Skip to main content
SpringerLink
Log in

Statistical Inference for Nonlinear Processes

  • Chapter
Book cover Long-Memory Processes

  • 4425 Accesses

Abstract

In this section, we consider nonlinear processes with long memory. We will mainly focus on volatility models: stochastic volatility (see Definitions 2.3–2.4 and Sect. 4.2.6 for limit theorems), ARCH(∞) processes (see Definition 2.1 and Sect. 4.2.7) and LARCH(∞) models (see (2.47) and (2.48), and Sect. 4.2.8). Statistical inference for traffic models is not well developed yet (see Faÿ et al. in Queueing Syst. 54(2):121–140, 2006, Bernoulli 13(2):473–491, 2007; Hsieh et al. in J. Econom. 141(2):913–949, 2007 for some results in this direction).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Adams, R. A., & Fournier, J. J. F. (2003). Sobolev spaces. Amsterdam: Academic Press.

    MATH  Google Scholar 

  • Arteche, J. (2004). Gaussian semiparametric estimation in long memory in stochastic volatility and signal plus noise models. Journal of Econometrics, 119(1), 131–154.

    Article  MathSciNet  Google Scholar 

  • Beran, J. (2006). On location estimation for LARCH processes. Journal of Multivariate Analysis, 97(8), 1766–1782.

    Article  MathSciNet  MATH  Google Scholar 

  • Beran, J., & Schützner, M. (2008). The effect of long memory in volatility on location estimation. Sankhya, Series B, 70(1), 84–112.

    MATH  Google Scholar 

  • Beran, J., & Schützner, M. (2009). On approximate pseudo maximum likelihood estimation for LARCH-processes. Bernoulli, 15(4), 1057–1081.

    Article  MathSciNet  MATH  Google Scholar 

  • Berkes, I., & Horváth, L. (2003). Asymptotic results for long memory LARCH sequences. The Annals of Applied Probability, 13, 641–668.

    Article  MathSciNet  MATH  Google Scholar 

  • Berkes, I., & Horváth, L. (2004). The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics, 32, 633–655.

    Article  MathSciNet  MATH  Google Scholar 

  • Berkes, I., Horváth, L., & Kokoszka, P. (2003). GARCH processes: structure and estimation. Bernoulli, 9, 201–228.

    Article  MathSciNet  MATH  Google Scholar 

  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.

    Article  MathSciNet  MATH  Google Scholar 

  • Breidt, F. J., Crato, N., & de Lima, P. (1998). On the detection and estimation of long memory in stochastic volatility. Journal of Econometrics, 83, 325–348.

    Article  MathSciNet  MATH  Google Scholar 

  • Dalla, V., Giraitis, L., & Hidalgo, J. (2006). Consistent estimation of the memory parameter for nonlinear time series. Journal of Time Series Analysis, 27, 211–251.

    Article  MathSciNet  MATH  Google Scholar 

  • Deo, R., & Hurvich, C. M. (2001). On the log periodogram regression estimator of the memory parameter in long memory stochastic volatility models. Econometric Theory, 17(4), 686–710.

    Article  MathSciNet  MATH  Google Scholar 

  • Doob, J. L. (1953). Stochastic processes. New York: Wiley.

    MATH  Google Scholar 

  • Drees, H. (1998). Optimal rates of convergence for estimates of the extreme value index. The Annals of Statistics, 26(1), 434–448.

    Article  MathSciNet  MATH  Google Scholar 

  • Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events. New York: Springer.

    Book  MATH  Google Scholar 

  • Faÿ, G., González-Arévalo, B., Mikosch, T., & Samorodnitsky, G. (2006). Modeling teletraffic arrivals by a Poisson cluster process. Queueing Systems, 54(2), 121–140.

    Article  MathSciNet  MATH  Google Scholar 

  • Faÿ, G., Roueff, F., & Soulier, P. (2007). Estimation of the memory parameter of the infinite source Poisson process. Bernoulli, 13(2), 473–491.

    Article  MathSciNet  MATH  Google Scholar 

  • Francq, C., & Zakoian, J.-M. (2008). Inconsistency of the QMLE and asymptotic normality of the weighted LSE for a class of conditionally heteroscedastic models (Working paper). Université Lille 3.

    Google Scholar 

  • Gelfand, I. M., & Shilov, G. E. (1966–1968). Generalized functions (Vols. 1–5). San Diego: Academic Press.

    Google Scholar 

  • Giraitis, L., & Robinson, P. M. (2001). Whittle estimation of ARCH models. Econometric Theory, 17, 608–631.

    Article  MathSciNet  MATH  Google Scholar 

  • Giraitis, L., Kokoszka, P., Leipus, R., & Teyssière, G. (2000b). Semiparametric estimation of the intensity of long memory in conditional heteroskedasticity. Statistical Inference for Stochastic Processes, 3(1–2), 113–128. 19th “Rencontres Franco-Belges de statisticiens” (Marseille, 1998).

    Article  MathSciNet  MATH  Google Scholar 

  • Giraitis, L., Kokoszka, P., Leipus, R., & Teyssiere, G. (2003). Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics, 112(2), 265–294.

    Article  MathSciNet  MATH  Google Scholar 

  • Giraitis, L., Leipus, R., & Surgailis, D. (2006). Recent advances in ARCH modelling. In G. Teyssière & A. P. Kirman (Eds.), Long memory in economics (pp. 3–38). Berlin: Springer.

    Google Scholar 

  • Hall, P., & Yao, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrika, 71, 285–317.

    Article  MathSciNet  MATH  Google Scholar 

  • Hosoya, Y. (1974). Estimation problems on stationary time series models. Ph.D. thesis, Yale.

