Underestimation of mean square error matrix in misspecified linear models

In this paper a necessary and sufficient condition is derived for the mean square error matrix to be overestimated on average by the covariance matrix of the OLS estimator when the linear model is misspecified by omitting at least one independent variable. A sufficient condition for this is also considered and the situation is illustrated by simple examples showing, as previously noted by Gupta and Maasoumi (1979), that underestimation is the prevailing case in practice.