Inference in models with omitted covariates: Cramér-type moderate deviations and applications to high-dimensional regression

Abstract

We study a score statistic for inference on an interest parameter in a linear model with omitted covariates, establishing Berry–Esseen and Cramér-type moderate deviation bounds on the associated normal approximation. This entails a coupling between well-behaved but unobservable random variables and observable ones to which standard results do not straightforwardly apply. The theory is of self-standing interest but also provides new insights on backwards reduction procedures used in high-dimensional regression. An example details how our results may be used to analyse the high-dimensional procedure proposed by Cox and Battey (Proceedings of the National Academy of Sciences, 114, 8592–8595, 2017).