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Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances


  • Anderson, T. W.
  • Kunitomo, Naoto


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.

Suggested Citation

  • Anderson, T. W. & Kunitomo, Naoto, 1992. "Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances," Journal of Multivariate Analysis, Elsevier, vol. 40(2), pages 221-243, February.
  • Handle: RePEc:eee:jmvana:v:40:y:1992:i:2:p:221-243

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    Cited by:

    1. Svetlana Borovkova & Hendrik P. LopuhaƤ & Budi Nurani Ruchjana, 2008. "Consistency and asymptotic normality of least squares estimators in generalized STAR models," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 62(4), pages 482-508.
    2. 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.
    3. Zhiyong Chen & Haibin Wang & Xuejun Wang, 2016. "The consistency for the estimator of nonparametric regression model based on martingale difference errors," Statistical Papers, Springer, vol. 57(2), pages 451-469, April.
    4. 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.
    5. 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.


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