This article develops a two-part finite mixture quantile regression model for semi-continuous longitudinal data. The proposed methodology allows heterogeneity sources that influence the model for the binary response variable to also influence the distribution of the positive outcomes. As is common in the quantile regression literature, estimation and inference on the model parameters are based on the asymmetric Laplace distribution. Maximum likelihood estimates are obtained through the EM algorithm without parametric assumptions on the random effects distribution. In addition, a penalized version of the EM algorithm is presented to tackle the problem of variable selection. The proposed statistical method is applied to the well-known RAND Health Insurance Experiment dataset which gives further insights on its empirical behaviour.
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