# American Institute of Mathematical Sciences

March  2019, 1(1): 61-85. doi: 10.3934/fods.2019003

## Particle filters for inference of high-dimensional multivariate stochastic volatility models with cross-leverage effects

 Department of Statistics & Applied Probability, National University of Singapore, Singapore, 117546, SG

* Corresponding author

Published  February 2019

Fund Project: AJ is supported by an AcRF tier 2 grant: R-155-000-161-112.

Multivariate stochastic volatility models are a popular and well-known class of models in the analysis of financial time series because of their abilities to capture the important stylized facts of financial returns data. We consider the problems of filtering distribution estimation and also marginal likelihood calculation for multivariate stochastic volatility models with cross-leverage effects in the high dimensional case, that is when the number of financial time series that we analyze simultaneously (denoted by $d$) is large. The standard particle filter has been widely used in the literature to solve these intractable inference problems. It has excellent performance in low to moderate dimensions, but collapses in the high dimensional case. In this article, two new and advanced particle filters proposed in [4], named the space-time particle filter and the marginal space-time particle filter, are explored for these estimation problems. The better performance in both the accuracy and stability for the two advanced particle filters are shown using simulation and empirical studies in comparison with the standard particle filter. In addition, Bayesian static model parameter estimation problem is considered with the advances in particle Markov chain Monte Carlo methods. The particle marginal Metropolis-Hastings algorithm is applied together with the likelihood estimates from the space-time particle filter to infer the static model parameter successfully when that using the likelihood estimates from the standard particle filter fails.

