Joe Cardinale and Larry W. Taylor 3157.3 The discrete-time hazard function
Duration data for economic cycles are invariably discrete since data are collected
at discrete intervals of time, for example, weekly, monthly, quarterly, or yearly.
Although data for markets such as housing or financial markets are available at
short intervals, the more interesting questions about such cycles are typically best
captured by intervals of at least a month. Consider also that a discrete-time dura-
tion analysis can be viewed as an approximation to a continuous-time analysis,
or vice versa, and general notions about one apply to the other.^6 A discrete-time
framework has the advantage of being more natural for the types of data encoun-
tered at the aggregate level; the framework is inherently semiparametric and is easy
to understand and implement. Since formal statistical inference depends on the
framework, it is best to adopt that of discrete-time if data are measured at long
intervals.
Consider a random sub-sample ofnobservations(T 1 ,T 2 ,...,Tn)from a discrete,
cumulative distributionF, such thatF(a) = 0 fora < 0.^7 The probability-
distribution function, ordensity function,isf(t)=P(T=t), and the discrete-time
hazard functionis:
h(t)=P(T=t|T≥t)=f(t)/G(t), (7.2)whereh(t)is the hazard function, andG(t)=P(T≥t)is thesurvival function.
The density function,f(t), gives the probability that a duration will lastexactly t
periods, the survival function,G(t), gives the probability that a duration will last
at least tperiods, and the hazard function,h(t), gives the conditional probability
that a phase will terminate in periodt, given that it has lastedt or moreperiods.
If rising or falling, the hazard provides useful information about the likelihood of
a change in phase. Using over 100 years of annual data, Mills (2001) finds several
instances of non-constant hazards in the business cycles of 22 countries.
The hazard may also be useful in the assessment of general market conditions.
For instance, Diebold and Rudebusch (1990) and Ohn, Taylor and Pagan (2004)
observe that post-World War II contractions are more prone to revert to expansion
than pre-World War II contractions. One explanation for this finding is that policy
makers are now much better able to manage potential economic crises. A second
explanation is that individuals and firms are better able to smooth shocks due to
innovation and financial deregulation. On the other hand, Watson (1994) finds
that, for most individual sectors of the economy, the average contraction and
expansion durations for the pre-war and post-war periods are similar; and, more
recently, Stock and Watson (2003) suggest that favorable market conditions in
the modern era are more likely due to good luck than to good management or
innovation.^8 This is especially true for recent times as there have been relatively
few long-lived supply disruptions since the 1970s.
7.3.1 Hazard plots
It is generally informative to plot the hazard function, with hazard rates easily
computed by the nonparametric life table method of Cutler and Ederer (1958). The