Nonlinear business cycle dynamics: cross-country evidence on the persistence of aggregate shocks.

Bradley, Michael D. ; Jansen, Dennis W.

I. INTRODUCTION

Economists have long been interested in the dynamics of business cycles, particularly regarding the symmetry of expansions and contractions. More recently, the specification of stochastic trends in economic growth has generated concerns about the relative persistence of positive and negative shocks to the economy.

Several papers have looked at the persistence of business cycles, including Nelson and Plosser [1982], Campbell and Mankiw [1987], Cochrane [1988], Diebold and Rudebusch [1989], Balke and Fomby [1990], Zelhorst and de Haan [1994], and Bradley and Jansen [1995]. While results vary, especially on the issue of a unit root in output, the overarching result is that output fluctuations are highly persistent.

One question not asked by the above authors is whether business cycle dynamics are asymmetric. This issue has been investigated by Neftci [1984], Falk [1986], DeLong and Summers [1986], Hamilton [1989], Diebold and Rudebusch [1990], and Terasvirta and Anderson [1991]. These authors take a variety of approaches but they reach a general conclusion that there is some evidence of asymmetric business cycle dynamics. However, most of these studies ask if contractions are shorter and steeper than expansions. Except Hamilton [1989], they do not ask if there is a difference in persistence due to the asymmetries.

In a recent paper, Beaudry and Koop [1993] propose a set of methods for simultaneously examining these two questions. Specifically, Beaudry and Koop (BK) recommend the augmentation of the standard ARMA representation of real output growth through the inclusion of a nonlinear term that captures the relative state of the economy. This augmentation provides for nonlinear dynamics and the possibility of asymmetric dynamic responses to positive and negative shocks. In addition, calculation of the implied impulse responses permits detection of the relative persistence implied by the combination of asymmetric cycle dynamics and stochastic trend.

Based upon their analysis of U.S. data, BK conclude that the response of the economy to positive and negative shocks is indeed asymmetric and "the imposition of symmetry on impulse responses has likely led to misleading measures of persistence." [BK, p. 162] As a result BK conclude "We find that the effects of negative shocks to be mainly temporary and the effects of positive shocks to be very persistent." [BK, p. 162]. If BK's results are correct, they have strong implications for the way macro economists view economic fluctuations and will, in their words, "contribute to a reappraisal of some macroeconomic theories." That is, BK's results suggest that macroeconomic models must be revised to provide for temporary effects of negative shocks but permanent effects of positive shocks.

Because of the potential importance of BK's results, we examine the robustness of those results by modifying and extending BK's proposed methods of testing. Not only do we extend the analysis to the other six G7 countries besides the United States, but also we redefine, in a sensible way, the nature of the nonlinear dynamics. Overall, our results from other economies provide only mixed support for the asymmetry of cyclical dynamics and the lack of persistence of negative shocks.

II. THE MODEL TO BE ESTIMATED

In the BK framework, the estimated model includes an asymmetric response term. The asymmetry is generated by a variable that measures the strength of a recession. The variable is calculated as the distance between current real GDP and the previous peak level of real GDP. That is, BK define a measure they call Current Depth of the Recession (CDR) where:

(1) [CDR.sub.t] = max [{[Y.sub.t-j]}.sub.j [greater than or equal to] 0 - [Y.sub.t].

With this definition in place, BK recommend augmenting an autoregressive model of real GDP growth as:

(2) [Phi](L) [Delta] [Y.sub.t] = [Delta] +[[Omega](L) - 1] [CDR.sub.t] + [[Epsilon].sub.t],

where [Delta] is the drift, [Phi](0) = 1, and [Omega](0) = 1. To interpret this equation, consider the case in which the lag on the CDR term is one. If the coefficient on the CDR variable is positive ([Omega](1) [greater than] 0), economic growth is greater when CDR is positive than when it is zero. Intuitively, this means that growth is faster as the economy recovers to its previous peak (loosely speaking, a recession) than when it is growing above its previous peak (an expansion).(1) In this case, economic growth responds more strongly to negative shocks than to positive ones. In addition, a positive CDR coefficient implies that positive shocks have more persistent effects on output than negative shocks. Clearly, if the coefficient on CDR is negative, just the opposite is true. In such a case, when the economy enters a recession it tends to be mired there.

