The Annals of Statistics

Asymptotic distribution of the reduced rank regression estimator under general conditions

T. W. Anderson

Full-text: Open access


In the regression model $\mathbf{Y} = \eta + \mathbf{BX} + \mathbf{Z}$ with $\mathbf{Z}$ unobserved, $\mathscr{E}\mathbf{Z} = \mathbf{0}$ and $\mathscr{E}\mathbf{ZZ}' = \mathbf{\Sigma}_{ZZ}$, the least squares estimator of $\mathbf{B}$ is $\hat{\mathbf{B}} = \mathbf{S}_{YX}\mathbf{S}_{XX}^{-1}$. If the rank of $\mathbf{B}$ is known to be $k$ less than the dimensions of $\mathbf{Y}$ and $\mathbf{X}$, the reduced rank regression estimator of $\mathbf{B}$, say $\mathbf{B}_k$, depends on the first $k$ canonical variates of $\mathbf{Y}$ and $\mathbf{X}$. The asymptotic distribution of $\hat{\mathbf{B}}_k$ is obtained and compared with the asymptotic distribution of $\hat{\mathbf{B}}$. The advantage of $\hat{\mathbf{B}}_k$ is characterized.

Article information

Ann. Statist., Volume 27, Number 4 (1999), 1141-1154.

First available in Project Euclid: 4 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H10: Distribution of statistics 62E20: Asymptotic distribution theory
Secondary: 62H12: Estimation

Canonical variates reduced rank regression maximum likelihood estimators


Anderson, T. W. Asymptotic distribution of the reduced rank regression estimator under general conditions. Ann. Statist. 27 (1999), no. 4, 1141--1154. doi:10.1214/aos/1017938918.

Export citation


  • AHN, S. K. and REINSEL, G. C. 1988. Nested reduced-rank autoregressive models for multiple time series. J. Amer. Statist. Assoc. 83 849 856. Z.
  • ANDERSON, T. W. 1951a. Estimating linear restrictions on regression coefficients for multivar
  • ANDERSON, T. W. 1951b. The asymptotic distribution of certain characteristic roots and vectors. Proc. Second Berkeley Symp. Math. Statist. Probab. 103 130 Univ. California Press, Berkeley. Z.
  • ANDERSON, T. W. 1984. An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York. Z.
  • ANDERSON, T. W. 1999. Asymptotic theory for canonical correlation analysis. J. Multivariate Anal. 70 1 29. Z.
  • BRILLINGER, D. R. 1975. Time Series: Data Analysis and Theory. Holt, Rinehart and Winston, New York.
  • IZENMAN, A. J. 1975. Reduced-rank regression for the multivariate linear model. J. Multivariate Anal. 5 248 264. Z.
  • JOHANSEN, S. 1995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford Univ. Press. Z.
  • LUTKEPOHL, H. 1993. Introduction to Multiple Time Series Analysis. Springer, New York. ¨ Z.
  • REINSEL, G. C. and VELU, R. P. 1998. Multivariate Reduced-rank Regression. Springer, New York. Z.
  • RYAN, D. A. J., HUBERT, J. J., CARTER, E. M., SPRAGUE, J. B. and PARROTT, J. 1992. A reduced-rank multivariate regression approach to joint toxicity experiments. Biometrics 48 155 162. Z.
  • SCHMIDLI, H. 1996. Reduced-rank Regression. Physica, Berlin. Z.
  • STOICA, P. and VIBERG, M. 1996. Maximum likelihood parameter and rank estimation in reduced-rank multivariate linear regressions. IEEE Trans. Signal Processing 44 3069 3078. Z.
  • TSAY, R. S. and TIAO, G. C. 1985. Use of canonical analysis in time series model identification. Biometrika 72 299 315. Z.
  • VELU, R. P., REINSEL, G. C. and WICHERN, D. W. 1986. Reduced rank models for multiple times series. Biometrika 73 105 118.