Does the US stock market information matter for European equity market volatility: a multivariate perspective?

ABSTRACT

This research investigates whether the US stock volatility index (S&P 500 index) has the forecasting ability to predict the volatility of CAC index (France), DAX index (Germany), and FTSE index (the UK) by employing a multivariate heterogeneous autoregressive realized volatility jump (MHAR-RV-CJ) model. Our empirical results provide consolidated comparisons using univariate and multivariate models. The in-sample results show us the US volatility will improve the long-term volatility regression coefficient. Moreover, our proposed model, the MHAR-RV-CJ model, nearly surpasses all competing models at out-of-sample forecasting, indicating that considering the multivariate DCC-GARCH information between US-France, US-Germany, and US-UK stock markets and jump component structures can help to predict individual European stock market volatility. Unsurprisingly, several forecasting evaluation tests and further analysis (high/low volatility) confirm the robustness of our results.

KEYWORDS:

• Volatility forecasting
• US market
• European market
• multivariate HAR-RV-type model
• DCC-GARCH

Acknowledgments

The authors are grateful to the Southwest Minzu University Research Startup Funds (Grant No. RQD2022012)..

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request. The data are not publicly available due to commercial restrictions.

Notes

¹ We choose the S&P 500 index, CAC 40 index, DAX index, and FTSE 100 index (hereinafter referred to as S&P 500, CAC, DAX, and FTSE) and represent the stock market indexes of the US, France, Germany, and the UK respectively. These four countries are all developed countries (all are members of the Group of Seven (G7) and the G7 is an alliance) and experience shows that the economic contagion and interaction between developed countries is more significant (Fountas and Karanasos 2007; Caporale, Gil‐alana, and Orlando 2016; Diaz, Molero, and de Gracia 2016).

² Of course, the added the US market volatility form is also composed of continuous sample path and the significant jumping component.

³ Typically, researchers depend on a multivariate generalized autoregressive conditional heteroskedasticity (GARCH) method to simulate a parametric of conditional variance and a conditional correlation for a multivariate financial time series. Engle (2002) originally proposed a multivariate GARCH with the familiar dynamic conditional correlations, the DCC-GARCH, which provided a very general framework for multivariate volatility models to solve the multivariate covariance in asset portfolios. In this research, we add this measure in our proposed model and we follow Bubák, Kočenda, and Žikeš (2011) by using DCC-GARCH to model the vector innovation term between two columns of a time series (US-UK, US-Germany and US-France bivariate stock market volatilities).

⁴ Correlative univariate models contain linear information regarding to the US stock market.

⁵ In this research, we set the US stock market as the explanatory variable and we only analyse and compare the relationship between the US and a single European country’s stock market (i.e. the S&P 500 index versus the CAC index, the S&P 500 index versus the DAX index, and the S&P 500 index versus the FTSE index), instead of observing four markets interacting. Additionally, we match the same trading days of the stock markets between individual European countries and the US, then match the daily, weekly, and monthly RVs.

⁶ Jarque-Bera statistical value proposed by Jarque and Bera (1987).

⁷ Q(5) statistic proposed by Ljung and Box (1978).

⁸ Researchers focus more on volatility forecasting by using the rolling window than the extending window. The volatility predicted by the rolling window can be closer to the latest fluctuation trend of the time series (e.g. Andreou and Ghysels 2002; Ma et al. 2015; Buncic and Gisler 2016; Tang et al. 2021).

⁹ Generally, the criteria for high prediction model accuracy is the p-value is greater than the significant level $\alpha$, and close to 1.

¹⁰ The degree of volatility in the next period is closely related to the volatility in the current period due to the clustering nature of volatility. Therefore, one-step-ahead forecasting can help investors better grasp transaction costs in short-term trading to obtain higher investment returns (Zhang, Ma, and Wei 2019; Bissoondoyal-Bheenick et al. 2020).

¹¹ In this research, concerning the high and low volatility of the out-of-sample, we choose the median realized volatility of the entire out-of-sample. Using France as an example, the observation of the entire period of the CAC index realized is 867 days. After sorting them in descending order, the first 434 observations are defined as high volatility and the remaining 433 observations are low volatility. Similarly, the high/low volatility of the DAX and FTSE indexes are classified with 398 high volatility observations and 397 low volatility observations for the DAX index of Germany, and 450 high volatility observations and 449 low volatility observations for the FTSE index of the UK.

¹² For France, the t is 867 days, the t is 795 days for Germany and the t is 899 days for the UK.