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Regimes and long memory in realized volatility

  • Elena Goldman EMAIL logo , Jouahn Nam , Hiroki Tsurumi and Jun Wang
Published/Copyright: July 23, 2013
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Studies in Nonlinear Dynamics and Econometrics
From the journal Studies in Nonlinear Dynamics and Econometrics Volume 17 Issue 5

Abstract

In this paper we model regimes and long memory in the dynamics of realized volatilities of intraday ETF and stock returns. We estimate threshold fractionally integrated (TARFIMA) models using Bayesian Markov Chain Monte Carlo (MCMC) algorithms with efficient jump. We also introduce a test based on posterior distributions of the mean squared forecast errors for model selection. Our findings are that the TARFIMA model that accounts for a different degree of long memory, persistence and variance in two regimes outperforms ARFIMA and other models using 5 day forecasts.

Keywords: Bayesian model selection; forecasting; realized volatility; threshold regimes
Corresponding author: Elena Goldman, Department of Finance and Economics, Lubin School of Business Pace University, One Pace Plaza, NY 10038, USA, Tel.: +212-618-6516, e-mail:
  1. 1

    See, e.g., Hafner and Wallmeier (2008) who analyzed variance swap positions for optimal portfolio investments.

  2. 2

    One could use past level of the return rather than volatility as a threshold variable. This choice of the threshold variable may reflect the leverage effect, which is the asymmetric effect of negative and positive returns on volatility.

  3. 3

    Many recent papers use threshold autoregressive (TAR) models in economic and financial time series to model the dynamics of short-term interest rates, real exchange rates, unemployment rate, stock prices, production, and inventories. See, e.g., Tsay (1989, 1998), Hansen (1997), Koop and Potter (1999), Phann, Schotman, and Tschering (1996), Forbes, Kalb, and Kofman (1999), Goldman and Agbeyegbe (2007), Dufrenot, Guegan, and Peguin-Feissolle (2008) among others.

  4. 4

    In additional results not reported in this paper we found that realized variance generally produces a better forecast than bipower variation for the stocks in the Dow Jones Index, since the former contains more information about the spikes incorporated in the dynamics of a high volatility regime.

  5. 5

    Higher frequencies may result in microstructure problems (see Ait-Sahalia, Mykland, and Zhang 2005).

  6. 6

    See Bollerslev, Law, and Tauchen (2008) for the data example illustrating this point.

  7. 7

    Alternatively past level of returns or any other observable variable could be determining the regimes.

  8. 8

    See, e.g., Ang and Bekaert (2002, 2004) showing the benefits of switching regime models for portfolio diversification.

  9. 9

    Although the logarithm of realized variance looks normal, statistical tests (e.g., Jarque Bera test) often reject normality.

  10. 10

    In the simplest case x is a constant term.

  11. 11

    One can use any lag of volatility or a combination of several lags, but our analysis shows that the previous day provided the best fit for the model using MBIC criterion.

  12. 12

    Note that in AR(1) process ρ=|φ|

  13. 13

    For example, the 95% HPDI for d(1) is (0.998577, 0.999990).

  14. 14

    Results of these simulated examples are available upon request.

  15. 15

    The mean squared errors are commonly used in the classical statistics. The Bayesian analogue of this popular measure is used in this paper.

  16. 16

    We note here the advantage of using a Bayesian approach where posterior distributions of any parameters of interest are easily obtained using MCMC draws as explained below. It is common in current forecasting literature to provide the distribution of forecasts rather that a point forecast.

  17. 17

    The mean is highly affected by skewness compared to median or mode of asymmetric distributions.

  18. 18

    We present only the forecasting results for individual stocks in the paper since the estimated parameters and stationarity tests are similar to ETFs. The stocks intra-daily data were only available to authors till December 2004.

  19. 19

    The 95% HPDIs are omitted here to save space. They are available from the authors on request.

  20. 20

    The results of orders are not surprising since the ARFIMA(p,d,q) model is equivalent to the infinite lag ARMA model. Thus, the fractional integration parameter d takes care of higher order lags.

  21. 21

    For VXX the 95% HPDI in regime 2 is [0.459, 0.994] with the upper bound close to 1. Since number of in-sample observations for VXX is relatively small and the estimation of TARFIMA model includes many parameters the results have higher standard deviations and wider confidence intervals.

  22. 22

    Due to smaller number of observations for VXX the standard errors are larger.

  23. 23

    Except for VXX.

  24. 24

    For VXX there is no difference in dynamics.

  25. 25

    However, considering the 95% HPDI the difference in parameters is not significant.

  26. 26

    However the standard deviations of the ME are large enough to include zero in the HPDIs.

  27. 27

    HPQ data starts in 2002.

