Mixture latent autoregressive models for longitudinal data

Many relevant statistical and econometric models for the analysis of longitudinal data include a latent process to account for the unobserved heterogeneity between subjects in a dynamic fashion. Such a process may be continuous (typically an AR(1)) or discrete (typically a Markov chain). In this paper, we propose a model for longitudinal data which is based on a mixture of AR(1) processes with different means and correlation coefficients, but with equal variances. This model belongs to the class of models based on a continuous latent process, and then it has a natural interpretation in many contexts of application, but it is more flexible than other models in this class, reaching a goodness-of-fit similar to that of a discrete latent process model, with a reduced number of parameters. We show how to perform maximum likelihood estimation of the proposed model by the joint use of an Expectation-Maximisation algorithm and a Newton-Raphson algorithm, implemented by means of recursions developed in the hidden Markov literature. We also introduce a simple method to obtain standard errors for the parameter estimates and a criterion to choose the number of mixture components. The proposed approach is illustrated by an application to a longitudinal dataset, coming from the Health and Retirement Study, about self-evaluation of the health status by a sample of subjects. In this application, the response variable is ordinal and time-constant and time-varying individual covariates are available.