Maximum likelihood estimation and specification test of the proportional hazard model using grouped durations

This article develops parametric and semi-parametric, maximum likelihood estimation methods of the proportional hazard model for the case when durations are grouped and heterogeneity is not fully measured. This article also extends Ryu's (1994b) specification tests for the proportional hazard model. The proposed test statistic is easy to implement, takes into account the form of alternative hypothesis to increase power, and identifies the source and direction of non-proportionality. This article identifies the nature of the bias resulting from neglected heterogeneity. This bias is essentially a sample selection bias, well-known in the literature. Gamma and discrete distributions are used to model unobserved heterogeneity. By updating the heterogeneity distribution at each stage of duration process, this article addresses the sample selection bias. The results obtained in this article are applied to tackle left-censoring problem.