Since the early 1980s, the econometric analysis of duration variables has become widespread. This chapter provides an overview of duration analysis, with an emphasis on the specification and identification of duration models, and with special attention to models for multiple durations. Most of the chapter deals with so-called reduced-form duration models, notably the popular Mixed Proportional Hazard (MPH) model and its multivariate extensions. The MPH model is often used to describe the relation between the empirical exit rate and “background variables” in a concise way. However, since the applications usually interpret the results in terms of some economic-theoretical model, we examine to what extent the deep structural parameters of some important theoretical models can be related to reduced-form parameters. We subsequently examine the specification and identification of the MPH model in great detail, we provide intuition on what drives identification, and we infer to what extent biases may occur because of misspecifications. This examination is carried out separately for the case of single-spell data and the case of multi-spell data. We also compare different functional forms for the unobserved heterogeneity distribution. Next, we examine models for multiple durations. In the applied econometric literature on the estimation of multiple-duration models, the range of different models is actually not very large. Typically, the models allow for dependence between the duration variables by way of their unobserved determinants, with each single duration following its own MPH model. In addition to this, the model may allow for an interesting “causal” effect of one duration on the other, as motivated by an underlying economic theory. For all these models we examine the conditions for identification. Some of these are intimately linked to particular estimation strategies. The multiple-duration model where the marginal duration distributions each satisfy an MPH specification, and the durations can only be dependent by way of their unobserved determinants, is called the Multivariate Mixed Proportional Hazard (MMPH) model. For this model, we address the issue of the dimensionality of the heterogeneity distribution and we compare the flexibility of different parametric heterogeneity distributions. On a number of occasions, we incorporate recent insights from the biostatistical literature on duration analysis, and we contrast points of view in this literature to those in the econometric literature. Finally, throughout the chapter, we discuss the importance of the possible collection of additional data.

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Duration Models: Specification, Identification, and Multiple Durations

The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.

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Asymptotic theory for the correlated gamma-frailty model

Introduction Log-Linear Anaylsis Log-Linear Anaylsis with Latent Variables and Missing Data Event History Analysis Event History Analysis with Latent Variables and Missing Data A: Computation of the Log-Linear Parameters When Using the IPF Algorithm B: The Log-Linear Model as One of the Generalized Linear Models C: The Newton-Raphson Algorithm D: The Uni-Dimensional Newton Algorithm E: Likelihood Equations for Modified Path Models F: The Estimation of Conditional Probabilities under Restrictions G: The Information Matrix in Modified Path Models with Missing Data

Log-Linear Models for Event Histories

jabbringgfeweb.vu.nl gberggfeweb.vu.nl SUMMARY In a large class of hazard models with proportional unobserved heterogeneity, the distribution of the heterogeneity among survivors converges to a gamma distribution. This convergence is often rapid. We derive this result as a general result for exponential mixtures and explore its implications for the specification and empirical analysis of univariate and multivariate duration models.

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The unobserved heterogeneity distribution in duration analysis

This paper exploits dynamic features of insurance contracts in the empirical analysis of moral hazard. We first show that experience rating implies negative occurrence dependence under moral hazard: individual claim intensities decrease with the number of past claims. We then show that dynamic insurance data allow to distinguish this moral-hazard effect from dynamic selection on unobservables. We develop nonparametric tests and estimate a flexible parametric model. We find no evidence of moral hazard in French car insurance. Our analysis contributes to a recent literature based on static data that has problems distinguishing between moral hazard and selection and dealing with dynamic features of actual insurance contracts. Methodologically, this paper builds on and extends the literature on state dependence and heterogeneity in event-history data. (JEL: D82, G22, C41, C14)

Moral Hazard and Dynamic Insurance Data

We give a new proof of the identifiably of the MPH model. This proof is constructive: it is a recipe for constructing the triple—regression function, base-line hazard, and distribution of the individual effect—from the observed cumulative distribution functions.



We then prove that the triples do not depend continuously on the observed cumulative distribution functions. Uniformly consistent estimators do not exist.



Finally we show that the MPH model is even identifiable from two-sided censored observations. This proof is constructive, too.

A. C. M. Rooij

Papers

Constructive identification of the mixed proportional hazards model