Uniform Inference in High-Dimensional Threshold Regression Models

Abstract: We develop uniform inference for high-dimensional threshold regression parameters, allowing for either cross-sectional or time series data. We first establish Oracle inequalities for prediction errors and $\ell_1$ estimation errors for the Lasso estimator of the slope parameters and the threshold parameter, accommodating heteroskedastic non-subgaussian error terms and non-subgaussian covariates. Next, we derive the asymptotic distribution of tests involving an increasing number of slope parameters by debiasing (or desparsifying) the Lasso estimator in cases with no threshold effect and with a fixed threshold effect. We show that the asymptotic distributions in both cases are the same, allowing us to perform uniform inference without specifying whether the true model is a linear or threshold regression. Finally, we demonstrate the consistent performance of our estimator in both cases through simulation studies, and we apply the proposed estimator to analyze two empirical applications.