On the Robustness of the LRT with Respect to Specification Errors in a Linear Model

We consider the linear model$(\mathbf{\mathit{Y}},\mathbf{\mathit{X}}\boldsymbol{\beta},\sigma ^{2}\mathbf{\mathit{I}})$and a set of estimable parametric functionals Aβ. In this paper, we consider alternative linear models which differ from$(\mathbf{\mathit{Y}},\mathbf{\mathit{X}}\boldsymbol{\beta},\sigma ^{2}\mathbf{\mathit{I}})$in the dispersion of the observations or expectation or both and obtain necessary and sufficient conditions for the F-test under$(\mathbf{\mathit{Y}},\mathbf{\mathit{X}}\boldsymbol{\beta},\sigma ^{2}\mathbf{\mathit{I}}$for testing$H_{0}$: Aβ = 0 to be valid under the alternative model also.