Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T16:54:37.214Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE POWER OF INVARIANT TESTS FOR HYPOTHESES ON A COVARIANCE MATRIX

Published online by Cambridge University Press:  14 December 2015

David Preinerstorfer  and
Benedikt M. Pötscher
David Preinerstorfer*
Affiliation:
University of Vienna
Benedikt M. Pötscher
Affiliation:
University of Vienna
*
*Address correspondence to David Preinerstorfer, Department of Statistics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The behavior of the power function of autocorrelation tests such as the Durbin–Watson test in time series regressions or the Cliff-Ord test in spatial regression models has been intensively studied in the literature. When the correlation becomes strong, Krämer (1985, Journal of Econometrics 28, 363–370.) (for the Durbin–Watson test) and Krämer (2005, Journal of Statistical Planning and Inference, 128, 489–496) (for the Cliff-Ord test) have shown that power can be very low, in fact can converge to zero, under certain circumstances. Motivated by these results, Martellosio (2010, Econometric Theory, 26, 152–186) set out to build a general theory that would explain these findings. Unfortunately, Martellosio (2010) does not achieve this goal, as a substantial portion of his results and proofs suffer from nontrivial flaws. The present paper now builds a theory as envisioned in Martellosio (2010) in an even more general framework, covering general invariant tests of a hypothesis on the disturbance covariance matrix in a linear regression model. The general results are then specialized to testing for spatial correlation and to autocorrelation testing in time series regression models. We also characterize the situation where the null and the alternative hypothesis are indistinguishable by invariant tests.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 1 , February 2017 , pp. 1 - 68
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank Michael Jansson, Federico Martellosio, Peter Phillips, and two referees for helpful comments on earlier versions of the paper.

References

REFERENCES

Anselin, L. (2001) Spatial econometrics. In A companion to theoretical econometrics. Blackwell Companions Contemporary Economics, 310–330.Google Scholar
Arnold, S.F. (1979) Linear models with exchangeably distributed errors. Journal of the American Statistical Association, 74, 194–199.CrossRefGoogle Scholar
Bartels, R. (1992) On the power function of the Durbin-Watson test. Journal of Econometrics, 51, 101–112.CrossRefGoogle Scholar
Bauer, H. (2001) Measure and integration theory, de Gruyter Studies in Mathematics, vol. 26. Walter de Gruyter.CrossRefGoogle Scholar
Cambanis, S., Huang, S., & Simons, G. (1981) On the theory of elliptically contoured distributions. Journal of Multivariate Analysis, 11, 368–385.Google Scholar
Gantmacher, F.R. (1959) The theory of matrices, vol. 2. AMS Chelsea Publishing.Google Scholar
Horn, R.A. & Johnson, C.R. (1985) Matrix analysis. Cambridge University Press.Google Scholar
Kadiyala, K.R. (1970) Testing for the independence of regression disturbances. Econometrica, 38, 97–117.CrossRefGoogle Scholar
Kariya, T. (1980) Note on a condition for equality of sample variances in a linear model. Journal of the American Statistical Association, 75, 701–703.CrossRefGoogle Scholar
King, M.L. (1985) A point optimal test for autoregressive disturbances. Journal of Econometrics, 27, 21–37.CrossRefGoogle Scholar
King, M.L. (1987) Testing for autocorrelation in linear regression models: A survey. In Specification analysis in the linear model. International Library of Economics, Routledge & Kegan Paul, 19–73.Google Scholar
King, M.L. & Hillier, G.H. (1985) Locally best invariant tests of the error covariance matrix of the linear regression model. Journal of the Royal Statistical Society. Series B (Methodological), 47, 98–102.CrossRefGoogle Scholar
Kleiber, C. & Krämer, W. (2005) Finite-sample power of the Durbin-Watson test against fractionally integrated disturbances. Econometrics Journal, 8, 406–417.CrossRefGoogle Scholar
Krämer, W. (1985) The power of the Durbin-Watson test for regressions without an intercept. Journal of Econometrics, 28, 363–370.CrossRefGoogle Scholar
Krämer, W. (2005) Finite sample power of Cliff-Ord-type tests for spatial disturbance correlation in linear regression. Journal of Statistical Planning and Inference, 128, 489–496.CrossRefGoogle Scholar
Krämer, W. & Zeisel, H. (1990) Finite sample power of linear regression autocorrelation tests. Journal of Econometrics, 43, 363–372.CrossRefGoogle Scholar
Lehmann, E.L. & Romano, J.P. (2005) Testing statistical hypotheses, 3rd ed. Springer Texts in Statistics.Google Scholar
Löbus, J.U. & Ritter, L. (2000) The limiting power of the Durbin-Watson test. Communications in Statistics-Theory and Methods, 29, 2665–2676.Google Scholar
Martellosio, F. (2010) Power properties of invariant tests for spatial autocorrelation in linear regression. Econometric Theory, 26, 152–186.CrossRefGoogle Scholar
Martellosio, F. (2011a) Efficiency of the OLS estimator in the vicinity of a spatial unit root. Statistics & Probability Letters, 81, 1285 – 1291.CrossRefGoogle Scholar
Martellosio, F. (2011b) Nontestability of equal weights spatial dependence. Econometric Theory, 27, 1369–1375.CrossRefGoogle Scholar
Martellosio, F. (2012) Testing for spatial autocorrelation: The regressors that make the power disappear. Econometric Reviews, 31, 215–240.CrossRefGoogle Scholar
Mynbaev, K. (2012) Distributions escaping to infinity and the limiting power of the Cliff-Ord test for autocorrelation. ISRN Probability and Statistics.CrossRefGoogle Scholar
Preinerstorfer, D. (2014) How to avoid the zero-power trap in testing for correlation. Working paper, in preparation.Google Scholar
Preinerstorfer, D. & Pötscher, B.M. (2013) On size and power of heteroskedasticity and autocorrelation robust tests. Econometric Theory, forthcoming.Google Scholar
Small, J.P. (1993) The limiting power of point optimal autocorrelation tests. Communications in Statistics–Theory and Methods, 22, 3907–3916.CrossRefGoogle Scholar
Stroock, D.W. (1999) A concise introduction to the theory of integration, 3rd ed. Birkhäuser Boston.Google Scholar
Tillman, J. (1975) The power of the Durbin-Watson test. Econometrica, 43, 959–974.CrossRefGoogle Scholar
Tyler, D.E. (1981) Asymptotic inference for eigenvectors. Annals of Statistics, 9, 725–736.CrossRefGoogle Scholar
Zeisel, H. (1989) On the power of the Durbin-Watson test under high autocorrelation. Communications in Statistics-Theory and Methods, 18, 3907–3916.CrossRefGoogle Scholar
Supplementary material: PDF

Preinerstorfer supplementary material

Preinerstorfer supplementary material 1

Download Preinerstorfer supplementary material(PDF)
PDF 145.5 KB