Asymptotics for threshold regression under general conditions

The inference of the threshold point in threshold models critically depends on the assumption that the density of the threshold variable at the true threshold point is continuous and bounded away from zero and infinity. However, violation of this assumption may arise in several econometric contexts such as treatment effects evaluation. This paper presents a thorough characterisation of the asymptotic distributions in the least‐squares estimation of such abnormal cases. First, the asymptotic results on the threshold point are different from the conventional case. For example, any convergence rate between zero and infinity is possible; the asymptotic distribution can be discrete, continuous or a mixture of discrete and continuous; the weak limits of the localised objective functions can be non‐homogeneous instead of homogeneous compound Poisson processes. Second, this paper distinguishes threshold regression from structural change models by studying a problem unique in threshold regression. Third, the asymptotic distributions of regular parameters are not affected by estimation of the threshold point irrespective of the density of the threshold variable. Numerical calculations and simulation results confirm the theoretical analysis, and the density of the threshold variable in an application is checked to illustrate the relevance of the study in this paper.