The Annals of Statistics

Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model

Takeaki Kariya and Hiroshi Kurata

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Abstract

In a general normal regression model, this paper first derives the least upper bound (LUB) for the covariance matrix of a generalized least squares estimator (GLSE) relative to the covariance matrix of the Gauss-Markov estimator. Second the result is applied to the (unrestricted) Zellner estimator in an N-equation seemingly unrelated regression (SUR) model and to the GLSE in a heteroscedastic model.

Article information

Source
Ann. Statist., Volume 24, Number 4 (1996), 1547-1559.

Dates
First available in Project Euclid: 17 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032298283

Digital Object Identifier
doi:10.1214/aos/1032298283

Mathematical Reviews number (MathSciNet)
MR1416648

Zentralblatt MATH identifier
0868.62060

Subjects
Primary: 62J05: Linear regression
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Nonlinear Gauss-Markov theorem efficiency of GLSE seemingly unrelated equation heteroscedastic model Kantorovich inequality

Citation

Kurata, Hiroshi; Kariya, Takeaki. Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model. Ann. Statist. 24 (1996), no. 4, 1547--1559. doi:10.1214/aos/1032298283. https://projecteuclid.org/euclid.aos/1032298283


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References

  • ANDERSON, T. W. 1971. The Statistical Analy sis of Time Series. Wiley, New York. Z.
  • BILODEAU, M. 1990. On the choice of the loss function in covariance estimation. Statist. Decisions 8 131 139. Z.
  • KARIy A, T. 1981. Bounds for the covariance matrices of Zellner's estimator in the SUR model and the 2SAE in a heteroscedastic model. J. Amer. Statist. Assoc. 76 975 979. Z.
  • KARIy A, T. and TOy OOKA, Y. 1985. Nonlinear versions of the Gauss Markov theorem and GLSE. In Multivariate Analy sis 6 345 354. North-Holland, Amsterdam. Z.
  • KHATRI, C. G. and SRIVASTAVA, M. S. 1971. On exact non-null distributions of likelihood ratio criteria for sphericity test and equality of two covariance matrices. Sankhy a Ser. A 33 201 206. Z.
  • REVANKAR, N. S. 1974. Some finite sample results in the context of two seemingly unrelated regression equations. J. Amer. Statist. Assoc. 69 187 190. Z.
  • SUGIy AMA, T. 1966. On the distribution of the largest latent root and the corresponding latent vector for principal component analysis. Ann. Math. Statist. 37 995 1001. Z.
  • SUGIy AMA, T. 1970. Joint distribution of the extreme roots of a covariance matrix. Ann. Math. Statist. 41 655 657.Z.
  • TOy OOKA, Y. and KARIy A, T. 1986. An approach to upper bound problems for risks of the generalized least squares estimators. Ann. Statist. 14 679 690. Z.
  • ZELLNER, A. 1962. An efficient method of estimating seemingly unrelated regression and tests for aggregation bias. J. Amer. Statist. Assoc. 57 348 368. Z.
  • ZELLNER, A. 1963. Estimators for seemingly unrelated regressions: some exact finite sample results. J. Amer. Statist. Assoc. 58 977 992.
  • KUNITACHI, TOKy O, 186 YAMAGUCHI-SHI, YAMAGUCHI-KEN JAPAN JAPAN E-MAIL: cr00055@srv.cc.hit-u.ac.jp