Faster Estimation of a Discrete-Time Proportional Hazards Model with Gamma Frailty

Abstract

Fitting a complementary log-log model that accounts for gamma-distributed unobserved heterogeneity often takes a significant amount of time. This is in part because numerical derivatives are used to approximate the gradient vector and Hessian matrix. The main contribution of this article is the use of Mata and a gf2 evaluator to express the gradient vector and Hessian matrix. Gradient vector expression allows one to use a few different options and postestimation commands. Furthermore, expression of the gradient vector and Hessian matrix increases the speed at which a likelihood function is maximized.

In this article, I present a complementary log-log model, show how the gamma distribution has been incorporated, and point out why the gradient vector and Hessian matrix can be expressed. I then discuss the speed at which a maximum is achieved, and I apply sampling weights that require an expression of the gradient vector. I introduce a new command for fitting this model. To demonstrate how this model can be applied, I will examine information on when young males first try marijuana.