Skip to main content
SpringerLink
Log in

Time Series with Stochastic Volatility

  • Chapter
  • First Online:
Book cover Statistics of Financial Markets

Part of the book series: Universitext ((UTX))

  • 3382 Accesses

Abstract

In the previous chapters we have already discussed that volatility plays an important role in modelling financial systems and time series. Unlike the term structure, volatility is unobservable and thus must be estimated from the data.

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 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight 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

  • Applebaum, D. (2004). Lévy processes and stochastic calculus. Cambridge University Press.

    Book  Google Scholar 

  • Bauwens, L., Laurent, S., & Rombouts, J. (2005). Multivariate GARCH models: A survey. Journal of Applied Econometrics, 21, 79–109.

    Article  MathSciNet  Google Scholar 

  • Bertoin, J. (1996). Lévy processes. Cambridge University Press.

    MATH  Google Scholar 

  • Black, F. (1976). Studies in stock price volatility changes. In Proceedings of the 1976 Meeting of the Business and Economic Statistics Section. American Statistical Association, pp. 177–181.

    Google Scholar 

  • Bollerslev, T., Engle, R., & Wooldridge, J. (1988). A capital asset pricing model with time varying covariances. Journal of Political Economy, 96, 116–131.

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Bollerslev, T. P. (1990). Modelling the coherence in short-run nominal exchange rates: A multivariate generalized arch model. Review of Economics and Statistics, 72, 498–505.

    Article  Google Scholar 

  • Bollerslev, T. P., & Wooldridge, J. M. (1992). Quasi maximum likelihood estimation of dynamic models with time-varying covariances. Econometric Reviews, 11, 143–172.

    Article  MathSciNet  Google Scholar 

  • Brockwell, P. J., Chadraa, E., & Lindner, A. (2006). Continuous time GARCH processes. Annals of Applied Probability, 16, 790–826.

    Article  MathSciNet  Google Scholar 

  • Buchmann, B., & Müller, G. (2008). Limit experiments of GARCH. Technical report. Monash University and Technische Universität München.

    Google Scholar 

  • Comte, F., & Lieberman, O. (2003). Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis, 84(1), 61–84.

    Article  MathSciNet  Google Scholar 

  • Czado, C., & Haug, S. (2007). Quasi maximum likelihood estimation and prediction in the compound poisson ECOGARCH(1,1) model. Technical report. Technische Universität München.

    Google Scholar 

  • Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1, 83–106.

    Article  Google Scholar 

  • Duan, J.-C. (1997). Augmented GARCH(p,q) process and its diffusion limit. Journal of Econometrics, 79, 97–127.

    Article  MathSciNet  Google Scholar 

  • Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events. Berlin: Springer-Verlag.

    Book  Google Scholar 

  • Engle, R. F. (1982a). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987–1008.

    Article  MathSciNet  Google Scholar 

  • Engle, R. F. (2002). Dynamic conditional correlation - a simple class of multivariate GARCH models. Journal of Business and Economic Statistics, 20, 339–350.

    Article  MathSciNet  Google Scholar 

  • Engle, R. F., Lilien, D. M., & Robins, R. P. (1987). Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica, 55, 391–407.

    Article  Google Scholar 

  • Engle, R. F., & Ng, V. K. (1993a). Measuring and testing the impact of news on volatility. Journal of Finance, 48, 1749–1778.

    Article  Google Scholar 

  • Engle, R., & Kroner, F. (1995). Multivariate simultaneous generalized arch. Econometric theory, 11, 122–150.

    Article  MathSciNet  Google Scholar 

  • Fama, E. F. (1965). The behavior of stock market prices. Journal of Business, 38, 34–105.

    Article  Google Scholar 

  • Fasen, V., Klüppelberg, C., & Lindner, A. (2005). Extremal behavior of stochastic volatility models. In A. N. Shiryaev, M. R. Grossinho, P. E. Oliviera, & M. L. Esquível (Eds.), Stochastic finance. Springer.

    Google Scholar 

  • Glosten, L., Jagannathan, R., & Runkle, D. (1993). On the relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779–1801.

    Article  Google Scholar 

  • Gouriéroux, C. (1997). ARCH models and financial applications. Springer-Verlag.

    Book  Google Scholar 

  • Gouriéroux, C., & Monfort, A. (1992). Qualitative threshold ARCH models. Journal of Econometrics, 52, 159–199.

    Article  MathSciNet  Google Scholar 

  • Gouriéroux, C., Monfort, A., & Trognon, A. (1984). Pseudo maximum likelihood methods: Theory. Econometrica, 52, 681–700.

    Article  MathSciNet  Google Scholar 

  • Hafner, C., & Herwartz, H. (2000). Testing linear autoregressive dynamics under heteroskedasticity. Econometrics Journal, 3, 177–197.

    Article  MathSciNet  Google Scholar 

  • Hafner, C. M., & Herwartz, H. (2008). Analytical quasi maximum likelihood inference in multivariate volatility models. Metrika, 67, 219–239.

    Article  MathSciNet  Google Scholar 

  • Härdle, W., Müller, M., Sperlich, S., & Werwatz, A. (2004). Non- and semiparametric modelling. Heidelberg: Springer-Verlag.

    Book  Google Scholar 

  • Haug, S., & Czado, C. (2007). An exponential continuous time GARCH process. Journal of Applied Probability, 44, 960–976.

    Article  MathSciNet  Google Scholar 

  • Haug, S., Klüppelberg, C., Lindner, A., & Zapp, M. (2007). Method of moments estimation in the COGARCH(1,1) model. Econometrics Journal, 10, 320–341.

    Article  MathSciNet  Google Scholar 

  • Haug, S., & Stelzer, R. (2010). Multivariate ECOGARCH processes. Econometric Theory.

