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Information Theoretic Optimality of Observation Driven Time Series Models

Author

Listed:
  • Francisco Blasques

    (VU University Amsterdam)

  • Siem Jan Koopman

    (VU University Amsterdam, the Netherlands, and CREATES, Aarhus University, Denmark)

  • André Lucas

    (VU University Amsterdam)

Abstract

This discussion paper led to a publication in 'Biometrika' , 2015, 102(2), 325-343. We investigate the information theoretic optimality properties of the score function of the predictive likelihood as a device to update parameters in observation driven time-varying parameter models. The results provide a new theoretical justification for the class of generalized autoregressive score models, which covers the GARCH model as a special case. Our main contribution is to show that only parameter updates based on the score always reduce the local Kullback-Leibler divergence between the true conditional density and the model implied conditional density. This result holds irrespective of the severity of model misspecification. We also show that the use of the score leads to a considerably smaller global Kullback-Leibler divergence in empirically relevant settings. We illustrate the theory with an application to time-varying volatility models. We show that th e reduction in Kullback-Leibler divergence across a range of different settings can be substantial in comparison to updates based on for example squared lagged observations.

Suggested Citation

  • Francisco Blasques & Siem Jan Koopman & André Lucas, 2014. "Information Theoretic Optimality of Observation Driven Time Series Models," Tinbergen Institute Discussion Papers 14-046/III, Tinbergen Institute.
  • Handle: RePEc:tin:wpaper:20140046
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    File URL: https://papers.tinbergen.nl/14046.pdf
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    References listed on IDEAS

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    1. Drew Creal & Bernd Schwaab & Siem Jan Koopman & Andr� Lucas, 2014. "Observation-Driven Mixed-Measurement Dynamic Factor Models with an Application to Credit Risk," The Review of Economics and Statistics, MIT Press, vol. 96(5), pages 898-915, December.
    2. Creal, Drew & Koopman, Siem Jan & Lucas, André, 2011. "A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(4), pages 552-563.
    3. Harvey,Andrew C., 2013. "Dynamic Models for Volatility and Heavy Tails," Cambridge Books, Cambridge University Press, number 9781107630024, September.
    4. Siem Jan Koopman & André Lucas & Marcel Scharth, 2016. "Predicting Time-Varying Parameters with Parameter-Driven and Observation-Driven Models," The Review of Economics and Statistics, MIT Press, vol. 98(1), pages 97-110, March.
    5. Robert Engle, 2002. "New frontiers for arch models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 425-446.
    6. Francisco Blasques & Siem Jan Koopman & Andre Lucas, 2014. "Maximum Likelihood Estimation for Score-Driven Models," Tinbergen Institute Discussion Papers 14-029/III, Tinbergen Institute, revised 23 Oct 2017.
    7. Andrew Harvey & Alessandra Luati, 2014. "Filtering With Heavy Tails," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1112-1122, September.
    8. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178.
    9. De Lira Salvatierra, Irving & Patton, Andrew J., 2015. "Dynamic copula models and high frequency data," Journal of Empirical Finance, Elsevier, vol. 30(C), pages 120-135.
    10. Richard A. Davis, 2003. "Observation-driven models for Poisson counts," Biometrika, Biometrika Trust, vol. 90(4), pages 777-790, December.
    11. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    12. Dong Hwan Oh & Andrew J. Patton, 2013. "Time-Varying Systemic Risk: Evidence from a Dynamic Copula Model of CDS Spreads," Working Papers 13-30, Duke University, Department of Economics.
    13. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    14. Andre Lucas & Bernd Schwaab & Xin Zhang, 2013. "Measuring Credit Risk in a Large Banking System: Econometric Modeling and Empirics," Tinbergen Institute Discussion Papers 13-063/IV/DSF56, Tinbergen Institute, revised 13 Oct 2014.
    15. Drew Creal & Siem Jan Koopman & André Lucas, 2008. "A General Framework for Observation Driven Time-Varying Parameter Models," Tinbergen Institute Discussion Papers 08-108/4, Tinbergen Institute.
    16. Francisco Blasques & Siem Jan Koopman & Andre Lucas, 2012. "Stationarity and Ergodicity of Univariate Generalized Autoregressive Score Processes," Tinbergen Institute Discussion Papers 12-059/4, Tinbergen Institute.
    17. Nelson, Daniel B & Foster, Dean P, 1994. "Asymptotic Filtering Theory for Univariate ARCH Models," Econometrica, Econometric Society, vol. 62(1), pages 1-41, January.
    18. Drew Creal & Siem Jan Koopman & André Lucas, 2013. "Generalized Autoregressive Score Models With Applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(5), pages 777-795, August.
    19. Ullah, Aman, 2002. "Uses of entropy and divergence measures for evaluating econometric approximations and inference," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 313-326, March.
    20. Maasoumi, Esfandiar, 1986. "The Measurement and Decomposition of Multi-dimensional Inequality," Econometrica, Econometric Society, vol. 54(4), pages 991-997, July.
    21. Shephard, Neil (ed.), 2005. "Stochastic Volatility: Selected Readings," OUP Catalogue, Oxford University Press, number 9780199257201.
    22. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Citations

