Bayesian Interpretations of Heteroskedastic Consistent Covariance Estimators Using the Informed Bayesian Bootstrap

Abstract

This article provides Bayesian interpretations for White's heteroskedastic consistent (HC) covariance estimator, and various modifications of it, in linear regression models. An informed Bayesian bootstrap provides a useful framework.

Keywords:

• Dirichlet
• Multinomial
• Semiparametric

JEL Classification:

• C1
• C11

ACKNOWLEDGMENT

I would like to thank two anonymous referees for their useful comments on an earlier draft.

Notes

¹Chamberlain and Imbens (2003, p. 12) note that because J can be arbitrarily large and our data are measured with finite precision, the finite support assumption is arguably not restrictive.

²Lancaster (2003) notes that β also has the WLS representation β = β(g) = [X′ diag {g}X]⁻¹ X′ diag{g}y. This leads to a slightly different approximation because the expansion in the next section is around θ not g. The effect is minimal: the first factor in (7b) becomes unity.

³Gilstein and Leamer (1983) characterize the set of WLS estimates as the union of the bounded convex polytopes formed by the hyperplanes on which residuals are zero.

⁴See Chesher and Jewitt (1987), Cribari-Neto et al. (2000), Cribari-Neto and Zarkos (1999, 2001), Godfrey (2006), and MacKinnon and White (1985).

Note: , (j = 1, 2,…, m), (j = m + 1, m + 2,…, J), , and .

Note: Only first 25 observations of Godfrey (2006) are used and these are replicated four times. MacKinnon and White (1985) data set doubles their initial 50 observations.

Note: Based on 10,000 replications of data sets with 100 observations.