Predictors of mean duration and an arbitrary quantile are given for the Weibull regression model for duration data. Associated prediction variances arising from maximum likelihood and least squares estimation are given. In an empirical example, based on duration of employment data, the uses of various model diagnostics and the predictors are illustrated.
KEY WORDS Prediction Duration model Weibull distribution
Duration of employment Weibull regression models are frequently used in explaining durations, e.g. the duration of unemployment (e.g. Lancaster, 1979). Predictions based on duration models can be used to support decision making at the individual level, e.g. in designing individual programmes for prisoners to be released (cf. Schmidt and Witte, 1984). The predictions can also be of great interest in missing data situations, and as another tool for model evaluation.
In this paper we consider the prediction of mean duration and of an arbitrary quantile. The unknown parameters of the Weibull model are estimated by the maximum likelihood and least squares methods. Prediction variances and confidence intervals for predictions are of particular interest in assessing the uncertainties of the predictions. Other measures, such as additional mean duration, can be predicted using the same approaches as considered here. For other distributions we can proceed in an analogous manner.
We let the duration f be Weibull distributed, such that y = l o g t = xfi*+aw* (1) where x = (1, xl), fi* = (fl,*, 1; )' and w* is an extreme value distributed disturbance term with E(w*) = $(I) = -0.5772 and V(w*) = $'(I) = x 2 / 6 ($(.) and $'(.) are digamma and trigamma functions, respectively). The k-vector x1 contains fixed and time invariant explanatory variables. From (1) the mean duration is obtained as E(t) = exp(xfi*)r(l + a) t , = exp (xfi*)[ -log (1 -p)]" ( 2 ) where r(.) is the gamma function. The quantile ( p = F(t,)*tp = F -' ( p ) ) is given as (3)
where p is the percentile, and F(.) is the distribution function of the duration t .