  We propose a M-quantile regression model for the analysis of multivariate,
continuous, longitudinal data. M-quantile regression represents an appealing
alternative to standard regression models, as it combines the robustness of
quantile and the efficiency of expectile regression, providing a complete
picture of the response variable distribution. Discrete, individual-specific,
random parameters are used to account for both dependence within the same
response recorded at different times and association between different
responses observed on the same sample unit at a given time. A suitable
parametrisation is also introduced in the linear predictor to account for
possible dependence between the individual specific random parameters and the
vector of observed covariates, that is to account for endogeneity of some
covariates. An extended EM algorithm is proposed to derive model parameter
estimates under a maximum likelihood approach. The model is applied to the
analysis of the strengths and difficulties questionnaire scores from the
Millennium Cohort Study in the UK.
