Smoothed gradient least squares estimator for linear threshold models

In the presence of fixed threshold effects, the least squares (LS) estimator of the threshold parameter poses challenges for statistical inference due to its nonstandard limiting distribution, which also presents challenges for bootstrap methods. To address this issue, we propose a novel estimator: a two-step smoothed gradient least squares (SGLS) estimator. Our proposed method achieves a normal limiting distribution for the threshold parameter with minimal efficiency loss compared to the LS estimator. Furthermore, our modified bootstrap method significantly enhances computational efficiency, leading to improved bootstrap confidence intervals (CIs) for the threshold parameter compared to asymptotic CIs. Our method is validated through a small Monte Carlo study and demonstrated with an empirical application.