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2013 | OriginalPaper | Buchkapitel

9. LLN, CLT and Ergodicity

verfasst von : Phoebus J. Dhrymes

Erschienen in: Mathematics for Econometrics

Verlag: Springer New York

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Abstract

We recall from Chap. 8 that discussion of random variables (r.v.) takes place in a probability space (Ω, \(\mathcal{A}\), \(\mathcal{P}\)), where Ω is the sample space, \(\mathcal{A}\) is the σ-algebra and \(\mathcal{P}\) is the probability measure.

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Vorheriges Kapitel Foundations of Probability

Nächstes Kapitel The General Linear Model

Part of this chapter is an adaptation of a set of Lectures given in the Spring of 2005, at the University of Cyprus, whose purpose was stated as “The purpose of these Lectures is to set forth, in a convenient fashion, the essential results from probability theory necessary to understand classical econometrics.” All page references, unless otherwise indicated, are to: Dhrymes (1989).

The notation a.c. means almost certainly; an alternative notation is a.s. which means almost surely. More generally, the notation, say, \(X_{n}\stackrel{\mathrm{a.c}}{\rightarrow }X_{0}\), or \(X_{n}\stackrel{\mathrm{P}}{\rightarrow }X_{0}\) means that the sequence of random variables {X _n: n ≥ 1} converges to the random variable X ₀, respectively, almost certainly or in probability. These concepts will be defined more precisely below.

This is a summary of relevant results presented earlier in Chap. 8.

The implication in item (3) is easily established using Proposition 8.12, Chap. 8, (Generalized Chebyshev Inequality) as follows: define \(Y _{n} = \vert X_{n} - X_{0}{\vert }^{p}\) and note that the Y-sequence consists of non-negative integrable rvs obeying EY _n = s _n, such that s _n converges to zero with n.

Strictly speaking, this and the preceding are implied by the following condition, which states that the sequence of rvs in question possesses finite (2 + δ)th moments, for arbitrary δ > 0. Moreover this last requirement implies the Lindeberg condition.

The function I in the equation below is the indicator function, which assumes the value 1, if \(\vert X_{in}\vert \geq \frac{1} {r}\) and the value zero otherwise.

This discussion owes a great deal to Billingsley (1995), Shiryayev (1984), and Stout (1974) which contain a very lucid description of the concepts and issues involved in ergodicity, including the role played by Kolmogorov’s extension theorem we discussed in Chap. 8 (Proposition 8.6), and Kolmogorov’s Zero-One Law, Proposition 8.25.

This is a property referred to under certain circumstances as mixing.

See Billingsley (1995), p. 325, who also gives a counterexample of an ergodic sequence which is not mixing.

Actually if we wished we could invoke the earlier discussion on ergodicity to argue that the convergence is actually with probability one, which of course implies convergence in probability.

Anderson, T.W. and H. Rubin (1949), Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations, Annals of Mathematical Statistics, pp. 46–63.

Anderson, T.W. and H. Rubin (1950), The Asymptotic Properties of Estimates of Parameters of in a Complete System of Stochastic Equations, Annals of Mathematical Statistics, pp. 570–582.

Balestra, P., & Nerlove, M. (1966). Pooling cross section time series data in the estimation of a dynamic model: The demand for natural gas. Econometrica, 34, 585–612.CrossRef

Bellman, R. G. (1960). Introduction to matrix analysis. New York: McGraw-Hill.MATH

Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.MATH

Billingsley, P. (1995). Probability and measure (3rd ed.). New York: Wiley.MATH

Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods (2nd ed.). New York: Springer-Verlag.CrossRef

Chow, Y. S., & Teicher, H. (1988). Probability theory (2nd ed.). New York: Springer-Verlag.CrossRefMATH

Dhrymes, P. J. (1969). Alternative asymptotic tests of significance and related aspects of 2SLS and 3SLS estimated parameters. Review of Economic Studies, 36, 213–226.CrossRef

Dhrymes, P. J. (1970). Econometrics: Statistical foundations and applications. New York: Harper and Row; also (1974). New York: Springer-Verlag.

Dhrymes, P. J. (1973). Restricted and Unrestricted Reduced Forms: Asymptotic Distributions and Relative Efficiencies, Econometrica, vol. 41, pp. 119–134.MathSciNetCrossRefMATH

Dhrymes, P. J. (1978). Introductory economics. New York: Springer-Verlag.CrossRef

Dhrymes, P.J. (1982) Distributed Lags: Problems of Estmation and Formulation (corrected edition) Amsterdam: North Holland

Dhrymes, P. J. (1989). Topics in advanced econometrics: Probability foundations. New York: Springer-Verlag.CrossRefMATH

Dhrymes, P. J. (1994). Topics in advanced econometrics: Volume II linear and nonlinear simultaneous equations. New York: Springer-Verlag.CrossRefMATH

Hadley, G. (1961). Linear algebra. Reading: Addison-Wesley.MATH

Kendall, M. G., & Stuart, A. (1963). The advanced theory of statistics. London: Charles Griffin.

Kendall M. G., Stuart, A., & Ord, J. K. (1987). Kendall’s advanced theory of statistics. New York: Oxford University Press.MATH

Kolassa, J. E. (1997). Series approximation methods in statistics (2nd ed.). New York: Springer-Verlag.CrossRefMATH

Sims, C.A. (1980). Macroeconomics and Reality, Econometrica, vol. 48, pp.1–48.

Shiryayev, A. N. (1984). Probability. New York: Springer-Verlag.CrossRefMATH

Stout, W. F. (1974). Almost sure convergence. New York: Academic.MATH

Theil, H. (1953). Estimation and Simultaneous Correlation in Complete Equation Systems, mimeograph, The Hague: Central Plan Bureau.

Theil, H. (1958). Economic Forecasts and Policy, Amsterdam: North Holland.

Titel: LLN, CLT and Ergodicity
verfasst von: Phoebus J. Dhrymes
Verlag: Springer New York
Buch: Mathematics for Econometrics
Print ISBN: 978-1-4614-8144-7

Electronic ISBN: 978-1-4614-8145-4

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-8145-4_9