Abstract
Empirical evidence shows that the dynamics of high frequency-based measures of volatility exhibit persistence and occasional abrupt changes in the average level. By looking at volatility measures for major indices, we notice similar patterns (including jumps at about the same time), with stronger similarities, the higher the degree of company capitalization represented in the indices. We adopt the recent Markov Switching asymmetric multiplicative error model to model the dynamics of the conditional expectation of realized volatility. This allows us to address the issues of a slow moving average level of volatility and of different dynamics across regimes. An extension sees a more flexible model combining the characteristics of Markov Switching and smooth transition dynamics.
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Notes
- 1.
Data are expressed as percentage annualized volatility, i.e. the square root of the realized variance series taken from the Oxford-Man Institute’s realised library version 0.1 [10], and multiplied by \(\sqrt{252} {\ast} 100\).
- 2.
Details about the reparameterization of \(\beta _{s_{t}}\) to guarantee a certain coherence between the regime and the level of volatility, and about the solution of possible estimation problems, are in [8]. In the same work another specification of the MS–AMEM is given, in which the asymmetry deriving from the sign of the returns may affect also the transition probabilities (the so-called Asymmetry in Probability MS–AMEM).
- 3.
Tests based on the likelihood function cannot be used to compare the AMEM with respect to the corresponding MS models because of the presence of nuisance parameters present only under the alternative hypothesis; in this case, with the proper caution, a classical information criterion could provide some information (see [12]); in particular the AIC seems to choose the correct state dimension more successfully than the BIC, provided that the parameter changes are not too small and the hidden Markov chain is fairly persistent.
- 4.
For MS models we have used the generalized residuals, introduced by [9] for latent variable models, defined as \(E(\hat{\epsilon }_{t}\vert \Psi _{t-1}) = \sum \nolimits _{i=1}^{3}\hat{\epsilon }_{s_{t},t}tPr(s_{t} = i\vert \Psi _{t-1})\), where \(\hat{\epsilon }_{s_{t},t}\) are the residuals at time t derived from the parameters of the model in state s t .
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Acknowledgments
Thanks are due to participants in the following conferences: ERCIM 2010 (London, December 10–12, 2010), ECTS 2011 (Monte Porzio Catone, June 13–14, 2011), SCO 2011 (Padova, September 19–21, 2011). Financial support from Italian MIUR under Grant 20087Z4BMK_002 is gratefully acknowledged.
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Gallo, G.M., Otranto, E. (2013). Volatility Swings in the US Financial Markets. In: Grigoletto, M., Lisi, F., Petrone, S. (eds) Complex Models and Computational Methods in Statistics. Contributions to Statistics. Springer, Milano. https://doi.org/10.1007/978-88-470-2871-5_11
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DOI: https://doi.org/10.1007/978-88-470-2871-5_11
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