A Degrees-of-freedom Approximation for a t-statistic with Heterogeneous Variance

Abstract

Ordinary least squares is a commonly used method for obtaining parameter estimates for a linear model. When the variance is heterogeneous, the ordinary least squares estimate is unbiased, but the ordinary least squares variance estimate may be biased. Various jackknife estimators of variance have been shown to be consistent for the asymptotic variance of the ordinary least squares estimator of the parameters. In small samples, simulations have shown that the coverage probabilities of confidence intervals using the jackknife variance estimators are often low. We propose a finite sample degrees-of-freedom approximation to use with the t-distribution, which, in simulations, appears to raise the coverage probabilities closer to the nominal level. We also discuss an ‘estimated generalized least squares estimator’, in which we model and estimate the variance, and use the estimated variance in a weighted regression. Even when the variance is modelled correctly, the coverage can be low in small samples, and, in terms of the coverage probability, one appears better off using ordinary least squares with a robust variance and our degrees-of-freedom approximation. An example with real data is given to demonstrate how potentially misleading conclusions may arise when the degrees-of-freedom approximation is not used.