Some properties of inferences in misspecified linear models

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 inferences based on the model μ = Xβ are considered without assuming that the model holds. It is shown that many of the usual properties continue to 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.