suggest that the time trend in GDP has changed over the last 60 years, there is not a clear break on or around 1973. Additionally, ALS estimates do not show any change at all around 1995. This leads us to question the use of trend breaks when estimating the long-term trend in GDP. Secondly, even allowing for a linear time trend that varies every period, we still cannot reject that the US real GDP time series is a integrated process with updrift. Because we cannot reject a unit root in GDP even using this very flexible structure our paper supports the literature that claims US real GDP does indeed follow a unit 5 The authors would like to thank Athanasios Orphanides for allowing us to use his data series. 22
root. Since we conclude that GDP follows a unit root process the output gap that we find, based on the specification used in this paper, is a statistical illusion. Our specification for the output gap was the deviations from a long-term trend. Since there is no long term trend, our findings for the output gap are not meaningful. We foresee future work in this area going two directions. The first is to use Monte Carlo methods to test a number of different things. We need to see how big a structural break needs to be in order to be picked up by ALS. It will also be interesting to see what ALS estimates look like in a world where there is a single structural break. Additionally, end of sample properties need to be tested using Monte Carlo methods. The second extension is to use a CBO like model estimated using ALS instead of a piecewise linear trend. If we can build an estimate of potential output using various time series including unemployment and capacity utilization rates then we can avoid the problems of statistical illusion reached in this paper. 23