Estimating the error variance in regression after a preliminary test of restrictions on the coefficients
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Maximum-likelihood estimation of the variance of the disturbances in a linear regression is considered in the context of exact linear restrictions on the coefficient vector. The exact risk function for the associated preliminary-test estimator of this variance is derived and evaluated numerically on the basis of relative quadratic loss. Our results indicate that in practical situations the risk of the pre-test estimator differs only slightly from that of the naive estimator which ignores the restrictions without testing. Applying the coefficient restrictions without testing their validity is not recommended.