Citation: Yaxian Xu, Ajay Jasra. Particle filters for inference of high-dimensional multivariate stochastic volatility models with cross-leverage effects. Foundations of Data Science, 2019, 1 (1) : 61-85. doi: 10.3934/fods.2019003
##### References:
 [1] C. Andrieu, A. Doucet and R. Holenstein, Particle Markov chain Monte Carlo methods (with discussion), J. R. Statist. Soc. Ser. B, 72 (2010), 269-342.  doi: 10.1111/j.1467-9868.2009.00736.x.  Google Scholar [2] M. Asai, M. McAleer and J. Yu, Multivariate stochastic volatility: A review, Econ. Rev., 25 (2006), 145-175.  doi: 10.1080/07474930600713564.  Google Scholar [3] L. Bauwens, S. Laurent and J. V. Rombouts, Multivariate GARCH models: A survey, J. Appl. Econ., 21 (2006), 79-109.  doi: 10.1002/jae.842.  Google Scholar [4] A. Beskos, D. Crisan, A. Jasra, K. Kamatani and Y. Zhou, A stable particle filter for a class of high-dimensional state-space models, Adv. Appl. Probab., 49 (2017), 24-48.  doi: 10.1017/apr.2016.77.  Google Scholar [5] P. Bickel, B. Li and T. Bengtsson, Sharp failure rates for the bootstrap particle filter in high dimensions, In Pushing the Limits of Contemporary Statistics: Contributions in Honor of J. Ghosh, IMS, 3 (2008), 318–329. doi: 10.1214/074921708000000228.  Google Scholar [6] S. Chib, F. Nadari and N. Shephard, Analysis of high-dimensional multivariate stochastic volatility models, J. Econ., 134 (2006), 341-371.  doi: 10.1016/j.jeconom.2005.06.026.  Google Scholar [7] N. Chopin, Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference, Ann. Statist., 32 (2004), 2385-2411.  doi: 10.1214/009053604000000698.  Google Scholar [8] P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, New York, 2004. doi: 10.1007/978-1-4684-9393-1.  Google Scholar [9] P. Del Moral, A. Doucet and A. Jasra, On adaptive resampling strategies for sequential Monte Carlo methods, Bernoulli, 18 (2012), 252-278.  doi: 10.3150/10-BEJ335.  Google Scholar [10] A. Doucet, On Sequential Simulation-based Methods for Bayesian Filtering, Technical Report, 1998. Google Scholar [11] A. Doucet and A. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, In Handbook of Nonlinear Filtering (eds. D. Crisan & B. Rozovsky), Oxford University Press, Oxford, (2011), 656–704.  Google Scholar [12] A. Doucet, M. K. Pitt, G. Deligiannidis and R. Kohn, Efficient Implementation of Markov chain Monte Carlo when Using an Unbiased Likelihood Estimator, Biometrika, 102 (2015), 295-313.  doi: 10.1093/biomet/asu075.  Google Scholar [13] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finan., 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.  Google Scholar [14] T. Ishihara and Y. Omori, Efficient Bayesian estimation of a multivariate stochastic volatility with cross leverage and heavy tailed errors, Comp. Statist. Data Anal., 56 (2012), 3674-3689.  doi: 10.1016/j.csda.2010.07.015.  Google Scholar [15] A. Jasra, D. A. Stephens, A. Doucet and T. Tsagaris, Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.  doi: 10.1111/j.1467-9469.2010.00723.x.  Google Scholar [16] N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin, An overview of sequential Monte Carlo methods for parameter estimation in general state-space sodels, IFAC Proc., 42 (2009), 774-785.   Google Scholar [17] N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin, On particle methods for parameter estimation in general state-space models, Statist. Sci., 30 (2015), 328-351.  doi: 10.1214/14-STS511.  Google Scholar [18] S. Kim, N. Shephard and S. Chib, Stochastic volatility: Likelihood inference and comparison with ARCH models, Rev. Econ. Stud., 65 (1998), 361-393.  doi: 10.1111/1467-937X.00050.  Google Scholar [19] G. Kitagawa, Monte Carlo filter and smoother for non-Gaussian nonlinear state-space models, J. Comp. Graph. Stat., 5 (1996), 1-25.  doi: 10.2307/1390750.  Google Scholar [20] M. Klaas, N. De Freitas and A. Doucet, Towards practical N2 Monte Carlo: The marginal particle filter, Uncert. A. I., (2005), 308–315. Google Scholar [21] A. Kong, J. S. Liu and W. H. Wong, Sequential imputations and Bayesian missing data problems, J. Amer. Statist. Assoc., 89 (1994), 278-288.  doi: 10.1080/01621459.1994.10476469.  Google Scholar [22] C. Naesseth, F. Lindten and T. Schön, Nested sequential Monte Carlo methods, ICML, (2015), 1292–1301. Google Scholar [23] J. Nakajima, Bayesian analysis of multivariate stochastic volatility with skew return distribution, Econ. Rev., 36 (2017), 546-562.  doi: 10.1080/07474938.2014.977093.  Google Scholar [24] S. S. Ozturk and J. F. Richard, Stochastic volatility and leverage: Application to a panel of S & P 500 stocks, Finan. Res. Lett., 12 (2015), 67-76.   Google Scholar [25] M. K. Pitt, R. Dos Santos Silva, P. Giordani and R. Kohn, On some properties of Markov chain Monte Carlo simulation methods based upon the particle filter, J. Econom., 171 (2012), 134-151.  doi: 10.1016/j.jeconom.2012.06.004.  Google Scholar [26] M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters, J. Amer. Statist. Assoc., 94 (1999), 590-599.  doi: 10.1080/01621459.1999.10474153.  Google Scholar [27] K. Platanioti, E. McCoy and D. A. Stephens, A Review of Stochastic Volatility Models, Technical Report, 2005. Google Scholar [28] C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering, Month. Weather Rev., 136 (2008), 4629-4640.  doi: 10.1175/2008MWR2529.1.  Google Scholar [29] C. Vergé, C. Duberry, P. Del Moral and E. Moulines, On parallel implementation of sequential Monte Carlo methods: The island particle filtering, Stat. Comp., 25 (2015), 243-260.  doi: 10.1007/s11222-013-9429-x.  Google Scholar

