Some properties of inferences in misspecified linear models

Let Y denote an n x 1 vector of observations such that Y = [mu] + [sigma][var epsilon] where [mu] is an unknown n x 1 vector, [sigma] > 0 is an unknown parameter, and [var epsilon] is an n x 1 vector of independent standard normal random variables. A linear regression analysis is often based on a model for [mu] such as [mu] =X[beta] where X is a known n x p matrix of independent variables and [beta] is a p x 1 vector of unknown parameters. When the assumption that [mu] = X[beta] for some [beta] holds, the results of the analysis can be interpreted as applying to [mu], the mean of Y. In this paper, the properties of interferences based on the model hold, although with respect to [mu]*, the vector of form X[beta] closest to [mu], rather than with respect to [mu]. Hence, the results of a linear regression analysis have a certain type of validity that applies whether or not the model is correctly specified.