III. ESTIMATING THE CDR MODEL FOR THE G7 COUNTRIES

Our first analysis is simple replication of the CDR-augmented autoregressive equation for the United States and the estimation of the same model for the other six G7 countries: Canada, France, Germany, Italy, Japan, and the United Kingdom. Post WWII data on real GDP, when available, were extracted from the IMF's International Financial Statistics data base. Real GNP was used for Germany and Japan. The data were available as indicated in Table I.

Equation (2) was fit for the first difference of the natural log of real GDP for each country. As in Terasvirta and Anderson [1993], the French series was adjusted for strikes in 1968 (1968.2), and the Italian series was adjusted for the widespread industrial action that took place in late 1969 (1969.4) and 1971 (1971.3).

The specifications presented in Table II result from a search over five possible lags of the autoregressive terms and five possible lags of the CDR terms for each country. The specification presented was based on the minimum Akaike Information Criterion statistic across the 36 different possible specifications, with the exception that the model for Italy was further simplified as indicated below.(2)

Table II shows that our specification search for the United States yielded one of BK's preferred models and our estimated coefficients are quite close to theirs. Given the differences in time periods (BK use 1947.1 through 1989.4) and given that our data series includes a recession that is not included in theirs, we find the results to be sufficiently close to convince us that we have replicated their findings for the United States.

TABLE I

Data Series for Each Country

Country Time Period

Canada 1950.1 - 1993.1
France 1965.1 - 1992.4
Germany 1960.1 - 1993.1
Italy 1960.1 - 1992.1
Japan 1955.2 - 1991.1
United Kingdom 1955.1 - 1993.1
United States 1950.1 - 1992.4

Results for the other six countries, however, show that the United States experience is not universal. Only half of the other countries display asymmetric dynamics like those found in the United States. Examination of the CDR coefficients in Table I reveals that Canada, France, and Japan did not show any evidence of asymmetry, as no CDR coefficients appear in the preferred model. In the remaining four cases, the U.K. stands out because the estimated CDR coefficient is negative. This means that in the U.K., negative shocks are substantially more persistent than positive shocks. This result suggests that the U.K. is likely to suffer from long recessions.

The final three countries do show evidence of asymmetry with positive CDR coefficients, like the United States. For Germany, the coefficient on the CDR term is much larger than in the United States, raising the possibility of "overshooting." This result implies that the level of real GDP is actually higher in the long run after a recession. The United States and Italy have coefficients on CDR terms that are similar in size, but in Italy the response to CDR involves long lags. The model specification search using the AIC resulted in a model with five lags of CDR. Subsequent investigation showed that the CDR at lags one through four could be eliminated, leaving the model in Table II with one CDR term, at lag five.

A last experiment in estimating the CDR model involves using our country-specific data as if it were panel data. A fixed-effects model was estimated under the usual restriction that the estimated coefficients on the AR and CDR term are the same across all seven countries. The statistical test of this restriction is rejected, however, at a high level of confidence. The calculated chi-square statistic for this restriction is 67.46 but the critical value at the 1% level is only 50.59. This means that the dynamics are different across the countries and a separate model should be estimated for each country. Nevertheless, if this result is ignored and a common model is estimated, it does produce evidence in favor of asymmetry. The estimated CDR coefficient from the fixed effects model is 0.28 with a t-statistic of 4.88.

IV. INVESTIGATING THE PERSISTENCE OF SHOCKS

The nature of the persistence can be better understood by examining of the impulse responses implied by the CDR models. Yet, we face the issue of defining an impulse response function for a nonlinear model. While linear models produce unique impulse response functions that describe the response of output to any shock, positive or negative, and from any arbitrary initial condition, the same is not true for nonlinear models. For nonlinear models, the response to a shock may depend on the size and sign of the shock, and on the state of the economy at the time of the shock. Because of this, we use the generalized impulse response function introduced by Potter [1994] and used in Pesaran and Potter [1994].

