Business cycle duration dependence reconsidered.

Sichel estimated a Weibull hazard model using the National Bureau of Economic Research business cycle chronology and found evidence of duration dependence only for prewar expansions and postwar contractions. The article updates the postwar sample through the end of the most recent expansion and uses a generalized Weibull model that provides much greater flexibility at the expense of one additional parameter. This model finds evidence of duration dependence for all samples and is statistically superior to the conventional Weibull model for all samples except postwar contractions.

KEY WORDS: Bootstrap; Generalized Weibull; Hazard function; Nonlinear.

1. INTRODUCTION

Sichel (1991, p. 254) asked "Are periods of expansion or contraction in economic activity more likely to end as they become older?" At the time, the United States was nearing the end of its longest peacetime expansion in history. The expansion of the 1990s lasted even longer. A recent National Bureau of Economic Research (NBER) decision dates the end of this cycle as March 2001, which corresponds to a 120-month-long expansion. Sichel's question is as relevant today as it was a decade earlier.

Sichel (1991) used a Weibull hazard model to test for duration dependence in the NBER business cycle chronology. The choice of the Weibull model was motivated by the relatively low power exhibited by nonparametric tests of duration dependence. Even with time series dating back to the mid-1800s, the sample size (which is determined by the number of distinct periods of expansion or contraction) is relatively small. Sichel found evidence of positive duration dependence in prewar expansions and postwar contractions, but no evidence of duration dependence in prewar contractions or postwar expansions. This suggests that in the postwar period, the probability that an expansion will end is unrelated to its current duration.

The fundamental assumption of the Weibull model is a linear relationship between the log of the hazard function and the log of duration. Diebold, Rudebusch, and Sichel (1993) used an exponential-quadratic hazard model, which assumes that the log-hazard function is quadratic in duration. The exponential-linear hazard model is a restricted case where the log-hazard function is linear in duration. One advantage of the exponential-quadratic model is that it allows nonlinear and nonmonotonic time profiles for the log-hazard function.

A comparison of the results of Sichel (1991) and Diebold et al. (1993) is complicated by several factors. First, the exponential-quadratic model does not nest the Weibull model. Consequently, the point estimates are not directly comparable. This is most obvious when comparing the Wiebull model with the exponential-linear model. With the Weibull model, the log-hazard function is linear in the log of duration and the duration elasticity is constant, whereas with the exponential-linear model, the log-hazard function is linear in duration and the duration elasticity is proportional to duration. Second, the method of correcting for censoring of cycles of relatively short duration differs. Both articles estimated a censoring threshold, de noted by [delta] here, as one period less than the minimum value of observed duration in the sample. They differ, however, in that Sichel estimated the censoring threshold from the NBER data, whereas Diebold et al. used a pooled sample of international data. Sichel then corrected for censoring by using the conditional density for duration given that it exceeds [delta], whereas Diebold et al. simply subtracted [delta] from observed duration and used the unconditional density function for duration. Finally, Sichel tested hypotheses using bootstrapped t statistics, whereas Diebold et al. reported p values for asymptotic Wald statistics.

Despite these differences, it is probably not surprising that the results obtained by Diebold et al. (1993) with the exponential-linear model are qualitatively similar to those of the Weibull model reported by Sichel (1991). Both studies find statistical evidence of a positive duration elasticity only for prewar expansions and postwar contractions. Furthermore, the exponential-quadratic model provides little evidence of a nonlinear time profile for the log-hazard function. The estimated coefficient of the quadratic term is not significant at conventional levels for any of the samples of U.S. data considered.

Diebold et al. (1993, p. …