The Great Moderation and the relationship between output growth and its volatility.

The estimate of [lambda] remains insignificant. That is, no relationship exists between growth and volatility measured by the two GARCH-M models with two different break dates. Two important consequences emerge from allowing for a structural change in the conditional variance. First, a large decline occurs in the estimated degree of persistence in the conditional variance. Each estimate in the variance equation in Table 3 falls below that in the model without the dummy in Table 2. The highly significant LR statistic in Table 3 proves no IGARCH effect. In addition, the estimates of [[alpha].sub.1] and [[beta].sub.1], or [[alpha].sub.1] and [[alpha].sub.2], not only fall in size but also become insignificant in the specification that includes the post-1984 dummy variable. The dummy variable replaces the (G)ARCH effects. Second, a strong interaction emerges between the dummy variable and the excess kurtosis, which previously proved significant (see Tables 1 and 2). This interaction now proves insignificant. These results suggest that the statistical evidence for time-varying variance and for excess kurtosis in the growth rate may reflect a shift in the unconditional variance caused by the Great Moderation. Figures 2-5 plot the conditional variances with and without a dummy for the four models, respectively. The solid line includes the dummy variable, while the dashed line excludes the dummy variable. One common characteristic appears in the four figures--a clear shift in the variance. The high volatility appears in the period before 1982 or 1984.

The structural break for the Great Moderation in 1982:I (or 1984:I) suggests that we should divide the sample into pre- and post-1982 (or post-1984) groups to estimate the relationship between the growth rate and its volatility separately for each period. Since the descriptive statistics indicate no time-varying variance in the growth series for three sub-samples (pre- and post-1984 and pre-1982), we constructed a moving-sample standard deviation to proxy for output volatility and used ordinary least squares (OLS) to estimate the relationship. For the post-1982 period, higher-order ARCH tests yield insignificant results in Table 1, suggesting the appropriateness of a simple ARCH(I) model. (9) Table 4 presents the results. For the two periods 1947:I to 1981:IV and 1947:I to 1983:IV, the coefficient of the autoregressive term at lag two proves insignificant. Thus, we report the estimation results for an autoregressive model with only one lag. The Ljung-Box diagnostic statistics show no evidence of first- and second-order autocorrelation in the residuals for the four subsamples, and the residuals reflect a normal distribution. The insignificant estimate of X again verifies our earlier finding that no relationship exists between the growth rate and its volatility in the United States. (10)

[FIGURE 3 OMITTED]

4. Further Evidence

This section considers two additional tests. First, we examined the possibility that the output growth rate affects its volatility, exploring whether an endogeneity bias exists in the GARCH and ARCH processes. Second, we studied whether a trend decrease in the volatility of output growth provided a better specification than the one-time shift considered previously. The first test follows the analysis of Fountas and Karanasos (2006), while the second test addresses the conclusion of Blanchard and Simon (2001).

Fountas and Karanasos (2006) recently found, using annual industrial production data from 1860 to 1999, that the output growth rate volatility exhibits no effect on the growth rate, but the output growth rate affects its volatility negatively in the United States, and bidirectional causality occurs between output growth and its volatility in Germany. The causal relationship between the output growth rate and its volatility suggests that the GARCH-M approach suffers from an endogeneity bias. Fountas and Karanasos (2006) included lagged growth in the conditional variance equation (the level effect) to test for the effect of growth on volatility in their GARCH(1,1)-M model. Following this specification, Table 5 reports the GARCH(1,1)-M estimation results, where we consider this level effect. The insignificance of [lambda] continues, even while the lagged growth estimate ([delta]) proves significant in the variance equation for either break date, 1982:I or 1984:I. All other estimates and diagnostic statistics closely mirror those in the models without this level effect. The findings that the output growth rate does not depend on changes in its volatility and that the output growth rate does affect its volatility negatively prove consistent with evidence in Fountas and Karanasos (2006), although they employed the long series of annual output data and we used quarterly data.

