318 Economic Cycles
7.4 Testing for duration dependence
The direction of duration dependence is easily obtained from a sample of dura-
tions. First, if the mean duration equals the sample standard deviation, there is no
evidence of duration dependence; that is, there isa constant hazard. Second, if the
mean duration is greater than the sample standard deviation, there is evidence of
positiveduration dependence, or a generally increasing hazard. Finally, if the mean
duration is less than the sample standard deviation, there is evidence ofnegative
duration dependence, or a generally decreasing hazard.
7.4.1 The nature of duration independence
Duration independence is considered the neutral case. Long expansions have no
greater chance of ending than short expansions; long bear markets have no greater
chance of ending than short ones; and long housing slumps have no greater chance
of ending than short slumps. The duration of the phase has no predictive power in
determining the end of the phase. Because of neutrality, the constant hazard is the
standard benchmark, and a graph of the hazard function is frequently employed
to see if the hazard function appears roughly constant. If so, there isduration inde-
pendence,and the hazard function does not depend ont. In other words, the null
hypothesis is:
H 0 :h(t)=p for some 0<p<1 and allt>0. (7.3)The densitymustbe geometric for constant hazards, and the above null hypothesis
is equivalent to:
H 0 :f(t)=P(T=t)=( 1 −p)tpfor 0≤t≤∞. (7.4)In other words, testing for duration independence is equivalent to testing whether
the durations follow the geometric density. Adirect,orstrong-form, test for the
geometric density is the usual chi-square goodness-of-fit test employed by Ohn,
Taylor and Pagan (2004).
Finally, for the geometric density,E(T)=( 1 −p)/pandV(T)=( 1 −p)/p^2 , leading
to a third null hypothesis for duration dependence:
H 0 :V(T)−[E(T)]^2 −E(T)=0. (7.5)7.4.2 Weak-form tests
Pagan (1998) and Mudambi and Taylor (1991, 1995) devise tests for duration
dependence based on theconsistency relationshipas in (7.5). Such tests are called
weak-formtests and have close links with continuous-time tests for theexponential
density. Although the exponential density is simply the continuous-time equiva-
lent of the discrete-time geometric density, there is one rather important difference
between the two. The consistency relationship for the exponential density is
V(T)−[E(T)]^2 =0 rather thanV(T)−[E(T)]^2 −E(T)=0, and thus the choice
between the discrete-time and continuous-time tests is important except whenp
is close to either 0 or 1, since thenV(T)≈[E(T)]^2.