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

Longitudinal studies have known widespread use in the last years in several fields of research, as they allow to distinguish between different sources of variation. We may observe differences at the beginning of the study that stay persistent through time, and changes in the response that are due to temporal dynamics in the observed covariates. Individual-specific, time-constant, effects are often included in the linear predictor to allow for unobserved individual-specific, time constant, heterogeneity motivated by omitted individual features. The random effect approach to estimation is based on considering such effects as random variables, usually with a specific parametric distribution. This approach has been frequently criticized, as it is often employed not considering correlation between observed (i.e., covariates) and unobserved (i.e., random effects) terms. To solve this issue, we may explicitly account for correlation between observed and unobserved heterogeneity, using the so-called correlated effects approach. In this article, we show that a more general solution may be developed by estimating the random effect conditional distribution non-parametrically via a discrete probability distribution on a finite number of locations. The approach we propose is assessed via a large-scale simulation study and illustrated by the analysis of a benchmark dataset.