Scalable Nonparametric Change-Plane Regression with Random Features: Estimation and Inference

Abstract

Threshold and change-plane models provide a useful framework for capturing regime-dependent dynamics, but most existing approaches typically rely on scalar thresholds or linear splitting boundaries. We propose a scalable semiparametric change-plane regression framework that allows the regime boundary to depend nonlinearly on multivariate state variables. Random Fourier features provide a flexible and computationally efficient approximation to the nonlinear boundary function. A two-stage procedure first estimates the induced regime classification using a smooth surrogate and then refits the structural parameters by hard-threshold OLS. We establish convergence of the first-stage regime classification and show that its estimation error is asymptotically negligible for the second-stage refit, yielding oracle equivalence. We further develop a dependent-multiplier bootstrap Sup-Wald test for detecting the existence of regime effects when the boundary is unidentified under the null, and establish its asymptotic validity. Simulations demonstrate accurate regime recovery, near-oracle performance of the proposed estimator, and satisfactory finite-sample behavior of the Sup-Wald test. An application to postwar U.S. unemployment data provides strong evidence of regime dependence, while the estimated nonlinear change plane identifies a distinct stress regime characterized by significant shifts in short-run unemployment dynamics.