An Approximate Test for Comparing Heteroscedastic Regression Models

Abstract

This article addresses the problem of testing whether the vectors of regression coefficients are equal for two independent regression models when the error variances are unequal. The usual Chow statistic, appropriate when equality of variances can be assumed, is modified by replacing the pooled residual variance in the denominator with a weighted average of the residual variances from each data set. The weights are functions of the mean of the eigenvalues of W = X′ ₁ X ₁(X′ ₁ X ₁ + X′ ₂ X ₂)^-1. Both numerator and denominator are then approximated by scalar multiples of chi-squared distributions. The parameters of these approximated distributions are chosen to equate their first two moments to those of the exact distribution. The resulting approximation for the modified Chow statistic, C*, is an F distribution with degrees of freedom that depend on the two sample sizes, the number of regressor variables, the average eigenvalue of W, and the true ratio of error variances. Since the latter is unknown, an approximate test is based on the F distribution with random degrees of freedom that result from substituting the mean squared errors of the two regressions for the respective variances. The Type I error probability of this approximation is evaluated by numerical integration. Conditioning on the estimated variance ratio, the probabilities are computed for selected configurations using a method due to J. P. Imhof; then the expectations of these probabilities with respect to the distribution of the estimated variance ratios are presented for comparison with the nominal significance level. Approximations to the power of the test are calculated for some configurations and are presented as a percentage of power of the likelihood ratio test. The results indicate that this test procedure yields a significance level closer to the nominal level than other suggested tests, and it has reasonably sufficient power across a wide range of configurations. Since the test is computationally easy to perform once the eigenvalues of W are determined, this procedure offers an excellent alternative for testing the equality of two vectors of regression coefficients.

Key Words:

• Behrens-Fisher problem
• Chow test
• Heteroscedastic variances
• Satterthwaite approximation
• Validation