IDEAS home Printed from https://ideas.repec.org/p/tin/wpaper/20230051.html
   My bibliography  Save this paper

Consistency, distributional convergence, and optimality of score-driven filters

Author

Listed:
  • Eric A. Beutner

    (Vrije Universiteit Amsterdam)

  • Yicong Lin

    (Vrije Universiteit Amsterdam)

  • Andre Lucas

    (Vrije Universiteit Amsterdam)

Abstract

We study the in-fill asymptotics of score-driven time series models. For general forms of model mis-specification, we show that score-driven filters are consistent for the Kullback-Leibler (KL) optimal time-varying parameter path, which minimizes the pointwise KL divergence between the statistical model and the unknown dynamic data generating process. This directly implies that for a correctly specified predictive conditional density, score-driven filters consistently estimate the time-varying parameter path even if the model is mis-specified in other respects. We also obtain distributional convergence results for the filtering errors and derive the filter that minimizes the asymptotic filter error variance. Score-driven filters turn out to be optimal under correct specification of the predictive conditional density. The results considerably generalize earlier findings on the continuous-time consistency of volatility filters under mis-specification: they apply to biased filters, use weaker assumptions, allow for more general forms of mis-specification, and consider general time-varying parameters in non-linear time series models beyond the volatility case. Several examples are used to illustrate the theory, including time-varying tail shape models, dynamic copulas, and time-varying regression models.

Suggested Citation

  • Eric A. Beutner & Yicong Lin & Andre Lucas, 2023. "Consistency, distributional convergence, and optimality of score-driven filters," Tinbergen Institute Discussion Papers 23-051/III, Tinbergen Institute.
  • Handle: RePEc:tin:wpaper:20230051
    as

