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Preliminary Test and Stein-Rule Estimators

  • Thomas B. Fomby
  • Stanley R. Johnson
  • R. Carter Hill

Abstract

In the previous chapter procedures for augmenting the available sample information were considered. Consequences of incorporating nonsample information were seen to depend on the quality of information introduced. As one would expect, only the use of good information provides positive benefits. Unfortunately, we seldom are sure of the quality of the information to be introduced. In this chapter we examine the consequences of that uncertainty. First of all, investigators are in the habit of checking their prior nonsample information against the data using statistical tests of the type outlined in Chapter 6. The nonsample information is then either adopted or not depending upon the outcome of the test. The resulting estimation rule is called a preliminary test estimator since its form depends upon the outcome of a (preliminary) hypothesis test. This estimator is superior to the estimator based on sample information alone only over a relatively small portion of the parameter space, which reflects the fact that classical statistical procedures are not designed to aid the choice of a model specification. These results are discussed in Section 7.2.

Keywords

Maximum Likelihood Estimator Risk Function Principal Component Regression Error Loss Noncentrality Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Aigner, D. J., and Judge, G. G. (1977). Application of pre-test and Stein estimators to economic data. Econometrica, 45, 1279–1280.CrossRefGoogle Scholar
  2. Akaike, H. (1974). A new look at the statistical identification model. I.E.E.E.: Transactions on Automatic Control, 19, 716–723.CrossRefGoogle Scholar
  3. Almon, S. (1965). The distributed lag between capital appropriations and expenditures. Econometrica, 33, 178–196.CrossRefGoogle Scholar
  4. Amemiya, T. (1980). Selection of regressors. International Economic Review, 21, 331–354.CrossRefGoogle Scholar
  5. Brook, R. J. (1976). On the use of a regret function to set significance points in prior tests of estimation. Journal of the American Statistical Association, 71, 126–131.CrossRefGoogle Scholar
  6. Draper, N. and Smith, H. (1966). Applied Regression Analysis. New York: Wiley. Efron, R. and Morris, C. (1977). Stein’s paradox in statistics. Scientific American, 236, 119–127.Google Scholar
  7. Farebrother, R. W. (1978). Estimating regression coefficients under conditional specification: comment. Communications in Statistics, A, 7, 193–196.Google Scholar
  8. Fomby, T. B. and Guilkey, D. K. (1978). On choosing the optimal level of significance for the Durbin-Watson test and the Bayesian alternative. Journal of Econometrics, 8, 203–214.CrossRefGoogle Scholar
  9. Greenberg, E. (1980). Finite sample moments of a preliminary test estimator in the case of possible heteroscedasticity. Econometrica, 48, 1805–1814.CrossRefGoogle Scholar
  10. Greenberg, E. and Webster, C. (1983). Advanced Econometrics: A Bridge to the Literature. New York: Wiley.Google Scholar
  11. Han, C. and Bancroft, T. A. (1978). Estimating regression coefficients under conditional specification. Communications in Statistics, A, 7, 47–56.Google Scholar
  12. Hill, R. C. and Ziemer, R. (1982). The application of generalized ridge and Stein-like general minimax rules to multicollinear data. Communications in Statistics, A, 11, 623–638.Google Scholar
  13. Hill, R. C, Judge, G. G., and Fomby, T. B. (1978). On testing the adequacy of the regression equation. Technometrics, 20, 491–494.CrossRefGoogle Scholar
  14. James, W. and Stein, C. (1961). Estimation with quadratic loss, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1, 361–379.Google Scholar
  15. Judge, G. and Bock, M. (1978). Statistical Implications of Pretest and Stein-Rule Estimators in Econometrics. Amsterdam: North-Holland.Google Scholar
  16. Judge, G., Yancey, T., and Bock, M. (1973). Properties of estimators after preliminary tests of significance when stochastic restrictions are used in regression. Journal of Econometrics, 1, 29–48.CrossRefGoogle Scholar
  17. Judge, G., Griffiths, W., Hill, R. and Lee, T. (1980). The Theory and Practice of Econometrics. New York: Wiley.Google Scholar
  18. Mallows, C. (1973). Some comments on Cp. Technometrics, 15, 661–676.Google Scholar
  19. Mittlehammer, R. (1981). Unpublished mimeo, Washington State University.Google Scholar
  20. Ohtani, K. and Toyoda, T. (1980). Estimation of regression coefficients after a preliminary test for homoscedasticity. Journal of Econometrics, 12, 151–160.CrossRefGoogle Scholar
  21. Oman, S. D. (1978). A Bayesian comparison of some estimators used in linear regression with multicollinear data. Communications in Statistics, A, 7, 517–534.Google Scholar
  22. Sawa, T. (1977). Information criteria for discriminating among alternative regression models. Faculty Working Paper 455, University of Illinois.Google Scholar
  23. Sawa, T. and Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis. Econometrica, 41, 1093–1101.CrossRefGoogle Scholar
  24. Sclove, S., Morris, C, and Radharkrishnan, R. (1972). Non-optimality of preliminary test estimators for the mean of a multivariate normal distribution. The Annals of Mathematical Statistics, 43, 1481–1490.CrossRefGoogle Scholar
  25. Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proceedings of the Third Berkeley Symposium, 1, 197–206.Google Scholar
  26. Toro-Vizcarrondo, C. and Wallace, T. (1968). A test of the mean square error criterion for restrictions in linear regression. Journal of the American Statistical Association, 63, 558–572.CrossRefGoogle Scholar
  27. Toyoda, T. and Wallace, T. (1976). Optimal critical values for pretesting in regression. Econometrica, 44, 365–376.CrossRefGoogle Scholar
  28. Trivedi, P. K. (1978). Estimation of a distributed lag model under quadratic loss. Econometrica, 46, 1181–1192.CrossRefGoogle Scholar
  29. Vinod, H. D. (1980). Improved Stein-rule estimator for regression problems. Journal of Econometrics, 12, 143–150.CrossRefGoogle Scholar
  30. Vinod, H. D. and Ullah, H. (1981). Recent Advances in Regression Methods. New York: Marcel Dekker.Google Scholar
  31. Wallace, T. (1972). Weaker criteria and tests for linear restrictions in regression. Econometrica, 40, 689–698.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • Thomas B. Fomby
    • 1
  • Stanley R. Johnson
    • 2
  • R. Carter Hill
    • 3
  1. 1.Department of EconomicsSouthern Methodist UniversityDallasUSA
  2. 2.The Center for Agricultural and Rural DevelopmentIowa State UniversityAmesUSA
  3. 3.Department of EconomicsLouisiana State UniversityBaton RougeUSA

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