The Lasso for High-Dimensional Regression with a Possible Change-Point

We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Under a sparsity assumption, we derive nonasymptotic oracle inequalities for both the prediction risk and the l_1 estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Therefore, the Lasso estimator not only selects covariates but also accomplishes model selection between the linear and threshold regression models. Furthermore, we establish conditions under which the unknown threshold parameter can be estimated at a rate of nearly 1/n when the number of regressors can be much larger than the sample size (n). We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.