show all references

##### References:
 [1] C. Andrieu, A. Doucet and R. Holenstein, Particle Markov chain Monte Carlo methods (with discussion), J. R. Statist. Soc. Ser. B, 72 (2010), 269-342.  doi: 10.1111/j.1467-9868.2009.00736.x.  Google Scholar [2] M. Asai, M. McAleer and J. Yu, Multivariate stochastic volatility: A review, Econ. Rev., 25 (2006), 145-175.  doi: 10.1080/07474930600713564.  Google Scholar [3] L. Bauwens, S. Laurent and J. V. Rombouts, Multivariate GARCH models: A survey, J. Appl. Econ., 21 (2006), 79-109.  doi: 10.1002/jae.842.  Google Scholar [4] A. Beskos, D. Crisan, A. Jasra, K. Kamatani and Y. Zhou, A stable particle filter for a class of high-dimensional state-space models, Adv. Appl. Probab., 49 (2017), 24-48.  doi: 10.1017/apr.2016.77.  Google Scholar [5] P. Bickel, B. Li and T. Bengtsson, Sharp failure rates for the bootstrap particle filter in high dimensions, In Pushing the Limits of Contemporary Statistics: Contributions in Honor of J. Ghosh, IMS, 3 (2008), 318–329. doi: 10.1214/074921708000000228.  Google Scholar [6] S. Chib, F. Nadari and N. Shephard, Analysis of high-dimensional multivariate stochastic volatility models, J. Econ., 134 (2006), 341-371.  doi: 10.1016/j.jeconom.2005.06.026.  Google Scholar [7] N. Chopin, Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference, Ann. Statist., 32 (2004), 2385-2411.  doi: 10.1214/009053604000000698.  Google Scholar [8] P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, New York, 2004. doi: 10.1007/978-1-4684-9393-1.  Google Scholar [9] P. Del Moral, A. Doucet and A. Jasra, On adaptive resampling strategies for sequential Monte Carlo methods, Bernoulli, 18 (2012), 252-278.  doi: 10.3150/10-BEJ335.  Google Scholar [10] A. Doucet, On Sequential Simulation-based Methods for Bayesian Filtering, Technical Report, 1998. Google Scholar [11] A. Doucet and A. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, In Handbook of Nonlinear Filtering (eds. D. Crisan & B. Rozovsky), Oxford University Press, Oxford, (2011), 656–704.  Google Scholar [12] A. Doucet, M. K. Pitt, G. Deligiannidis and R. Kohn, Efficient Implementation of Markov chain Monte Carlo when Using an Unbiased Likelihood Estimator, Biometrika, 102 (2015), 295-313.  doi: 10.1093/biomet/asu075.  Google Scholar [13] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finan., 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.  Google Scholar [14] T. Ishihara and Y. Omori, Efficient Bayesian estimation of a multivariate stochastic volatility with cross leverage and heavy tailed errors, Comp. Statist. Data Anal., 56 (2012), 3674-3689.  doi: 10.1016/j.csda.2010.07.015.  Google Scholar [15] A. Jasra, D. A. Stephens, A. Doucet and T. Tsagaris, Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.  doi: 10.1111/j.1467-9469.2010.00723.x.  Google Scholar [16] N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin, An overview of sequential Monte Carlo methods for parameter estimation in general state-space sodels, IFAC Proc., 42 (2009), 774-785.   Google Scholar [17] N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin, On particle methods for parameter estimation in general state-space models, Statist. Sci., 30 (2015), 328-351.  doi: 10.1214/14-STS511.  Google Scholar [18] S. Kim, N. Shephard and S. Chib, Stochastic volatility: Likelihood inference and comparison with ARCH models, Rev. Econ. Stud., 65 (1998), 361-393.  doi: 10.1111/1467-937X.00050.  Google Scholar [19] G. Kitagawa, Monte Carlo filter and smoother for non-Gaussian nonlinear state-space models, J. Comp. Graph. Stat., 5 (1996), 1-25.  doi: 10.2307/1390750.  Google Scholar [20] M. Klaas, N. De Freitas and A. Doucet, Towards practical N2 Monte Carlo: The marginal particle filter, Uncert. A. I., (2005), 308–315. Google Scholar [21] A. Kong, J. S. Liu and W. H. Wong, Sequential imputations and Bayesian missing data problems, J. Amer. Statist. Assoc., 89 (1994), 278-288.  doi: 10.1080/01621459.1994.10476469.  Google Scholar [22] C. Naesseth, F. Lindten and T. Schön, Nested sequential Monte Carlo methods, ICML, (2015), 1292–1301. Google Scholar [23] J. Nakajima, Bayesian analysis of multivariate stochastic volatility with skew return distribution, Econ. Rev., 36 (2017), 546-562.  doi: 10.1080/07474938.2014.977093.  Google Scholar [24] S. S. Ozturk and J. F. Richard, Stochastic volatility and leverage: Application to a panel of S & P 500 stocks, Finan. Res. Lett., 12 (2015), 67-76.   Google Scholar [25] M. K. Pitt, R. Dos Santos Silva, P. Giordani and R. Kohn, On some properties of Markov chain Monte Carlo simulation methods based upon the particle filter, J. Econom., 171 (2012), 134-151.  doi: 10.1016/j.jeconom.2012.06.004.  Google Scholar [26] M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters, J. Amer. Statist. Assoc., 94 (1999), 590-599.  doi: 10.1080/01621459.1999.10474153.  Google Scholar [27] K. Platanioti, E. McCoy and D. A. Stephens, A Review of Stochastic Volatility Models, Technical Report, 2005. Google Scholar [28] C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering, Month. Weather Rev., 136 (2008), 4629-4640.  doi: 10.1175/2008MWR2529.1.  Google Scholar [29] C. Vergé, C. Duberry, P. Del Moral and E. Moulines, On parallel implementation of sequential Monte Carlo methods: The island particle filtering, Stat. Comp., 25 (2015), 243-260.  doi: 10.1007/s11222-013-9429-x.  Google Scholar
Plot of Scaled Effective Sample Size (ESS) averaged over 20 runs when standard particle filter is applied to MSV model with dimension 200
Plots of Scaled Effective Sample Size (ESS) averaged over 20 runs
Plots of mean of estimates for the $1^{st}$ component of the mean of the filters across 20 runs
Plots of SD of estimates for the $1^{st}$ component of the mean of the filters across 20 runs
Plots of SD of the estimated log-likelihoods across 20 runs
Time Comparison Study Plots of Scaled Effective Sample Size (ESS) averaged over 20 runs
Time Comparison Study Plots of Relative SD of the estimated log-likelihoods (w.r.t. the SD of the STPF) across 20 runs
Trace plots, histograms and ACF plots of the parameter estimates using PMMH for $\rho_{ij,\varepsilon\varepsilon}$
Trace plots, histograms and ACF plots of the parameter estimates using PMMH for $\sigma_{i,\eta\eta}$
Plot of Scaled Effective Sample Size (ESS)
Plots of Scaled Effective Sample Size (ESS) averaged over 20 runs
Plots of SD of the estimated log-likelihoods across 20 runs
Time Comparison Study Plots of Scaled Effective Sample Size (ESS) averaged over 20 runs
Time Comparison Study Plots of Relative SD of the estimated log-likelihoods (w.r.t. the SD of the STPF) across 20 runs
Number of particles used in each algorithm
 $d$ Standard PF STPF Marginal STPF 25 $N=1000$ $N=50$, $M_d=20$ $N=1$, $M_d=1000$ 50 $N=1000$ $N=50$, $M_d=20$ $N=1$, $M_d=1000$ 100 $N=1000$ $N=50$, $M_d=20$ N.A. 200 $N=1000$ $N=50$, $M_d=20$ N.A.
 $d$ Standard PF STPF Marginal STPF 25 $N=1000$ $N=50$, $M_d=20$ $N=1$, $M_d=1000$ 50 $N=1000$ $N=50$, $M_d=20$ $N=1$, $M_d=1000$ 100 $N=1000$ $N=50$, $M_d=20$ N.A. 200 $N=1000$ $N=50$, $M_d=20$ N.A.
Computation time (in minutes) per 50 time points for each algorithm
 $d$ Standard PF STPF Marginal STPF 25 $0.3$ $2$ $46$ 50 $0.6$ $3$ $110$ 100 $2$ $16.7$ N.A. 200 $5$ $120$ N.A.
 $d$ Standard PF STPF Marginal STPF 25 $0.3$ $2$ $46$ 50 $0.6$ $3$ $110$ 100 $2$ $16.7$ N.A. 200 $5$ $120$ N.A.
Number of particles and computation time (in minutes) per 50 time points for each algorithm
 $d$ Standard PF STPF Computation Time 100 $N=25000$ $N=50$, $M_d=20$ $16.7$ 200 $N=40000$ $N=50$, $M_d=20$ $120$
 $d$ Standard PF STPF Computation Time 100 $N=25000$ $N=50$, $M_d=20$ $16.7$ 200 $N=40000$ $N=50$, $M_d=20$ $120$
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