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Extreme Value GARCH modelling with Bayesian Inference

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Abstract

Extreme value theory is widely used financial applications such as risk analysis, forecasting and pricing models. One of the major difficulties in the applications to finance and economics is that the assumption of independence of time series observations is generally not satisfied, so that the dependent extremes may not necessarily be in the domain of attraction of the classical generalised extreme value distribution. This study examines a conditional extreme value distribution with the added specification that the extreme values (maxima or minima) follows a conditional autoregressive heteroscedasticity process. The dependence has been modelled by allowing the location and scale parameters of the extreme distribution to vary with time. The resulting combined model, GEV-GARCH, is developed by implementing the GARCH volatility mechanism in these extreme value model parameters. Bayesian inference is used for the estimation of parameters and posterior inference is available through the Markov Chain Monte Carlo (MCMC) method. The model is firstly applied to relevant simulated data to verify model stability and reliability of the parameter estimation method. Then real stock returns are used to consider evidence for the appropriate application of the model. A comparison is made between the GEV-GARCH and traditional GARCH models. Both the GEV-GARCH and GARCH show similarity in the resulting conditional volatility estimates, however the GEV-GARCH model differs from GARCH in that it can capture and explain extreme quantiles better than the GARCH model because of more reliable extrapolation of the tail behaviour.

Suggested Citation

  • Les Oxley & Marco Reale & Carl Scarrott & Xin Zhao, 2009. "Extreme Value GARCH modelling with Bayesian Inference," Working Papers in Economics 09/05, University of Canterbury, Department of Economics and Finance.
  • Handle: RePEc:cbt:econwp:09/05
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    File URL: http://www.econ.canterbury.ac.nz/RePEc/cbt/econwp/0905.pdf
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    References listed on IDEAS

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    1. Scharth, Marcel & Medeiros, Marcelo C., 2009. "Asymmetric effects and long memory in the volatility of Dow Jones stocks," International Journal of Forecasting, Elsevier, vol. 25(2), pages 304-327.
    2. Bali, Turan G. & Weinbaum, David, 2007. "A conditional extreme value volatility estimator based on high-frequency returns," Journal of Economic Dynamics and Control, Elsevier, vol. 31(2), pages 361-397, February.
    3. McNeil, Alexander J. & Frey, Rudiger, 2000. "Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 271-300, November.
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    Cited by:

    1. Xin Zhao & Carl John Scarrott & Marco Reale & Les Oxley, 2009. "Bayesian Extreme Value Mixture Modelling for Estimating VaR," Working Papers in Economics 09/15, University of Canterbury, Department of Economics and Finance.

    More about this item

    Keywords

    Extreme value distribution; dependency; Bayesian; MCMC; Return quantile;

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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