Does Regression Approximate the Influence of the Covariates or Just Measurement Errors? A Model Validity Test

Abstract: A criterion is proposed for testing hypothesis about the nature of the error variance in the dependent variable in linear model, which separates correctly and incorrectly specified models. In the former only measurement errors determine the variance (i.e., dependent variable is correctly explained by independent ones, up to measurement errors), while in the latter the model lacks some independent covariates (or has nonlinear structure). The proposed MEM-V (Measurement Error Model Validity) test checks the validity of the model when both dependent and independent covariates are measured with errors. The criterion has asymptotic character, but numerical simulations outlined approximate boundaries where estimates make sense. A practical example of the implementation of the test is discussed in detail; it shows ability of the test to detect wrong specification even in seemingly perfect models. This type of relation between measurement errors and model specification has not been studied before, and the proposed criterion may stimulate future research in this important area.