| Measurement error (ME) models are used in situations where at least one independent variable in the model is imprecisely measured. Having at least one independent variable measured with error leads to an unidentified model and a bias in the naive estimate of the effect of the variable that is measured with error. One way to correct these problems is through the use of an instrumental variable (IV). An IV is one that is correlated with the unknown, or latent, true variable, but uncorrelated with the measurement error of the unknown truth and the model error. An IV provides the identifying information in our method of estimating the parameters for generalized simple measurement error (GSME) models. The GSME model is developed and it is shown how many well studied ME models with one predictor can fit into its framework. Included in these are linear, generalized linear, nonlinear, multinomial, multivariate regression, and other ME models. The GSME model, by design, can handle situation for continuous, discrete, and categorical observable, or manifest, variables. We provide theorems that give conditions under which the GSME model is identified. The initial step in our estimation method is to "categorize" all continuous and discrete variables. Categorical variables remain unchanged. Assuming conditional independence given the latent variable, the joint distribution of the categorized manifest variables and any that were already categorical is the product of the conditional cell probabilities and conditional distributions of the categorized continuous and discrete manifest variables summing over the categorical values of the latent variable. Maximum likelihood estimates of the joint categorical distribution are used to solve nonlinear equations for the parameters of interest which enter through the conditional probabilities. Estimated generalized nonlinear least squares is used to solve the equations for the parameters of interest. We show that our estimators have favorable asymptotic properties and develop methods of inference for them. We show how many commonly studied ME model problems fit into the general framework developed and how they can be solved using our method. |
