Advertisement

Journal of Statistical Theory and Practice

, Volume 1, Issue 2, pp 149–166 | Cite as

Weighted Averages and Local Polynomial Estimation for Fractional Linear ARCH Processes

  • Jan Beran
  • Yuanhua Feng
Article

Abstract

We consider local polynomial regression estimation for time series defined by Yi = μ(ti)+Ui where Ui is a stationary zero mean process with Wold decomposition Ui = C(B)Zi. The innovations Zi are assumed to be generated by a LARCH process, and may thus exhibit long-range correlations in volatility. The linear dependence structure, defined by the filter C(B) includes short memory, long memory and antipersistence. A central limit theorem for weighted averages, with triangular arrays of weights, and a limit theorem for local polynomial estimates of μ(ν)(i) are derived. The asymptotic distribution of μ(ν)(ti) turns out to be unaffected by long-range dependence in volatility. The question of optimal regression weights is also addressed.

AMS Subject Classification

62G08 and 62M10 

Keywords

long memory LARCH process volatility central limit theorem location estimation local polynomial estimation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arcones, M.A., Bin Yu, 1994. Limit theorems for empirical processes under dependence. In: Chaos, Expansions, Multiple Wiener-Itô-Integrals and their Applications. CRC Press, London.Google Scholar
  2. Avram, F., Taqqu, M.S., 1987. Noncentral limit theorems and Appell polynomials. Annals of Probability, 15, 767–775.MathSciNetCrossRefGoogle Scholar
  3. Balkema, A.A., Klüppelberg, C., Resnick, S.I., 1993. Densities with Gaussian tails. Proc. Lond. Math. Soc., III., Ser. 66, No. 3, 568–588.MathSciNetCrossRefGoogle Scholar
  4. Barndorff-Nielsen, O.E., Klüppelberg, C., 1992. A note on the tail accuracy of the univariate saddlepoint approximation. Annales de la Faculté des Sciences de Toulouse, Série 6, Vol. I, No. 1, 5–14.MathSciNetCrossRefGoogle Scholar
  5. Beran, J., 1994. Statistics for long-memory processes. Chapman & Hall, London.zbMATHGoogle Scholar
  6. Beran, J., Feng, Y., 2001. Local polynomial estimation with a FARIMA-GARCH error process. Bernoulli, 7, 733–750.MathSciNetCrossRefGoogle Scholar
  7. Beran, J., Feng, Y., 2002. Local polynomial fitting with long-memory, short-memory and antipersistent errors. Annals of the Institute of Statistical Mathematics, 54, 291–311.MathSciNetCrossRefGoogle Scholar
  8. Berkes, I., Horvath, L., 2003. Asymptotic results for long memory LARCH sequences. Annals of Applied Probability, 13, No. 2, 641–668.MathSciNetzbMATHGoogle Scholar
  9. Dehling, H., Taqqu, M.S., 1989. The empirical process of some long-range dependent sequences with applications to U-statistics. Annals of Statistics, 17, No. 4, 1767–1783.MathSciNetCrossRefGoogle Scholar
  10. Dobrushin, R.L., Major, P., 1979. Non-central limit theorems for non-linear functionals of Gaussian fields. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 50, 27–52.MathSciNetCrossRefGoogle Scholar
  11. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling extremal events. Springer, New York.CrossRefGoogle Scholar
  12. Fan, J., Gasser, Th., Gijbels, I., Brockmann, M., Engel, J., 1997. Local polynomial regression: optimal kernels and asymptotic minimax efficiency. Annals of the Institute of Statistical Mathematics, 49, 79–99.MathSciNetCrossRefGoogle Scholar
  13. Feng, Y., 2005. On the asymptotic variance in nonparametric regression with fractional time series errors. Preprint, Heriot-Watt University.Google Scholar
  14. Giraitis, L., 1983. Convergence of some nonlinear transformations of a Gaussian sequence to self-similar processes. Litovskii Matematicheskii Sbornik, 23, No. 1, 58–68.MathSciNetGoogle Scholar
  15. Giraitis, L., 1985. Central limit theorems for functionals of a linear process. Litivskii Matematickii Sbornik, 25, 43–57.MathSciNetGoogle Scholar
  16. Giraitis, L., Robinson, P. M., Surgailis, D., 2000. A model for long memory conditional heteroskedasticity. Annals of Applied Probability, 10, 1002–1024.MathSciNetCrossRefGoogle Scholar
  17. Giraitis, L., Surgailis, D., 1985. Central limit theorems and other limit theorems for functionals of Gaussian processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 70, 191–212.MathSciNetCrossRefGoogle Scholar
  18. Giraitis, L., Surgailis, D., 1986. Multivariate Appell polynomials and the central limit theorem. Dependence in Probability and Statistics, E. Eberlein and M.S. Taqqu (eds.), Birkhäuser, Boston, pp. 21–27.CrossRefGoogle Scholar
  19. Giraitis, L., Surgailis, D., 1999. Central limit theorem for the empirical process of a linear sequence with long memory. Journal of Statistical Planning and Inference, 80, 81–93.MathSciNetCrossRefGoogle Scholar
  20. Granger, C.W.J., Joyeux, R., 1980. An introduction to long-range time series models and fractional differencing. Journal of Time Series Analysis, 1, 15–30.MathSciNetCrossRefGoogle Scholar
  21. Haiman, G., Habach, L., 1999. Almost sure behaviour of extremes of m-dependent stationary sequences. Comptes Rendus de l’Académie des Sciences de Paris, Series I, Mathematics, Volume 329, Issue 10, 887–892.MathSciNetzbMATHGoogle Scholar
  22. Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and Its Application. Academic Press, New York.zbMATHGoogle Scholar
  23. Ho, H.-C., Hsing, T., 1996. On the asymptotic expansion of the empirical process of long-memory moving averages. Annals of Statistics, 24, 11725–1739.MathSciNetGoogle Scholar
  24. Ho, H.-C., Hsing, T., 1997. Limit theorems for functionals of moving averages. Annals of Probability, 25, no. 4, 1636–1669.MathSciNetCrossRefGoogle Scholar
  25. Hosking, J.R.M., 1981. Fractional differencing. Biometrika, 68, 165–176.MathSciNetCrossRefGoogle Scholar
  26. Müller, H.G., 1987. Weighted local regression and kernel methods for nonparametric curve fitting. Journal of the American Statistical Association, 82 231–238.Google Scholar
  27. Robinson, P. M., 1991. Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47, 67–84.MathSciNetCrossRefGoogle Scholar
  28. Rosenblatt, M., 1961. Independence and dependence. In: Proc. 4th Berkeley Symposium, J. Neyman (ed.), Univ. California, Berkeley Univ. Press, Berkeley, pp. 431–443.Google Scholar
  29. Ruppert, D., Wand, M.P., 1994. Multivariate locally weighted least squares regression. Annals of Statistics, 22 1346–1370.MathSciNetCrossRefGoogle Scholar
  30. Surgailis, D., 1981. Convergence of sums of nonlinear functions of moving averages to self-similar processes. Soviet Math. Doklady, 23, 247–250.zbMATHGoogle Scholar
  31. Surgailis, D., 1982. Zones of attraction of self-similar multiple integrals. Lithuanian Mathematical Journal, 22, 327–340.CrossRefGoogle Scholar
  32. Taqqu, M.S., 1975. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 31, 287–302.MathSciNetCrossRefGoogle Scholar
  33. Taqqu, M.S., 1979. Weak convergence of integrated processes of arbitrary Hermite rank. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 50, 53–83.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany
  2. 2.of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghUK

Personalised recommendations