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LARCH, Leverage, and Long Memory

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  • Liudas Giraitis

Abstract

We consider the long-memory and leverage properties of a model for the conditional variance V-sub-t-super-2 of an observable stationary sequence X-sub-t, where V-sub-t-super-2 is the square of an inhomogeneous linear combination of X-sub-s, s < t, with square summable weights b-sub-j. This model, which we call linear autoregressive conditionally heteroskedastic (LARCH), specializes, when V-sub-t-super-2 depends only on X-sub-t - 1, to the asymmetric ARCH model of Engle (1990, Review of Financial Studies 3, 103--106), and, when V-sub-t-super-2 depends only on finitely many X-sub-s, to a version of the quadratic ARCH model of Sentana (1995, Review of Economic Studies 62, 639--661), these authors having discussed leverage potential in such models. The model that we consider was suggested by Robinson (1991, Journal of Econometrics 47, 67--84), for use as a possibly long-memory conditionally heteroskedastic alternative to i.i.d. behavior, and further studied by Giraitis, Robinson and Surgailis (2000, Annals of Applied Probability 10, 1002--1004), who showed that integer powers X-sub-t-super-ℓ, ℓ ≥ 2 can have long-memory autocorrelations. We establish conditions under which the cross-autocovariance function between volatility and levels, h-sub-t = covV-sub-t-super-2,X-sub-0, decays in the manner of moving average weights of long-memory processes on suitable choice of the b-sub-j. We also establish the leverage property that h-sub-t < 0 for 0 < t ≤ k, where the value of k (which may be infinite) again depends on the b-sub-j. Conditions for finiteness of third and higher moments of X-sub-t are also established. Copyright 2004, Oxford University Press.

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Bibliographic Info

Article provided by Society for Financial Econometrics in its journal Journal of Financial Econometrics.

Volume (Year): 2 (2004)
Issue (Month): 2 ()
Pages: 177-210

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Handle: RePEc:oup:jfinec:v:2:y:2004:i:2:p:177-210

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Citations

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Cited by:
  1. Michael McAller & Marcelo C. Medeiros, 2007. "A multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries," Textos para discussão 544, Department of Economics PUC-Rio (Brazil).
  2. Christian Francq & Jean-Michel Zakoïan, 2010. "Inconsistency of the MLE and inference based on weighted LS for LARCH models," Post-Print hal-00732536, HAL.
  3. Feng, Yuanhua & Beran, Jan & Yu, Keming, 2006. "Modelling financial time series with SEMIFAR-GARCH model," MPRA Paper 1593, University Library of Munich, Germany.
  4. Mohamed Boutahar & Rabeh Khalfaoui2, 2011. "Estimation of the long memory parameter in non stationary models: A Simulation Study," Working Papers halshs-00595057, HAL.
  5. Bildirici, Melike & Ersin, Özgür, 2012. "Nonlinear volatility models in economics: smooth transition and neural network augmented GARCH, APGARCH, FIGARCH and FIAPGARCH models," MPRA Paper 40330, University Library of Munich, Germany, revised May 2012.
  6. Tomasz Wojtowicz & Henryk Gurgul, 2009. "Long memory of volatility measures in time series," Operations Research and Decisions, Wroclaw University of Technology, Institute of Organization and Management, vol. 1, pages 37-54.
  7. Axel Groß‐KlußMann & Nikolaus Hautsch, 2013. "Predicting Bid–Ask Spreads Using Long‐Memory Autoregressive Conditional Poisson Models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 32(8), pages 724-742, December.
  8. Agnieszka Jach & Piotr Kokoszka, 2010. "Empirical wavelet analysis of tail and memory properties of LARCH and FIGARCH models," Computational Statistics, Springer, vol. 25(1), pages 163-182, March.
  9. Conrad, Christian & Karanasos, Menelaos, 2006. "The impulse response function of the long memory GARCH process," Economics Letters, Elsevier, vol. 90(1), pages 34-41, January.
  10. Josu Arteche, 2012. "Standard and seasonal long memory in volatility: an application to Spanish inflation," Empirical Economics, Springer, vol. 42(3), pages 693-712, June.
  11. Beran, Jan, 2006. "On location estimation for LARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1766-1782, September.
  12. Doukhan, Paul & Wintenberger, Olivier, 2008. "Weakly dependent chains with infinite memory," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 1997-2013, November.
  13. Iqbal Owadally, 2014. "Tail risk in pension funds: an analysis using ARCH models and bilinear processes," Review of Quantitative Finance and Accounting, Springer, vol. 43(2), pages 301-331, August.

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