Models of duration dependence

Abstract

The paper discusses models for longitudinal data in which a sample of individuals or firms is followed over time. Since the typical panel has a large number of individuals observed over a short time period, the relevant limiting distributions have the number of individuals increasing but not the time periods. If we allow for individual specific parameters, then maximizing the joint likelihood function does not in general provide a consistent estimator of the parameters common to all individuals. The paper discusses procedures that are valid in general, based on conditional, marginal and posterior likelihood functions. As for models with discrete data in discrete time, the paper argues that the distinction between serial correlation and state dependence cannot be made just on the basis of the multinomial distribution of the binary sequence. Some distinctions are possible if we consider the distribution of the binary sequence conditional on observed variables. These discrete time models are appropriate if the time interval has some natural significance. In many problems, however, the basic data is the amount of time spent in a state. In studying labor force participation, a complete description of the process is the duration of the first spell of participation and the date it began, the duration of the following spell of non-participation, and so on. This complete history will generate a binary sequence when it is cut up into fixed length periods, but these periods may have little to do with the underlying process. In particular, the measurement of serial correlation depends on the interval between observations. As this interval becomes shorter, the probability that a person who participated last period will participate this period approaches one. So a more fruitful question is whether an individual's history helps to predict his future given his current state. A stationary version of this Markov property implies that the durations of the spells are independently distributed with exponential distributions. With heterogeneity, each individual has his own exponential rate parameters. Departures from this model constitute duration dependence. The paper shows the binary sequences generated by various sampling schemes do allow us to test for duration dependence in the continuous time model. A particularly interesting and relevant case is the sampling procedure that asks whether the individual was ever in a particular state (participation, for example) during the previous period. The paper develops methods for using the underlying data on duration when it is available. We model the distribution of duration conditional on other variables, allowing for duration dependence. Using models with log-linear hazard functions, we discuss the restrictive features of Cox's conditional likelihood analysis, including the role fo strict exogeneity and the sense in which age and time are not strictly exogenous. The restrictive features are relaxed in a posterior likelihood analysis. We also consider marginal and conditional likelihood methods in models based on the Weibull, gamma, and log-normal distributions for duration.