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Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

Author

Listed:
  • Anderson, T. W.
  • Kunitomo, Naoto

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.

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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    Citations

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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, November.
    2. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2008. "On the Asymptotic Optimality of the LIML Estimator with Possibly Many Instruments," CIRJE F-Series CIRJE-F-542, CIRJE, Faculty of Economics, University of Tokyo.
    3. Mynbayev, Kairat, 2007. "OLS Asymptotics for Vector Autoregressions with Deterministic Regressors," MPRA Paper 101688, University Library of Munich, Germany, revised 2018.
    4. 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.
    5. Takamitsu Kurita & Bent Nielsen, 2019. "Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms," Econometrics, MDPI, vol. 7(4), pages 1-35, October.
    6. Aiting Shen, 2019. "Asymptotic properties of LS estimators in the errors-in-variables model with MD errors," Statistical Papers, Springer, vol. 60(4), pages 1193-1206, August.
    7. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2006. "A New Light from Old Wisdoms : Alternative Estimation Methods of Simultaneous Equations with Possibly Many Instruments," CIRJE F-Series CIRJE-F-399, CIRJE, Faculty of Economics, University of Tokyo.
    8. Naoto Kunitomo & T. W. Anderson, 2007. "On Likelihood Ratio Tests of Structural Coefficients: Anderson-Rubin (1949) revisited," CIRJE F-Series CIRJE-F-499, CIRJE, Faculty of Economics, University of Tokyo.
    9. 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.
    10. 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.
    11. Takamitsu Kurita & B. Nielsen, 2018. "Partial cointegrated vector autoregressive models with structural breaks in deterministic terms," Economics Papers 2018-W03, Economics Group, Nuffield College, University of Oxford.

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