A two-stage Generalized Method of Moments model for feedback with time-dependent covariates

Correlated observations in longitudinal studies are often due to repeated measures on the subjects. Additionally, correlation may be realized due to the association between responses at a particular time and the predictors at earlier times. There are also feedback effects (relation between responses in the present and the covariates at a later time), though these are not always relevant and are often ignored. All these cases of correlation must be accounted for as they can have different effects on the regression coefficients. Several authors have provided models that reflect the direct and delayed impact of covariates on the response, utilizing valid moment conditions to estimate the relevant regression coefficients. However, there are applications when one cannot ignore the effect of the responses on future covariates. A two-stage model to account for the feedback, modeling the direct as well as the delayed effects of the covariates on future responses and vice versa is presented. The use of the two-stage model is demonstrated by revisiting child morbidity and its impact on future values of body mass index using Philippines health data. Also, obesity status and its feedback effects on physical activity and depression levels using the Add Health dataset are analyzed.