On the Simultaneous Effects of Model Misspecification and Errors-in-Variables

Misspecified models and noisy covariate measurements are two common sources of bias in statistical inferences. While there is considerable literature on the consequences of each problem in isolation, this article investigates the effect of both problems in tandem. In the context of linear models, the large-sample error in estimating the regression function is partitioned into two terms, one resulting from model misspecification and the other from covariate imprecision. While trivial to establish, this decomposition proves interesting. Specifically, it reveals tradeoffs between the deleterious effects of model misspecification and covariate imprecision in a number of scenarios. A finite-sample version of the decomposition is also presented. This permits study of the relative impacts of model misspecification, covariate imprecision, and sampling variability, with reference to the detectability of the model misspecification via diagnostic plots.