    Download full text from publisher

    File URL: https://papers.tinbergen.nl/23051.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    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. Peter Reinhard Hansen & Matthias Schmidtblaicher, 2021. "A Dynamic Model of Vaccine Compliance: How Fake News Undermined the Danish HPV Vaccine Program," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(1), pages 259-271, January.
    4. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    5. Dong Hwan Oh & Andrew J. Patton, 2018. "Time-Varying Systemic Risk: Evidence From a Dynamic Copula Model of CDS Spreads," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 36(2), pages 181-195, April.
    6. 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, Oxford University Press, vol. 17(1), pages 1-32.
    7. Danielsson, Jon, 1994. "Stochastic volatility in asset prices estimation with simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 375-400.
    8. Blasques, F. & Francq, Christian & Laurent, Sébastien, 2023. "Quasi score-driven models," Journal of Econometrics, Elsevier, vol. 234(1), pages 251-275.
    9. Enzo D’Innocenzo & André Lucas & Bernd Schwaab & Xin Zhang, 2024. "Modeling Extreme Events: Time-Varying Extreme Tail Shape," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 42(3), pages 903-917, July.
    10. Blasques, Francisco & van Brummelen, Janneke & Koopman, Siem Jan & Lucas, André, 2022. "Maximum likelihood estimation for score-driven models," Journal of Econometrics, Elsevier, vol. 227(2), pages 325-346.
    11. 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.
    12. André Lucas & Bernd Schwaab & Xin Zhang, 2014. "Conditional Euro Area Sovereign Default Risk," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(2), pages 271-284, April.
    13. Patton, Andrew J. & Ziegel, Johanna F. & Chen, Rui, 2019. "Dynamic semiparametric models for expected shortfall (and Value-at-Risk)," Journal of Econometrics, Elsevier, vol. 211(2), pages 388-413.
    14. André Lucas & Julia Schaumburg & Bernd Schwaab, 2019. "Bank Business Models at Zero Interest Rates," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(3), pages 542-555, July.
    15. Han Li & Kai Yang & Dehui Wang, 2019. "A threshold stochastic volatility model with explanatory variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 73(1), pages 118-138, February.
    16. Angelini, Giovanni & Gorgi, Paolo, 2018. "DSGE Models with observation-driven time-varying volatility," Economics Letters, Elsevier, vol. 171(C), pages 169-171.
    17. Linton, Oliver & Wu, Jianbin, 2020. "A coupled component DCS-EGARCH model for intraday and overnight volatility," Journal of Econometrics, Elsevier, vol. 217(1), pages 176-201.
    18. Yu, Jun, 2005. "On leverage in a stochastic volatility model," Journal of Econometrics, Elsevier, vol. 127(2), pages 165-178, August.
    19. Nelson, Daniel B., 1996. "Asymptotic filtering theory for multivariate ARCH models," Journal of Econometrics, Elsevier, vol. 71(1-2), pages 1-47.
    20. Davide Delle Monache & Ivan Petrella & Fabrizio Venditti, 2021. "Price Dividend Ratio and Long-Run Stock Returns: A Score-Driven State Space Model," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 1054-1065, October.
    21. Engle, Robert F. & Gallo, Giampiero M., 2006. "A multiple indicators model for volatility using intra-daily data," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 3-27.
    22. Nelson, Daniel B., 1992. "Filtering and forecasting with misspecified ARCH models I : Getting the right variance with the wrong model," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 61-90.
    23. Nelson, Daniel B & Foster, Dean P, 1994. "Asymptotic Filtering Theory for Univariate ARCH Models," Econometrica, Econometric Society, vol. 62(1), pages 1-41, January.
    24. 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.
    25. Francisco Blasques & André Lucas & Erkki Silde, 2018. "A stochastic recurrence equations approach for score driven correlation models," Econometric Reviews, Taylor & Francis Journals, vol. 37(2), pages 166-181, February.
    26. Jensen Anders Tolver & Lange Theis, 2010. "On Convergence of the QMLE for Misspecified GARCH Models," Journal of Time Series Econometrics, De Gruyter, vol. 2(1), pages 1-31, June.
    27. Siem Jan Koopman & Rutger Lit & André Lucas & Anne Opschoor, 2018. "Dynamic discrete copula models for high‐frequency stock price changes," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(7), pages 966-985, November.
    28. 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.
    29. F. Blasques & S. J. Koopman & A. Lucas, 2015. "Information-theoretic optimality of observation-driven time series models for continuous responses," Biometrika, Biometrika Trust, vol. 102(2), pages 325-343.
    30. Wei Lin & Shenghong Li & Shane Chern & Jin E. Zhang, 2019. "Pricing VIX derivatives with free stochastic volatility model," Review of Derivatives Research, Springer, vol. 22(1), pages 41-75, April.
    31. Bram van Os, 2023. "Information-Theoretic Time-Varying Density Modeling," Tinbergen Institute Discussion Papers 23-037/III, Tinbergen Institute.
    32. Babii, Andrii & Chen, Xi & Ghysels, Eric, 2019. "Commercial and Residential Mortgage Defaults: Spatial Dependence with Frailty," Journal of Econometrics, Elsevier, vol. 212(1), pages 47-77.
    33. Anne Opschoor & Pawel Janus & André Lucas & Dick Van Dijk, 2018. "New HEAVY Models for Fat-Tailed Realized Covariances and Returns," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 36(4), pages 643-657, October.
    34. Igor Custodio João & Julia Schaumburg & André Lucas & Bernd Schwaab, 2024. "Dynamic Nonparametric Clustering of Multivariate Panel Data," Journal of Financial Econometrics, Oxford University Press, vol. 22(2), pages 335-374.