[TABULAR DATA FOR TABLE II OMITTED]

The generalized impulse response functions are calculated as follows. Initially, the relevant history, at the time of the shock, is summarized by lags of the changes in real output and lags of the CDR term. Then, the impulse response function at any moment in history can be calculated in four steps. First, the model is estimated over the sample period and the residuals are saved for later use. Second, for any point in time during the sample period, say time t, we calculate the baseline forecast by drawing randomly from the history of residuals and using the draw to calculate the forecast of output from time t forward. We calculate these forecasts for 24 quarters forward of time t. This calculation is repeated 10,000 times, and the average of output at each point in time from period t to period (t + 24) defines the baseline path of output.

Third, we impose impulses at time t, of various sizes, and then from time period (t + 1) through time period (t + 24) we draw shocks, randomly, from the history of residuals. We use these draws to calculate the forecast of output for all future periods in response to the realization of the shock in period t. This process is repeated 10,000 times, and the average of output at each point in time from period t through period (t + 24) defines the path of output after the shock at time t. Fourth, the generalized impulse response function is calculated as the value of output at time t + I, given a specific shock at time t, minus the value of output at time t + I in the baseline case. This generalized impulse response function is what we presented in the tables and figures that follow. We calculate these responses for both positive and negative shocks of size 0.5%, 1.0%, 1.5%, and 2.0% of real GDP.

Beaudry and Koop calculate impulse response functions assuming that all future shocks are zero. This is not an innocuous assumption, as we show in Table III. There, we present impulse response for the United States CDR model. In the columns with the heading "BK Method," we report impulse responses as in BK, where the initial conditions are an economy in the steady state. Shocks of various sizes are imposed, along with the stipulation [TABULAR DATA FOR TABLE III OMITTED] that future shocks are zero. The path of the economy after the shock is traced out, and the impulse response is calculated. It is calculated as the difference in output after the shock relative to the output level in the baseline (i.e., steady state) case.

The numbers presented in Table III represent the distance real GDP would be from its base line value at 12 and 24 quarters after a shock. The numbers are calculated as the difference between real GDP after the shock and baseline real GDP, divided by the size of the shock (in absolute value). A positive number indicates how much real GDP would be above its base line value, as a percentage of the original shock. A negative number indicates how much real GDP would be below its baseline, again as a percentage of the original shock. For example, with a shock of -2%, the impulse response value of -0.77 indicates that real GDP is 1.54% (2 [multiplied by] 0.77) below the value it would have had in the steady state.

Table Ill shows the BK result of an asymmetric response to negative shocks. Large negative shocks are less persistent; the impulse response for negative shocks is smaller in absolute value. Positive shocks exhibit the same invariance to the size of the shock as one gets from a linear model.

How do these results compare to those from Potter's generalized impulse response functions? We give three examples. One is a "steady state" initial condition, identical to the initial condition of the BK model. The difference between these impulse response values and those of the BK method are that the Potter method does not assume that all future shocks are zero. Instead, in both the baseline forecast and the forecasted response to shocks, the Potter method has future shocks drawn randomly from the history of residuals.

Clearly, the Potter generalized impulse response function gives different results than the BK method. We still find that large negative shocks are less persistent than smaller negative shocks. But positive shocks exhibit a different pattern, and do not show any in-variance to the size of the shock. This result is due to the nature of the model, which has an asymmetric response to positive and negative shocks. Although the initial positive shock does not generate any asymmetric response, the future shocks are drawn from a history that does contain many negative shocks. Thus, the path of output will be influenced by these negative shocks, which result in less persistence than positive shocks. With a symmetric distribution of historical residuals about zero - an accurate characterization of this series - the generalized impulse response function averages both positive and negative future shocks, whereas the BK method sets all future shocks to their expected value of zero. By Jensen's inequality and the asymmetric nature of the response to positive and negative shocks, we would expect to see [TABULAR DATA FOR TABLE IV OMITTED] the nonuniqueness of the generalized impulse response function to positive shocks.

