316 Economic Cycles
computer package LIMDEP 7.0 constructs such life tables, with an approximate,
but intuitive, explanation of the procedure as follows:
- Place the contractions in ascending order by length.
- Construct the hazard rate at time “t” as the ratio of the number of contractions
terminating in month “t” over the number of contractions lasting at least “t”
months.
Sample information is thus used to estimateP(T =t)/P(T ≥t)in a straight-
forward manner. A contraction that terminates is said toexitthe sample, while
those contractions lasting at leasttmonths are said to be stillat risk. Of course,
the pool of contractions stillat riskdecreases witht. This implies that the effec-
tive sample size for estimating the hazard rates for relatively long contractions is
less than for short contractions. Formal statistical tests are thus necessary to avoid
spurious conclusions from inspecting the graphs alone.
7.3.2 Benchmark hazards
Nevertheless, graphs convey important information about the general shape of the
hazard function. Hollander and Proschan (1975) and Hollander and Wolfe (1999)
provide some important benchmarks:
- Constant Failure Rate: CFR
- Increasing (Decreasing) Failure Rate: IFR, DFR
- Increasing (Decreasing) Failure Rate on Average: IFRA, DFRA
- New Better (Worse) than Used: NBU, NWU.
Figure 7.1 depicts hazard functions from CFR, IFR, IFRA, and NBU life
distributions.^9 Because there is a one-to-one relationship between the hazard func-
tion and the probability density function, a comparison of hazard rates is a natural
way of analyzing the nature of exit probabilities, more so than a comparison of
density functions. In particular, the CFR hazard is almost always given special
consideration in any duration analysis.
CFR hazards correspond to the geometric density. New economic expansions are
no more or less likely to terminate than mature ones. In contrast, if expansions are
IFR, the hazard, orfailure,rate is never decreasing, and our illustrated IFR hazard
implies an ever more likely chance of termination, ormortality.
IFR hazards are not the only ones that have a tendency to rise. Although the
depicted IFRA hazard has periods of decline, IFRA has the same overall upward
trend. For example, militarized interstate disputes initially have a decreasing haz-
ard rate for a short period, but then exhibit increasing hazards over most of the
duration due, perhaps, to a more concentrated effort to either negotiate or impose
settlement.
In contrast to IFRA, the depicted NBU hazard function rises above and then falls
back to the initial value. There is no overall trend in either direction. Because the
hazard function never falls below the initial value, new phases have the greatest
chance of surviving anadditionalweek or month. An NBU hazard may arise due to