Volatility dynamics under an endogenous Markov-switching framework: a cross-market approach

Abstract

This study uses an endogenous Markov-switching framework to examine the interrelatedness of the volatility dynamics of the US and Korean markets. Previous literature assumes that the US market implied volatility index is exogenous to the Korean implied volatility index. Here, we allow for correlations between the US and Korean variables and suggest two types of endogeneity, namely endogeneity in the regressors and in the regime-switching probabilities. The estimation results show that both types of endogeneity are present in the US variables and that the parameter estimates are quite different when endogeneity is considered, indicating a serious endogeneity bias in the parameter estimates. The results of the endogeneity test for the regressors show that the effects of global shocks are often persistent and may last for as long as six periods. Sub-period analyses indicate that the degrees of endogeneity were especially strong during the global financial crisis.

Keywords:

• Cross-market approach
• Endogenous Markov-switching
• Implied volatility index
• VIX
• VKOSPI

JEL Classification:

• C22
• E44
• F36
• G17
• G19

Acknowledgements

We are grateful for the helpful comments and suggestions from Chang-Jin Kim, Ke Tang, Ziran Li, and Xingguo Luo, and all participants at the fifth international conference on futures and other derivatives (ICFOD) in Shenzhen, China, in December, 2016.

Notes

¹ Although Song et al. (2016) examine the effects of overseas (US) factors in explaining VKOSPI dynamics, they assume that the US factors are exogenous variables and only allow unilateral effects from the United States to Korea. In contrast, this study allows the variables in the two countries to be correlated. One possible source of correlation is the effect of global shocks. In particular, our sample period includes the GFC, during which especially strong cross-market relations are present. The global shocks cause the variables in the two countries to be correlated, which necessarily leads to endogeneity issues between the variables. We explicitly consider the possibility of correlated variables and determine the effects of global shocks on the estimates. In this respect, our study offers significant improvements over Song et al. (2016).

² The time series graphs of the VKOSPI and the VIX, as well as their correlation structures, are available on request. A visual comparison of the time series suggests that they may move together. The similarity in the movements of the VIX and VKOSPI suggests that they are affected by common global shocks, and/or that the Korean and the US business cycles are highly correlated. Either way, the apparent high degree of correlation between the time series raises the question of whether the series are determined endogenously.

³ The heavy trading volume and the interrelatedness of the KOSPI200 options and futures markets are noted by Guo et al. (2013), Lee et al. (2015), Ryu (2011, 2015), Ryu and Yang (2017), Webb et al. (2016), and Yang et al. (forthcoming).

⁴ Previous studies that analyze intraday data from the KOSPI200 spot and/or index derivatives markets consistently find that domestic individuals are primarily speculators, and are more easily affected by sentiment and market psychology than are institutional investors. This makes domestic individuals noisy net losers in this market (Sim et al. 2016, Yang et al. 2017).

⁵ Endogeneity may be caused by either an omitted variable bias (when omitted common factors or global shocks affect both the US and the Korean economies) or a simultaneity bias (when the US and the Korean economies affect each other at the same time). In this study, our models implicitly assume that endogeneity is caused by the omitted global shocks. However, this is just for expositional convenience and does not exclude the possibility of simultaneity between the US and the Korean variables. Thus, we focus on the existence of an endogeneity bias but do not explore the sources of endogeneity bias, which is beyond the scope of this study.

⁶ For details on the derivation of the transformed model and on the estimation procedure, see Kim (2004).

⁷ We define lnL _R and lnL _U as the log-likelihood values from restricted and unrestricted maximum likelihood estimations, respectively. Then, the LR test statistic is LR = 2(lnL _U –lnL _R ) ~ χ ²(J), where J is the dimension of .

⁸ We define the LR test statistic as follows: LR = 2(lnL _U –lnL _R )~χ ²(1), where lnL _U and lnL _R are the maximized value of the log-likelihood function without and with the restriction of ρ = 0, respectively. If we reject the null hypothesis of ρ = 0, we conclude that endogeneity exists in the regime switching, and that parameters must be estimated accounting for the endogeneity. Kim et al. (2008) also suggest using the t-statistic to test for endogeneity but the simulation results show that the LR test appears to be a better test for endogenous switching.

⁹ Detailed results are shown in table 4 in section 6.

¹⁰ Song et al. (2016) estimate a three-regime Markov-switching model for a similar dataset. However, we estimate two-regime models because (i) we want to focus on the endogeneity issue using more parsimonious models, and (ii) the cost of estimating the three-regime model is significant. For example, Model 2 has 52 estimated parameters with two regimes. In the case of three regimes, this increases to 81 parameters. Therefore, it is very difficult to achieve convergence for Model 2 with three regimes and, thus, we do not consider three-regime models here.

¹¹ The VIX also has a high correlation with the disturbance term in the lag t–7 (not shown in the table).

¹² We do not attempt to identify and analyze global shocks here because this would require developing structural economic models, which is beyond the scope of the study.

¹³ If the regressors in Model 3 are endogenous, the biases in the estimated coefficients are even more severe. This result is available from the authors upon request.

Additional information

Funding

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea [NRF-2017S1A5A2A01025583].