Fixed and Random Covariance Regression Analyses

TL;DR

This paper develops a new theoretical framework for covariance regression analysis considering random explanatory variables, establishing estimator properties and model assessment methods, with applications to stock return data.

Contribution

It introduces the first comprehensive theory for Random-$X$ covariance regression, including estimator properties and bias-variance analysis for model assessment.

Findings

Both quasi-maximum likelihood and weighted least squares estimators are consistent and asymptotically normal.

Moving from Fixed-$X$ to Random-$X$ increases bias and variance in test errors.

Proposed estimators effectively assess model performance in simulations and empirical stock return data.

Abstract

Covariance regression analysis is an approach to linking the covariance of responses to a set of explanatory variables $X$, where $X$ can be a vector, matrix, or tensor. Most of the literature on this topic focuses on the "Fixed-$X$" setting and treats $X$ as nonrandom. By contrast, treating explanatory variables $X$ as random, namely the "Random-$X$" setting, is often more realistic in practice. This article aims to fill this gap in the literature on the estimation and model assessment theory for Random-$X$ covariance regression models. Specifically, we construct a new theoretical framework for studying the covariance estimators under the Random-$X$ setting, and we demonstrate that the quasi-maximum likelihood estimator and the weighted least squares estimator are both consistent and asymptotically normal. In addition, we develop pioneering work on the model assessment theory of covariance regression. In particular, we obtain the bias-variance decompositions for the expected test errors under both the Fixed-$X$ and Random-$X$ settings. We show that moving from a Fixed-$X$ to a Random-$X$ setting can increase both the bias and the variance in expected test errors. Subsequently, we propose estimators of the expected test errors under the Fixed-$X$ and Random-$X$ settings, which can be used to assess the performance of the competing covariance regression models. The proposed estimation and model assessment approaches are illustrated via extensive simulation experiments and an empirical study of stock returns in the US market.