Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-15T00:58:57.196Z Has data issue: false hasContentIssue false
  • English
  • Français

The Bias of Bootstrapped Versus Conventional Standard Errors in the General Linear and SUR Models

Published online by Cambridge University Press:  18 October 2010

Scott E. Atkinson  and
Paul W. Wilson
Scott E. Atkinson
Affiliation:
University of Georgia
Paul W. Wilson
Affiliation:
University of Texas
Get access Rights & Permissions [Opens in a new window]

Abstract

When estimating the seemingly unrelated regression (SUR) model in small samples, the bootstrap feasible generalized least-squares (FGLS) covariance estimator has been widely advocated as less biased than the conventional FGLS covariance estimator obtained by evaluating the asymptotic covariance matrix. Assuming multivariate normal errors and an unbiased estimator of the error covariance, Eaton proves that the conventional estimator is biased downward for a general SUR model. Ignoring terms O(T–2) for this model, we prove that the bootstrap estimator is also biased downward. However, from these results, the relative magnitude of these two biases is indeterminant in general. By ignoring terms O(T–2) for Zellner's two-equation, orthogonal regressor model with bivariate normal errors, we show that the bias of both estimators is downward and that the bootstrap estimator exhibits a smaller bias than the conventional estimator. Monte Carlo simulation results indicate that, in general, neither estimator uniformly dominates the other.

Type
Articles
Information
Econometric Theory , Volume 8 , Issue 2 , June 1992 , pp. 258 - 275
Copyright
Copyright © Cambridge University Press 1992

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.)

References

REFERENCES

1.Amemiya, T.Advanced Econometrics. Cambridge: Harvard University Press, 1985.Google Scholar
2.Bhattacharya, R. & Qumsiyeh, M.. Second order and Lp-comparisons between the bootstrap and empirical edgeworth expansion methodologies. The Annals of Statistics 17 (1989): 160–169.CrossRefGoogle Scholar
3.Binkley, J.K. & Nelson, C.N.. A note on the efficiency of seemingly unrelated regression.The American Statistician 42 (1988): 137–139.CrossRefGoogle Scholar
4.Breusch, T.R.Useful invariance results for generalized regression models. Journal of Econometrics 13 (1980): 327–340.CrossRefGoogle Scholar
5.Eaton, M.L. The Gauss-Markov theorem in multivariate analysis. In Krishnaiah, P. R. (ed.) Multivariate Analysis VI, Chapter 10. Amsterdam: North Holland, 1985.Google Scholar
6.Efron, B.Bootstrap methods: Another look at the jackknife. Annals of Statistics 1 (1979): 1–16.Google Scholar
7.Efron, B.The Jackknife, the Bootstrap, and Other Resampling Plans. Philadelphia: Society for Industrial and Applied Mathematics, 1982.CrossRefGoogle Scholar
8.Efron, B.Better bootstrap confidence intervals. Journal of the American Statistical Association 82 (1987): 171–185.CrossRefGoogle Scholar
9.Efron, B. & Tibshirani, R.. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science 1 (1986): 54–75.Google Scholar
10.Freedman, D.A.Bootstrapping regression models. The Annals of Statistics 9 (1981): 1218–1228.CrossRefGoogle Scholar
11.Freedman, D.A.On bootstrapping two-stage least-squares estimates in stationary linear models. The Annals of Statistics 12 (1984): 827–842.CrossRefGoogle Scholar
12.Freedman, D.A. & Peters, S.C.. Bootstrapping a regression equation: Some empirical results. Journal of the American Statistical Association 79 (1984): 97–106.CrossRefGoogle Scholar
13.Freedman, D.A. & Peters, S.C.. Bootstrapping an econometric model: Some empirical results. Journal of Business and Economic Statistics 2 (1984): 150–158.CrossRefGoogle Scholar
14.Judge, G.G., Hill, R.C., Griffiths, W.E., Lutkepohl, H., & Lee, T.. Introduction to the Theory and Practice of Econometrics, 2nd ed. New York: Wiley, 1988.CrossRefGoogle Scholar
15.Marais, M.L. On the Finite Sample Performance of Estimated Generalized Least Squares in Seemingly Unrelated Regressions: Nonnormal Disturbances and Alternative Standard Error Estimators. Working paper, Graduate School of Business, University of Chicago (1986).Google Scholar
16.Peters, S.C. & Freedman, D.A.. Some notes on the bootstrap in regression problems. Journal of Business and Economic Statistics 2 (1984): 406–409.CrossRefGoogle Scholar
17.Phillips, P.C.B.An approximation of the finite sample distribution of Zellner's seemingly unrelated regression estimator. Journal of Econometrics 6 (1977): 147–164.CrossRefGoogle Scholar
18.Revankar, N.S.Use of restricted residuals in SUR systems: Some finite sample results. Journal of the American Statistical Association 71 (1976): 183–188.CrossRefGoogle Scholar
19.Rothenberg, T.J.Approximate normality of generalized least squares estimates. Econo-metrica 52 (1984): 811–826.CrossRefGoogle Scholar
20.Wu, C.F.J. Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics 14 (1986): 1261–1295.Google Scholar
21.Zellner, A.An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association 57 (1962): 348–368.CrossRefGoogle Scholar
22.Zellner, A.Estimators for seemingly unrelated regression equations: Some exact finite sample results. Journal of the American Statistical Association 58 (1963): 977–992.CrossRefGoogle Scholar
23.Zellner, A.Corrigenda. Journal of the American Statistical Association 67 (1972): 255.Google Scholar
24.Zellner, A.Finite sample estimation of covariance matrices of SUR coefficient estimators.Working paper. Graduate School of Business, University of Chicago (1986).Google Scholar