In empirical research we may wish to investigate the dependence of individual hazard functions on exogenous variables. These variables, called the control variates, depict in general various individual characteristics. Let us point out a few examples. In the job search analysis, a typical control variate is the amount of unemployment benefits, which influences the effort of unemployed individuals devoted to the job search and consequently the duration of unemployment. Empirical findings also suggest that family support provided by the state influences the birth rate, and that the expected increase of the insurance premium has an effect on the frequency of declared car accidents. As well, there is evidence indicating that the lengths of hospital stays depend on the cost incurred by patients, or else, the duration of an outstanding balance on a credit card is in part determined by the interest paid by the cardholder. Some explanatory variables
differ accross individuals, and are invariant in time (e. g. gender), while others (e. g. age) are individual and time dependent. Such variables need to be doubly indexed by individual and time (see Hsiao, Chapter 16, in this volume). Other variables may have a common impact on all individuals in the sample and vary in time, like the rate of inflation or the global rate of unemployment.
Parametric duration models can accommodate the effect of observable or unobservable individual characteristics on durations. Let us denote by xi the observable explanatory variables and by p; a latent heterogeneity factor. We proceed in two steps to define the extended duration model. First, we consider the conditional distribution of the duration variable Y given the observable covariates and heterogeneity. It is characterized by either the conditional pdf f (yi | xi, p;), or the conditional hazard function X( y t |x„ p;). Next, we introduce a heterogeneity distribution n(p;) (say), which is used to derive the conditional distribution of the duration variable given the observable covariates only. This latter distribution is characterized by either the conditional pdf f (yi1 x), or the conditional hazard function X(yi | x;).
In the first subsection we describe the exponential duration model without heterogeneity and its estimation by the maximum likelihood. In the following subsection we introduce a gamma distributed heterogeneity factor, which leads us to the Pareto regression model. The effect of unobservable heterogeneity and its relationship with the negative duration dependence are covered in the third part of this section. Finally, we discuss the problem of partial observability of duration variables due to truncation or censoring effects.