Linear regression under model uncertainty

TL;DR

This paper explores linear regression when faced with covariate missingness and unknown, changing measurement error variance, using sublinear expectation theory to model uncertainty and develop robust estimators.

Contribution

It introduces a novel framework employing sublinear expectation to handle mean and variance uncertainty in linear regression, providing consistent estimators under mild conditions.

Findings

Developed a family of estimators for uncertain regression parameters.

Proved estimator consistency under mild data conditions.

Applied the approach to forecasting the S&P Index.

Abstract

We reexamine the classical linear regression model when the model is subject to two types of uncertainty: (i) some of covariates are either missing or completely inaccessible, and (ii) the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician. By following the recent theory of sublinear expectation, we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables, which encompass an infinite family of probability distributions for the response variable in the sense of (linear) classical probability theory. The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty. The consistency of the estimators is established under mild conditions on the data generation process. Three applications are introduced to assess the quality of the approach including a forecasting model for the S\&P Index.