Handling initial conditions and endogenous covariates in dynamic/transition models for binary data with unobserved heterogeneity

Distinguishing between longitudinal dependence due to the effects of previous responses on subsequent responses and dependence due to unobserved heterogeneity is important in many disciplines. For example, wheezing is an inflammatory reaction that may 'remodel' a child's airway structure and thereby affect the probability of future wheezing (state dependence). Alternatively, children could vary in their susceptibilities because of unobserved covariates such as genes (unobserved heterogeneity). For binary responses, distinguishing between state dependence and unobserved heterogeneity is typically accomplished by using dynamic/transition models that include both a lagged response and a random intercept. Naive maximum likelihood estimators can be severely inconsistent because of two kinds of endogeneity problem: lack of independence of the initial response and the random intercept (the initial conditions problem) and lack of independence of the covariates and the random intercept (the endogenous covariates problem). We clarify and unify previous work on handling these problems in the disconnected literatures of statistics and econometrics, suggest improved methods, investigate the asymptotic performance of competing methods and provide practical recommendations. The recommended methods are applied to longitudinal data on children's wheezing, where we investigate the extent of state dependence and unobserved heterogeneity and whether there is an effect of maternal smoking.