    Google Scholar 

  • Hosoya, Y., & Taniguchi, M. (1982). A central limit theorem for stationary processes and the parameter estimation of linear processes. The Annals of Statistics, 10(1), 132–153.

    Article  MathSciNet  MATH  Google Scholar 

  • Hsieh, M.-C., Hurvich, C. M., & Soulier, P. (2007). Asymptotics for duration-driven long range dependent processes. Journal of Econometrics, 141(2), 913–949.

    Article  MathSciNet  Google Scholar 

  • Hurvich, C. M., & Soulier, P. (2002). Testing for long memory in volatility. Econometric Theory, 18(6), 1291–1308.

    Article  MathSciNet  MATH  Google Scholar 

  • Hurvich, C. M., Moulines, E., & Soulier, P. (2005b). Estimating long memory in volatility. Econometrica, 73(4), 1283–1328.

    Article  MathSciNet  MATH  Google Scholar 

  • Jach, A., & Kokoszka, P. (2008). Wavelet-domain test for long-range dependence in the presence of a trend. Statistics: A Journal of Theoretical and Applied Statistics, 42, 101–113.

    MathSciNet  MATH  Google Scholar 

  • Jach, A., McElroy, T., & Politis, D. N. (2012). Subsampling inference for the mean of heavy-tailed long memory time series. Journal of Time Series Analysis, 33, 96–111.

    Article  MathSciNet  Google Scholar 

  • Kanwal, R. P. (2004). Generalized functions: theory and applications (3rd ed.). Boston: Birkhäuser.

    MATH  Google Scholar 

  • Krengel, U. (1985) De Gruyter studies in mathematics: Vol. 6. Ergodic theorems.

    Book  MATH  Google Scholar 

  • Kulik, R., & Soulier, P. (2011). The tail empirical process for long memory stochastic volatility sequences. Stochastic Processes and Their Applications, 121(1), 109–134.

    Article  MathSciNet  MATH  Google Scholar 

  • Lee, S.-W., & Hansen, B. E. (1994). Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econometric Theory, 10, 29–52.

    Article  MathSciNet  MATH  Google Scholar 

  • Lighthill, M. J. (1958). Cambridge monographs on mechanics. An introduction to fourier analysis and generalised functions. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Lumsdaine, R. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1, 1) and covariance stationary GARCH(1, 1) models. Econometrica, 64, 575–596.

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, L. (2011). High quantile estimation for some stochastic volatility models. M.Sc. thesis, University of Ottawa.

    Google Scholar 

  • Petersen, K. (1989). Cambridge studies in advanced mathematics: Vol. 2. Ergodic theory. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Resnick, S. I. (1997). Heavy tail modelling and teletraffic data: special invited paper. The Annals of Statistics, 25(5), 1805–1869.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, P. M., & Zaffaroni, P. (1997). Modelling nonlinearity and long memory in time series. Fields Institute Communications, 11, 161–170.

    MathSciNet  Google Scholar 

  • Robinson, P. M., & Zaffaroni, P. (1998). Nonlinear time series with long memory: a model for stochastic volatility. Journal of Statistical Planning and Inference, 68, 359–371.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, P. M., & Zaffaroni, P. (2006). Pseudo-maximum likelihood estimation of ARCH(∞) models. The Annals of Statistics, 34(3), 1049–1074.

    Article  MathSciNet  MATH  Google Scholar 

  • Stout, W. F. (1974). Almost sure convergence. New York: Academic Press.

    MATH  Google Scholar 

  • Straumann, D. (2004). Lecture notes in statistics: Vol. 181. Estimation in conditionally heteroskedastic time series models. New York: Springer.

    Google Scholar 

  • Strichartz, R. (1994). A guide to distribution theory and Fourier transforms. Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Teyssière, G., & Abry, P. (2006). Wavelet analysis of nonlinear long–range dependent processes. Applications to financial time series. In G. Teyssiere & A. Kirman (Eds.), Long memory in economics (pp. 173–238). Berlin: Springer.

    Google Scholar 

  • Truquet, L. (2008). A new smoothed QMLE for AR processes with LARCH errors (Working paper). Université Panthéon-Sorbonne Paris I.

    Google Scholar 

  • Vladimirov, V. S. (2002). Methods of the theory of generalized functions. London: Taylor & Francis.

    MATH  Google Scholar 

  • Walters, P. (2000). An introduction to ergodic theory. New York: Springer.

    MATH  Google Scholar 

  • Weiss, A. A. (1986). Asymptotic theory for ARCH models: estimation and testing. Econometric Theory, 2, 107–131.

    Article  Google Scholar 

  • Wright, J. H. (2002). Log-periodogram estimation of long memory volatility dependencies with conditionally heavy tailed returns. Econometric Reviews, 21(4), 397–417.

    Article  MathSciNet  MATH  Google Scholar 

  • Zaffaroni, P. (2009). Whittle estimation of EGARCH and other exponential volatility models. Journal of Econometrics, 151, 190–200.

    Article  MathSciNet  Google Scholar 

  • Zemanian, A. H. (2010). Distribution theory and transform analysis: an introduction to generalized functions, with applications. New York: Dover.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics and Statistics, University of Konstanz, Konstanz, Germany

    Jan Beran

  2. Faculty of Business Administration and Economics, University of Paderborn, Paderborn, Germany

    Yuanhua Feng

  3. Swiss Federal Research Institute WSL, Birmensdorf, Switzerland

    Sucharita Ghosh

  4. Dept. of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada

    Rafal Kulik

Authors
  1. Jan Beran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuanhua Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sucharita Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rafal Kulik
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Beran, J., Feng, Y., Ghosh, S., Kulik, R. (2013). Statistical Inference for Nonlinear Processes. In: Long-Memory Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35512-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation