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Non-Normal Errors and the Distribution of OLS and 2SLS Structural Estimators

Published online by Cambridge University Press:  18 October 2010

John L. Knight
John L. Knight*
Affiliation:
The University of New South Wales and Indiana University
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Abstract

This paper examines the sensitivity of the distributions of OLS and 2SLS estimators to the assumption of normality of disturbances in a structural equation with two included endogenous variables. The approach taken is that ofimposing Edgeworth distributed errors on the reduced form equations and deriving the pdf of the estimators via the technique of Davis [11]. The sensitivity of the pdf s to changes in the non-normality parameters, i.e., skewness and kurtosis i s examined via extensive numerical computations.

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Type
Research Article
Information
Econometric Theory , Volume 2 , Issue 1 , April 1986, pp. 75 - 106
Copyright
Copyright © Cambridge University Press 1986 

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References

1. Baker, G. A. Random sampling from non-homogeneous populations. Metron 1 (1931): 67–87.Google Scholar
2. Baker, G. A. Distribution of the means of samples of n drawn from a population represented by a Gram-Charlier series. Annals of Mathematical Statistics 1 (1931): 199–204.Google Scholar
3. Baker, G. A. The relation between the means and variances, mean-squares errors and variances in samples from combinations of normal populations. Annals of Mathematical Statistics 2 (1931): 333–354.10.1214/aoms/1177732977Google Scholar
4. Baker, G. A. Distribution of the means divided by the standard deviations of samples from non-homogeneous populations. Annals of Mathematical Statistics 3 (1932): 1–9.Google Scholar
5. Baker, G. A. Note on the distributions of the standard deviations and second moments of samples from a Gram-Charlier population. Annals of Mathematical Statistics 6 (1935): 127–130.10.1214/aoms/1177732587Google Scholar
6. Bartlett, M. S. The effect of non-normality on the t-distribution. Proceedings of the Cambridge Philosophical Society 31 (1935): 223–231.10.1017/S0305004100013311Google Scholar
7. Barton, D. E. and Dennis, K. E.. The conditions under which the Gram-Charlie r and Edgeworth curves are positive definite and unimodal. Biometrika 39 (1952): 425–427.10.1093/biomet/39.3-4.425Google Scholar
8. Box, G.E.P. Non-normality and test on variances. Biometrika 40 (1953): 318–335.10.1093/biomet/40.3-4.318Google Scholar
9. Chandra, R. Estimation of econometric models when disturbances are not necessarily normal.Unpublished Ph.D. Thesis, Department of Statistics, Lucknow University, India, 1983.Google Scholar
10. Cramer, H. Mathematical Methods of Statistics. Princeton: Princeton University Press, 1974.Google Scholar
11. Davis, A. W. Statistical distributions in univariate and multivariate Edgeworth populations. Biometrika 63 (1976): 661–670.10.1093/biomet/63.3.661Google Scholar
12. Davis, A. W. On the effects of moderate non-normality on Wilk's likelihood ratio criterion. Biometrika 67 (1980): 419–427.10.1093/biomet/67.2.419Google Scholar
13. Davis, A. W. On the distribution of Hotelling's one sample T2 under moderate non-normality. Journal of Applied Probability 19 A (1982): 207–216.Google Scholar
14. Draper, N. R. and Tierney, D. E.. Regions of positive and unimodal series expansion of the Edgeworth and Gram-Charlier approximations. Biometrika 59 (1972): 463–465.10.1093/biomet/59.2.463Google Scholar
15. Gayen, A. K. The distribution of Student's t in random samples of any size drawn from non-normal universes. Biometrika 36 (1949): 353–369.Google Scholar
16. Gayen, A. K. The distribution of variance ratio in random samples of any size drawn from non-normal universes. Biometrika 37 (1950): 236–255.10.1093/biomet/37.3-4.236Google Scholar
17. Gayen, A. K. The frequency distribution of the product-moment correlation in random samples of any size drawn from non-norma l universes. Biometrika 38 (1951): 219–247.10.1093/biomet/38.1-2.219Google Scholar
18. Geary, R. C. The distribution of Student's ratio for non-normal samples. Supplement to the Journal of the Royal Statistical Society 3 (1936): 178–184.Google Scholar
