Two-Stage Regularization of Pseudo-Likelihood Estimators with Application to Time Series

Abstract: Estimators derived from score functions that are not the likelihood are in wide use in practical and modern applications. Their regularization is often carried by pseudo-posterior estimation, equivalently by adding penalty to the score function. We argue that this approach is suboptimal, and propose a two-staged alternative involving estimation of a new score function which better approximates the true likelihood for the purpose of regularization. Our approach typically identifies with maximum a-posteriori estimation if the original score function is in fact the likelihood. We apply our theory to fitting ordinary least squares (OLS) under contemporaneous exogeneity, a setting appearing often in time series and in which OLS is the estimator of choice by practitioners.