| Regression usually assumes exactly known values for the covariates, with random error in the response only. In some situations the covariates themselves must be estimated using proxy variables and models of instrumental variables. The following study seeks to extend methods for estimating regression parameters and inferential statistics under conditions of longitudinal data when interactions between covariates are involved. Longitudinal data introduces random subject effects and correlated error terms into models for the covariate and the response. Interaction introduce second order terms and cross terms. Standard errors and confidence intervals for the parameters of interest are studied. Substituting instrumental models and back transforming, with some approximations, yields acceptable results in a range of cases. In addition, for some situations a non-parametric surface fit is desired. Use of local likelihood methods is explored for longitudinal data for both normal and count outcomes, and an algorithm is proposed. |
