Cramer-Rao Bounds for Misspecified Models

In this paper, we derive some lower bounds of the Cramer-Rao type for the covariance matrix of any unbiased estimator of the pseudo-true parameters in a parametric model that may be misspecified. We obtain some lower bounds when the true distribution belongs either to a parametric model that may differ from the specified parametric model or to the class of all distributions with respect to which the model is regular. As an illustration, we apply our results to the normal linear regression model. In particular, we extend the Gauss-Markov Theorem by showing that the OLS estimator has minimum variance in the entire class of unbiased estimators of the pseudo-true parameters when the mean and the distribution of the errors are both misspecified.