We also calculate the generalized impulse response function for two historical dates, a period of expansion (1991.2) and a period of recession (1990.4). The period of expansion mimics the pattern of the steady state, but with somewhat smaller magnitudes of the impulse response values. The period of recession also mimics the pattern of the steady state, but with much smaller magnitudes of the impulse response values. It appears that in both periods the persistence of large negative shocks was less than the persistence of small negative shocks, but the effect is more pronounced during recessions. For positive shocks, larger shocks are more persistent, although again the effect is most pronounced during expansions.

Because of the importance of initial conditions, we report two generalized impulse responses for each of the four countries with asymmetric dynamics due to CDR terms. One will be for a period of economic expansion, in which the economy was growing for two or more quarters, and the other will be for a period of economic contraction, in which the economy was contracting for two or more quarters. These impulse responses are presented in Table IV for expansions and Table V for recessions.

Japan, France, and Canada: Symmetric responses to all shocks

For Japan, France, and Canada, the lack of asymmetry in the autoregressive model for output growth implies that positive and negative shocks have equal persistence. At both 12 and 24 quarters, these economies will be as far below the baseline following a negative shock as they would be above the baseline following a positive shock.

Germany, Italy, and the United States: Asymmetric responses

For Germany, Italy, and the United States, Table IV indicates that expansions are characterized as follows: For negative shocks, the larger the shock, the smaller the persistence. For positive shocks, the larger the shock, the larger the persistence. It is interesting that for all three countries, small negative shocks are more persistent than small positive shocks. Indeed, for Germany a shock of -.5% results in a greater than 2% decline in real GDP after 24 quarters. Larger negative shocks are not nearly so persistent, as they generate declines in real GDP that, in turn, lead to larger CDR values and larger future increases in output growth. For Italy and the United States small negative shocks are not as persistent as in Germany, but the effect is still present.

[TABULAR DATA FOR TABLE V OMITTED]

During recessions, the results for Germany and Italy diverge from those of the United States In the United States, the pattern of responses is the same as during an expansion, but the magnitude of the responses is smaller. For Germany and Italy, however, large negative shocks lead to an increase in output relative to the baseline. Thus, a -2% shock in Germany leads to output being .38% higher than the baseline after 12 quarters. This result occurs because of the strong response of growth rates to a recession. Remember that these results are generated for an economy already in recession. The large negative shocks make the existing recession worse, and the sharper the recession, the stronger the effect on future growth in real GDP. This effect was pointed out by BK for extremely large negative shocks to United States real GDP that occur when the economy is in a steady state. Here, we see it showing up in the impulse response function for Germany and Italy when the initial state is a recession. For Italy, the effect is even stronger than in Germany. A -2% shock to real GDP in Italy results, after 12 quarters, in real GDP being about 1.4% above the baseline value of real GDP. This is similar to the effect of a +2% shock, which results in output 1.3% above the baseline. Notice that small negative or positive shocks have the biggest negative effect on real GDP. In these cases, the initial condition of a recession runs its course, and the shocks have little effect on the path of the economy. When measured against output levels far in the future, a negative shock in the midst of a recession may not have much effect and may increase output substantially, as happens with Italy.

The U.K.: More persistent negative shocks

The U.K. exhibits an impulse response that is very different in pattern to the results for the United States, Italy, and Germany. In the U.K., negative shocks have more persistence at 12 and 24 quarters than do positive shocks, and this is true for both a recession period and an expansion period. The magnitude of this difference also increases with the magnitude of the negative shock. That is, large negative shocks have more persistence than smaller negative shocks. The greater persistence for large shocks holds for both positive and negative shocks, and in both expansions and recession.

