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On the Stationary Version of the Generalized Hyperbolic ARCH Model

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Abstract

This paper finds conditions under which the generalized hyperbolic ARCH-type model is strictly stationary. Properties of the model are investigated and in particular an estimation procedure is proposed. The resulting stationary model provides with a robust non-Gaussian ARCH-type alternative.

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References

  • Abramowitz M., Stegun I.A. (eds) (1992). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications Inc., New York Reprint of the 1972 edition.

    Google Scholar 

  • Andersson J. (2001). On the normal inverse Gaussian stochastic volatility model. Journal of Business Economic Statistics 19(1):44–54

    Article  MathSciNet  Google Scholar 

  • Bain L.J. (1969). Moments of non-central t and non-central F-distribution. The American Statistician. 23(4):33–34

    Article  Google Scholar 

  • Barndorff-Nielsen O.E. (1977). Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society. London A. 353:401–419

    Google Scholar 

  • Barndorff-Nielsen O.E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics 24(1):1–13

    Article  MATH  MathSciNet  Google Scholar 

  • Barndorff-Nielsen O.E., Shephard N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society. Series B. Statistical Methodology 63:167–241

    Article  MATH  MathSciNet  Google Scholar 

  • Bollerslev T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. The Review of Economics and Statistics. 69:542–47

    Article  Google Scholar 

  • Bollerslev T., Chou R.C., Kroner K.F. (1992). ARCH modelling in finace: a review of the theory and empirical evidence. Journal of Econometrics 52:5–56

    Article  MATH  Google Scholar 

  • Cox D.R. (1981). Statistical analysis of time series: some recent developments. Scandinavian Journal of Statistics 8:93–115

    MATH  Google Scholar 

  • Eberlein, E., Hammerstein, E.A. (2004). Generalized hyperbolic and inverse Gaussian distributions: Limiting cases and approximation of processes. In: R.C. Dalang, M. Dozzi, F. Russo (Eds.), Seminar on stochastic analysis, random fields and applications IV, progress in probability, vol. 58, Birkhuser, pp. 221–264

  • Engle R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica 50(4):987–1008

    Article  MATH  MathSciNet  Google Scholar 

  • Gaver D.P., Lewis P.A.W. (1980). First-order autoregressive gamma sequences and point processes. Advances in Applied Probability 12:727–745

    Article  MATH  MathSciNet  Google Scholar 

  • Jacobs P.A., Lewis P.A.W. (1977). A mixed autoregressive-moving average exponential sequence and point process (EARMA(1, 1)). Advances in Applied Probability 9:87–104

    Article  MATH  MathSciNet  Google Scholar 

  • Jensen M.B., Lunde A. (2001). The NIG-S & ARCH model: a fat tailed, stochastic, and autoregressive conditional heteroskedastic volatility model. Econometrics Journal 4:319–342

    Article  MATH  MathSciNet  Google Scholar 

  • Joe H. (1996). Time series models with univariate margins in the convolution-closed infinitely divisible class. Journal of Applied Probability 33:664–77

    Article  MATH  MathSciNet  Google Scholar 

  • Jørgensen, B. (1982). Statistical properties of the generalized inverse Gaussian distribution. Lecture notes in statistics, vol. 9. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Jørgensen B., Song P. (1998). Stationary time series models with exponentials dispersion model margins. Journal of Applied Probability 35:78–92

    Article  MATH  MathSciNet  Google Scholar 

  • Lawrance A.J. (1982). The innovation distribution of a gamma distributed autoregressive process. Scandinavian Journal of Statistics 9:234–36

    MATH  Google Scholar 

  • Lawrance A.J. (1991). Directionality and reversivility in time series. International Statistical Review 59:67–79

    Article  Google Scholar 

  • Lawrance A.J., Lewis P.A.W. (1977). An exponential moving-average sequence and point process (EMA 1). Journal of Applied Probability 14:98–113

    Article  MATH  MathSciNet  Google Scholar 

  • Lawrance A.J., Lewis P.A.W. (1980). The exponential autoregressive-moving average EARMA (p, q) process. Journal of the Royal Statistical Society. Series B. Statistical Methodology 42:150–61

    MATH  MathSciNet  Google Scholar 

  • Liu J.S., Wong W.H., Kong A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimation and augmentation schemes. Biometrika 81:27–40

    Article  MATH  MathSciNet  Google Scholar 

  • Louis T.A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society. Series B. Statistical Methodology 44:226–33

    MATH  MathSciNet  Google Scholar 

  • McKenzie E. (1986). Autoregressive moving-average processes with negative-binomial and geometric marginal distributions. Advances in Applied Probability 18:679–705

    Article  MATH  MathSciNet  Google Scholar 

  • McKenzie E. (1988). Some ARMA models for dependent sequences of Poisson counts. Advances in Applied Probability 20:822–835

    Article  MATH  MathSciNet  Google Scholar 

  • Mencía, F.J., Sentana, E. (2004). Estimation and testing of dynamic models with generalised hyperbolic innovations. Technical Report 0411, CEMFI.

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

    Article  MathSciNet  Google Scholar 

  • Pitt M.K., Walker S.G. (2005). Constructing stationary time series models using auxiliary variables with applications. Journal of the American Statistical Association 100:554–564

    Article  MathSciNet  MATH  Google Scholar 

  • Pitt M.K., Chatfield C., Walker S.G. (2002). Constructing first order autoregressive models via latent processes. Scandinavian Journal of Statistics 29:657–663

    Article  MATH  MathSciNet  Google Scholar 

  • Prause, K. (1999). The generalized hyperbolic model: estimation, financial derivatives, and risk measures. Ph.D. Thesis, Universität Freiburg.

  • Press, W., Teukolsky, S., Vetterling, W., Flannery, B. (1992). Numerical Recipes. C. Cambridge University Press.

  • Raible, S. (2000). Lévy processes in finance: theory, numerics, and empirical facts. Ph.D. Thesis, Universität Freiburg.

  • Robert C.P., Casella G. (2002). Monte Carlo statistical methods, 3rd edn. Springer, Berlin Heidelberg NewYork

    Google Scholar 

  • Shephard N. (1996). Statistical aspects of ARCH and stochastic volatility. In: Cox D.R., Hinkley D.V., Barndorff-Nielsen O.E. (eds). Time series models in econometrics finance and other fields. Chapman and Hall, London, pp. 1–67

    Google Scholar 

  • Walker S.G. (1996). An EM algoritm for nonlinear random effects models. Biometrics 52:934–944

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autonoma de México (UNAM), A.P. 20-726, 01000, Mexico, D.F. México

    Ramsés H. Mena

  2. Institute of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, Kent, CT2 7NZ, UK

    Stephen G. Walker

Authors
  1. Ramsés H. Mena
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  2. Stephen G. Walker
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Correspondence to Ramsés H. Mena.

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Mena, R.H., Walker, S.G. On the Stationary Version of the Generalized Hyperbolic ARCH Model. AISM 59, 325–348 (2007). https://doi.org/10.1007/s10463-006-0052-x

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  • DOI: https://doi.org/10.1007/s10463-006-0052-x

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