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

Christophe Gaillac (1, 2) , Eric Gautier (1, 3)

1 TSE-R - Toulouse School of Economics
2 CREST - Centre de Recherche en Économie et Statistique
3 UT Capitole - Université Toulouse Capitole

We consider a linear model where the coefficients - intercept and slopes - are random with a law in a nonparametric class and independent from the regressors. Identification often requires the regressors to have a support which is the whole space. This is hardly ever the case in practice. Alternatively, the coefficients can have a compact support but 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 (not the intercept) do not have heavy-tails. Lower bounds on the supremum risk for the estimation of the joint density of the random coefficients density are obtained for a wide range of smoothness, where some allow for polynomial and nearly parametric rates of convergence. We present a minimax optimal estimator, a data-driven rule for adaptive estimation, and made available a R package.

Mots clés: Minimax; Random Coefficients; Adaptation; Ill-posed Inverse Problem

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Economies et finances; Méthodes et statistiques

Christophe Gaillac, Eric Gautier. Adaptive estimation in the linear random coefficients model when regressors have limited variation. 2020. ⟨hal-02130472v4⟩