Estimation of the Cholesky Multivariate Stochastic Volatility Model Using Iterated Filtering

• Authors: Piotr Szczepocki

Title variants

PL

Estymacja wielowymiarowego modelu stochastycznej zmienności z dekompozycją Choleskiego przy użyciu iterowanej filtracji

Abstracts

PL

Cel: Celem artykułu jest zaproponowanie nowej metody estymacji dla wielowymiarowego modelu stochastycznej zmienności z dekompozycją Choleskiego w oparciu o algorytm iterowanej filtracji (Ionides et al., 2006, 2015). Metodyka: Iterowana filtracja jest metodą należącą do klasycznego częstościowego wnioskowania, która poprzez wielokrotne powtórzenia procesu filtrowania zapewnia sekwencję aktualizowanych oszacowań parametrów zbieżnych do estymatora największej wiarygodności. Wyniki: Efektywność zaproponowanej metody estymacji została pokazana na przykładzie empirycznym, w którym wykorzystano wielowymiarowy model stochastyczny zmienności z dekompozycją Choleskiego w badaniu aktywów bezpiecznej przystani dla jednego indeksu rynkowego: Standard and Poor's 500 oraz trzech kandydatów na aktywa bezpiecznej przystani: złota, Bitcoina i Ethereum. Implikacje i rekomendacje: W dalszych badaniach metodę iterowanej filtracji można zastosować do bardziej zaawansowanych wielowymiarowych modeli zmienności stochastycznej, które uwzględniają np. efekt dźwigni (Ishihara et al., 2016) oraz rozkłady gruboogonowe (Ishihara i Omori, 2012). Oryginalność/Wartość: Głównym osiągnięciem artykułu jest propozycja nowej metody estymacji wielowymiarowego modelu stochastycznej zmienności z dekompozycją Choleskiego w oparciu o iterowany algorytm filtrowania. Jest to jedna z niewielu metod klasycznego częstościowego wnioskowania dla wielowymiarowych modeli stochastycznej zmienności.

EN

Aim: The paper aims to propose a new estimation method for the Cholesky Multivariate Stochastic Volatility Model based on the iterated filtering algorithm (Ionides et al., 2006, 2015). Methodology: The iterated filtering method is a frequentist-based technique that through multiple repetitions of the filtering process, provides a sequence of iteratively updated parameter estimates that converge towards the maximum likelihood estimate. Results: The effectiveness of the proposed estimation method was shown in an empirical example in which the Cholesky Multivariate Stochastic Volatility Model was used in a study on safe-haven assets of one market index: Standard and Poor’s 500 and three safe-haven candidates: gold, Bitcoin and Ethereum. Implications and recommendations: In further research, the iterating filtering method may be used for more advanced multivariate stochastic volatility models that take into account, for example, the leverage effect (as in Ishihara et al., 2016) and heavy-tailed errors (as in Ishihara and Omori, 2012). Originality/Value: The main contribution of the paper is the proposition of a new estimation method for the Cholesky Multivariate Stochastic Volatility Model based on iterated filtering algorithm This is one of the few frequentist-based statistical inference methods for multivariate stochastic volatility models.

Keywords

EN

multivariate stochastic volatility; iterated filtering; particle filters; the Cholesky Multivariate Stochastic Volatility (ChMSV) Model

PL

wielowymiarowe modele stochastycznej zmienności; iterowana filtracja; filtry cząsteczkowe

• Publisher: Wydawnictwo Uniwersytetu Ekonomicznego we Wrocławiu
• Journal: Econometrics. Ekonometria. Advances in Applied Data Analytics
• Year: 2023
• Volume: 27
• Issue: 4
• Pages: 44-58

Dates

published

2023

Contributors

author

Piotr Szczepocki

• University of Lodz, Lodz, Poland

• https://orcid.org/0000000183773831

References

• Aftab, M., Shah, S. Z. A., and Ismail, I. (2019). Does Gold Act as a Hedge or a safe Haven Against Equity and Currency in Asia? Global Business Review, 20(1), 105-118. https://doi.org/10.1177/0972150918803993
• Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle Markov Chain Monte Carlo Methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3), 269-342. https://doi.org/10.1111/j.1467-9868.2009.00736.x
• Asai, M., McAleer, M., and Yu, J. (2006). Multivariate Stochastic Volatility: A Review. Econometric Reviews, 25(2-3). https://doi.org/10.1080/07474930600713564
• Baur, D. G., and Lucey, B. M. (2009). Flights and Contagion - An Empirical Analysis of Stock-Bond Correlations. Journal of Financial Stability, 5(4), 339-352. https://doi.org/10.1016/j.jfs.2008.08.001
• Baur, D. G., and McDermott, T. K. (2010). Is Gold a Safe Haven? International Evidence. Journal of Banking & Finance, 34(8), 1886-1898. https://doi.org/10.1016/j.jbankfin.2009.12.008
• Bejger, S., and Fiszeder, P. (2021). Forecasting Currency Covariances Using Machine Learning Tree-based Algorithms with Low and High Prices. Przegląd Statystyczny, 68(3), 1-15. https://doi.org/10.5604/01.3001.0015.5582
• Będowska-Sójka, B., and Kliber, A. (2021). Is There One Safe-haven for Various Turbulences? The Evidence from Gold, Bitcoin and Ether. The North American Journal of Economics and Finance, 56, 101390. https://doi.org/10.1016/j.najef.2021. 101390
• Bhadra, A., Ionides, E. L., Laneri, K., Pascual, M., Bouma, M., and Dhiman, R. C. (2011). Malaria in Northwest India: Data Analysis via Partially Observed Stochastic Differential Equation Models Driven by Lévy Noise. Journal of the American Statistical Association, 106(494). https://doi.org/10.1198/jasa.2011.ap10323
• Bollerslev, T., Patton, A. J., and Quaedvlieg, R. (2018). Modeling and Forecasting (Un)reliable Realized Covariances for More Reliable Financial Decisions. Journal of Econometrics, 207(1). https://doi.org/10.1016/j.jeconom.2018.05.004
• Bouri, E., Molnár, P., Azzi, G., Roubaud, D., and Hagfors, L. I. (2017). On the Hedge and Safe Haven Properties of Bitcoin: Is it Really More than a Diversifier? Finance Research Letters, 20, 192-198.
• Bretó, C. (2014). On Idiosyncratic Stochasticity of Financial Leverage Effects. Statistics & Probability Letters, (91). 20-26. https://doi.org/10.1016/j.spl.2014.04.003
• Chib, S., Omori, Y., and Asai, M. (2009). Multivariate Stochastic Volatility. In T. Mikosch, J.-P. Kreiß, R. A. Davis, and T. G. Andersen (Eds.), Handbook of Financial Time Series (pp. 365-400). Springer. https://doi.org/10.1007/978-3-540-71297-8_16
• Choudhury, T., Kinateder, H., and Neupane, B. (2022). Gold, Bonds, and Epidemics: A Safe Haven Study. Finance Research Letters, 48, 102978. https://doi.org/10.1016/j.frl.2022.102978
• Chopin, N., Jacob, P. E., and Papaspiliopoulos, O. (2013). SMC2: An Efficient Algorithm for Sequential Analysis of State Space Models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(3). https://doi.org/10.1111/j.1467- 9868.2012.01046.x
• Chopin, N., and Papaspiliopoulos, O. (2020). An Introduction to Sequential Monte Carlo. Springer Nature.
• Engle, R. (2002). Dynamic Conditional Correlation. Journal of Business & Economic Statistics, 20(3), 339-350. https://doi.org/ 10.1198/073500102288618487
• Engle, R. F., and Kroner, K. F. (1995). Multivariate Simultaneous Generalized ARCH. Econometric Theory, 11(1), 122-150. https://doi.org/10.1017/S0266466600009063
• Fiszeder, P., and Orzeszko, W. (2021). Covariance Matrix Forecasting Using Support Vector Regression. Applied Intelligence, 51(10). https://doi.org/10.1007/s10489-021-02217-5
• Gordon, N. J., Salmond, D. J., and Smith, A. F. M. (1993). Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation. IEE Proceedings F Radar and Signal Processing, 140(2), Art. 2. https://doi.org/10.1049/ip-f-2.1993.0015
• Hartwig, B. (2019). Robust Inference in Time-varying Vector Autoregression: The DC-Cholesky Multivariate Stochastic Volatility Model. (Deutsche Bundesbank Discussion Paper No 34/2020). http://dx.doi.org/10.2139/ssrn.3665125
• Harvey, A., Ruiz, E., and Shephard, N. (1994). Multivariate Stochastic Variance Models. The Review of Economic Studies, 61(2). https://doi.org/10.2307/2297980
• He, D., Ionides, E. L., and King, A. A. (2009). Plug-and-play Inference for Disease Dynamics: Measles in Large and Small Populations as a Case Study. Journal of The Royal Society Interface, 7(43), 271-283. https://doi.org/10.1098/rsif. 2009.0151
• Ionides, E. L., Bretó, C., and King, A. A. (2006). Inference for Nonlinear Dynamical Systems. Proceedings of the National Academy of Sciences of the United States of America, 103(49). https://doi.org/10.1073/pnas.0603181103
• Ionides, E. L., Bhadra, A., Atchadé, Y., and King, A. (2011). Iterated Filtering. The Annals of Statistics, 39(3). https://doi.org/ 10.1214/11-AOS886
• Ionides, E. L., Nguyen, D., Atchadé, Y., Stoev, S., and King, A. A. (2015). Inference for Dynamic and Latent Variable Models via Iterated, Perturbed Bayes Maps. Proceedings of the National Academy of Sciences, 112(3). https://doi.org/10.1073/ pnas.1410597112
• Ishihara, T., and Omori, Y. (2012). Efficient Bayesian Estimation of a Multivariate Stochastic Volatility Model with Cross Leverage and Heavy-tailed Errors. Computational Statistics & Data Analysis, 56(11), 3674-3689. https://doi.org/10.1016/ j.csda.2010.07.015
• Ishihara, T., Omori, Y., and Asai, M. (2016). Matrix Exponential Stochastic Volatility with Cross Leverage. Computational Statistics & Data Analysis, 100, 331-350. https://doi.org/10.1016/j.csda.2014.10.012
• Jin, X., and Maheu, J. M. (2013). Modeling Realized Covariances and Returns. Journal of Financial Econometrics, 11(2). https://doi.org/10.1093/jjfinec/nbs022
• Jungbacker, B., and Koopman, S. J. (2006). Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models. Econometric Reviews, 25(2-3), 385-408. https://doi.org/10.1080/07474930600712848
• Kantas, N., Doucet, A., Singh, S. S., Maciejowski, J., and Chopin, N. (2015). On Particle Methods for Parameter Estimation in State-Space Models. Statistical Science, 328-351. https://doi.org/10.1214/14-STS511
• King, A. A., Ionides, E. L., Pascual, M., and Bouma, M. J. (2008). Inapparent Infections and Cholera Dynamics. Nature, 454(7206), https://doi.org/10.1038/nature07084
• King, A. A., Nguyen, D., and Ionides, E. L. (2016). Statistical Inference for Partially Observed Markov Processes via the R Package Pomp. Journal of Statistical Software, 69, 1-43. https://doi.org/10.18637/jss.v069.i12
• Kliber, A., Marszałek, P., Musiałkowska, I., and Świerczyńska, K. (2019). Bitcoin: Safe Haven, Hedge or Diversifier? Perception of Bitcoin in the context of a Country's Economic Situation - A Stochastic Volatility Approach. Physica A: Statistical Mechanics and Its Applications, 524, 246-257. https://doi.org/10.1016/j.physa.2019.04.145
