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ON THE PROPERTIES OF THE t- AND F-RATIOS IN LINEAR REGRESSIONS WITH NONNORMAL ERRORS

Published online by Cambridge University Press:  01 August 2004

Huaizhen Qin  and
Alan T.K. Wan
Huaizhen Qin
Affiliation:
Peking University
Alan T.K. Wan
Affiliation:
City University of Hong Kong
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Abstract

In this paper we derive the necessary and sufficient conditions for the t-ratio to be Student's t distributed. In particular, it is demonstrated for a special case that under conditions of nonnormality characterized by elliptical symmetry, the t-ratio remains Student's t distributed provided that the random vector forming the t-ratio has a diagonal covariance structure. Our results also show that the findings of Magnus (2002, in A. Ullah, A.T.K. Wan, & A. Chaturvedi (eds.), Handbook of Applied Econometrics and Statistical Inference, 277–285) on the sensitivity of the t-ratio remain invariant in the elliptically symmetric distribution setting. Extension to the linear model is considered. Exact results giving finite sample justification for the t-statistic under nonnormal error terms are derived. Furthermore, the distribution of the F-ratio assuming elliptical errors is examined. Our results reject the argument by Zaman (1996, Statistical Foundations for Econometric Techniques) that nonnormality of disturbances has in general no effect on the F-statistic.The authors thank Anurag Banerjee, Judith Clarke, Kazuhiro Ohtani, and Guohua Zou for their helpful suggestions and advice during the course of this work. In particular, we are very grateful to Judith Clarke for bringing to our attention the unpublished work of D.H. Thomas and to Guohua Zou for his careful reading of an earlier version of this paper. Thanks also go to the co-editor Benedikt Pötscher and two anonymous referees for their valuable comments. The second author gratefully acknowledges financial support from the Hong Kong Research Grant Council.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 4 , August 2004 , pp. 690 - 700
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Breusch, T.S., J.C. Robertson, & A.H. Welsh (1997) The emperor's new clothes: A critique of the multivariate t regression model. Statistica Neerlandica 51, 269–286.Google Scholar
Chib, S., R.C. Tiwari, & S.R. Jammalamadaka (1988) Bayes prediction in regressions with elliptical errors. Journal of Econometrics 38, 349–360.Google Scholar
Fang, K.T., S. Kotz, & K.W. Ng (1990) Symmetric Multivariate and Related Distributions. Chapman and Hall.
Fang, K.T., R. Li, & J. Liang (1998) A multivariate version of Ghosh's T3-plot to detect non-multinormality. Computational Statistics and Data Analysis 28, 371–386.Google Scholar
Fang, K.T. & Y.T. Zhang (1990) Generalized Multivariate Analysis. Springer-Verlag.
Giles, J.A. (1991) Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances. Journal of Econometrics 50, 377–398.Google Scholar
Greene, W.H. (2000) Econometric Analysis, 4th ed. Prentice-Hall.
Jammalamadaka, S.R., R.C. Tiwari, & S. Chib (1987) Bayes prediction in the linear model with spherically symmetric errors. Economics Letters 24, 39–44.Google Scholar
Läuter, J. (1996) Exact t and F tests for analyzing studies with multiple endpoints. Biometrics 52, 964–970.Google Scholar
Läuter, J., E. Glimm, & S. Kropf (1996) New multivariate tests for data with an inherent structure. Biometrical Journal 38, 5–23.Google Scholar
Li, R., K.T. Fang, & L. Zhu (1997) Some probability plots to test spherical and elliptical symmetry. Journal of Computational and Graphical Statistics 6, 435–450.Google Scholar
Liang, J., R. Li, H. Fang, & K.T. Fang (2000) Testing multinormality based on low-dimensional projection. Journal of Statistical Planning and Inference 86, 129–141.Google Scholar
Magnus, J.R. (2002) On the sensitivity of the t-statistic. In A. Ullah, A.T.K. Wan, & A. Chaturvedi (eds.), Handbook of Applied Econometrics and Statistical Inference, pp. 277–285. Marcel Dekker.
Ohtani, K. & J.A. Giles (1993) Testing linear restrictions on coefficients in a linear regression model with proxy variables and spherically symmetric disturbances. Journal of Econometrics 57, 393–406.Google Scholar
Osiewalski, J. (1991) A note on Bayesian inference in a regression model with elliptical errors. Journal of Econometrics 48, 183–193.Google Scholar
Qin, H., H. Cui, & Y. Li (1998) The N.S. conditions for the ratio of two quadratic forms to have an F distribution and its application. Communications in Statistics, Theory and Methods 27, 453–471.Google Scholar
Smith, M.D. (1992) Comparing the exact distribution of the t-statistic to Student's distribution when its constituent normal and chi-squared variables are dependent. Communications in Statistics, Theory and Methods 21, 3589–3600.Google Scholar
Thomas, D.H. (1970) Some Contributions to Radial Probability Distributions, Statistics and the Operational Calculi. Ph.D. thesis, Wayne State University.
Ullah, A. & P.C.B. Phillips (1986) Distribution of the F-ratio. Econometric Theory 2, 449–452.Google Scholar
Ullah, A. & V.K. Srivastava (1994) Moments of the ratio of quadratic forms in non-normal variables with econometric examples. Journal of Econometrics 62, 129–141.Google Scholar
Ullah, A. & V. Zinde-Walsh (1985) Estimation and testing in a regression model with spherically symmetric errors. Economics Letters 17, 127–132.Google Scholar
Zaman, A. (1996), Statistical Foundations for Econometric Techniques. Academic Press.