Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T22:41:31.025Z Has data issue: false hasContentIssue false
  • English
  • Français

Determination of Estimators with Minimum Asymptotic Covariance Matrices

Published online by Cambridge University Press:  11 February 2009

Charles E. Bates  and
Halbert White
Charles E. Bates
Affiliation:
KPMG Peat Marwick Policy Economics Group
Halbert White
Affiliation:
University of California, San Diego
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a straightforward condition sufficient for determining the minimum asymptotic variance estimator in certain classes of estimators relevant to econometrics. These classes are relatively broad, as they include extremum estimation with smooth or nonsmooth objective functions; also, the rate of convergence to the asymptotic distribution is not required to be n−½. We present examples illustrating the content of our result. In particular, we apply our result to a class of weighted Huber estimators, and obtain, among other things, analogs of the generalized least-squares estimator for least Lp-estimation, 1 ≤ p < ∞.

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 4 , August 1993 , pp. 633 - 648
Copyright
Copyright © Cambridge University Press 1993

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

1.Aitken, A.C.On least squares and linear combinations of observations. Proceedings of the Royal Society of Edinburgh 55 (1935): 42–48.Google Scholar
2.Amemiya, T.The nonlinear two-stage least-squares estimator. Journal of Econometrics 2 (1979): 105–110.CrossRefGoogle Scholar
3.Amemiya, T.The maximum likelihood and the nonlinear three-stage least squares estimator in the general nonlinear simultaneous equation model. Econometrica 45 (1977): 955–968.CrossRefGoogle Scholar
4.Andrews, D.W.K.Asymptotics for semiparametric econometric models: III. Yale Cowles Foundation Discussion Paper 910, 1989.Google Scholar
5.Bahadur, R.R.On Fisher's bound for asymptotic variances. Annals of Mathematical Statistics 35 (1964): 1545–1552.Google Scholar
6.Bates, C.E. & White, H.. A unified theory of consistent estimation for parametric models. Econometric Theory 1 (1985): 151–178.Google Scholar
7.Bates, C.E. & White, H.. Efficient estimation of parametric models. UCSD Department of Economics Discussion Paper, 1987.Google Scholar
8.Bates, C.E. & White, H.. Efficient instrumental variables estimation for systems of implicit heterogeneous nonlinear dynamic equations with nonspherical errors. In Barnett, W., Berndt, E., and White, H. (eds.), Dynamic Econometric Modeling, pp. 3–25. New York: Cambridge University Press, 1990.Google Scholar
9.Begun, J., Hall, W., Huang, W., & Wellner, J.. Information and asymptotic efficiency in parametric-nonparametric models. Annals of Statistics 11 (1985): 432–452.Google Scholar
10.Brundy, J.M. & Jorgenson, D.W.. Consistent and efficient estimation of systems of simultaneous equations by means of instrumental variables. In Zarembka, P. (ed.), Frontiers in Econometrics, pp. 215–244. New York: Academic Press, 1974.Google Scholar
11.Chamberlain, G.Asymptotic efficiency in semiparametric models with censoring. Journal of Econometrics 32 (1986): 189–218.Google Scholar
12.Chiang, C.L.On regular best asymptotically normal estimates. Annals of Mathematical Statistics 21 (1956): 336–351.CrossRefGoogle Scholar
13.Cragg, J.More efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 51 (1983): 751–764.Google Scholar
14.Cramér, H.Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.Google Scholar
15.Darmois, G.Sur le lois limites de la dispersion de certaines estimations. Review of the Institute of the International Statistical Society 13 (1945): 9–15.CrossRefGoogle Scholar
16.Domowitz, I. & White, H.. Misspecified models with dependent observations. Journal of Econometrics 20 (1982): 35–58.Google Scholar
17.Engle, R.Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflations. Econometrica 50 (1982): 987–1008.Google Scholar
18.Fisher, R.A.The mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society, London Series A 222 (1921): 309–368.Google Scholar
19.Fisher, R.A.Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society 22 (1925): 700–725.CrossRefGoogle Scholar
20.Frechet, M.Sur l'extension de certaines evaluations statistiques de petits echantillons. Review of the Institute of the International Statistical Society 11 (1943): 182–205.CrossRefGoogle Scholar
21.Hajek, J.A characterization of limiting distributions of regular estimates. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandete Gebiete 14 (1970): 323–330.CrossRefGoogle Scholar
