  The identification of factors associated with mental and behavioral disorders
in early childhood is critical both for psychopathology research and the
support of primary health care practices. Motivated by the Millennium Cohort
Study, in this paper we study the effect of a comprehensive set of covariates
on children's emotional and behavioural trajectories in England. To this end,
we develop a Quantile Mixed Hidden Markov Model for joint estimation of
multiple quantiles in a linear regression setting for multivariate longitudinal
data. The novelty of the proposed approach is based on the Multivariate
Asymmetric Laplace distribution which allows to jointly estimate the quantiles
of the univariate conditional distributions of a multivariate response,
accounting for possible correlation between the outcomes. Sources of unobserved
heterogeneity and serial dependency due to repeated measures are modeled
through the introduction of individual-specific, time-constant random
coefficients and time-varying parameters evolving over time with a Markovian
structure, respectively. The inferential approach is carried out through the
construction of a suitable Expectation-Maximization algorithm without
parametric assumptions on the random effects distribution.
