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On Consistency and Inconsistency of Estimating Equations

Published online by Cambridge University Press:  18 October 2010

Martin Crowder
Martin Crowder
Affiliation:
Surrey University
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Abstract

The primary concern is to establish a fairly general framework in which estimators resulting from estimating equations gnθ = 0 are not consistent. This leads on to consistency by an intuitive route. Asymptotic distributions of consistent estimators are also touched upon, and the results are applied to various examples.

Type
Articles
Information
Econometric Theory , Volume 2 , Issue 3 , December 1986 , pp. 305 - 330
Copyright
Copyright © Cambridge University Press 1986

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References

REFERENCES

1.Aitchison, J. and Silvey, S. D.. Maximum likelihood estimation of parameters subject to restraints. Annals of Mathematical Statistics 29 (1958): 813–828.CrossRefGoogle Scholar
2.Bates, C. and White, H.. A unified theory of consistent estimation for parametric models. Econometric Theory 1 (1985).CrossRefGoogle Scholar
3.Billingsley, P. Convergence of Probability Measures. New York: Wiley, 1968.Google Scholar
4.F., Burguete J., Gallant, A. R., and Souza, G.. On the unification of the asymptotic theory of nonlinear econometric models. Econometric Reviews 1 (1982): 151–190.Google Scholar
5.Cramér, H. Mathematical Methods of Statistics. Princeton: Princetown University Press, 1946.Google Scholar
6.J, Crowder M.. Maximum likelihood estimation for dependent observations. Journal of the Royal Statistical Society B38 (1976): 45–53.Google Scholar
7.J, Crowder M.. On the asymptotic properties of least-squares estimators in autoregression. Annals of Statistics 8 (1980): 132–146.Google Scholar
8.J, Crowder M.. On weighted least-squares and some variants. Technical Report No. 13 (1981): Maths Department, Surrey University.Google Scholar
9.P, Dempster A.. Elements of Continuous Multivariate Analysis. Reading, Mass., U.S.A.: Addison-Wesley, 1969.Google Scholar
10.L, Doob J.. Probability and statistics. Transactions of the American Mathematical Society 36 (1934): 759–775.Google Scholar
11.L, Doob J.. Stochastic Processes. New York: Wiley, 1953.Google Scholar
12.Drygas, H.Weak and strong consistency of the least-squares estimators in regression models. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976): 119–127.Google Scholar
13.Feller, W. An Introduction to Probability Theory and its Applications Vol. II, 2nd Ed. New York: Wiley, 1971.Google Scholar
14.Frydman, R. A proof of the consistency of maximum likelihood estimators of nonlinear regression models with autocorrelated errors. Econometrica 48 (1980): 853–860.CrossRefGoogle Scholar
15.R., Gallant A. and White, H.. A unified theory of estimation and inference for nonlinear dynamic models. UCSD Department of Economics Discussion Paper, 1985.Google Scholar
16.Hall, P. and Heyde, C. C.. Martingale Limit Theory and its Application. London: Academic Press, 1980.Google Scholar
17.R.D.H., Heijmans and Magnus, J. R.. Consistent maximu m likelihood estimation with dependent observations: the general (non-normal) case and the normal case. Journal of Econometrics (to appear).Google Scholar
18.Hoadley, B. Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case. Annals of Mathematical Statistics 42 (1971): 1977–1991.CrossRefGoogle Scholar
19.J, Huber P.. The behaviour of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics and Probability. Berkeley: University of California Press, 1967.Google Scholar
20.Inagaki, N. Asymptotic relations between the likelihood estimating function and the maximum likelihood estimator. Annals of the Institute of Statistical Mathematics 25 (1973): 1–26.CrossRefGoogle Scholar
21.I, Jennrich R.. Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40 (1969): 633–643.Google Scholar
22.Kaffes, D. and Bhaskara, Rao M.. Weak consistency of least-squares estimators in linear models. Journal of Multivariate Analysis 12 (1982): 186–198.CrossRefGoogle Scholar
23.A., Klimko L. and Nelson, P. I.. On conditional least-squares estimation for stochastic processes. Annals of Statistics 6 (1978): 629–642.Google Scholar
24.L., Lai T., Robbins., H. and Wei, C. Z.. Strong consistency of least-squares estimates in multiple regression II. Journal of Multivariate Analysis 9 (1979): 343–361.Google Scholar
25.LeCam, L. On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates. University of California Publications in Statistics. Vol. I (1953): 277–300.Google Scholar
26.Malinvaud, E. The consistency of nonlinear regressions. Annals of Mathematical Statistics 41 (1970): 956–969.CrossRefGoogle Scholar
27.Rao, C.R.Linear Statistical Inference and its Applications. New York: Wiley, 1965.Google Scholar
28.M, Robinson P.. On consistency in time series analysis. Annals of Statistics 6 (1978): 215–223.Google Scholar
29.Rubin, H. Uniform convergence of random functions with applications to statistics. Annals of Mathematical Statistics 27 (1956): 200–203.CrossRefGoogle Scholar
30.D, Silvey S.. A note on maximum likelihood in the case of dependent random variables. Journal of the Royal Statistical Society B23 (1961): 444–452.Google Scholar
31.Wald, A. Asymptotic properties of the maximum likelihood estimate of a n unknown parameter of a discrete stochastic process. Annals of Mathematical Statistics 19 (1948): 40–46.CrossRefGoogle Scholar
32.Wald, A. Note on the consistency of the maximum likelihood estimate. Annals of Mathematical Statistics 20 (1949): 595–601.CrossRefGoogle Scholar
33.R.W.M, Wedderburn. Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61 (1974): 439–447.Google Scholar
34.White, H. and Domowitz, I.. Nonlinear regression with dependent observations. Econo-metrica 52 (1984): 143–161.CrossRefGoogle Scholar
35.Whittle, P. Gaussian estimation in stationary time series. Bulletin of the International Statistical Institute 39 (1961): 1–26.Google Scholar
36.Wu, C. F.. Characterizing the consistent directions of least-squares estimates. Annals of Statistics 8 (1980): 789–801.CrossRefGoogle Scholar
37.Wu, C. F.. Asymptotic theory of nonlinear least-squares estimation. Annals of Statistics 9 (1981): 501–513.CrossRefGoogle Scholar