A validity test for a multivariate linear measurement error model

Abstract

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