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Bayesian forecasting of Value at Risk and Expected Shortfall using adaptive importance sampling


  • Hoogerheide, Lennart
  • van Dijk, Herman K.


An efficient and accurate approach is proposed for forecasting the Value at Risk (VaR) and Expected Shortfall (ES) measures in a Bayesian framework. This consists of a new adaptive importance sampling method for the Quick Evaluation of Risk using Mixture of t approximations (QERMit). As a first step, the optimal importance density is approximated, after which multi-step 'high loss' scenarios are efficiently generated. Numerical standard errors are compared in simple illustrations and in an empirical GARCH model with Student-t errors for daily S&P 500 returns. The results indicate that the proposed QERMit approach outperforms alternative approaches, in the sense that it produces more accurate VaR and ES estimates given the same amount of computing time, or, equivalently, that it requires less computing time for the same numerical accuracy.

Suggested Citation

  • Hoogerheide, Lennart & van Dijk, Herman K., 2010. "Bayesian forecasting of Value at Risk and Expected Shortfall using adaptive importance sampling," International Journal of Forecasting, Elsevier, vol. 26(2), pages 231-247, April.
  • Handle: RePEc:eee:intfor:v:26:y::i:2:p:231-247

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    References listed on IDEAS

    1. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-1339, November.
    2. Hoogerheide, Lennart F. & Kaashoek, Johan F. & van Dijk, Herman K., 2007. "On the shape of posterior densities and credible sets in instrumental variable regression models with reduced rank: An application of flexible sampling methods using neural networks," Journal of Econometrics, Elsevier, vol. 139(1), pages 154-180, July.
    3. Luc Bauwens & Michel Lubrano, 1998. "Bayesian inference on GARCH models using the Gibbs sampler," Econometrics Journal, Royal Economic Society, vol. 1(Conferenc), pages 23-46.
    4. Kloek, Tuen & van Dijk, Herman K, 1978. "Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo," Econometrica, Econometric Society, vol. 46(1), pages 1-19, January.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. HOOGERHEIDE, Lennart F. & VAN DIJK, Herman K. & VAN OEST, Rutger D., 2007. "Simulation based Bayesian econometric inference: principles and some recent computational advances," CORE Discussion Papers 2007015, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    8. Andrews, Donald W K & Monahan, J Christopher, 1992. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, vol. 60(4), pages 953-966, July.
    9. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    10. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    11. 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.
    12. van Dijk, H. K. & Kloek, T., 1980. "Further experience in Bayesian analysis using Monte Carlo integration," Journal of Econometrics, Elsevier, vol. 14(3), pages 307-328, December.
    13. Ardia, David & Hoogerheide, Lennart F. & van Dijk, Herman K., 2008. "AdMit: Adaptive Mixtures of Student-t Distributions," DQE Working Papers 10, Department of Quantitative Economics, University of Freiburg/Fribourg Switzerland, revised 07 Jan 2009.
    14. Nakatsuma, Teruo, 2000. "Bayesian analysis of ARMA-GARCH models: A Markov chain sampling approach," Journal of Econometrics, Elsevier, vol. 95(1), pages 57-69, March.
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    Cited by:

    1. Stavros Degiannakis & Pamela Dent & Christos Floros, 2014. "A Monte Carlo Simulation Approach to Forecasting Multi-period Value-at-Risk and Expected Shortfall Using the FIGARCH-skT Specification," Manchester School, University of Manchester, vol. 82(1), pages 71-102, January.
    2. Ardia, David & Hoogerheide, Lennart F. & van Dijk, Herman K., 2009. "Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 29(i03).
    3. Lennart F. Hoogerheide & David Ardia & Nienke Corre, 2011. "Stock Index Returns' Density Prediction using GARCH Models: Frequentist or Bayesian Estimation?," Tinbergen Institute Discussion Papers 11-020/4, Tinbergen Institute.
    4. Hoogerheide, Lennart F. & Ardia, David & Corré, Nienke, 2012. "Density prediction of stock index returns using GARCH models: Frequentist or Bayesian estimation?," Economics Letters, Elsevier, vol. 116(3), pages 322-325.
    5. Ardia, David & Hoogerheide, Lennart F., 2014. "GARCH models for daily stock returns: Impact of estimation frequency on Value-at-Risk and Expected Shortfall forecasts," Economics Letters, Elsevier, vol. 123(2), pages 187-190.
    6. Hoogerheide, Lennart & Opschoor, Anne & van Dijk, Herman K., 2012. "A class of adaptive importance sampling weighted EM algorithms for efficient and robust posterior and predictive simulation," Journal of Econometrics, Elsevier, vol. 171(2), pages 101-120.
    7. Chen, Qian & Gerlach, Richard & Lu, Zudi, 2012. "Bayesian Value-at-Risk and expected shortfall forecasting via the asymmetric Laplace distribution," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3498-3516.
    8. Lennart Hoogerheide & Anne Opschoor & Herman K. van Dijk, 2011. "A Class of Adaptive EM-based Importance Sampling Algorithms for Efficient and Robust Posterior and Predictive Simulation," Tinbergen Institute Discussion Papers 11-004/4, Tinbergen Institute.
    9. Lennart F. Hoogerheide & Francesco Ravazzolo & Herman K. van Dijk, 2011. "Backtesting Value-at-Risk using Forecasts for Multiple Horizons, a Comment on the Forecast Rationality Tests of A.J. Patton and A. Timmermann," Tinbergen Institute Discussion Papers 11-131/4, Tinbergen Institute.
    10. Begen, Mehmet A. & Pun, Hubert & Yan, Xinghao, 2016. "Supply and demand uncertainty reduction efforts and cost comparison," International Journal of Production Economics, Elsevier, vol. 180(C), pages 125-134.
    11. Gerlach, Richard & Abeywardana, Sachin, 2016. "Variational Bayes for assessment of dynamic quantile forecasts," International Journal of Forecasting, Elsevier, vol. 32(4), pages 1385-1402.
    12. repec:kap:compec:v:50:y:2017:i:3:d:10.1007_s10614-016-9588-x is not listed on IDEAS
    13. Lukasz Gatarek & Lennart Hoogerheide & Koen Hooning & Herman K. van Dijk, 2013. "Censored Posterior and Predictive Likelihood in Left-Tail Prediction for Accurate Value at Risk Estimation," Tinbergen Institute Discussion Papers 13-060/III, Tinbergen Institute, revised 06 Mar 2014.
    14. Nieto, Maria Rosa & Ruiz, Esther, 2016. "Frontiers in VaR forecasting and backtesting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 475-501.

    More about this item


    Value at Risk Expected Shortfall Numerical standard error Importance sampling Mixture of Student-t distributions Variance reduction technique;

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty


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