Fast Bootstrap Confidence Intervals for Continuous Threshold Linear Regression

Continuous threshold regression is a common type of nonlinear regression that is attractive to many practitioners for its easy interpretability. More widespread adoption of threshold regression faces two challenges: (i) the computational complexity of fitting threshold regression models and (ii) obtaining correct coverage of confidence intervals under model misspecification. Both challenges result from the nonsmooth and nonconvex nature of the threshold regression model likelihood function. In this article we first show that these two issues together make the ideal approach for making model-robust inference in continuous threshold linear regression an impractical one. The need for a faster way of fitting continuous threshold linear models motivated us to develop a fast grid search method. The new method, based on the simple yet powerful dynamic programming principle, improves the performance by several orders of magnitude. Supplementary materials for this article are available online.