L1-Regularized MLE and Inference of High-Dimensional Correlated Random Effects Probit

Abstract

We propose a computationally simple two-step procedure for inference in sparse correlated random effects probit and fractional probit models where the number of regressors can exceed the sample size n. In particular, our procedure can be applied to situations where we want to model the unobserved effect with a very large number of approximating terms, which essentially generalizes the Chamberlain's approach. We establish the root-n-asymptotic normality of our second-step estimator for any given finite set of elements in the parameter vector. This result requires neither perfect selection nor sparsity of our first-step estimator, and it does not require the invertibility of the entire population Hessian matrix. We also provide estimators for the Average Partial Effect with respect to any covariate and establish the asymptotic normality of these estimators. Moreover, we provide consistent estimators for the asymptotic variances. To the best of our knowledge, this paper is the first to provide inference results for a high-dimensional (p>n) nonlinear panel data model with a short time series.