Omitted covariates bias and finite mixtures of regression models for longitudinal responses

Individual-specific, time-constant, random effects are often used to model dependence and/or to account for omitted covariates in regression models for longitudinal responses. Longitudinal studies have known a huge and widespread use in the last few years as they allow to distinguish between so-called age and cohort effects; these relate to differences that can be observed at the beginning of the study and stay persistent through time, and changes in the response that are due to the temporal dynamics in the observed covariates. While there is a clear and general agreement on this purpose, the random effect approach has been frequently criticized for not being robust to the presence of correlation between the observed (i.e. covariates) and the unobserved (i.e. random effects) heterogeneity. Starting from the so-called correlated effect approach, we argue that the random effect approach may be parametrized to account for potential correlation between observables and unobservables. Specifically, when the random effect distribution is estimated non-parametrically using a discrete distribution on finite number of locations, a further, more general, solution is developed. This is illustrated via a large scale simulation study and the analysis of a benchmark dataset.