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Generic Uniform Convergence

Published online by Cambridge University Press:  18 October 2010

Donald W.K. Andrews
Donald W.K. Andrews
Affiliation:
Yale University
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Abstract

This paper presents several generic uniform convergence results that include generic uniform laws of large numbers. These results provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence. The results are useful for establishing asymptotic properties of estimators and test statistics.

The results given here have the following attributes, (1) they extendresults of Newey to cover convergence almost surely as well as convergence in probability, (2) they apply to totally bounded parameter spaces (rather than just to compact parameter spaces), (3) they introduce a set of conditions for a generic uniform law of large numbers that has the attribute of giving the weakest conditions available for i.i.d. contexts, but which apply in some dependent nonidentically distributed contexts as well, and (4) they incorporate and extend themain results in the literature in a parsimonious fashion.

Type
Articles
Information
Econometric Theory , Volume 8 , Issue 2 , June 1992 , pp. 241 - 257
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

1.Andrews, D.W.K.Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econornetrica 55 (1987): 1465–1471.CrossRefGoogle Scholar
2.Andrews, D.W.K.Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory 4 (1988): 458–467.CrossRefGoogle Scholar
3.Andrews, D.W.K. Asymptotics for semiparametric econometric models: I. Estimation and testing. Cowles Foundation Discussion Paper No. 908Rs Yale University, 1988.Google Scholar
4.Andrews, D.W.K. Asymptotics for semiparametric econometric models: II. Stochastic equicontinuity and nonparametric kernel estimation. Cowles Foundation Discussion Paper No. 909R, Yale University, 1989.Google Scholar
5.Bierens, H.J.Model specification testing of time series regressions. Journal ofEconometrics 26 (1984): 323–353.CrossRefGoogle Scholar
6.Bierens, H.J. “Least Squares Estimation of Linear and Nonlinear ARMAX Models under Data Heterogeneity,” unpublished manuscript, Department of Econometrics, Free University, Amsterdam.Google Scholar
7.Billingsley, P.Convergence of Probability Measures. Wiley: New York, 1968.Google Scholar
8.Gallant, A.R.Nonlinear Statistical Models. Wiley: New York, 1987.CrossRefGoogle Scholar
9.Gallant, A.R. & White, H.. A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Basil Blackwell: Oxford, U.K., 1988.Google Scholar
10.Hannan, E.J.Nonlinear time series regression. Journal of Applied Probability 8 (1971): 767–780.CrossRefGoogle Scholar
11.Hansen, B.E.Strong laws for dependent heterogeneous processes. Econometric Theory 1(1991): 213–221.CrossRefGoogle Scholar
12.Jennrich, R.I.Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40 (1969): 633–643.CrossRefGoogle Scholar
13.Lukacs, E.Stochastic Convergence, 2nd ed. (1969): Academic Press: New York, 1975.Google Scholar
14.McLeish, D.L.A maximal inequality and dependent strong laws. Annals of Probability 3 (1975): 829–839.CrossRefGoogle Scholar
15.Newey, W.K.Uniform convergence in probability and stochastic equicontinuity. Econometrica 59 (1991): 1161–1167.CrossRefGoogle Scholar
16.Pollard, D.Empirical Processes: Theory and Applications. CBMS Regional Conference Series in Probability and Statistics, Volume 2, Institute of Mathematical Statistics: Hayward, CA, 1990.Google Scholar
17.Pötscher, B.M. & Prucha, I.R.. Consistency in nonlinear econometrics: A generic uniform law of large numbers and some comments on recent results, working paper no. 86–9, Department of Economics, University of Maryland, 1986.Google Scholar
18.Pötscher, B.M. & Prucha, I.R.. A uniform law of large numbers for dependent and heterogeneous dataprocesses. Econometrica 57 (1989): 675–683.CrossRefGoogle Scholar
19.Pötscher, B.M. & Prucha, I.R.. Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: Consistency and approximation concepts. Econometric Reviews 10 (1991): 125–216.CrossRefGoogle Scholar
20.Pötscher, B.M. & Prucha, I.R.. Generic uniform convergence and equicontinuity concepts for random functions: An exploration of the basic structure. Working paper no. 90–14, Department of Economics, University of Maryland, 1990.Google Scholar
21.Prohorov, Yu. V.Convergence of random processes and limit theorems in probability theory. Theory of Probability and Its Applications 1 (1956): 157214.CrossRefGoogle Scholar