Abstract
We provide a novel approach of estimating a regime-switching nonlinear and non-Gaussian state-space model based on a particle learning scheme. In particular, we extend the particle learning method in Liu, J., and M. West. 2001. “Combined Parameter and State Estimation in Simulation-Based Filtering.” In Sequential Monte Carlo Methods in Practice, 197–223. Springer. by constructing a new proposal distribution for the latent regime index variable that incorporates all available information contained in the current and past observations. The Monte Carlo simulation result implies that our approach categorically outperforms a popular existing algorithm. For empirical illustration, the proposed algorithm is used to analyze the underlying dynamics of US excess stock return.
Acknowledgement
We are grateful to two anonymous referees and the editor for helpful suggestions and constructive comments that greatly improved the paper. All errors are our own.
Appendix A
Volatility MSE and QPS comparison
Appendix B
Comparison of parameter MSE: case 1
Appendix C
Comparison of parameter MSE: case 2
Appendix D
Table 2 Robust Check
Model A | Model B | Model C | Model D | ||||||
---|---|---|---|---|---|---|---|---|---|
Regime Δ in Vol. | No | No | Yes | Yes | Yes | Yes | Yes | Yes | |
Regime Δ in Mean | No | No | No | No | No | No | Yes | Yes | |
Regime Δ in Leverage | No | No | No | No | Yes | Yes | No | No | |
Prior | Mean | S.D. | Mean | S.D. | Mean | S.D. | Mean | S.D. | |
−2.314 | 0.376 | −1.990 | 0.282 | −2.204 | 0.372 | −3.891 | 0.357 | ||
− | − | − | − | − | − | 2.053 | 0.360 | ||
1.269 | 0.082 | 0.812 | 0.068 | 0.917 | 0.083 | 0.828 | 0.085 | ||
− | − | 1.026 | 0.118 | 1.017 | 0.124 | 1.155 | 0.121 | ||
0.528 | 0.131 | 0.324 | 0.152 | 0.861 | 0.032 | 0.861 | 0.030 | ||
−0.218 | 0.090 | −0.046 | 0.120 | 0.002 | 0.077 | 0.011 | 0.104 | ||
−0.205 | 0.073 | 0.032 | 0.114 | −0.063 | 0.067 | −0.020 | 0.087 | ||
0.830 | 0.052 | 0.843 | 0.040 | 0.881 | 0.024 | 0.824 | 0.049 | ||
−0.198 | 0.020 | −0.231 | 0.023 | −0.182 | 0.030 | −0.262 | 0.033 | ||
− | − | − | − | −0.256 | 0.042 | − | − | ||
ρ | −0.026 | 0.217 | −0.104 | 0.236 | 0.020 | 0.259 | −0.050 | 0.252 | |
0.042 | 0.010 | 0.044 | 0.012 | 0.013 | 0.004 | 0.016 | 0.005 | ||
0.068 | 0.013 | 0.065 | 0.017 | 0.013 | 0.003 | 0.012 | 0.003 | ||
− | − | − | − | − | − | 0.975 | 0.006 | ||
− | − | − | − | − | − | 0.975 | 0.007 | ||
− | − | 0.992 | 0.002 | 0.992 | 0.002 | 0.991 | 0.003 | ||
− | − | 0.989 | 0.004 | 0.984 | 0.005 | 0.987 | 0.004 | ||
log(ML) | −4105.9 | −4099.6 | −4105.2 | −4105.0 |
Refer to footnotes in Table 2.
References
Ang, A., and A. Timmermann. 2012. “Regime Changes and Financial Markets.” Annual Review of Financial Economics 4: 313–337.10.3386/w17182Search in Google Scholar
Beltratti, A., and C. Morana. 2006. “Breaks and Persistency: Macroeconomic Causes of Stock Market Volatility.” Journal of Econometrics 131: 151–177.10.1016/j.jeconom.2005.01.007Search in Google Scholar
Bernardo, J., M. Bayarri, J. Berger, A. Dawid, D. Heckerman, A. Smith, and M. West. 2011. “Particle Learning for Sequential Bayesian Computation.” Bayesian Statistics 9 9: 317.10.1093/acprof:oso/9780199694587.001.0001Search in Google Scholar
Bollen, N. P., S. F. Gray, and R. E. Whaley. 2000. “Regime Switching in Foreign Exchange Rates:: Evidence from Currency Option Prices.” Journal of Econometrics 94: 239–276.10.1016/S0304-4076(99)00022-6Search in Google Scholar
Brandt, M. W., and Q. Kang. 2004. “On the Relationship between the Conditional Mean and Volatility of Stock Returns: A Latent var Approach.” Journal of Financial Economics 72: 217–257.10.3386/w9056Search in Google Scholar
Carvalho, C. M., and H. F. Lopes. 2007. “Simulation-Based Sequential Analysis of Markov Switching Stochastic Volatility Models.” Computational Statistics & Data Analysis 51: 4526–4542.10.1016/j.csda.2006.07.019Search in Google Scholar
Carvalho, C., M. S. Johannes, H. F. Lopes, and N. Polson. 2010. “Particle Learning and Smoothing.” Statistical Science 25: 88–106.10.1214/10-STS325Search in Google Scholar
Chib, S. 1998. “Estimation and Comparison of Multiple Change-Point Models.” Journal of Econometrics 86: 221–241.10.1016/S0304-4076(97)00115-2Search in Google Scholar
Christie, A. A. 1982. “The Stochastic Behavior of Common Stock Variances: Value, Leverage and Interest Rate Effects.” Journal of Financial Economics 10: 407–432.10.1016/0304-405X(82)90018-6Search in Google Scholar
Ghysels, E., P. Santa-Clara, and R. Valkanov. 2005. “There is a Risk-Return Trade-off after All.” Journal of Financial Economics 76: 509–548.10.1016/j.jfineco.2004.03.008Search in Google Scholar
Glosten, L. R., R. Jagannathan, and D. E. Runkle. 1993. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance 48: 1779–1801.10.1111/j.1540-6261.1993.tb05128.xSearch in Google Scholar
Gordon, N. J., D. J. Salmond, and A. F. Smith. 1993. “Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation.” in IEE Proceedings F-Radar and Signal Processing volume 140, IET, 140: 107–113.10.1049/ip-f-2.1993.0015Search in Google Scholar
Kass, R. E., and A. E. Raftery. 1995. “Bayes Factors.” Journal of the American Statistical Association 90: 773–795.10.1080/01621459.1995.10476572Search in Google Scholar
Kim, C.-J. 1994. “Dynamic Linear Models with Markov-Switching.” Journal of Econometrics 60: 1–22.10.1016/0304-4076(94)90036-1Search in Google Scholar
Kim, C.-J., and C. R. Nelson. 1999. State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications, Edn. 1, Vol. 1. Cambridge, MA: MIT Press Books.Search in Google Scholar
Liu, J., and M. West. 2001. “Combined Parameter and State Estimation in Simulation-Based Filtering.” In Sequential Monte Carlo Methods in Practice, edited by Arnaud Doucet, Nando de Freitas, and Neil Gordon. 197–223. New York, NY: Springer.10.1007/978-1-4757-3437-9_10Search in Google Scholar
Lopes, H. F., and R. S. Tsay. 2011. “Particle Filters and Bayesian Inference in Financial Econometrics.” Journal of Forecasting 30: 168–209.10.1002/for.1195Search in Google Scholar
Ludvigson, S. C., and S. Ng. 2007. “The Empirical Risk–Return Relation: A Factor Analysis Approach.” Journal of Financial Economics 83: 171–222.10.3386/w11477Search in Google Scholar
Lundblad, C. 2007. “The Risk Return Tradeoff in the Long Run: 1836–2003.” Journal of Financial Economics 85: 123–150.10.1016/j.jfineco.2006.06.003Search in Google Scholar
Marcucci, J. 2005. “Forecasting Stock Market Volatility with Regime-Switching Garch Models.” Studies in Nonlinear Dynamics & Econometrics 9: 1–55.10.2202/1558-3708.1145Search in Google Scholar
Morana, C., and A. Beltratti. 2002. “The Effects of the Introduction of the Euro on the Volatility of European Stock Markets.” Journal of Banking & Finance 26: 2047–2064.10.1016/S0378-4266(01)00182-0Search in Google Scholar
Nelson, D. B. 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica: Journal of the Econometric Society 59: 347–370.10.2307/2938260Search in Google Scholar
Pitt, M. K., and N. Shephard. 1999. “Filtering via Simulation: Auxiliary Particle Filters.” Journal of the American Statistical Association 94: 590–599.10.1080/01621459.1999.10474153Search in Google Scholar
Raftery, A. E. 1995. “Bayesian Model Selection in Social Research.” Sociological Methodology 25: 111–163.10.2307/271063Search in Google Scholar
Rios, M. P., and H. F. Lopes. 2013. “The Extended Liu and West Filter: Parameter Learning in Markov Switching Stochastic Volatility Models.” In State-Space Models, 23–61. Springer.10.1007/978-1-4614-7789-1_2Search in Google Scholar
So, M. E. P., K. Lam, and W. K. Li. 1998. “A Stochastic Volatility Model with Markov Switching.” Journal of Business & Economic Statistics 16: 244–253.Search in Google Scholar
Yang, B., J. R. Stroud, and G. Huerta. 2017. “Sequential Monte Carlo Smoothing with Parameter Estimation.” Bayesian Analysis 13: 1133–1157.10.1214/17-BA1088Search in Google Scholar
Supplementary Material
The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2018-0016).
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