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Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

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  • Anderson, T. W.
  • Kunitomo, Naoto
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    Abstract

    In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is yt = Bzt + vt. The unobserved error sequence {vt} is a sequence of martingale differences with conditional covariance matrices {[Sigma]t} and satisfying supt=1,..., n3{v'tvtI(v'tvt>a) zt, vt-1, zt-1, ...} 0 0 as a --> [infinity]. The sample covariance of the independent variables z1, ..., zn, is assumed to have a probability limit M, constant and nonsingular; maxt=1,...,nz'tzt/n0 0. If (1/n)[Sigma]t=1n[Sigma]t0[Sigma], constant, then [radical sign]nvec(Bn-B)0N(0,M-1[circle times operator][Sigma]) and [Sigma]n0[Sigma]. The autoregression model is xt = Bxt - 1 + vt with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {vt}, and (1/n)[Sigma]t=max(r,s)+1([Sigma]t[circle times operator]vt-1-rv't-1-s)0 [delta]rs([Sigma][circle times operator][Sigma]), where [delta]rs is the Kronecker delta. Then [radical sign]nvec(Bn-B)0N(0,[Gamma]-1[circle times operator][Sigma]), where [Gamma] = [Sigma]s = 0[infinity]Bs[Sigma](B')s.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 40 (1992)
    Issue (Month): 2 (February)
    Pages: 221-243

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    Handle: RePEc:eee:jmvana:v:40:y:1992:i:2:p:221-243

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    Keywords: asymptotic distribution Lindeberg condition central limit theorem regression coefficients autoregression coefficients;

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    Cited by:
    1. Anderson, T.W. & Kunitomo, Naoto & Matsushita, Yukitoshi, 2010. "On the asymptotic optimality of the LIML estimator with possibly many instruments," Journal of Econometrics, Elsevier, vol. 157(2), pages 191-204, August.
    2. Mynbaev, Kairat, 2003. "Asymptotic properties of OLS estimates in autoregressions with bounded or slowly growing deterministic trends," MPRA Paper 18448, University Library of Munich, Germany, revised 2005.
    3. Kunitomo, Naoto & Sato, Seisho, 1996. "Asymmetry in economic time series and the simultaneous switching autoregressive model," Structural Change and Economic Dynamics, Elsevier, vol. 7(1), pages 1-34, March.

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