Heteroskedasticity–robust tests with minimum size distortion

ABSTRACT

Asymptotically valid inference in linear regression models is easily achieved under mild conditions using the well-known Eicker–White heteroskedasticity–robust covariance matrix estimator or one of its variant. In finite sample however, such inferences can suffer from substantial size distortion. Indeed, it is well established in the literature that the finite sample accuracy of a test may depend on which variant of the Eicker–White estimator is used, on the underlying data generating process (DGP) and on the desired level of the test.

This paper develops a new variant of the Eicker–White estimator which explicitly aims to minimize the finite sample null error in rejection probability (ERP) of the test. This is made possible by selecting the transformation of the squared residuals which results in the smallest possible ERP through a numerical algorithm based on the wild bootstrap. Monte Carlo evidence indicates that this new procedure achieves a level of robustness to the DGP, sample size and nominal testing level unequaled by any other Eicker–White estimator based asymptotic test.

KEYWORDS:

• Heteroskedasticity–robust inference
• size distortion
• wild bootstrap.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G35

Notes

1 We classify as “asymptotic” any inference method that uses critical values taken from an analytical asymptotic approximation to a given statistics’ distribution.

2 Results of tests using HC0 and HC1 are omitted to save space and because they overreject in every case considered.

3 Simulations carried-out with DGPs 2, 3, and 4 yielded very similar results.

4 In DGP 1, a change of γ affects the severity of the heteroskedasticity without affecting the leverage of each observation because the values taken by the elements of the matrix X do not depend on γ. Thus, moving from the left to the right of Figure 1 means increasing heteroskedasticity while keeping the regression design fixed. For instance, the ratio of the maximum to the minimum σ_i is 9.28 with γ = 1 and 102.75 with γ = 2.

5 Similar results were obtained for tests at 1% and 10% significance levels. They are not reported to save space.