Adaptive estimation in the linear random coefficients model when regressors have limited variation

Abstract

We consider a linear model where the coefficients – intercept and slopes – are random and independent from the regressors. The law of the coefficients is nonparametric. Without further restriction, nonparametric identification requires the regressors to have a support which is the whole space. This is hardly ever the case in practice. It is possible to handle regressors with limited variation when the coefficients can have a compact support. This is not compatible with unbounded error terms as usual in regression models. In this paper, the regressors can have a support which is a proper subset but the slopes do not have heavy-tails. Lower bounds on the minimax risk for the estimation of the joint density of the random coefficients density are obtained for a wide range of smoothness. Some allow for polynomial and nearly parametric rates of convergence. We present a minimax optimal estimator and a data-driven rule for adaptive estimation. A $\mathsf{R}$ package is available to implement this estimator.