Model Selection in Linear Measurement Error Models

Abstract

We investigate model (variable) selection in linear measurement error model. The focus is on Akaike information criterion (AIC). In normal functional model, the maximum likelihood estimation breaks down. We use the idea of unbiased estimating function to create quasi AIC. For the normal structural model, it is a typical IID parametric model. In principle, all the statistical inferences based on log-likelihood work well. However, we find that the resulting AIC has unusual behavior and it tends to select the simplest model. The problem is traced back to ordinary regression model with random regressor. Surprisingly, there is no literature discussing the AIC in this case. We argue that, using the idea of S-sufficiency, the AIC of the random regressor should be the same as that of fixed regressor. Using the parameter transformation proposed by Gleser (1992), the normal structural model can be transformed into an regression model with random regressor. Because the transformation is one-to- one, then the AIC is invariant. In other words, the AIC of the structural model is the same as that of the fixed regression model (This is joint work with professor Chih-Ling Tsai).