Models as Approximations: How Random Predictors and Model Violations Invalidate Classical Inference in Regression

We review and interpret 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," that is, under the assumption that models are approximations rather than generative truths. This form of inference, which is pervasive in econometrics, relies on the "sandwich estimator" of standard error. Whereas linear models theory in statistics assumes models to be true and predictors to be fixed, White's theory permits models to be approximate and predictors to be random. Careful reading of his work shows that the deepest consequences for statistical inference arise from a synergy --- a "conspiracy" --- of nonlinearity and randomness of the predictors which invalidates the ancillarity argument that justifies conditioning on the predictors when they are random. Unlike the standard error of linear models theory, the sandwich estimator provides asymptotically correct inference in the presence of both nonlinearity and heteroskedasticity. An asymptotic comparison of the two types of standard error shows that discrepancies between them can be of arbitrary magnitude. If there exist discrepancies, standard errors from linear models theory are usually too liberal even though occasionally they can be too conservative as well. A valid alternative to the sandwich estimator is provided by the "pairs bootstrap"; in fact, the sandwich estimator can be shown to be a limiting case of the pairs bootstrap. We conclude by giving meaning to regression slopes when the linear model is an approximation rather than a truth. --- In this review we limit ourselves to linear least squares regression, but many qualitative insights hold for most forms of regression.