  28. 28

    The 95% HPDIs not presented here include actual data except for the flash crash.

  29. 29

    The data for daily VIX is from CBOE website

  30. 30

    One of the potential applications of threshold type models is to estimate this model for the spread and see when the spread is above or below threshold levels making trading in options profitable. We leave this for future research.

  31. 31

    Using modes of MSE instead of medians provides similar results.

  32. 32

    We thank Evgeny Goldman for suggesting the following way to provide a common grid for two CDF graphs. Let ai and bi denote samples of xs for two CDF graphs. The grid of common x’s is given by xi=Δxi, i=1,…,nt, where number of points in the grid

    nt=min((amax–amin)/Δ, (bmax–bmin)/Δ)). The first degree Taylor approximation for two CDFs is given by and correspondingly.

  33. 33

    Generally these results hold for longer term forecasts such as 10 days as well.

  34. 34

    The GAUSS codes are available from the authors.

  35. 35

    m depends on the number of observations, if the sample size is small higher m is recommended. For example, Koop and Potter (1999) used 15% as a minimum sample size in each regime.

  36. 36

    This priors are standard in the literature for ARMA models [see Chib and Greenberg (1994) among others].

  37. 37

    Compared to existing methods of estimating multiple thresholds this method is more efficient, since we avoid estimation on a grid of points. Simultaneous estimation of multiple thresholds and difference parameters using many dimensions of grid search is virtually impossible. The Bayesian analogue of a grid search is a Griddy Gibbs sampler withing MCMC which was done for a single threshold parameter in Phann, Schotman, and Tschering (1996).

  38. 38

    We construct an arranged ARFIMA model in a similar way as Tsay (1989) and others constructed arranged autoregressive model sorting data yt by regimes.

  39. 39

    We found that nlags=40 which corresponds to 40 days was sufficient. Since in this paper we generally work with large samples sizes the problem of truncation does not arise.

  40. 40

    For example, we use Kolmogorov Smirnov and the filtered fluctuation tests [these tests are studied and compared in Goldman, Valieva, and Tsurumi (2008)].

  41. 41

    Alternatively one can use Chib and Jeliazkov (2001) estimator of marginal likelihood.

  42. 42

    We estimated models with two and three regimes, but using both in-sample and out-of-sample criteria all results were in favor of two regimes. Therefore, we present only results with two regimes in the paper.

References

Ait-Sahalia, and Mancini. 2008. “Out of Sample Forecasts of Quadratic Variation.” Journal of Econometrics 18: 17–33.10.1016/j.jeconom.2008.09.015Search in Google Scholar

Ait-Sahalia, Y., P. Mykland, and L. Zhang. 2005. “How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise.” Review of Financial Studies 18: 351–416.10.1093/rfs/hhi016Search in Google Scholar

Andersen, T. G., and T. Bollerslev. 1997. “Intraday Periodicity and Volatility Persistence in Financial Markets.” Journal of Empirical Finance 4: 115–158.10.1016/S0927-5398(97)00004-2Search in Google Scholar

Andersen, T. G., and T. Bollerslev. 1998. “Deutsche Mark-Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies.” The Journal of Finance 53: 219–265.10.1111/0022-1082.85732Search in Google Scholar

Andersen, T. G., T. Bollerslev, and F. X. Diebold. 2007. “Roughing it Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility.” Review of Economics and Statistics 89 (4): 701–720.10.1162/rest.89.4.701Search in Google Scholar

Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2001a. “The Distribution of Exchange Rate Volatility.” Journal of the American Statistical Association 96: 42–55.10.1198/016214501750332965Search in Google Scholar

Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens. 2001b. “The Distribution of Realized Stock Volatility.” Journal of Financial Economics 61: 43–76.10.1016/S0304-405X(01)00055-1Search in Google Scholar

Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2003. “Modeling and Forecasting Realized Volatility.” Econometrica 71: 579–625.10.1111/1468-0262.00418Search in Google Scholar

Ang, A., and G. Bekaert. 2002. “International Asset Allocation with Regime Shifts.” Review of Financial Studies 15: 1137–1187.10.1093/rfs/15.4.1137Search in Google Scholar

Ang, A., and G. Bekaert. 2004. “How do Regimes Affect Asset Allocation?” Financial Analysts Journal 60 (2): 86–99.10.2469/faj.v60.n2.2612Search in Google Scholar

Baillie, R. T. 1996. “Long Memory Processes and Fractional Integration in Econometrics.” Journal of Econometrics 73: 5–59.10.1016/0304-4076(95)01732-1Search in Google Scholar

Bandi, F. M., J. R. Russell, and C. Yang. 2008. “Realized Volatility Forecasting and Option Pricing.” Journal of Econometrics 147: 34–46.10.1016/j.jeconom.2008.09.002Search in Google Scholar