19. Ghurye, S. G. On the use of Student's f-test in an asymmetrical population. Biometrika 36 (1949): 426–430.10.1093/biomet/36.3-4.426Google Scholar
20. Hale, C, Mariano, R. S., and Ramage, J. G.. Finite sample analysis of misspecification in simultaneous equation models. Journal of the American Statistical Association 75 (1980): 418–427.10.1080/01621459.1980.10477488Google Scholar
21. Kariya, T. A robustness property of the tests for serial correlation. Annals of Statistics 5 (1977): 1212–1220.10.1214/aos/1176344005Google Scholar
22. King, M. L. Small sample properties of econometric estimators and tests assuming elliptically symmetric disturbances. Working Paper No. 15/80, Department of Econometrics and Operations Research, Monash University, 1980.Google Scholar
23. Knight, J. L. Non-Normality of disturbances and the k-class structural estimator. Discussion Paper No. 40, School of Economics, The University of New South Wales, 1981.Google Scholar
24. Knight, J. L. The moments of OLS and 2SLS when the disturbances are non-normal. Journal of Econometrics 27 (1985): 39–60.10.1016/0304-4076(85)90043-0Google Scholar
25. Knight, J. L. The joint characteristic function of linear and quadratic forms of non-normal variables. Forthcoming, Sankhya (1985).Google Scholar
26. Knight, J. L. Non-normal disturbances and the distribution of the Durbin-Watson Statistic. Paper presented at the European Meeting of the Econometric Society, Pisa, September 1983.Google Scholar
27. Phillips, P.C.B. The exact finite sample density of instrumental variable estimators in an equation with n + 1 endogenous variables Econometrica 48 (1980): 861–878.10.2307/1912937Google Scholar
28. Phillips, P.C.B. Small sample distribution theory in econometric models of simultaneous equations. Cowles Foundation Discussion Paper No. 617, Yale University, 1982.Google Scholar
29. Phillips, P.C.B. Exact small sample theory. In Griliches, Z. and Intriligator, M. D. (eds.), Handbook of Econometrics, Chapter 8 and pp. 449–516. New York: North Holland, 1983.10.1016/S1573-4412(83)01012-0Google Scholar
30. Phillips, P.C.B. The exact distributio n of LIML I. International Economic Review 25 (1984): 249–261.10.2307/2648878Google Scholar
31. Pearson, E. S. and Adyanthaya, N. K.. The distribution of frequency constants in small samples from non-normal symmetrical and skew populations. Biometrika 21 (1929): 259–286.10.1093/biomet/21.1-4.259Google Scholar
32. Raj, B. Monte Carlo study of simultaneous equations estimators under normal and non-normal errors. Journal of the American Statistical Association 75 (1980): 221–229.Google Scholar
33. Rhodes, G. F. and Westbrook, M. D.. A study of estimator densities and performance under misspecification. Journal of Econometrics 16 (1981): 311–337.10.1016/0304-4076(81)90033-6Google Scholar
34. Richardson, D. H. The exact distribution of a structural coefficient estimator. Journal of the American Statistical Association. 63 (1968): 1214–1226.Google Scholar
35. Richardson, D. H. and Wu, De Min. A note on the comparison of ordinary and two-stage least squares estimators. Econometrica 39 (1971): 973–981.Google Scholar
36. Sawa, T. Th e exact finite sampling distribution of ordinary least squares and two stage least squares estimators. Journal of the American Statistical Association 64 (1969): 923–936.Google Scholar
37. Slater, L. J. Confluent Hypergeometric Functions. Cambridge: Cambridge University Press, 1960.Google Scholar
38. Srivastava, A.B.L. Effect of non-normality on the power function of the t-test. Biometrika 45 (1958): 421–429.Google Scholar
39. Srivastava, A.B.L. Effect of non-normality on the power function of the analysis of variance test. Biometrika 46 (1959): 114–122.Google Scholar
40. Subrahmaniam, K. Some contributions to non-normality 1 (Univariate Case). Sankhya Series A 28 (1966): 389–406.Google Scholar
41. Subrahmaniam, K. Some contributions to non-normality II. Sankhya Series A 30 (1968): 411–432.Google Scholar
42. Ullah, A., Srivastava, V. K., and Chandra, R.. Properties of shrinkage estimators when disturbances are not normal. Journal of Econometrics 21 (1983): 389–402.10.1016/0304-4076(83)90053-2Google Scholar