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Auxiliary likelihood-based approximate Bayesian computation in state space models. (English) Zbl 07499073

Summary: A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a “match” between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set ofsufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach – including a computationally efficient version based on the auxiliary score – achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multiparameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the article. Supplementary materials for this article are available online.

MSC:

62-XX Statistics

Software:

R; abc; Rugarch; abctools
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Full Text: DOI Link

References:

[1] Andersen, T. G.; Benzoni, L.; Lund, J., “An Empirical Investigation of Continuous-Time Equity Return Models,”, The Journal of Finance, 57, 1239-1284 (2002) · doi:10.1111/1540-6261.00460
[2] Anderson, T. W., The Statistical Analysis of Time Series (1968), New York: Wiley, New York
[3] Andrieu, C.; Doucet, A.; Holenstein, R., “Particle Markov Chain Monte Carlo Methods,”, Journal of the Royal Statistical Society, Series B, 72, 269-342 (2010) · Zbl 1411.65020 · doi:10.1111/j.1467-9868.2009.00736.x
[4] Beaumont, M. A.; Cornuet, J.-M.; Marin, J.-M.; Robert, C. P., “Adaptive Approximate Bayesian Computation,, Biometrika, 96, 983-990 (2009) · Zbl 1437.62393 · doi:10.1093/biomet/asp052
[5] Beaumont, M. A.; Zhang, W.; Balding, D. J., “Approximate Bayesian Computation in Population Genetics,”, Genetics, 162, 2025-2035 (2002)
[6] Blum, M. G. B., “Approximate Bayesian Computation: A Nonparametric Perspective,”, Journal of the American Statistical Association, 105, 1178-1187 (2010) · Zbl 1390.62052 · doi:10.1198/jasa.2010.tm09448
[7] Blum, M. G. B.; Francois, O., “Non-linear Regression Models for Approximate Bayesian Computation,”, Statistics and Computing, 20, 63-73 (2010) · doi:10.1007/s11222-009-9116-0
[8] Blum, M. G. B.; Nunes, M. A.; Prangle, D.; Sisson, S. A., “A Comparative Review of Dimension Reduction Methods in Approximate Bayesian Computation,”, Statistical Science, 28, 189-208 (2013) · Zbl 1331.62123 · doi:10.1214/12-STS406
[9] Bollerslev, T.; Chou, R. Y.; Kroner, K. F., “ARCH Modelling in Finance: A Review of the Theory and Empirical Evidence, Journal of Econometrics, 52, 5-59 (1992) · Zbl 0825.90057 · doi:10.1016/0304-4076(92)90064-X
[10] Calvet, C.; Czellar, V., “Accurate Methods for Approximate Bayesian Computation Filtering,”, Journal of Financial Econometrics, 13, 798-838 (2015) · doi:10.1093/jjfinec/nbu019
[11] Carr, P.; Wu, L., “The Finite Moment Log Stable Process and Option Pricing,”, Journal of Finance, LVIII, 753-777 (2003) · doi:10.1111/1540-6261.00544
[12] Chambers, J. M.; Mallows, C.; Stuck, B. W., “A Method for Simulating Stable Random Variables,”, Journal of the American Statistical Association, 71, 340-344 (1976) · Zbl 0341.65003 · doi:10.1080/01621459.1976.10480344
[13] Creel, M.; Gao, J.; Hong, H.; Kristensen, D., Bayesian Indirect Inference and the ABC of GMM, arXiv no. 1512.07385 (2015)
[14] Creel, M.; Kristensen, D., “ABC of SV: Limited Information Likelihood Inference in Stochastic Volatility Jump-Diffusion Models,”, Journal of Empirical Finance, 31, 85-108 (2015) · doi:10.1016/j.jempfin.2015.01.002
[15] Csillery, K.; Francois, O.; Blum, M. G. B., “abc: An R Package for Approximate Bayesian Computation,”, Methods in Ecology and Evolution, 3, 475-479 (2012) · doi:10.1111/j.2041-210X.2011.00179.x
