The Conspiracy of Random Predictors and Model Violations against Classical Inference in Regression

We review the early insights of Halbert White who over thirty years ago inaugurated a form of statistical inference for regression models that is asymptotically correct even under “model misspecification.” This form of inference, which is pervasive in econometrics, relies on the “sandwich estimator” or “heteroskedasticity-consistent estimator” of standard error. Whereas common practice in statistics assumes models to be correct and inference to be conditional on the predictors, White permits models to be “incorrect” and predictors to be random. Careful reading of his theory shows that it is in fact a synergistic effect — a “conspiracy” — of model misspecification and randomness of the predictors that has the deepest consequences for statistical inference. In this review we limit ourselves to linear least squares regression as the demonstration object, but the qualitative insights hold for all forms of regression. We will see that the term “heteroskedasticity-consistent estimator” is misleading because nonlinearity is a more consequential form of model deviation than heteroskedasticity, and both forms are handled asymptotically correctly by the sandwich estimator. The same analysis shows that a valid alternative to the sandwich estimator is provided by the “pairs bootstrap.” We continue with a novel asymptotic comparison of the sandwich estimator and the standard error estimator from classical linear models theory. The comparison shows that when standard errors from linear models theory deviate from their sandwich analogs, they are usually too liberal, but occasionally they can be too conservative as well. We conclude by answering questions that would occur to statisticians acculturated to the assumption of model correctness and conditionality on the predictors: (1) Why should we be interested in inference for models that are not correct? (2) What are the arguments for conditioning on predictors, and why might they not be valid?