[FIGURE 4 OMITTED]

Blanchard and Simon (2001) argued that a trend decline in the volatility of output growth provides a better explanation of output growth volatility than does the one-time shift. Tables 6 and 7 present the evidence. Table 6 introduces a time trend in specifications of the GARCH and ARCH processes, but without a one-time shift dummy variable. The coefficient of the time trend proves negative for both specifications, although it is only significantly negative at the 5% level in the ARCH process. All other coefficients and diagnostic statistics closely mirror those in the models estimated without the time trend or the one-time shift dummy variable (see Table 2). One exception exists: The LR statistic now proves significant at the 5% level when the time trend appears, which rejects the null hypothesis of an IGARCH. Furthermore, although the volatility persistence falls substantially with the time trend, excess kurtosis remains. Thus, the time trend captures some, but not all, of the time-varying property of the variance. Finally, the volatility measure remains insignificant in the growth rate equation, matching the results of Tables 2 and 3.

[FIGURE 5 OMITTED]

Table 7 includes the time trend and the one-time shift dummy variable together in the GARCH and ARCH processes. In all four models, the coefficient of the time trend proves insignificant. Moreover, the coefficient of the one-time shift dummy variable proves significantly negative in each specification. All remaining coefficients and diagnostic statistics nearly match those in Table 3, including the insignificant coefficient of the variance measure in the growth rate equation.

In summary, the one-time shift dummy variable dominates the time trend across our various tests. That is, based on the log-likelihood value, the corresponding specifications in Tables 3 and 7 do not exhibit significant differences, whereas the corresponding specifications in Tables 2 and 6 do exhibit significant differences compared to those in Tables 3 and 7.

5. Conclusion

This paper examines the effect of the Great Moderation on the relationship between quarterly real GDP growth rate and its volatility in the United States over the period 1947:I to 2006:II. We began by considering the possible effects, if any, of structural change on the volatility process. Our initial results, based on either a GARCH-M or an ARCH-M model of the conditional variance of the residuals, showed strong evidence of volatility persistence and excess kurtosis in the growth rate. Subsequent analysis revealed that this conclusion was not robust to a one-time shift in output variability due to the Great Moderation. First, the findings of a time-varying variance measured by the GARCH-M or ARCH-M model disappeared in the specifications that included the post-1984 dummy variable. That is, the GARCH effect proved spurious. In any case, no GARCH-M effect emerged. Second, excess kurtosis vanished in the specifications that included either the 1982 or the 1984 dummy variable in either the GARCH or the ARCH process. Both the data analysis and the OLS estimates generally suggested no relationship between U.S. output volatility and growth, favoring macroeconomic models that dichotomize the determination of output volatility and growth. In sum, our results add to the conclusion that the relationship between the output growth rate and its volatility in the United States proves weak, at best.

The independence between the output growth and its volatility needs careful interpretation. Endogenous growth theory, for example, does not imply any importance for the second moment. Blackburn and Galindev (2003) and Blackburn and Pelloni (2004) modeled the link between the mean and variance of the output growth rate explicitly. Different mechanisms of endogenous technological change and nominal or real shocks can lead to a positive or negative relationship between growth and volatility. In his model, Blackburn (1999) showed that, for a linear endogenous learning function, the effect of the output growth rate volatility on the output growth rate equals zero. A concave (convex) learning function generates a negative (positive) effect. That is, an independent relationship may exist with or without the Great Moderation. The discrepancy of our findings from those in previous studies highlights the sensitivity of the results to the country considered, the time period examined, the frequency of the data, and the methodology employed. This apparent inconclusiveness warrants further investigation of the relationship between growth and its volatility. Moreover, since studies generally focus on developed countries, additional analysis from developing countries may prove illuminating. For example, the Asian newly industrializing countries may provide totally different scenarios because of their high growth rates.