The purpose of this paper is to consider the problem of statistical inference about a hazard rate function that is specified as the product of a parametric regression part and a non‐parametric baseline hazard. Unlike Cox's proportional hazard model, the baseline hazard not only depends on the duration variable, but also on the starting date of the phenomenon of interest. We propose a new estimator of the regression parameter which allows for non‐stationarity in the hazard rate. We show that it is asymptotically normal at root‐n and that its asymptotic variance attains the information bound for estimation of the regression coefficient. We also consider an estimator of the integrated baseline hazard, and determine its asymptotic properties. The finite sample performance of our estimators are studied.