Barndorff-Nielsen, O., and N. Shephard. 2004. “Power and Bi-Power Variation with Stochastic Volatility and Jumps.” Journal of Financial Econometrics 2 (2): 1–37.Search in Google Scholar

Bollerslev, T., T. H. Law, and G. Tauchen. 2008. “Risk, Jumps, and Diversification.” Journal of Econometrics 144: 234–256.10.1016/j.jeconom.2008.01.006Search in Google Scholar

Brockwell, P. J., and R. A. Davis. 1987. Time Series: Theory and Methods, 464–465. New York: Springer-Verlag.10.1007/978-1-4899-0004-3Search in Google Scholar

Chib, S., and E. Greenberg. 1994. ’Bayes Inference in Regression Models with ARMA(p,q) errors.’ Journal of Econometrics 64: 183–206.Search in Google Scholar

Chib, S., and E. Greenberg. 1995. ’Understanding the Metropolis-Hastings Algorithm.’ The American Statistician 49 (4): 327–335.Search in Google Scholar

Chib, S., and I. Jeliazkov. 2001. ’Marginal Likelihood from the Metropolis-Hastings Output.’ Journal of the American Statistical Association 96: 270–281.10.1198/016214501750332848Search in Google Scholar

Chung, C. F. 2001. “Calculating and Analyzing Impulse Responses for the Vector ARFIMA Model.” Economics Letters 71: 17–25.10.1016/S0165-1765(00)00399-2Search in Google Scholar

Corsi, F. 2009. “A Simple Approximate Long-memory Model of Realized Volatility.” Journal of Financial Econometrics 7: 174–196.10.1093/jjfinec/nbp001Search in Google Scholar

Diebold, F. X., and R. Mariano. 1995. “Comparing Predictive Accuracy.” Journal of Business and Economic Statistics 13: 253–263.Search in Google Scholar

Diebold, F. X., and A. Inoue. 2001. “Long Memory and Regime Switching.” Journal of Econometrics 105: 131–159.10.1016/S0304-4076(01)00073-2Search in Google Scholar

Diebold, F. X., S. Husted, and G. D. Rudebusch. 1991. “Real Exchange Rates under the Gold Standard.” Journal of Political Economy 99: 1252–1271.10.1086/261799Search in Google Scholar

Dufrenot, G., D. Guegan, and A. Peguin-Feissolle. 2008.“Changing-Regime Volatility: A Fractionally Integrated SETAR Model.” Applied Financial Economics 18: 519–526.10.1080/09603100600993778Search in Google Scholar

Fleming, J., C. Kirby, and B. Ostdiek. 2003. ’The Economic Value of Volatility Timing using ‘Realized’ Volatility.’ Journal of Financial Economics 67 (3): 473–509.10.1016/S0304-405X(02)00259-3Search in Google Scholar

Ferguson, T. S. 1973. “A Bayesian Analysis of Some Nonparametric Problems.” Annals of Statistics 2: 615–629.10.1214/aos/1176342360Search in Google Scholar

Forbes, C. S., G. Kalb, and P. Kofman. 1999. “Bayesian Arbitrage Threshold Analysis.” Journal of Business and Economic Statistics, American Statistical Association 17 (3): 364–372.Search in Google Scholar

Geweke, J., and S. Porter-Hudak. 1983. “The Estimation and Application of Long Memory Time Series Models.” Journal of Time Series Analysis 4: 221–238.10.1111/j.1467-9892.1983.tb00371.xSearch in Google Scholar

Gil-Alana, L. A. 2001. “Estimation of Fractionally ARIMA Models for the UK Unemployment.” Annales D’Economice et de Statistique 62: 128–137.10.2307/20076284Search in Google Scholar

Goldman, E., and H. Tsurumi. 2005. “Bayesian Analysis of a Doubly Truncated ARMA-GARCH Model.” Studies in Nonlinear Dynamics and Econometrics 9 (2): Article 5.10.2202/1558-3708.1166Search in Google Scholar

Goldman, E., and T. Agbeyegbe. 2007. “Estimation of Threshold Time Series Models using Efficient Jump MCMC.” In Bayesian Statistics and its Applications, edited by S. K. Upadhyay, U. Singh, and Dipak Dey, 241–253. New Delhi: Anamaya Publishers.Search in Google Scholar

Goldman, E., S. Radchenko, T. Nakatsuma, and H. Tsurumi. 2001. “A Bayesian Test of Stationarity in a Regression Model with an ARMA Error Term.” Proceedings of the Annual Meeting of the American Statistical Association, August 5–9.Search in Google Scholar

Gouskova, E. 2002. “ARFIMA Modeling of Financial Assets Volatility.” Ph.D. dissertation, Rutgers University.Search in Google Scholar

