Modelling and forecasting volatility with high-frequency and VIX information: a component realized EGARCH model with VIX

ABSTRACT

This paper studies the joint use of high-frequency and VIX information to model and forecast volatility. Our framework relies on an extension of the realized EGARCH (REGARCH) model, namely the component REGARCH model with VIX (hereafter REGARCH(C)-VIX). The REGARCH(C)-VIX model facilitates exploitation of the high-frequency and VIX information through the inclusion of realized measure and VIX for modelling and forecasting volatility. Moreover, the model features a component volatility structure, which has the ability to capture the long memory volatility. An empirical investigation with the S&P 500 index shows that the REGARCH(C)-VIX model outperforms a variety of competing models in both empirical fit and out-of-sample volatility forecasting. Our findings provide strong evidence for including the high-frequency and VIX information as well as the component volatility structure to model and forecast volatility.

KEYWORDS:

• Realized EGARCH
• components volatility structure
• high-frequency information
• VIX
• volatility forecasting

JEL CLASSIFICATION:

• C32
• C50
• C51
• C52
• C53
• C58

Acknowledgements

We would like to thank the Editor, Mark Taylor, and an anonymous referee for their valuable and insightful comments and suggestions that greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ We thank a referee for pointing this out.

² It is worth pointing out that the leverage functions, ${\nu }_1{z}_t+{\nu }_2(z_t^2-1)$ and ${\tau }_1{z}_t+{\tau }_2(z_t^2-1)$, capture the long-run and short-run leverage effects, respectively.

³ It should be noted that this model is a modified version of the classical EGARCH model of Nelson (1991), in which the leverage function has a flexible quadratic form instead of the functional form used in Nelson (1991).

⁴ To simplify the model and improve the out-of-sample performance, we follow Hansen and Huang (2016) and Banulescu-Radu et al. (2019) by imposing the restrictions $\mu =0$ and $\phi =1$.

⁵ Inspired by the heterogeneous market hypothesis, Corsi (2009) develops the HAR model that provides a convenient framework to directly model and forecast the realized measure of volatility (RV). The HAR model as well as its extension, LHAR model, are given respectively by

$\mathrm{H}\mathrm{A}\mathrm{R}:\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}{\mathrm{V}}_t=c+{\beta }_d\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}{\mathrm{V}}_{t-1}+{\beta }_w{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}\mathrm{V}}_{{ }_{t-1}}^w+{\beta }_m{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}\mathrm{V}}_{{ }_{t-1}}^m+{\epsilon }_t,$

$\mathrm{L}\mathrm{H}\mathrm{A}\mathrm{R}:\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}{\mathrm{V}}_t=c+{\beta }_d\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}{\mathrm{V}}_{t-1}+{\beta }_w{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}\mathrm{V}}_{{ }_{t-1}}^w+{\beta }_m{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{R}\mathrm{V}}_{{ }_{t-1}}^m+{\gamma }_d{r}_{{ }_{t-1}}^{\mathrm{\_}}+{\epsilon }_t,$

where $log{\mathrm{R}\mathrm{V}}_{t-1}^w=\frac{1}{5}{\sum }_{i=1}^{5}log\mathrm{R}{\mathrm{V}}_{t-i}$ and $log{\mathrm{R}\mathrm{V}}_{t-1}^m=\frac{1}{22}{\sum }_{i=1}^{22}log\mathrm{R}{\mathrm{V}}_{t-i}$ are the weekly and monthly averages of log-RV, respectively, $r_{t-1}^{-}=min[r_{t-1},0]$, and ${\epsilon }_t$ is the error term. We employ the logarithmic HAR models as benchmarks, since the logarithmic specification can reduce the heteroscedasticity of the error terms. Introducing the VIX into the HAR and LHAR models yields the following HAR-VIX and LHAR-VIX models:

$\mathrm{H}\mathrm{A}\mathrm{R}-\mathrm{V}\mathrm{I}\mathrm{X}:logR{V}_t=c+{\beta }_{d}logR{V}_{t-1}+{\beta }_{w}logR{V}_{t-1}^w+{\beta }_{m}logR{V}_{t-1}^m+\theta logVI{X}_{t-1}+{\epsilon }_t,$

$\mathrm{L}\mathrm{H}\mathrm{A}\mathrm{R}-\mathrm{V}\mathrm{I}\mathrm{X}:logR{V}_t=c+{\beta }_{d}logR{V}_{t-1}+{\beta }_{w}logR{V}_{t-1}^w+{\beta }_{m}logR{V}_{t-1}^m+{\gamma }_d{r}_{t-1}^{-}+\theta logVI{X}_{t-1}+{\epsilon }_t.$

We use the adjusted RK defined by the following Equation 36(36) $h_t=\kappa \mathrm{R}{\mathrm{K}}_t,\kappa =\frac{\sum _t{r}_t^2}{\sum _t\mathrm{R}{\mathrm{K}}_t}.$(36) as the RV in the four HAR-class models.

⁶ To save space, the volatility forecasts from the other models are not presented in the figures. The results are available upon request.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 71971001; University Natural Science Research Project of Anhui Province under Grant No. KJ2019A0659; Southern Jiangsu Capital Markets Research Center under Grant No. 2017ZSJD020; Anhui University Collaborative Innovation Project under Grant No. GXXT-2021-078