  We consider a multivariate linear response regression in which the number of
responses and predictors is large and comparable with the number of
observations, and the rank of the matrix of regression coefficients is assumed
to be small. We study the distribution of singular values for the matrix of
regression coefficients and for the matrix of predicted responses. For both
matrices, it is found that the limit distribution of the largest singular value
is a rescaling of the Tracy-Widom distribution. Based on this result, we
suggest algorithms for the model rank selection and compare them with the
algorithm suggested by Bunea, She and Wegkamp. Next, we design two consistent
estimators for the singular values of the coefficient matrix, compare them, and
derive the asymptotic distribution for one of these estimators..