    Google Scholar 

  • He, C., & Teräsvirta, T. (1999). Properties of moments of a family of GARCH processes. Journal of Econometrics, 92, 173–192.

    Article  MathSciNet  Google Scholar 

  • Hill, B. (1975). A simple general approach to inference about the tail of a distribution. Annals of Statistics, 3, 1163–1174.

    Article  MathSciNet  Google Scholar 

  • Jeantheau, T. (1998). Strong consistency of estimators for multivariate arch models. Econometric Theory, 14, 70–86.

    Article  MathSciNet  Google Scholar 

  • Kallsen, J., & Vesenmayer, B. (2009). COGARCH as a continuous-time limit of GARCH(1,1). Stochastic Processes and Their Applications, 119, 74–98.

    Article  MathSciNet  Google Scholar 

  • Klüppelberg, C., Lindner, A., & Maller, R. A. (2004). A continuous-time GARCH process driven by a lévy process: stationarity and second-order behaviour. Journal of Applied Probability, 41, 601–622.

    Article  MathSciNet  Google Scholar 

  • Klüppelberg, C., Lindner, A., & Maller, R. A. (2006). Continuous time volatility modelling: COGARCH versus ornsteinuhlenbeck models. In Y. Kabanov, R. Lipster, & J. Stoyanov (Eds.), From stochastic calculus to mathematical finance. Springer.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Lubrano, M. (1998). Smooth transition GARCH models: A Bayesian perspective. Technical report. CORE, Louvain-la-Neuve.

    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  Google Scholar 

  • Maercker, G. (1997). Statistical inference in conditional heteroskedastic autoregressive models. Aachen: Shaker Verlag.

    Google Scholar 

  • Maller, R. A., Müller, G., & Szimayer, A. (2008). GARCH modelling in continuous time for irregularly spaced time series data. Bernoulli, 14, 519–542.

    Article  MathSciNet  Google Scholar 

  • Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36, 394–419.

    Article  Google Scholar 

  • Mikosch, T., & Starica, C. (2004). Non-stationarities in financial time series, the long range dependence and the IGARCH effects. Review of Economics and Statistics, 86(1), 389–390.

    Article  Google Scholar 

  • Müller, G. (2010). MCMC estimation of the COGARCH(1,1) model. Journal of Financial Econometrics, 1, 1–30.

    Google Scholar 

  • Nelson, D. B. (1990a). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory, 6, 318–334.

    Article  MathSciNet  Google Scholar 

  • Nelson, D. B. (1990b). Arch models as diffusion approximations. Journal of Econometrics, 45, 7–38.

    Article  MathSciNet  Google Scholar 

  • Nelson, D. B. (1991a). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347–370.

    Article  MathSciNet  Google Scholar 

  • Protter, P. E. (2005). Stochastic integration and differential equations. Springer.

    Book  Google Scholar 

  • Rabemananjara, R., & Zakoian, J. M. (1993). Threshold ARCH models and asymmetries in volatility. Journal of Applied Econometrics, 8, 31–49.

    Article  Google Scholar 

  • Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press.

    MATH  Google Scholar 

  • Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility models. In D. R. Cox, D. V. Hinkley, & O. E. Barndorff-Nielsen (Eds.), Time series models in econometrics, finance and other fields. London: Chapman & Hall.

    Google Scholar 

  • Starica, C. (2003). Is GARCH(1,1) as good a model as the accolades of the Nobel prize would imply? working paper.

    Google Scholar 

  • Stelzer, R. (2009). First jump approximation of a lévy-driven SDE and an application to multivariate ECOGARCH processes. Stochastic Processes and Their Applications, 119, 1932–1951.

    Article  MathSciNet  Google Scholar 

  • Stelzer, R. (2010). Multivariate COGARCH(1,1) processes. Bernoulli, 16, 80–115.

    Article  MathSciNet  Google Scholar 

  • Szimayer, A., & Maller, R. A. (2007). Finite approximation schemes for lévy processes, and their application to optimal stopping problems. Stochastic Processes and Their Applications, 117, 1422–1447.

    Article  MathSciNet  Google Scholar 

  • Teräsvirta, T. (1994). Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association, 89, 208–218.

    MATH  Google Scholar 

  • Tsay, R. S. (2002). Analysis of financial time series. Wiley.

    Book  Google Scholar 

  • Tse, Y. K., & Tsui, A. (2002). A multivariate GARCH model with time-varying correlations. Journal of Business and Economic Statistics, 20, 351–362.

    Article  MathSciNet  Google Scholar 

  • Wand, M., & Jones, M. (1995). Kernel smoothing. London: Chapman and Hall.

    Book  Google Scholar 

  • Wang, Y. (2002). Asymptotic nonequivalence of GARCH models and diffusions. The Annals of Statistics, 30, 754–783.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25.

    Article  MathSciNet  Google Scholar 

  • Zakoian, J. M. (1991). Threshold heteroskedastic models. Technical report. INSEE.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Jürgen Franke

  2. Ladislaus von Bortkiewicz Chair of Statistics, Humboldt-Universität Berlin, Berlin, Germany

    Wolfgang Karl Härdle

  3. Louvain Institute of Data Analysis and Modeling in Economics and Statistics, UCLouvain, Louvain-la-Neuve, Belgium

    Christian Matthias Hafner

Authors
  1. Jürgen Franke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang Karl Härdle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Matthias Hafner
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Franke, J., Härdle, W.K., Hafner, C.M. (2019). Time Series with Stochastic Volatility. In: Statistics of Financial Markets. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-13751-9_13

Download citation

Publish with us

Policies and ethics

Search

Navigation