Bounded support in linear random coefficient models: Identification and variable selection

TL;DR

This paper studies the identification of parameters in linear random coefficient models with finite support regressors and proposes a variable selection method using adaptive LASSO to ensure valid covariance estimates.

Contribution

It provides conditions for identifiability of moments in finite support models and demonstrates the variable selection consistency of adaptive LASSO for covariance parameters.

Findings

Identification of means, variances, and covariances from first two moments under support conditions

Adaptive LASSO achieves variable selection consistency in finite and moderate dimensions

Estimated covariance matrices are positive semidefinite and valid

Abstract

We consider linear random coefficient regression models, where the regressors are allowed to have a finite support. First, we investigate identifiability, and show that the means and the variances and covariances of the random coefficients are identified from the first two conditional moments of the response given the covariates if the support of the covariates, excluding the intercept, contains a Cartesian product with at least three points in each coordinate. We also discuss ientification of higher-order mixed moments, as well as partial identification in the presence of a binary regressor. Next we show the variable selection consistency of the adaptive LASSO for the variances and covariances of the random coefficients in finite and moderately high dimensions. This implies that the estimated covariance matrix will actually be positive semidefinite and hence a valid covariance matrix, in contrast to the estimate arising from a simple least squares fit. We illustrate the proposed method in a simulation study.