Specification and identification
This paper analyzes the specification and identification of causal multivariate duration models. We focus on the case in which one duration concerns the point in time a treatment is initiated and we are interested in the effect of this treatment on some outcome duration. We define “no anticipation of treatment” and relate it to a common assumption in biostatistics. We show that (i) no anticipation and (ii) randomized treatment assignment can be imposed without restricting the observational data. We impose (i) but not (ii) and prove identification of models that impose some structure. We allow for dependent unobserved heterogeneity and we do not exploit exclusion restrictions on covariates. We provide results for both single-spell and multiple-spell data. The timing of events conveys useful information on the treatment effect.
Abbring, Jaap H., and Gerard J. Van Den Berg (2003), "The identifiability of the mixed proportional hazards competing risks model", Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65, 701–710.
We prove identification of dependent competing risks models in which each risk has a mixed proportional hazard specification with regressors, and the risks are dependent by way of the unobserved heterogeneity, or frailty, components. We show that the conditions for identification given by Heckman and Honoré can be relaxed. We extend the results to the case in which multiple spells are observed for each subject.
Ridder (1990, Review of Economic Studies 57, 167–182) provides an identification result for the Generalized Accelerated Failure-Time (GAFT) model. We point out that Ridder’s proof of this result is incomplete, and provide an amended proof with an additional necessary and sufficient condition that requires that a function varies regularly at 0 and ∞. We also give more readily interpretable sufficient conditions on the tails of the error distribution or the asymptotic behavior of the transformation of the dependent variable. The sufficient conditions are shown to encompass all previous results on the identification of the Mixed Proportional Hazards (MPH) model. Thus, this paper not only clarifies, but also unifies the literature on the non-parametric identification of the GAFT and MPH models.