322 Economic Cycles
de Gracia (2003) segment the time line for Latin American and Asian countries
to determine the effect of financial liberalization on stock market cycles. Mod-
els that employ covariates have not been as popular and the studies that do use
them generally assume duration independence. For instance, Estrella and Mishkin
(1998) employ a discrete-time analysis to examine various financial variables as
predictors of US recessions, and Chin, Geweke and Miller (2000) use a similar anal-
ysis to predict turning points in the civilian unemployment rate. Conditional upon
the right-hand-side variables, however, both studies assume the hazard function is
independent of time.^11
The strong assumption of duration independence in models with covariates is
defensible in some circumstances. In the political science literature, Bennett (1999,
p. 262) goes so far as to argue that including covariates to effectively eliminate
duration dependence is a laudable goal:
Unless we can anthropomorphize and assume that the phenomenon we are
examining truly has a life of its own, then the pattern or covariation over
time that we observe is somehow, somewhere, driven by a variable or set
of variables that characterizes the world. If the causal factor driving duration
dependence is measured and included in the model as an independent variable,
then unexplained duration dependence...may disappear.Bennett’s view aligns with the concept ofprobabilistic reductionthat Spanos (1995)
traces back to the biometric tradition of Galton and Pearson; see also Spanos (2006)
and Hoover (2006). In the regression framework, probabilistic reduction implies
that a complete theory must induce white-noise disturbances in the model. In
practice, whether covariates can account for the observed duration dependence in
any binary series is surely an empirical question that cannot be assumed away for
the model at hand. Complete theories are ideal but rare.
Another reason that duration dependence is frequently ignored for discrete-time
analysis is that many researchers apparently believe that it is not possible to incor-
porate such dependence. For instance, Bennett (1999, p. 259) argues that the logit
model is insufficient for the analysis of duration data because “it assumes that no
duration dependence exists.” Below we show that a slightly modified logit model
is suitable to capture autonomous changes in the discrete-time hazard as well as
changes due to covariates.
Probit models are alternatives to logit models. There is no need, however, to
adopt any type of latent structure for either probit or logit. In other words, it is
neither necessary nor desirable to insist that there exists some type of latent vari-
able,y∗t, such thatSt =1ify∗t >0, andSt=0ify∗t ≤0. Although such an
assumptionisdesirable in the discrete-choice literature, whereyt∗is interpreted as
a utility function, it only unnecessarily complicates a duration analysis. In fact,
to mark time for economic cycles, we frequently map anobservedseries,{yt},to
{St}. Thus, the unobservables of true interest in either the logit or probit probabil-
ity models are the estimable parameters controlling the termination probabilities
of{St}.^12