Asymptotic expansion and asymptotic robustness of the normal-theory estimators in the random regression model
Asymptotic expansions of the distributions of the sample regression coefficients and residual variance in the multiple regression model with random independent variables are derived under normality and non-normality using Edgeworth expansions up to order O(1/n). It is shown that the asymptotic variances and biases of the sample regression coefficients, and the asymptotic bias of the sample residual variance with the assumption of normality have asymptotic robustness against the violation of the assumption when the explanatory variables are independently distributed with the residual. The results of the simulations show, in finite samples, the accuracy of the asymptotic cumulants and higher-order asymptotic variances used in the asymptotic expansions and illustrate the asymptotic robustness described above. The simulations also indicate that the usual normal approximations to the distributions of the sample regression coefficients are fairly good, when non-normality is not excessive and the normal approximation to the distribution of the sample residual variance is not satisfactory.
asymptotic robustness random regression model residual variance non-normality edgeworth expansion covariance structure