Canonical correlation analysis and reduced rank regression in autoregressive models

Abstract

When the rank of the autoregression matrix is unrestricted, the maximum likelihood estimator under normality is the least squares estimator. When the rank is restricted, the maximum likelihood estimator is composed of the eigenvectors of the effect covariance matrix in the metric of the error covariance matrix corresponding to the largest eigenvalues [Anderson, T. W. (1951). Ann. Math. Statist. 22 327-351]. The asymptotic distribution of these two covariance matrices under normality is obtained and is used to derive the asymptotic distributions of the eigenvectors and eigenvalues under normality. These asymptotic distributions differ from the asymptotic distributions when the regressors are independent variables. The asymptotic distribution of the reduced rank regression is the asymptotic distribution of the least squares estimator with some restrictions; hence the covariance of the reduced rank regression is smaller than that of the least squares estimator. This result does not depend on normality.