Some properties of inferences in misspecified linear models

Abstract

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