17 - On the So-Called “Huber Sandwich Estimator” and “Robust Standard Errors”

Summary

Abstract. The “Huber Sandwich Estimator” can be used to estimate the variance of the MLE when the underlying model is incorrect. If the model is nearly correct, so are the usual standard errors, and robustification is unlikely to help much. On the other hand, if the model is seriously in error, the sandwich may help on the variance side, but the parameters being estimated by the MLE are likely to be meaningless–except perhaps as descriptive statistics.

Introduction

This chapter gives an informal account of the so-called “Huber Sandwich Estimator,” for which Peter Huber is not to be blamed. We discuss the algorithm and mention some of the ways in which it is applied. Although the chapter is mainly expository, the theoretical framework outlined here may have some elements of novelty. In brief, under rather stringent conditions the algorithm can be used to estimate the variance of the MLE when the underlying model is incorrect. However, the algorithm ignores bias, which may be appreciable. Thus, results are liable to be misleading.

To begin the mathematical exposition, let i index observations whose values are y_i. Let θ ∈ R^p be a p × 1 parameter vector. Let y → f_i (y|θ) be a positive density. If y_i takes only the values 0 or 1, which is the chief case of interest here, then f_i (0|θ) > 0, f_i (1|θ) > 0, and f_i (0|θ) + f_i (0|θ) + f_i (1|θ) = 1. Some examples involve real- or vector-valued y_i, and the notation is set up in terms of integrals rather than sums.