    35. Andrew J. Patton, 2006. "Modelling Asymmetric Exchange Rate Dependence," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(2), pages 527-556, May.
    36. Paolo Gorgi, 2020. "Beta–negative binomial auto‐regressions for modelling integer‐valued time series with extreme observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1325-1347, December.
    37. Buccheri, Giuseppe & Corsi, Fulvio & Flandoli, Franco & Livieri, Giulia, 2021. "The continuous-time limit of score-driven volatility models," Journal of Econometrics, Elsevier, vol. 221(2), pages 655-675.
    38. Hetland, Simon & Pedersen, Rasmus Søndergaard & Rahbek, Anders, 2023. "Dynamic conditional eigenvalue GARCH," Journal of Econometrics, Elsevier, vol. 237(2).
    39. Andrew J. Patton & Yasin Simsek, 2023. "Generalized Autoregressive Score Trees and Forests," Papers 2305.18991, arXiv.org.
    40. Giuseppe Buccheri & Giacomo Bormetti & Fulvio Corsi & Fabrizio Lillo, 2021. "A Score-Driven Conditional Correlation Model for Noisy and Asynchronous Data: An Application to High-Frequency Covariance Dynamics," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 920-936, October.
    41. Daniele Massacci, 2017. "Tail Risk Dynamics in Stock Returns: Links to the Macroeconomy and Global Markets Connectedness," Management Science, INFORMS, vol. 63(9), pages 3072-3089, September.
    42. Giovanni Angelini & Paolo Gorgi, 2018. "DSGE Models with Observation-Driven Time-Varying parameters," Tinbergen Institute Discussion Papers 18-030/III, Tinbergen Institute.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Blasques, Francisco & van Brummelen, Janneke & Koopman, Siem Jan & Lucas, André, 2022. "Maximum likelihood estimation for score-driven models," Journal of Econometrics, Elsevier, vol. 227(2), pages 325-346.
    2. Mauro Bernardi & Leopoldo Catania, 2015. "Switching-GAS Copula Models With Application to Systemic Risk," Papers 1504.03733, arXiv.org, revised Jan 2016.
    3. 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.
    4. Opschoor, Anne & Lucas, André, 2021. "Observation-driven models for realized variances and overnight returns applied to Value-at-Risk and Expected Shortfall forecasting," International Journal of Forecasting, Elsevier, vol. 37(2), pages 622-633.
    5. Blasques, Francisco & Ji, Jiangyu & Lucas, André, 2016. "Semiparametric score driven volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 58-69.
    6. Creal, Drew & Koopman, Siem Jan & Lucas, André & Zamojski, Marcin, 2024. "Observation-driven filtering of time-varying parameters using moment conditions," Journal of Econometrics, Elsevier, vol. 238(2).
    7. 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.
    8. Buccheri, Giuseppe & Corsi, Fulvio & Flandoli, Franco & Livieri, Giulia, 2021. "The continuous-time limit of score-driven volatility models," Journal of Econometrics, Elsevier, vol. 221(2), pages 655-675.
    9. Marco Bazzi & Francisco Blasques & Siem Jan Koopman & Andre Lucas, 2017. "Time-Varying Transition Probabilities for Markov Regime Switching Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(3), pages 458-478, May.
    10. Drew Creal & Siem Jan Koopman & André Lucas & Marcin Zamojski, 2015. "Generalized Autoregressive Method of Moments," Tinbergen Institute Discussion Papers 15-138/III, Tinbergen Institute, revised 06 Jul 2018.
    11. Harvey, A., 2021. "Score-driven time series models," Cambridge Working Papers in Economics 2133, Faculty of Economics, University of Cambridge.
    12. Caballero, Diego & Lucas, André & Schwaab, Bernd & Zhang, Xin, 2020. "Risk endogeneity at the lender/investor-of-last-resort," Journal of Monetary Economics, Elsevier, vol. 116(C), pages 283-297.
    13. 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.
    14. Michel Ferreira Cardia Haddad & Szabolcs Blazsek & Philip Arestis & Franz Fuerst & Hsia Hua Sheng, 2023. "The two-component Beta-t-QVAR-M-lev: a new forecasting model," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 37(4), pages 379-401, December.
    15. Andre Lucas & Anne Opschoor, 2016. "Fractional Integration and Fat Tails for Realized Covariance Kernels and Returns," Tinbergen Institute Discussion Papers 16-069/IV, Tinbergen Institute, revised 07 Jul 2017.
    16. Giuseppe Buccheri & Stefano Grassi & Giorgio Vocalelli, 2021. "Estimating Risk in Illiquid Markets: a Model of Market Friction with Stochastic Volatility," CEIS Research Paper 506, Tor Vergata University, CEIS, revised 08 Nov 2021.
    17. Anne Opschoor & André Lucas, 2019. "Observation-driven Models for Realized Variances and Overnight Returns," Tinbergen Institute Discussion Papers 19-052/IV, Tinbergen Institute.
    18. Anne Opschoor & André Lucas & István Barra & Dick van Dijk, 2021. "Closed-Form Multi-Factor Copula Models With Observation-Driven Dynamic Factor Loadings," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 1066-1079, October.
    19. 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.
    20. Francisco (F.) Blasques & Andre (A.) Lucas & Andries van Vlodrop, 2017. "Finite Sample Optimality of Score-Driven Volatility Models," Tinbergen Institute Discussion Papers 17-111/III, Tinbergen Institute.

    More about this item

    Keywords

    score-driven models; information theoretic optimality; Kullback-Leibler divergence; pseudo true time-varying parameters; in-fill asymptotics.;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tin:wpaper:20230051. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Tinbergen Office +31 (0)10-4088900 (email available below). General contact details of provider: https://edirc.repec.org/data/tinbenl.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.