[16] Dean, T. A.; Singh, S. S.; Jasra, A.; Peters, G. W., “Parameter Estimation or Hidden Markov Models With Intractable Likelihoods,”, Scandinavian Journal of Statistics, 41, 970-987 (2014) · Zbl 1305.62303 · doi:10.1111/sjos.12077
[17] Douc, R.; Moulines, E., “Asymptotic Properties of the Maximum Likelihood Estimation in Misspecified Hidden Markov Models,”, Annals of Statistics, 40, 2697-2732 (2012) · Zbl 1373.62436 · doi:10.1214/12-AOS1047
[18] Drovandi, C. C.; Pettitt, A. N.; Faddy, M. J., “Approximate Bayesian Computation Using Indirect Inference,”, Journal of the Royal Statistical Society, Series C, 60, 1-21 (2011)
[19] Drovandi, C. C.; Pettitt, A. N.; Lee, A., “Bayesian Indirect Inference Using a Parametric Auxiliary Model,”, Statistical Science, 30, 72-95 (2015) · Zbl 1332.62088 · doi:10.1214/14-STS498
[20] Eraker, B., “Do Stock Prices and Volatility Jump? Reconciling Evidence From Spot and Option Prices,”, The Journal of Finance, LIX, 1367-1403 (2004) · doi:10.1111/j.1540-6261.2004.00666.x
[21] Fearnhead, P.; Prangle, D., “Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic Approximate Bayesian Computation,”, Journal of the Royal Statistical Society, Series B, 74, 419-474 (2012) · Zbl 1411.62057 · doi:10.1111/j.1467-9868.2011.01010.x
[22] Frazier, D. T.; Martin, G. M.; Robert, C. P.; Rousseau, J., “Asymptotic Properties of Approximate Bayesian Computation,”, Biometrika, 105, 593-607 (2018) · Zbl 1499.62112 · doi:10.1093/biomet/asy027
[23] Garcia, R.; Renault, E.; Veredas, D., “Estimation of Stable Distributions With Indirect Inference,”, Journal of Econometrics, 161, 325-337 (2011) · Zbl 1441.62699 · doi:10.1016/j.jeconom.2010.12.007
[24] Gallant, A. R.; Tauchen, G., “Which Moments to Match,”, Econometric Theory, 12, 657-681 (1996) · doi:10.1017/S0266466600006976
[25] Gallant, A. R.; Tauchen, G., Handbook of Financial Econometrics, 1, “Simulated Score Methods and Indirect Inference for Continuous-Time Models,” (2010), Amsterdam: Elsevier, Amsterdam
[26] Ghalanos, A. (2018), “rugarch: Univariate GARCH Models,” R Package Version 1.4-0.
[27] Gouriéroux, C.; Monfort, A., Statistics and Econometric Models (1995), Cambridge: CUP, Cambridge · Zbl 1420.91461
[28] Gouriéroux, C.; Monfort, A.; Renault, E., “Indirect Inference,”, Journal of Applied Econometrics, 85, S85-S118 (1993) · doi:10.1002/jae.3950080507
[29] Heston, S. L., “A Closed-Form Solution for Options With Stochastic Volatility With Applications to Bond and Currency Options,”, The Review of Financial Studies, 6, 327-343 (1993) · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[30] Jabot, T.; Faure, T.; Dumoulin, D.; Albert, C. (2015)
[31] Jasra, A., “Approximate Bayesian Computation for a Class of Time Series Models,”, International Statistical Review, 83, 405-435 (2015) · Zbl 07763454 · doi:10.1111/insr.12089
[32] Jasra, A.; Singh, S.; Martin, J.; McCoy, E., “Filtering via Approximate Bayesian Computation,”, Statistics and Computing, 22, 1223-1237 (2012) · Zbl 1252.62093 · doi:10.1007/s11222-010-9185-0
[33] Johannes, M.; Polson, N. G.; Stroud, J. R., “Optimal Filtering of Jump-Diffusions: Extracting Latent States From Asset Prices,”, Review of Financial Studies, 22, 2759-2799 (2009) · doi:10.1093/rfs/hhn110
[34] Joyce, P.; Marjoram, P., “Approximately Sufficient Statistics and Bayesian Computation,”, Statistical Applications in Genetics and Molecular Biology, 7, 1-16 (2008) · Zbl 1276.62077
[35] Julier, S. J.; Uhlmann, J. K.; Durrant-Whyte, H. F., A New Approach for Filtering Nonlinear Systems, 1628-1632 (1995)
[36] Li, W.; Fearnhead, P., “On the Asymptotic Efficiency of ABC Estimators,”, Biometrika, 105, 285-299 (2018) · Zbl 07072413 · doi:10.1093/biomet/asx078
[37] Li, W.; Fearnhead, P., “Convergence of Regression-Adjusted Approximate Bayesian Computation,”, Biometrika, 105, 301-318 (2018) · Zbl 07072414
[38] Lombardi, M. J.; Calzolari, G., “Indirect Estimation of α-Stable Stochastic Volatility Models,”, Computational Statistics and Data Analysis, 53, 2298-2308 (2009) · Zbl 1453.62143 · doi:10.1016/j.csda.2008.11.016
[39] Marin, J.-M.; Pudlo, P.; Robert, C.; Ryder, R., “Approximate Bayesian Computation Methods,”, Statistics and Computing, 21, 289-291 (2011)
[40] Marjoram, P.; Molitor, J.; Plagonal, V.; Tavaré, S., “Markov Chain Monte Carlo Without Likelihoods,”, Proceedings of the National Academy of Sciences of the United States of America, 100, 15324-15328 (2003) · doi:10.1073/pnas.0306899100
[41] Ng, J.; Forbes, C. S.; Martin, G. M.; McCabe, B. P. M., “Non-parametric Estimation of Forecast Distributions in Non-Gaussian, Non-linear State Space Models,”, International Journal of Forecasting, 29, 411-430 (2013) · doi:10.1016/j.ijforecast.2012.10.005
[42] Nott, D.; Fan, Y.; Marshall, L.; Sisson, S., “Approximate Bayesian Computation and Bayes Linear Analysis: Towards High-Dimensional ABC,”, Journal of Computational and Graphical Statistics, 23, 65-86 (2014) · doi:10.1080/10618600.2012.751874
[43] Nunes, M. A.; Prangle, D., “abctools: An R Package for Tuning Approximate Bayesian Computation Analyses,”, The R Journal, 7, 189-205 (2015) · doi:10.32614/RJ-2015-030
[44] Peters, G. W.; Sisson, S. A.; Fan, Y., “Likelihood-Free Bayesian Inference for α-Stable Models,”, Computational Statistics and Data Analysis, 65, 3743-3756 (2012) · Zbl 1255.62071 · doi:10.1016/j.csda.2010.10.004
[45] Pitt, M. K.; dos Santos Silva, R.; Giordani, P.; Kohn, R., “On Some Properties of Markov Chain Monte Carlo Simulation Methods Based on the Particle Filter,”, Journal of Econometrics, 171, 134-151 (2012) · Zbl 1443.62499 · doi:10.1016/j.jeconom.2012.06.004
[46] Pritchard, J. K.; Seilstad, M. T.; Perez-Lezaun, A.; Feldman, M. W., “Population Growth of Human Y Chromosomes: A Study of Y Chromosome Microsatellites,”, Molecular Biology and Evolution, 16, 1791-1798 (1999) · doi:10.1093/oxfordjournals.molbev.a026091
[47] Core Team, R., R: A Language and Environment for Statistical Computing (2013), Vienna, Austria: R Foundation for Statistical Computing, Vienna, Austria
[48] Samoradnitsky, G.; Taqqu, M. S., Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance, 1 (1994), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 0925.60027
[49] Sisson, S.; Fan, Y.; Brooks, S. P.; Gelman, A.; Jones, G.; Meng, X.-L., Handbook of Markov Chain Monte Carlo, “Likelihood-Free Markov Chain Monte Carlo,” (2011), Boca Raton, FL: Chapman and Hall/CRC Press, Boca Raton, FL · Zbl 1218.65001
[50] Sisson, S.; Fan, Y.; Tanaka, M., “Sequential Monte Carlo Without Likelihoods,”, Proceedings of the National Academy of Sciences of the United States of America, 104, 1760-1765 (2007) · Zbl 1160.65005 · doi:10.1073/pnas.0607208104
[51] Tavaré, S.; Balding, D. J.; Griffiths, R. C.; Donnelly, P., “Inferring Coalescence Times From DNA Sequence Data,”, Genetics, 145, 505-518 (1997)
[52] Wegmann, D.; Leuenberger, C.; Excoffier, L., “Efficient Approximate Bayesian Computation Coupled With Markov Chain Monte Carlo With Likelihood,”, Genetics, 182, 1207-1218 (2009) · doi:10.1534/genetics.109.102509
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