Estimation of the scale parameter after a pre-test for homogeneity in a mis-specified regression model

The risk properties of estimators of the scale parameter after a pre-test for homogeneity of the error variances in the two sample linear regression model has received quite an amount of attention in the literature. This literature typically assumes normal disturbances and a properly specified model. In this paper we consider that both equations may be mis-specified by the omission of relevant regressors and that the error distributions may belong to a wider class than the normal distribution. We derive and analyse the exact risk (under quadratic loss) of the pre-test estimator of the scale parameter for the first sub-sample.