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STATIONARY INTEGRATED ARCH(∞) AND AR(∞) PROCESSES WITH FINITE VARIANCE

Published online by Cambridge University Press:  17 October 2017

Liudas Giraitis ,
Donatas Surgailis  and
Andrius Škarnulis
Liudas Giraitis*
Affiliation:
Queen Mary, London University
Donatas Surgailis
Affiliation:
Vilnius University
Andrius Škarnulis
Affiliation:
Vilnius University
*
*Address correspondence to Liudas Giraitis, School of Economics and Finance, QMUL, Mile End Road, London E1 4NS, UK; e-mail: l.giraitis@qmul.ac.uk.
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Abstract

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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Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 6 , December 2018 , pp. 1159 - 1179
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

The authors are grateful to three anonymous referees, the Co-editor, and the Editor for insightful comments, suggestions, and useful criticisms.

References

REFERENCES

Abadir, K.M., Distaso, W., Giraitis, L., & Koul, H.L. (2014) Asymptotic normality for weighted sums of linear processes. Econometric Theory 30, 252–284.Google Scholar
Baillie, R.T., Bollerslev, T., & Mikkelsen, H.O. (1996) Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74, 3–30.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer-Verlag.CrossRefGoogle Scholar
Davidson, J. (2004) Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business & Economic Statistics 22, 16–29.CrossRefGoogle Scholar
Ding, Z. & Granger, C.W.J. (1996) Modeling volatility persistence of speculative returns: A new approach. Journal of Econometrics 73, 185–215.CrossRefGoogle Scholar
Douc, R., Roueff, F., & Soulier, P. (2008) On the existence of some ARCH(∞) processes. Stochastic Processes and their Applications 118, 755–761.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.Google Scholar
Giraitis, L., Kokoszka, P., & Leipus, R. (2000a) Stationary ARCH models: Dependence structure and central limit teorem. Econometric Theory 16, 3–22.Google Scholar
Giraitis, L., Robinson, P.M., & Surgailis, D. (2000b) A model for long memory conditional heteroscedasticity. Annals of Applied Probabability 10, 1002–1024.Google Scholar
Giraitis, L. & Surgailis, D. (2002) ARCH-type bilinear models with double long memory. Stochastic Processes and their Applications 100, 275–300.Google Scholar
Giraitis, L., Leipus, R., Robinson, P.M., & Surgailis, D. (2004) LARCH, leverage and long memory. Journal of Financial Econometrics 2, 177–210.CrossRefGoogle Scholar
Giraitis, L., Leipus, R., & Surgailis, D. (2007) Recent advances in ARCH modelling. In Teyssière, G. & Kirman, A.P. (eds.), Long Memory in Economics, pp. 3–38. Springer-Verlag.Google Scholar
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. 71–84. Springer-Verlag.Google Scholar
Giraitis, L., Koul, H.L., & Surgailis, D. (2012) Large Sample Inference for Long Memory Processes. Imperial College Press.Google Scholar
Granger, C.W.J. & Andersen, A.P. (1978) An Introduction to Bilinear Time Series Models. Vandenhoek and Ruprecht.Google Scholar
Granger, C.W.J. & Ding, Z. (1995) Some properties of absolute return: An alternative measure of risk. Annales d’ Economie et de Statistique 40, 67–91.CrossRefGoogle Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.Google Scholar
Ibragimov, I.A. & Linnik, Yu.V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff.Google Scholar
Kazakevičius, V. & Leipus, R. (2002) On stationarity in the ARCH(∞) model. Econometric Theory 18, 1–16.Google Scholar
Kazakevičius, V. & Leipus, R. (2003) A new theorem on existence of invariant distributions with applications to ARCH processes. Journal of Applied Probability 40, 147–162.CrossRefGoogle Scholar
Koulikov, D. (2003) Long memory ARCH(∞) models: Specification and quasi-maximum likelihood estimation. Working paper 163, Centre for Analytical Finance, Univ. Aarhus. Available at www.cls.dk/caf/wp/wp-163.pdf.Google Scholar
Lieberman, O. & Phillips, P.C.B. (2008) Refined inference on long memory in realized volatility. Econometric Reviews 27, 254–267.Google Scholar
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. 149–168. Risk Books.Google Scholar
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. 439–459. Birkhäuser.Google Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.Google Scholar
Robinson, P.M. & Zaffaroni, P. (2006) Pseudo-maximum likelihood estimation of ARCH(1) models. Annals of Statististics 34, 1049–1074.Google Scholar
Rudin, W. (1987) Real and Complex Analysis. McGraw-Hill Book Company.Google Scholar
Subba Rao, T. (1981) On the theory of bilinear time series models. Journal of Royal Statistical Society B 43, 244–255.Google Scholar