On the singular values of the reduced-rank multivariate response regression

We consider a multivariate linear response regression in which the number of responses and predictors is large and comparable with the number of observations. The number of the model factors is assumed to be small. First, we study the distribution of singular values for the matrices of regression coefficients and predicted responses. We find that in both cases the distribution of the largest singular value is related to the Tracy-Widom distribution. Based on this result, we suggest algorithms for the model rank selection and compare it 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.