  In this paper, we propose a novel variable selection approach in the
framework of multivariate linear models taking into account the dependence that
may exist between the responses. It consists in estimating beforehand the
covariance matrix of the responses and to plug this estimator in a Lasso
criterion, in order to obtain a sparse estimator of the coefficient matrix. The
properties of our approach are investigated both from a theoretical and a
numerical point of view. More precisely, we give general conditions that the
estimators of the covariance matrix and its inverse have to satisfy in order to
recover the positions of the null and non null entries of the coefficient
matrix when the size of the covariance matrix is not fixed and can tend to
infinity. We prove that these conditions are satisfied in the particular case
of some Toeplitz matrices. Our approach is implemented in the R package
MultiVarSel available from the Comprehensive R Archive Network (CRAN) and is
very attractive since it benefits from a low computational load. We also assess
the performance of our methodology using synthetic data and compare it with
alternative approaches. Our numerical experiments show that including the
estimation of the covariance matrix in the Lasso criterion dramatically
improves the variable selection performance in many cases.