Visualizing the results

We provide graphs of the impulse response functions to supplement the tables. Figures 1 and 2 provide graphs of the impulse response functions for expansions and recessions, respectively. They show the asymmetry of the responses to positive and negative shocks, and the different behavior between countries, and between expansions and contractions. They also provide information on the impulse responses for horizons other than 12 and 24 quarters.

In sum, the results of estimating the CDR model for the G7 countries show that the United States experience is not generally representative. Only Italy and Germany exhibit responses to shocks that are roughly similar to the pattern found for the United States The remaining countries, furthermore, show great diversity in the effects that positive and negative shocks have on the economy. Japan, Canada, and France exhibit no evidence of nonlinearities, and the U.K. exhibits greater persistence of large shocks.

V. AN ALTERNATIVE RECESSION MODEL

The BK measure of recession, CDR, covers the entire period during which output does not exceed its pre-recession peak level of output. As a result, it covers periods both of economic contraction and of economic expansion - it includes both the recession and the recovery. Our previous results showing that economic growth is faster for some countries in the period between peaks raise the question whether economic performance is homogeneous in the inter-peak period. Specifically, is it true that the growth-enhancement effect is the same in the recession phase of the inter-peak period as it is in the recovery phase of that period?

To allow for the possibility that the role of the recession phase is different from the role of the recovery phase, we propose a more general specification of CDR that more closely follows the traditional definition of a recession: A recession occurs when the growth in GDP is negative, and a recovery occurs when GDP is growing toward its previous peak. This change leads to a new, two-part definition of CDR:

(3) [Mathematical Expression Omitted]

and,

(4) [Mathematical Expression Omitted]

With these definitions, Equation (2) is respecified as:

(5) [Phi](t) [Delta][Y.sub.t] = [Delta]

+ [[Omega](L) - 1][CDR1.sub.t] + [[Psi](L) - 1][CDR2.sub.t] + [[Epsilon].sub.t],

where [Phi](0) = [Omega](0) = [Psi](0) = 1. If the coefficients on CDR1 are positive and significant, then the growth enhancement effect begins immediately following the arrival of the recession. If the coefficients are zero, the existence of an asymmetry depends upon the CDR2 coefficients. If these terms are positive and significant, the growth enhancement effect occurs only during the recovery phase of the inter-peak period. Finally, if the coefficients on both CDR1 and CDR2 terms are positive and significant, the relative magnitudes of the coefficients determine the relative strength of the growth enhancement effect in the two parts of the inter-peak period. In the case that the coefficients are the same size, the simpler specification of CDR applies.

TABLE VI

Estimates Of GDP Growth Equations Including New CDR Terms

 Germany Italy UK USA

Drift -0.00303 0.00405 0.00634 0.00323
 (1.08) (2.87) (6.28) (3.17)

AR(1) 0.27556 0.27465 0.28906
 (2.15) (3.77) (3.76)

AR(2) 0.19532 0.15156 0.16268
 (1.91) (2.31) (1.87)

AR(3) 0.30933
 (3.22)

AR(4) 0.21622
 (2.37)

AR(5)

CDR1(-1) 0.66870(a) -0.14711
 (2.61) (1.90)

CDR1(-2)

CDR1(-3)

CDR1(-4)

CDR1(-5) 0.35303
 (3.07)

CDR2(-1) 1.05725(a) 0.37602
 (3.26) (2.55)

CDR2(-2)

CDR2(-3)

CDR2(-4)

CDR2(-5)

# of OBS 127 123 147 166

AIC -8.9146 -9.0212 -8.957 -9.4303

a The F-statistic for the hypothesis that these
coefficients are equal has a marginal significance level of
.237.

This extended specification was re-estimated for all seven countries and the results are presented in Table VI. A search over the 216 possible specifications led to the preferred models presented in that table. In the case of Italy, we further investigated the significance of the CDR1 terms at lags one through four, which we found insignificant and removed from the estimated model of real GDP growth.

The results in Table VI are, in general, stable across this redefinition. The orders of the AR models are the same in both Table II and Table VI for all countries and the orders of the split CDR terms replicates that of the single CDR term.

