  The vast majority of market impact studies assess each product individually,
and the interactions between the different order flows are disregarded. This
strong approximation may lead to an underestimation of trading costs and
possible contagion effects. Transactions in fact mediate a significant part of
the correlation between different instruments. In turn, liquidity shares the
sectorial structure of market correlations, which can be encoded as a set of
eigenvalues and eigenvectors. We introduce a multivariate linear propagator
model that successfully describes such a structure, and accounts for a
significant fraction of the covariance of stock returns. We dissect the various
dynamical mechanisms that contribute to the joint dynamics of assets. We also
define two simplified models with substantially less parameters in order to
reduce overfitting, and show that they have superior out-of-sample performance.
