Linear-quadratic quantile regression model with a change point due to a threshold covariate

This article proposes a novel quantile regression model, called the linear-quadratic quantile regression model, which can capture the nonlinear effect of a covariate on the response variable. It is actually a two-segmented quantile regression model, in the sense that the linear term associated with a continuous exposure in standard quantile regression is replaced by a linear term and a quadratic term below and above an unknown change point, respectively. A two-step estimation procedure is proposed to estimate the regression coefficients and the change point at a given quantile level. The asymptotic properties of the proposed estimator are derived by the empirical process theory, and the change point estimator is shown to achieve standard root-n consistency. A sup-Wald test is developed to test the existence of a change-point at a given quantile level. Furthermore, estimation and inference procedures for linear-quadratic quantile regression across multiple quantile indices are presented. Particularly, the test and inference for the common threshold across different quantile indices are proposed. Monte Carlo simulations and an empirical application to GDP per capita illustrate the practical usefulness of the proposed method.