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    Cited by:

    1. Francisco Blasques & Siem Jan Koopman & André Lucas, 2014. "Maximum Likelihood Estimation for correctly Specified Generalized Autoregressive Score Models: Feedback Effects, Contraction Conditions and Asymptotic Properties," Tinbergen Institute Discussion Papers 14-074/III, Tinbergen Institute.
    2. Lucas, André & Zhang, Xin, 2016. "Score-driven exponentially weighted moving averages and Value-at-Risk forecasting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 293-302.
    3. Mauro Bernardi & Leopoldo Catania, 2015. "Switching-GAS Copula Models With Application to Systemic Risk," Papers 1504.03733, arXiv.org, revised Jan 2016.
    4. André Lucas & Bernd Schwaab & Xin Zhang, 2017. "Modeling Financial Sector Joint Tail Risk in the Euro Area," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(1), pages 171-191, January.
    5. Siem Jan Koopman & Rutger Lit & André Lucas, 2015. "Intraday Stock Price Dependence using Dynamic Discrete Copula Distributions," Tinbergen Institute Discussion Papers 15-037/III/DSF90, Tinbergen Institute.
    6. Blasques, Francisco & Koopman, Siem Jan & Lucas, Andre & Schaumburg, Julia, 2016. "Spillover dynamics for systemic risk measurement using spatial financial time series models," Journal of Econometrics, Elsevier, vol. 195(2), pages 211-223.
    7. Olusanya E. Olubusoye & OlaOluwa S. Yaya, 2016. "Time series analysis of volatility in the petroleum pricing markets: the persistence, asymmetry and jumps in the returns series," OPEC Energy Review, Organization of the Petroleum Exporting Countries, vol. 40(3), pages 235-262, September.
    8. P Gorgi & P R Hansen & P Janus & S J Koopman, 2019. "Realized Wishart-GARCH: A Score-driven Multi-Asset Volatility Model," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 17(1), pages 1-32.
    9. Francisco Blasques & Siem Jan Koopman & André Lucas, 2014. "Optimal Formulations for Nonlinear Autoregressive Processes," Tinbergen Institute Discussion Papers 14-103/III, Tinbergen Institute.
    10. Andre Lucas & Bernd Schwaab & Xin Zhang, 2013. "Measuring Credit Risk in a Large Banking System: Econometric Modeling and Empirics," Tinbergen Institute Discussion Papers 13-063/IV/DSF56, Tinbergen Institute, revised 13 Oct 2014.
    11. Pawel Janus & André Lucas & Anne Opschoor & Dick J.C. van Dijk, 2014. "New HEAVY Models for Fat-Tailed Returns and Realized Covariance Kernels," Tinbergen Institute Discussion Papers 14-073/IV, Tinbergen Institute, revised 19 Aug 2015.

    More about this item

    Keywords

    generalized autoregressive models; information theory; optimality; Kullback-Leibler distance; volatility models;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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