[TABULAR DATA FOR TABLE VII OMITTED]

As before, Japan, France, and Canada continue to exhibit symmetry in the response to positive and negative shocks. Consequently, the results for the extended CDR definition are the same for these countries as those presented in Table II and they are not repeated in Table VI. For the United States, however, the asymmetry is again found, but the CDR term is only active for the expansion portion of the inter-peak period. The asymmetry in response to shocks found by BK and replicated above arises because economic growth during recoveries is faster than it is during recessions or during expansions in GDP above its previous peak. In other words, the asymmetric response to the negative shocks does not occur until the trough of the recession has been passed.

In contrast, the excess persistence of negative shocks for the U.K. and Italy come solely from the recession phase of the inter-peak period. No CDR2 terms enter the model for these nations, suggesting that dynamics of growth are reinforced for recessionary periods. As in the original CDR specification, the estimated coefficient is negative for the U.K. and positive for Italy. For the U.K., this indicates that negative shocks cause longer and deeper deviations from the baseline than positive shocks. However, our generalization of CDR indicates that this reinforcement of negative shocks lasts only until the trough of the recession. It is solely a peak-to-trough phenomenon. Meanwhile for Italy, we see that the CDR1 term has a positive coefficient, though this term is lagged five periods. Thus, in Italy, negative shocks will be less persistent than positive shocks and this effect occurs due to the positive, offsetting response of output growth to recessions. With five lags, it is solely a peak-to-trough phenomenon.

The growth models for Germany do not show this bifurcation into models involving only CDR1 or only CDR2. In fact, for Germany the model with the split CDR definition includes one lag of CDR1 and one lag of CDR2 in place of the one lag of the original CDR. Furthermore, the magnitudes of the coefficients estimated across the redefinition are .67 for CDR1 and 1.06 for CDR2, both significantly different from zero but not significantly different from each other. An F-test for the null hypothesis that these two coefficients are the same has a marginal significance level of 24%. If one accepts the null hypothesis of equality between the coefficients, then the model should be estimated under the constraint of equality. Those results were already presented in Table II.

Despite the statistical tests, the standard errors are large enough, and the point estimates are far enough apart, to raise the concern of the role of the accuracy of the estimation in the statistical test. Consequently, we present the additional results for Germany in Table VI without the coefficients on CDR1 and CDR2 constrained to equality, and we compute the associated impulse response functions. The reader can choose which of the models is more appealing.

[TABULAR DATA FOR TABLE VIII OMITTED]

As expected, the impulse response functions presented in Table VII for expansions and Table VIII for recessions are quite different for some countries when computed for the split CDR model. For the United States, the redefinition of CDR somewhat reduces the gap between the persistence of positive and negative shocks. For the initial definition of CDR, a positive 2% shock during an expansion caused output to be above its baseline value by more than twice (1.70) the size of the shock. A negative shock of the same size caused output to be below its baseline by only about half (.53) the size of the shock.(3) When we estimate the model with the more general definition of CDR, this difference falls. A positive shock of 2% causes output to exceed its baseline by 1.59 times the size of the shock, while the same-sized negative shock causes output to fall below its baseline by .71 times the size of the shock. Thus, for the United States during expansions we see a small decrease in the effect of a positive shock relative to the effect on a negative shock of the same size. Moreover, a similar story is found for recessions.

For the U.K. the change in impulse response function from the CDR model to the CDR1 model is slight. This is true for any shock that we investigate, and for both expansion and recession periods.

Splitting CDR makes more difference for Germany and Italy. For Germany, the persistence of large shocks, especially positive shocks, is smaller when we use the split CDR model. A negative 2% shock, during an expansion, has an impulse response value of -1.18 after 24 quarters in the original CDR model and an impulse response value of -1.03 in the split CDR model. However, a positive 2% shock has an impulse response of 3.25 in the original CDR model, but only 1.37 in the split CDR model. During a recession a negative 2% shock in the single CDR model leads to an increase in output after 24 quarters (the impulse response value is .32), while the split CDR model has output falling, with a persistence of -.56.

