Adaptive estimation of the threshold point in threshold regression

This paper studies semiparametric efficient estimation of the threshold point in threshold regression. The classical literature of semiparametric efficient estimation rests on the fact that the maximum likelihood estimator is efficient in any parametric submodel for a large class of loss functions. However, in threshold regression, the maximum likelihood estimator is not efficient, while the Bayes estimators are efficient and different loss functions induce different efficient estimators. For an additively separable loss function that separates the efficiency problem of the threshold point from that of other parameters, we show that the semiparametric and parametric efficiency risk bounds coincide. Then we design a semiparametric empirical Bayes estimator to achieve this bound. In consequence, the threshold point can be adaptively estimated even under conditional moment restrictions. We also provide a valid confidence interval called the nonparametric posterior interval for the threshold point. Simulation studies show that the semiparametric empirical Bayes approach is substantially better than existing methods. To illustrate our procedure in practice, we apply it to an economic growth model for detecting different growth patterns.