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ARCH/GARCH with persistent covariate: Asymptotic theory of MLE

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  • Han, Heejoon
  • Park, Joon Y.

Abstract

The paper considers a volatility model which introduces a persistent, integrated or near-integrated, covariate to the standard GARCH(1, 1) model. For such a model, we derive the asymptotic theory of the quasi-maximum likelihood estimator. In particular, we establish consistency and obtain limit distribution. The limit distribution is generally non-Gaussian and represented as a functional of Brownian motions. However, it becomes Gaussian if the covariate has innovation uncorrelated with the squared innovation of the model or the volatility function is linear in parameter. We provide a simulation study to demonstrate the relevance and usefulness of our asymptotic theory.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 167 (2012)
Issue (Month): 1 ()
Pages: 95-112

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Handle: RePEc:eee:econom:v:167:y:2012:i:1:p:95-112

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Web page: http://www.elsevier.com/locate/jeconom

Related research

Keywords: ARCH; GARCH; Persistent covariate; Maximum likelihood estimator; Asymptotic distribution theory;

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  1. Han, Heejoon & Park, Joon Y., 2008. "Time series properties of ARCH processes with persistent covariates," Journal of Econometrics, Elsevier, vol. 146(2), pages 275-292, October.
  2. Francq, Christian & Zakoian, Jean-Michel, 2007. "Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1265-1284, September.
  3. R. F. Engle & A. J. Patton, 2001. "What good is a volatility model?," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 237-245.
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  5. He, Changli & Ter svirta, Timo, 1999. "FOURTH MOMENT STRUCTURE OF THE GARCH(p,q) PROCESS," Econometric Theory, Cambridge University Press, vol. 15(06), pages 824-846, December.
  6. Lawrence R. Glosten & Ravi Jagannathan & David E. Runkle, 1993. "On the relation between the expected value and the volatility of the nominal excess return on stocks," Staff Report 157, Federal Reserve Bank of Minneapolis.
  7. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
  8. Park, Joon Y., 2002. "Nonstationary nonlinear heteroskedasticity," Journal of Econometrics, Elsevier, vol. 110(2), pages 383-415, October.
  9. Jensen, S ren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(06), pages 1203-1226, December.
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  11. Park, Joon, 2003. "Weak Unit Roots," Working Papers 2003-17, Rice University, Department of Economics.
  12. Liudas Giraitis & Peter M Robinson, 2000. "Whittle Estimation of ARCH Models," STICERD - Econometrics Paper Series /2000/406, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  13. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  14. Nelson, Daniel B., 1990. "Stationarity and Persistence in the GARCH(1,1) Model," Econometric Theory, Cambridge University Press, vol. 6(03), pages 318-334, September.
  15. Gray, Stephen F., 1996. "Modeling the conditional distribution of interest rates as a regime-switching process," Journal of Financial Economics, Elsevier, vol. 42(1), pages 27-62, September.
  16. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(01), pages 29-52, March.
  17. Hagiwara, May & Herce, Miguel A, 1999. "Endogenous Exchange Rate Volatility, Trading Volume and Interest Rate Differentials in a Model of Portfolio Selection," Review of International Economics, Wiley Blackwell, vol. 7(2), pages 202-18, May.
  18. Lumsdaine, Robin L, 1996. "Consistency and Asymptotic Normality of the Quasi-maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models," Econometrica, Econometric Society, vol. 64(3), pages 575-96, May.
  19. Søren Tolver Jensen & Anders Rahbek, 2004. "Asymptotic Normality of the QMLE Estimator of ARCH in the Nonstationary Case," Econometrica, Econometric Society, vol. 72(2), pages 641-646, 03.
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Cited by:
  1. Heejoon Han & Dennis Kristensen, 2013. "Asymptotic theory for the QMLE in GARCH-X models with stationary and non-stationary covariates," CeMMAP working papers CWP18/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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