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Asymptotic Theory for a Vector ARMA-GARCH Model

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  • Shiqing Ling
  • Michael McAleer

Abstract

This paper investigates the asymptotic theory for a vector ARMA-GARCH model. The conditions for the strict stationarity, ergodicity, and the higherorder moments of the model are established. Consistency of the quasi- maximum likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate ARCH and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors, and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models, as well as a consistent estimator of the asymptotic covariance.

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File URL: http://www.iser.osaka-u.ac.jp/library/dp/2001/dp0549.pdf
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Paper provided by Institute of Social and Economic Research, Osaka University in its series ISER Discussion Paper with number 0549.

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Date of creation: Jul 2001
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Handle: RePEc:dpr:wpaper:0549

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  1. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  2. Tse, Y. K., 2000. "A test for constant correlations in a multivariate GARCH model," Journal of Econometrics, Elsevier, vol. 98(1), pages 107-127, September.
  3. Shiqing Ling & Michael McAleer, 2001. "On Adaptive Estimation in Nonstationary ARMA Models with GARCH Errors," ISER Discussion Paper 0548, Institute of Social and Economic Research, Osaka University.
  4. J. Pfanzagl, 1969. "On the measurability and consistency of minimum contrast estimates," Metrika, Springer, vol. 14(1), pages 249-272, December.
  5. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
  6. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(01), pages 70-86, February.
  7. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(01), pages 107-131, April.
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