Markov regime-switching (MRS) autoregressive model is a widely used approach to model the economic and financial data with potential structural breaks. The innovation series of such MRS-type models are usually assumed to follow a Normal distribution, which cannot accommodate fat-tailed properties commonly present in empirical data. Many theoretical studies suggest that this issue can lead to inconsistent estimates. In this paper, we consider the tempered stable distribution, which has the attractive stability under aggregation property missed in other popular alternatives like Student’s t-distribution and General Error Distribution (GED). Through systematically designed simulation studies with the MRS autoregressive models, our results demonstrate that the model with tempered stable distribution uniformly outperforms those with Student’s t-distribution and GED. Our empirical study on the implied volatility of the S&P 500 options (VIX) also leads to the same conclusions. Therefore, we argue that the tempered stable distribution could be widely used for modelling economic and financial data in general contexts with an MRS-type specification.
Ardia, D. 2009. “Bayesian Estimation of a Markov-Switching Threshold Asymmetric GARCH Model with Student-t Innovations.” The Econometrics Journal 12: 105–126.
Bianchi, M. L., S. T. Rachev, Y. S. Kim, and F. J. Fabozzi. 2010. “Tempered Stable Distributions and Processes in Finance: Numerical Analysis.” In: Mathematical and Statistical Methods for Actuarial Sciences and Finance, edited by M. Corazza and C. Pizzi, 33–42. Italy: Springer.
Bollerslev, T. 1987. “A Conditional Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” Review of Economics and Statistics 69: 542–547.
Calzolari, G., R. Halbleib, and A. Parrini. 2014. “Estimating GARCH-type Models with Symmetric Stable Innovations: Indirect Inference Versus Maximum Likelihood.” Computational Statistics & Data Analysis 76: 158–171.
Carr, P., H. Geman, D. B. Madan, and M. Yor. 2002. “The Fine Structure of Asset Returns: An Empirical Investigation*.” The Journal of Business 75: 305–333.
Carrasco, M., L. Hu, and W. Ploberger. 2014. “Optimal Test for Markov Switching Parameters.” Econometrica 82: 765–784.
Cho, J. S., and H. White. 2007. “Testing for Regime Switching.” Econometrica 75: 1671–1720.
Constantinides, A., and S. Savel’ev. 2013. “Modelling Price Dynamics: A Hybrid Truncated Lévy Flight-GARCH Approach.” Physica A 392: 2072–2078.
Cont, R., and P. Tankov. 2004. Financial Modelling with Jump Processes. United Kingdom: Chapman and Hall/CRS Press.
Diebold, F. X., and A. Inoue. 2001. “Long Memory and Regime Switching.” Journal of Econometrics 105: 131–159.
Douc, R., E. Moulines, and T. Ryden. 2004. “Asymptotic Properties of the Maximum Likelihood Estimator in Autoregressive Models with Markov Regime.” Annals of Statistics 32: 2254–2304.
Douc, R., and E. Moulines. 2012. “Asymptotic Properties of the Maximum Likelihood Estimation in Misspecified Hidden Markov Models.” The Annals of Statistics 40: 2697–2732.
Feng, L., and Y. Shi. 2016. “Fractionally Integrated Garch Model with Tempered Stable Distribution: A Simulation Study.” Journal of Applied Statistics 44: 2837–2857.
Haas, M. 2009. “Value-at-Risk via Mixture Distributions Reconsidered.” Applied Mathematics and Computation 215: 2103–2119.
Haas, M., and M. S. Paolella. 2012. “Mixture and Regime-Switching GARCH Models.” In: Handbook of Volatility Models and Their Applications, edited by L. Bauwens, C. Hafner, and S. Laurent, 71–102. United Kingdom: Wiley.
Haas, M., S. Mittnik, and M. S. Paolella. 2004. “A New Approach to Markov-Switching GARCH Models.” Journal of Financial Econometrics 2: 493–530.
Hamilton, J. D. 1988. “Rational-Expectations Econometric Analysis of Changes in Regime: An Investigation of the Term Structure of Interest Rates.” Journal of Economic Dynamics and Control 12: 385–423.
Hamilton, J. D. 1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.” Econometrica 57: 357–384.
Hamilton, J. D. 1994. Time Series Analysis. Princeton: Princeton University Press.
Ho, K. Y., Y. Shi, and Z. Zhang. 2013. “How does News Sentiment Impact Asset Volatility? Evidence from Long Memory and Regime-Switching Approaches.” North American Journal of Economics and Finance 26: 436–456.
Kim, Y. S., S. T. Rachev, M. L. Bianchi, and F. J. Fabozzi. 2008. “Financial Market Models with Lévy Processes and Time-Varying Volatility.” Journal of Banking & Finance 32: 1363–1378.
Klaassen, F. 2002. “Improving GARCH Volatility Forecasts with Regime-Switching GARCH.” Empirical Economics 27: 363–394.
Koponen, I. 1995. “Analytic Approach to the Problem of Convergence of Truncated Lévy Flights Towards the Gaussian Stochastic Process.” Physical Review E 52: 1197.
Küchler, U., and S. Tappe. 2013. “Tempered Stable Distributions and Processes.” Stochastic Processes and their Applications 123: 4256–4293.
Marcucci, J. 2005. “Forecasting Stock Market Volatility with Regime-Switching GARCH Models.” Studies in Nonlinear Dynamics & Econometrics 9: 1–55.
Mevel, L., and L. Finesso. 2004. “Asymptotical Statistics of Misspecified Hidden Markov Models.” IEEE Transactions on Automatic Control 49: 1123–1132.
Mittnik, S., T. Doganoglu, and D. Chenyao. 1999. “Computing the Probability Density Function of the Stable Paretian Distribution.” Mathematical and Computer Modelling 29: 235–240.
Pouzo, D., Z. Psaradakis, and M. Sola. 2016. Maximum Likelihood Estimation in Possibly Misspecified Dynamic Models with Time Inhomogeneous Markov Regimes. Available at SSRN: https://ssrn.com/abstract=2887771.
Ross, G. J., N. M. Adams, D. K. Tasoulis, and D. J. Hand. 2011. “A Nonparametric Change Point Model for Streaming Data.” Technometrics 53: 379–389.
Shi, Y., and K. Y. Ho. 2015. “Long Memory and Regime Switching: A Simulation Study on the Markov Regime-Switching ARFIMA Model.” Journal of Banking & Finance 61: S189–S204.
Shi, Y., and L. Feng. 2016. “A Discussion on the Innovation Distribution of the Markov Regime-Switching GARCH Model.” Economic Modelling 53: 278–288.
Stanley, H. E., V. Plerou, and X. Gabaix. 2008. “A Statistical Physics View of Financial Fluctuations: Evidence for Scaling and Universality.” Physica A 387: 3967–3981.
Susmel, R., and R. F. Engle. 1994. “Hourly Volatility Spillovers between International Equity Markets.” Journal of International Money and Finance 13: 3–25.
Tankov, P. 2003. Financial Modelling with Jump Processes. Vol. 2. Boca Raton, Florida: CRC press.
Wilfling, B. 2009. “Volatility Regime-Switching in European Exchange Rates Prior to Monetary Unification.” Journal of International Money and Finance 28: 240–270.
SNDE recognizes that advances in statistics and dynamical systems theory can increase our understanding of economic and financial markets. The journal seeks both theoretical and applied papers that characterize and motivate nonlinear phenomena. Researchers are required to assist replication of empirical results by providing copies of data and programs online. Algorithms and rapid communications are also published.