Estimation and testing of nonproportional Weibull hazard models

Most applications of the Weibull hazard model specify a common shape parameter. This is a proportional hazard model that imposes a common rate of duration dependence. A wide class of nonproportional Weibull models may be estimated by making the shape parameter a linear function of observable regressors. The log-likelihood function for these models is well behaved. The conditions under which this generalization is useful are essentially the same conditions under which interaction terms are useful in classical regression. Since the nonproportional model nests the proportional model, a formal test for nonproportionality may be conducted by likelihood ratio test. Estimation and testing of nonproportional models is illustrated with data sets for housing sales, out-of-court settlements and oil field exploration. Finally, estimation of a proportional Weibull model after adding temporal interaction terms to the regressors that specify the scale parameter is shown to be a fundamental misspecification. The standard log-likelihood function fails to recognize the stochastic nature of temporal interaction terms and the resulting estimates often fall outside the parameter space of the Weibull.