Guidolin, M., and A. Timmermann. 2002. “Optimal Portfolio Choice under Regime Switching, Skew and Kurtosis Preferences.” Working Paper, UC San Diego.10.2139/ssrn.406720Search in Google Scholar

Hafner, R., and M. Wallmeier. 2008. “Optimal Investments in Volatility.” Financial Markets and Portfolio Management 22: 147–167.10.1007/s11408-008-0076-8Search in Google Scholar

Hansen, B. E. 1997. “Inference in TAR Models.” Studies in Nonlinear Dynamics and Econometrics 2 (1): 1–14.10.2202/1558-3708.1024Search in Google Scholar

Hauser, M. A. 1999. “Maximum Likelihood Estimation for ARFIMA Models: A Monte Carlo Study.” Journal of Statistical Planning and Inference 80 (1): 229–255.10.1016/S0378-3758(98)00252-3Search in Google Scholar

Huang, X., and G. Tauchen. 2005. “The Relative Contributions of Jumps to Total Variance.” Journal of Financial Econometrics 3: 456–499.10.1093/jjfinec/nbi025Search in Google Scholar

Koop, G., and S. M. Potter. 1999. “Dynamic Asymmetries in U.S. Unemployment.” Journal of Business and Economic Statistics 17: 298–312.Search in Google Scholar

Koop, G., E. Ley, J. Osiewalski, and M. F. J. Steel. 1997. “Bayesian Analysis of Long Memory and Persistence using ARFIMA Models.” Journal of Econometrics 76: 149–169.10.1016/0304-4076(95)01787-9Search in Google Scholar

Lee, S. S., and P. A. Mykland. 2008. “Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics.” The Review of Financial Studies 21: 2535–2562.10.1093/rfs/hhm056Search in Google Scholar

Liu, C., and J. M. Maheu. 2008. “Are There Structural Breaks in Realized Volatility?” Journal of Financial Econometrics 6: 326–360.10.1093/jjfinec/nbn006Search in Google Scholar

Maheu, J. M., and T. H. McCurdy. 2002. “Non-Linear Features of Realized FX Volatility.” Review of Economics and Statistics 84: 668–681.10.1162/003465302760556486Search in Google Scholar

McAleer, M., and M. C. Medeiros. 2008. “A Multiple Regime Smooth Transition Heterogeneous Autoregressive Model for Long Memory and Asymmetries.” Journal of Econometrics 147: 104–119.10.1016/j.jeconom.2008.09.032Search in Google Scholar

Phann, G. A., P. C. Schotman, and R. Tschering. 1996. “Non-Linear Interest Rate Dynamics and Implications for the Term Structure. Journal of Econometrics 74: 149–176.10.1016/0304-4076(95)01754-2Search in Google Scholar

Ramchand, L., and R. Susmel. 1998. “Volatility and Cross Correlations Across Major International Markets.” Journal of Empirical Finance 5: 397–416.10.1016/S0927-5398(98)00003-6Search in Google Scholar

Shimizu and Tsurumi. 2009. “Comparison of Several Likelihood Functions used in Fractionally Integrated ARFIMA Models.” Working Paper, Rutgers University.Search in Google Scholar

Sowell, F. 1992. “Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models.” Journal of Econometrics 53: 165–188.10.1016/0304-4076(92)90084-5Search in Google Scholar

Thomakos, D. D., and T. Wang. 2003. “Realized Volatility in the Futures Markets.” Journal of Empirical Finance 10: 321–353.10.1016/S0927-5398(02)00052-XSearch in Google Scholar

Tsay, R. S. 1989. “Testing and Modeling Threshold Autoregressive Processes.” Journal of the American Statistical Association 84: 231–240.10.1080/01621459.1989.10478760Search in Google Scholar

Tsay, R. S. 1998. “Testing and Modeling Multivariate Threshold Models.” Journal of the American Statistical Association 93: 1188–1202.10.1080/01621459.1998.10473779Search in Google Scholar

Published Online: 2013-07-23
Published in Print: 2013-12-01

©2013 by Walter de Gruyter Berlin Boston

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https://doi.org/10.1515/snde-2012-0018
Keywords for this article
Bayesian model selection; forecasting; realized volatility; threshold regimes

Articles in the same Issue

  1. Masthead
  2. Masthead
  4. Reproducing business cycle features: are nonlinear dynamics a proxy for multivariate information?
  5. Stochastic volatility model with regime-switching skewness in heavy-tailed errors for exchange rate returns
  6. Regimes and long memory in realized volatility
  7. Estimating C-CAPM and the equity premium over the frequency domain
  8. Determining the number of global and country-specific factors in the euro area
  9. A maximum score test for binary response models
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