Duration Models
Duration Models
Duration models are used to describe the amount of time that elapses until a given event, or the length of time spent in a given state. Duration models have been used to examine many phenomena, including labor-market outcomes, by modeling the length of time spent in unemployment; industry consolidation, by modeling time elapsed until a firm is acquired by another firm; criminal recidivism, by modeling time elapsed until conviction for a criminal offense for persons released from prison; and the viability of medical procedures, by modeling time until death, relapse, or recurrence after surgery.
Duration models share a duality relationship with count-data models. In count-data models, one models the number of occurrences of some event within a specified interval of time. If, for example, counts of some event are distributed Poisson, then the intervals between events can be shown to have an exponential distribution. One can model either counts over a time interval, or the intervals between the events comprising the counts. Duration models are often used in situations where only one interval is observed for each agent, as in the case of unemployment spells or the lives of firms.
In many cases, data on durations are censored. Suppose one collects data on durations in a given state by conducting surveys at calendar times tA and tB. Figure 1 illustrates four possibilities. At time tA, agent 1 has not yet entered the state of interest, but by time tB, he has entered and exited the state at times between tA and tB. Hence if the survey at time tB asks the right questions, it will be possible for the researcher to observe this agent’s time in the spell completely. Agent 2 enters the state of interest prior to the time of the first survey, and exits prior to the time of the second survey; again, if the right questions are asked, it will be possible for the researcher to observe this agent’s complete duration as well. In some cases, however, depending on how the first survey is conducted, agent 2’s time of entry into the state may be unobserved, in which case this agent’s duration will be left-censored; all the researcher would know in this case is that the agent was already in the state of interest at time tA, and the time of exit before time tB. Agents 3 and 4 remain in the given state until after the time of the second survey; for these agents, the researcher can know only that they remained in the state of interest at time tB and were still waiting to exit. Observations for these agents are right-censored; in addition, if the entry time for agent 3 is unknown, the observation for this agent will be both left- and rightcensored.
The maximum likelihood method is often used to estimate duration models after specifying a distribution function
F (t ) = Pr(T ≤ t )
for the random variable T describing the length of time spent in a state. Because durations are necessarily nonnegative, one must specify a one-sided distribution for T ; examples include the exponential, Weibull, Gompertz, and log-normal distributions. The distribution function
F (t ) implies a density function f (t ) = dF (t ), a survivor function S (t ) = 1 – F (t ) which gives the probability of remaining in the state up to time t, and a hazard function
giving the rate at which exits occur at time t. The goal of researchers is often to estimate the marginal effects of various covariates on the hazard rate.
John Kalbfleisch and Ross Prentice (2002), Tony Lancaster (1990), and Colin Cameron and Pravin Trivedi (2005) give extensive details on specification, censoring in, and estimation of duration models, as well as examples of applications.
SEE ALSO Censoring, Left and Right; Maximum Likelihood Regression; Regression Analysis; Variables, Random
BIBLIOGRAPHY
Cameron, A. Colin, and Pravin K. Trivedi. 2005. Microeconometrics: Methods and Applications. Cambridge, U.K., and New York: Cambridge University Press.
Kalbleisch, John D., and Ross L. Prentice. 2002. The Statistical Analysis of Failure Time Data. 2nd ed. Hoboken, NJ: J. Wiley.
Lancaster, Tony. 1990. The Econometric Analysis of Transition Data. Cambridge, U.K., and New York: Cambridge University Press.
Paul W. Wilson