Inference for Misspecified Models With Fixed Regressors

Abstract

Following the work by Eicker, Huber, and White it is common in empirical work to report standard errors that are robust against general misspecification. In a regression setting, these standard errors are valid for the parameter that minimizes the squared difference between the conditional expectation and a linear approximation, averaged over the population distribution of the covariates. Here, we discuss an alternative parameter that corresponds to the approximation to the conditional expectation based on minimization of the squared difference averaged over the sample, rather than the population, distribution of the covariates. We argue that in some cases this may be a more interesting parameter. We derive the asymptotic variance for this parameter, which is generally smaller than the Eicker–Huber–White robust variance, and propose a consistent estimator for this asymptotic variance. Supplementary materials for this article are available online.

• Bootstrap
• Conditional inference
• Confidence intervals
• Robust standard errors

SUPPLEMENTARY MATERIALS

The supplementary materials contain the proof of Theorem 2 and Corollary 1 under asymptotic equicontinuity condition and an application to quantile regression.