22.Hajek, J. Local asymptotic minimax and admissibility in estimation. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 175–194. Berkeley: University of California Press, 1972.Google Scholar
23.Hall, P. & Heyde, C.C.. Martingale Limit Theory and Its Application. New York: Academic Press, 1980.Google Scholar
24.Hausman, J.An instrumental variable approach to full information estimators for linear and certain nonlinear econometric models. Econometrica 43 (1975): 727–738.Google Scholar
25.Hansen, L.P.Large sample properties of generalized method of moments estimators. Econometrica 50 (1982): 1029–1054.CrossRefGoogle Scholar
26.Hansen, L.P.A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators. Journal of Econometrics 30 (1985): 203–238.Google Scholar
27.Huber, P. J.Robust estimation of a location parameter. Annals of Mathematical Statistics 35 (1964): 73–101.CrossRefGoogle Scholar
28.Huber, P.J.Robust regression: asymptotics, conjectures and Monte Carlo. Annals of Statistics 1 (1972): 799–821.Google Scholar
29.Ibragimov, I.A. & Has'minskii, R.Z.. Statistical Estimation: Asymptotic Theory. New York: Springer-Verlag, 1981.Google Scholar
30.Jorgenson, D.W. & Laffont, J.. Efficient estimation of nonlinear simultaneous equations with additive disturbances. Annals of Economics and Social Measurement 3 (1974): 615–640.Google Scholar
31.Kaufman, S.Asymptotic efficiency of the maximum likelihood estimator. Annals of the Institute of Statistics and Mathematics 18 (1966): 155–170.CrossRefGoogle Scholar
32.Le Cam, L.On the assumptions used to prove asymptotic normality of maximum likelihood estimators. Annals of Mathematical Statistics 41 (1970): 802–828.Google Scholar
33.Phillips, P.C.B.Partially identified econometric models. Econometric Theory 5 (1989): 181–240.CrossRefGoogle Scholar
34.Powell, J. Efficient estimation of monotonic regression models using quantile restrictions. In Barnett, W., Powell, J., and Tauchen, G. (eds.), Methods in Econometrics and Statistics, pp. 357–386. New York: Cambridge University Press, 1991.Google Scholar
35.Rao, C.R.Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society 37 (1945): 81–91.Google Scholar
36.Rao, C.R.Criteria of estimation of large samples. Sankhyā A 25 (1963): 189–206.Google Scholar
37.Rao, C.R.Efficiency of an estimator and Fisher's lower-bound to asymptotic variance. Bulletin of the International Statistical Society 41 (1965): 55–63.Google Scholar
38.Robinson, P.Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55 (1987): 875–891.Google Scholar
39.Rothenberg, T.J. & Leenders, C.T.. Efficient estimation of simultaneous equation systems. Econometrica 32 (1964): 57–76.Google Scholar
40.Roussas, G.G.Contiguity of Probability Measures: Some Applications in Statistics. Cambridge: Cambridge University Press, 1972.CrossRefGoogle Scholar
41.Sargan, J.D.The estimation of relationships with autocorrelated residuals by means of instrumental variables. Journal of the Royal Statistical Society Series B 21 (1959): 91–105.Google Scholar
42.Sargan, J.D.Three stage least squares and full information maximum likelihood estimates. Econometrica 32 (1964): 77–81.Google Scholar
43.Weiss, L.Asymptotic properties of maximum likelihood estimators in some nonstandard cases. Journal of the American Statistical Association 66 (1971): 345–350.CrossRefGoogle Scholar
44.Weiss, L.Asymptotic properties of maximum likelihood estimators in some nonstandard cases II. Journal of the American Statistical Association 68 (1973): 428–430.Google Scholar
45.White, H.Instrumental variables regression with independent observations. Econometrica 50 (1982): 483–500.Google Scholar
46.White, H.Asymptotic Theory for Econometricians. New York: Academic Press, 1984.Google Scholar
47.White, H.Instrumental variables analogs of generalized least squares estimators. Journal of Advances in Statistical Computing and Statistical Analysis 1 (1986): 173–227.Google Scholar
48.White, H. and Stinchcombe, M.. Adaptive efficient weighted least squares with dependent observations. In Stahel, W. and Weisberg, S. (eds.), Directions in Robustness and Diagnostics in Statistics. IMA Volumes in Mathematics and Its Applications, pp. 337–365. New York: Springer-Verlag, 1991.CrossRefGoogle Scholar
49.Wolfowitz, J.Asymptotic efficiency of the maximum likelihood estimator. Theory of Probability and Its Applications 10 (1965): 247–260.Google Scholar
50.Wooldridge, J.Asymptotic properties of econometric estimators. University of California San Diego Department of Economics Doctoral Dissertation, 1986.Google Scholar
51.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.Google Scholar
52.Zellner, A. & Theil, H.. Three stage least squares: simultaneous estimation of simultaneous equations. Econometrica 30 (1962): 54–78.CrossRefGoogle Scholar