We consider a multivariate linear response regression in which the number of responses and predictors is large and comparable with the number of observations, while the number of the model factors is 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 values 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. Finally, we design a consistent estimator for the singular values of the coefficient matrix and derive the asymptotic distribution for this estimator.
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