Generalised residuals and heterogeneous duration models: With applications to the Weilbull model

This paper deals with models for the duration of an event that are misspecified by the neglect of random multiplicative heterogeneity in the hazard function. This type of misspecification has been widely discussed in the literature [e.g., Heckman and Singer (1982), Lancaster and Nickell (1980)], but no study of its effect on maximum likelihood estimators has been given. This paper aims to provide such a study with particular reference to the Weibull regression model which is by far the most frequently used parametric model [e.g., Heckman and Borjas (1980), Lancaster (1979)]. In this paper we define generalised errors and residuals in the sense of Cox and Snell (1968, 1971) and show how their use materially simplifies the analysis of both true and misspecified duration models. We show that multiplicative heterogeneity in the hazard of the Weibull model has two errors in variables interpretations. We give the exact asymptotic inconsistency of M.L. estimation in the Weibull model and give a general expression for the inconsistency of M.L. estimators due to neglected heterogeneity for any duration model to O(σ²), where σ² is the variance of the error term. We also discuss the information matrix test for neglected heterogeneity in duration models and consider its behaviour when σ²>0.