For Italy, the pattern of impulse responses during an expansion period is roughly the same in the CDR and split CDR models. However, there are some differences during a recession period. For the CDR model, the response to a -2% shock is +.64 at 24 quarters, and the response to a +2% shock is +.72 at 24 quarters. For the split CDR model, the response to a -2% shock is .29 at 24 quarters, and the response to a +2% shock is .96 at 24 quarters. Thus, for a recession period the split CDR model has a lower persistence of negative shocks and a greater persistence of positive shocks than the CDR model.

In sum, the decomposition of the CDR variable into its recession and recovery portions has provided some additional insights into the cyclical dynamics of the countries studied. For example, we see that some countries experience the nonlinearity during the recession itself, as output declines below the previous peak (Italy and the U.K.), while other countries experience the nonlinearity during the recovery of the economy to the previous peak of output (the United States), and still others experience the nonlinearity in both the recession and recovery period (Germany). Of course, still other countries do not exhibit this type of nonlinearity (Canada, France, and Japan). The results of the extension to a split CDR model serve to reinforce and underscore the heterogeneity of these dynamics across the G7 economies.(4)

VI. CONCLUSION

We have examined the response to both positive and negative real GDP shocks for the G7 countries under a variety of assumptions about the nature of possible asymmetries in that response. We propose a more general model of inter-peak economic performance and we account for low frequency, high magnitude innovations in real GDP growth. An overall assessment of our results indicates that we find mixed evidence for asymmetric business cycles and for greater persistence of positive innovations.

Our results clearly suggest that there is little commonality in the dynamics of real GDP growth across industrialized countries. We found it difficult to replicate Beaudry and Koop's United States results using data from other industrialized countries. The Italian and, to a lesser degree, the German economies appear to most closely match United States dynamics. Three countries, Canada, France, and Japan showed no evidence of asymmetries and the U.K. economy exhibited seemingly perverse results: negative innovations have much longer persistence than positive innovations.

Editorial decisions on this paper were made by Richard Sweeney. The authors thank Heather Anderson, Nathan Balke, Alison Butler, Fred Joutz, and two anonymous referees for helpful comments. Jansen thanks the Private Enterprise Research Center for its support.

1. These are not the traditional definitions of recessions and expansions. In a later section, we redefine CDR to make it consistent with the traditional definitions of the phases of the business cycle, so that a recession occurs as long as the growth in GDP is negative, and a recovery occurs while GDP is growing until it reaches its previous peak.

2. The formula for the AIC is given as: In [[Sigma].sup.2] + 2K/n where K is the number of parameters and n is the number of observations.

3. These numbers are for the 12 quarter horizon. The results for the 24 quarter horizon are virtually identical.

4. Although there has been general recognition that the growth rate in real GDP may be stochastic, there has been disagreement about the nature of the generating mechanism. Recent research has questioned the characterization of real GDP growth as a simple random walk (Balke and Fomby [1991], Zivot and Andrews [1992], Zelhorst and de Haan [1994], and Bradley and Jansen [1995]).

In Bradley and Jansen [1995], we used Tsay's outlier detection technique to identify these low frequency, high magnitude shocks to real GDP growth in the G7 countries. In an earlier version of this paper we tested if accounting for those shocks alters our conclusions about the symmetry and persistence of positive and negative shocks. Generally speaking, although they are important for modeling real output growth, inclusion of the innovations does not affect the results from the original CDR definition. In particular, it does not eliminate the importance of including the CDR terms.

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Michael D. Bradley: Professor, Department of Economics George Washington University, Washington, D.C. Phone 1-202-994-8089, Fax 1-202-994-6147 E-mail mdbrad@gwis2.circ.gwu.edu

Dennis W. Jansen: Professor and Head, Department of Economics Texas A&M University, College Station Phone 1-409-845-7358, Fax 1-409-847-8757 E-mail d-jansen@tamu.edu