  A family of parsimonious Gaussian cluster-weighted models is presented. This
family concerns a multivariate extension to cluster-weighted modelling that can
account for correlations between multivariate responses. Parsimony is attained
by constraining parts of an eigen-decomposition imposed on the component
covariance matrices. A sufficient condition for identifiability is provided and
an expectation-maximization algorithm is presented for parameter estimation.
Model performance is investigated on both synthetic and classical real data
sets and compared with some popular approaches. Finally, accounting for linear
dependencies in the presence of a linear regression structure is shown to offer
better performance, vis-\`{a}-vis clustering, over existing methodologies.
