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  • Liudas Giraitis (a1), Donatas Surgailis (a2) and Andrius Škarnulis (a2)

We prove the long standing conjecture of Ding and Granger (1996) about the existence of a stationary Long Memory ARCH model with finite fourth moment. This result follows from the necessary and sufficient conditions for the existence of covariance stationary integrated AR(∞), ARCH(∞), and FIGARCH models obtained in the present article. We also prove that such processes always have long memory.

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*Address correspondence to Liudas Giraitis, School of Economics and Finance, QMUL, Mile End Road, London E1 4NS, UK; e-mail:
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The authors are grateful to three anonymous referees, the Co-editor, and the Editor for insightful comments, suggestions, and useful criticisms.

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Abadir K.M., Distaso W., Giraitis L., & Koul H.L. (2014) Asymptotic normality for weighted sums of linear processes. Econometric Theory 30, 252284.
Baillie R.T., Bollerslev T., & Mikkelsen H.O. (1996) Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74, 330.
Bollerslev T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307327.
Brockwell P.J. & Davis R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer-Verlag.
Davidson J. (2004) Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business & Economic Statistics 22, 16–29.
Ding Z. & Granger C.W.J. (1996) Modeling volatility persistence of speculative returns: A new approach. Journal of Econometrics 73, 185215.
Douc R., Roueff F., & Soulier P. (2008) On the existence of some ARCH(∞) processes. Stochastic Processes and their Applications 118, 755761.
Engle R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 9871008.
Giraitis L., Kokoszka P., & Leipus R. (2000a) Stationary ARCH models: Dependence structure and central limit teorem. Econometric Theory 16, 322.
Giraitis L., Robinson P.M., & Surgailis D. (2000b) A model for long memory conditional heteroscedasticity. Annals of Applied Probabability 10, 10021024.
Giraitis L. & Surgailis D. (2002) ARCH-type bilinear models with double long memory. Stochastic Processes and their Applications 100, 275300.
Giraitis L., Leipus R., Robinson P.M., & Surgailis D. (2004) LARCH, leverage and long memory. Journal of Financial Econometrics 2, 177210.
Giraitis L., Leipus R., & Surgailis D. (2007) Recent advances in ARCH modelling. In Teyssière G. & Kirman A.P. (eds.), Long Memory in Economics, pp. 338. Springer-Verlag.
Giraitis L., Leipus R., & Surgailis D. (2009) ARCH(∞) models and long memory properties. In Andersen T.G., Davis R.A., Kreiss J.-P., & Mikosch T. (eds.), Handbook of Financial Time Series, pp. 7184. Springer-Verlag.
Giraitis L., Koul H.L., & Surgailis D. (2012) Large Sample Inference for Long Memory Processes. Imperial College Press.
Granger C.W.J. & Andersen A.P. (1978) An Introduction to Bilinear Time Series Models. Vandenhoek and Ruprecht.
Granger C.W.J. & Ding Z. (1995) Some properties of absolute return: An alternative measure of risk. Annales d’ Economie et de Statistique 40, 6791.
Hosking J.R.M. (1981) Fractional differencing. Biometrika 68, 165176.
Ibragimov I.A. & Linnik Yu.V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff.
Kazakevičius V. & Leipus R. (2002) On stationarity in the ARCH(∞) model. Econometric Theory 18, 116.
Kazakevičius V. & Leipus R. (2003) A new theorem on existence of invariant distributions with applications to ARCH processes. Journal of Applied Probability 40, 147162.
Koulikov D. (2003) Long memory ARCH(∞) models: Specification and quasi-maximum likelihood estimation. Working paper 163, Centre for Analytical Finance, Univ. Aarhus. Available at
Lieberman O. & Phillips P.C.B. (2008) Refined inference on long memory in realized volatility. Econometric Reviews 27, 254267.
Mikosch T. & Stărică C. (2000) Is it really long memory we see in financial returns? In Embrechts P. (ed.), Extremes and Integrated Risk Management, London, pp. 149168. Risk Books.
Mikosch T. & Stărică C. (2003) Long-range dependence effects and ARCH modeling. In Doukhan P., Oppenheim G., & Taqqu M. S. (eds.), Theory and Applications of Long-Range Dependence, pp. 439459. Birkhäuser.
Robinson P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 6784.
Robinson P.M. & Zaffaroni P. (2006) Pseudo-maximum likelihood estimation of ARCH(1) models. Annals of Statististics 34, 10491074.
Rudin W. (1987) Real and Complex Analysis. McGraw-Hill Book Company.
Subba Rao T. (1981) On the theory of bilinear time series models. Journal of Royal Statistical Society B 43, 244255.
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Econometric Theory
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