Bayesian Analysis of Hazard Regression Models under Order Restrictions on Covariate Effects and Ageing

We propose Bayesian inference in hazard regression models where the baseline hazard is unknown, covariate effects are possibly age-varying (non-proportional), and there is multiplicative frailty with arbitrary distribution. Our framework incorporates a wide variety of order restrictions on covariate dependence and duration dependence (ageing). We propose estimation and evaluation of age-varying covariate effects when covariate dependence is monotone rather than proportional. In particular, we consider situations where the lifetime conditional on a higher value of the covariate ages faster or slower than that conditional on a lower value; this kind of situation is common in applications. In addition, there may be restrictions on the nature of ageing. For example, relevant theory may suggest that the baseline hazard function decreases with age. The proposed framework enables evaluation of order restrictions in the nature of both covariate and duration dependence as well as estimation of hazard regression models under such restrictions. The usefulness of the proposed Bayesian model and inference methods are illustrated with an application to corporate bankruptcies in the UK

A Simple Test for the Absence of Covariate Dependence in Hazard Regression Models

This paper extends commonly used tests for equality of hazard rates in a two-sample or k-sample setup to a situation where the covariate under study is continuous. In other words, we test the hypothesis that the conditional hazard rate is the same for all covariate values, against the omnibus alternative as well as more specific alternatives, when the covariate is continuous. The tests developed are particularly useful for detecting trend in the underlying conditional hazard rates or changepoint trend alternatives. Asymptotic distribution of the test statistics are established and small sample properties of the tests are studied. An application to the e¤ect of aggregate Q on corporate failure in the UK shows evidence of trend in the covariate e¤ect, whereas a Cox regression model failed to detect evidence of any covariate effect. Finally, we discuss an important extension to testing for proportionality of hazards in the presence of individual level frailty with arbitrary distribution

Bayesian Analysis of Hazard Regression Models under Order Restrictions on Covariate Effects and Ageing

We propose Bayesian inference in hazard regression models where the baseline hazard is unknown, covariate effects are possibly age-varying (non-proportional), and there is multiplicative frailty with arbitrary distribution. Our framework incorporates a wide variety of order restrictions on covariate dependence and duration dependence (ageing). We propose estimation and evaluation of age-varying covariate effects when covariate dependence is monotone rather than proportional. In particular, we consider situations where the lifetime conditional on a higher value of the covariate ages faster or slower than that conditional on a lower value; this kind of situation is common in applications. In addition, there may be restrictions on the nature of ageing. For example, relevant theory may suggest that the baseline hazard function decreases with age. The proposed framework enables evaluation of order restrictions in the nature of both covariate and duration dependence as well as estimation of hazard regression models under such restrictions. The usefulness of the proposed Bayesian model and inference methods are illustrated with an application to corporate bankruptcies in the UK.Bayesian nonparametrics; Nonproportional hazards; Frailty; Age-varying covariate e¤ects; Ageing

Models of Firm Dynamics and the Hazard Rate of Exits: Reconciling Theory and Evidence using Hazard Regression Models

This paper considers empirical work relating to models of firm dynamics. We show that a hazard regression model for firm exits, with a modification to accommodate age-varying covariate effects, provides an empirical framework accommodating many of the features of interest in studies on firm dynamics. Modelling implications of some of the popular theoretical models are considered and a set of empirical procedures for verifying testable implications of the theoretical models are proposed. The proposed hazard regression models can accommodate negative effects of initial size that go to zero with age (active learning model), negative initial size effects that fall with age but stay permanently negative (passive learning model), conditional and unconditional hazard rates that decrease with age at higher ages, and adverse effects of macroeconomic shocks that decrease with age of the firm. The methods are illustrated using data on quoted UK firms. Consistent with the active learning model, the effect of initial size is significantly negative for a young firm and falls to zero with age. The hazard function conditional on size, other firm- and industry-level characteristics, and macroeconomic conditions decreases with age only at higher ages, but shows the weaker property of Increasing Mean Residual Life over its entire life-duration. Instability in exchange rates affects survival of very young firms strongly, and the effect decreases to insignificant levels for older firms.Firm exit, Learning, Firm Dynamics, Non-proportional hazards, Hazard regression models

A Simple Test for the Absence of Covariate Dependence in Hazard Regression Models

This paper extends commonly used tests for equality of hazard rates in a two-sample or k-sample setup to a situation where the covariate under study is continuous. In other words, we test the hypothesis that the conditional hazard rate is the same for all covariate values, against the omnibus alternative as well as more specific alternatives, when the covariate is continuous. The tests developed are particularly useful for detecting trend in the underlying conditional hazard rates or changepoint trend alternatives. Asymptotic distribution of the test statistics are established and small sample properties of the tests are studied. An application to the e¤ect of aggregate Q on corporate failure in the UK shows evidence of trend in the covariate e¤ect, whereas a Cox regression model failed to detect evidence of any covariate effect. Finally, we discuss an important extension to testing for proportionality of hazards in the presence of individual level frailty with arbitrary distribution.Covariate dependence; Continuous covariate; Two-sample tests; Trend tests; Proportional hazards; Frailty/ unobserved heterogeneity; Linear transformation model