Abstract: Bayes and empirical Bayes estimators for the estimation of error variances in a heteroscedastic linear model are proposed. We concentrate primarily on the situation in which only a few replicates are available at each design point but the total number of observations N is relatively large. Some properties of the Bayes estimators are discussed. Asymptotic expansions of the bias and variance of the empirical Bayes estimators are also given. Based on these expansions, the empirical Bayes estimator is shown to have smaller mean squared error than the customary variance estimator, i. e., the within group sample variance, if N is large.