Bounding parameters in a linear regression model with a mismeasured regressor using additional information

This paper discusses a linear regression model with a mismeasured regressor in which the measurement error is correlated with both the latent variable and the regression error. We use a linear structure to capture the correlation between the measurement error and the latent variable. This paper shows that the variance of the latent variable is very useful for revealing information on the parameters which otherwise cannot be obtained with such a nonclassical measurement error. The main result is that the finite bounds on the parameters can be found using the variance of the latent variable, regardless of how severely the measurement error and the regression error are correlated, if the mismeasured regressor contains enough information on the latent one. This paper also discusses the special but interesting case of the latent variable being dichotomous. In this case, the mean of the latent variable may even reveal information on the correlation between the measurement error and the regression error. All the bounds developed in the paper are tight.