• Lele, S. R., Dennis, B., and Lutscher, F. (2007). Data Cloning: Easy Maximum Likelihood Estimation for Complex Ecological Models Using Bayesian Markov Chain Monte Carlo Methods. Ecology Letters, 10(7), 551-563. https://doi.org/10.1111/ j.1461-0248.2007.01047.x
• Liu, J., and West, M. (2001). Combined Parameter and State Estimation in Simulation-based Filtering. In Sequential Monte Carlo Methods in Practice (pp. 197-223). Springer New York.
• Lopes, H. F., McCulloch, R. E., and Tsay, R. (2012). Cholesky Stochastic Volatility Models for High-dimensional Time Series. Vienna University of Economics and Business Discussion Papers. Retrieved January 13, 2023 from https://www.wu.ac.at/ fileadmin/wu/d/i/statmath/Research_Seminar/WS_2011-12/slideslopes.pdf
• Malik, S., and Pitt, M. K. (2011). Particle Filters for Continuous Likelihood Evaluation and Maximization. Journal of Econometrics, 165(2). https://doi.org/10.1016/j.jeconom.2011.07.006
• Nguyen, D. (2016). Another Look at Bayes Map Iterated Filtering. Statistics & Probability Letters, 118, 32-36. https://doi.org/ 10.1016/j.spl.2016.05.0134
• Pajor, A. (2010). Wielowymiarowe procesy wariancji stochastycznej w ekonometrii finansowej: ujęcie bayesowskie. Zeszyty Naukowe/Uniwersytet Ekonomiczny w Krakowie. Seria Specjalna, Monografie (195).
• Philipov, A., and Glickman, M. E. (2006a). Factor Multivariate Stochastic Volatility via Wishart Processes. Econometric Reviews, 25(2-3), 311-334. https://doi.org/10.1080/07474930600713366
• Philipov, A., and Glickman, M. E. (2006b). Multivariate Stochastic Volatility via Wishart Processes. Journal of Business and Economic Statistics, 24, 313-328. https://doi.org/10.1198/073500105000000306
• R Development Core Team. (2010). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org
• Stocks, T., Britton, T., and Höhle, M. (2020). Model Selection and Parameter Estimation for Dynamic Epidemic Models via Iterated Filtering: Application to Rotavirus in Germany. Biostatistics, 21(3). https://doi.org/10.1093/biostatistics/kxy057
• Szczepocki, P. (2020). Application of Iterated Filtering to Stochastic Volatility Models Based on non-Gaussian Ornstein-Uhlenbeck Process. Statistics in Transition New Series, 21(2). https://doi.org/10.21307/stattrans-2020-019
• Szczepocki, P. (2022). Estimation of Yu and Meyer Bivariate Stochastic Volatility Model by Iterated Filtering. Przegląd Statystyczny. Statistical Review, 69(4), 1-19. https://doi.org/10.59139/ps.2022.04.1
• Toni, T., Welch, D., Strelkowa, N., Ipsen, A., and Stumpf, M. P. H. (2008). Approximate Bayesian Computation Scheme for Parameter Inference and Model Selection in Dynamical Systems. Journal of The Royal Society Interface, 6(31), 187-202. https://doi.org/10.1098/rsif.2008.0172
• Tsay, R. S. (2005). Analysis of Financial Time Series. John Wiley & Sons.
• You, C., Deng, Y., Hu, W., Sun, J., Lin, Q., Zhou, F., Pang, C. H., Zhang, Y., Chen, Z., and Zhou, X.-H. (2020). Estimation of the Time-varying Reproduction Number of COVID-19 Outbreak in China. International Journal of Hygiene and Environmental Health, 228, 113555. https://doi.org/10.1016/j.ijheh.2020.113555
• Yu, J., and Meyer, R. (2006). Multivariate Stochastic Volatility Models: Bayesian Estimation and Model Comparison. Econometric Reviews, 25(2-3). https://doi.org/10.1080/07474930600713465
• Zaharieva, M. D., Trede, M., and Wilfling, B. (2020). Bayesian Semiparametric Multivariate Stochastic Volatility with Application. Econometric Reviews, 39(9), 947-970. https://doi.org/10.1080/07474938.2020.1761152

Identifiers

DOI

10.15611/eada.2023.4.04

Biblioteka Nauki

28407803