Crude oil and gasoline volatility risk into a Realized-EGARCH model

Abstract

This paper disentangles oil volatility risk to two components. The first component is attributed to crude oil, while the second is related to gasoline. This disentanglement serves the purpose of investigating the extent to which crude oil and gasoline are complementary in impacting return and variance residuals. The Realized-EGARCH model of Hansen et al. (J Appl Econom 29(5):774–799, 2014) is used to test the hypothesis that stock markets show some delay in incorporating oil information. This study shows that both crude oil- and gasoline-based information impact stock markets contemporaneously in a complementary fashion. Unlike the underreaction hypothesis, which is suggested as an explanation to the negative lagged effect of crude oil price change on return, the sequential information hypothesis explains better the ways information about oil is disseminated among U.S. industry portfolios.

Appendices

Appendix 1

See Table 8.

Table 8 U.S. industry portfolio names

Appendix 2: Decomposition of the oil risk factor

Start with

$$r_{c,t} = \alpha_{c} + \beta_{cx} r_{x,t} + \varepsilon_{c,t} ,$$

(15)

$$r_{g,t} = \alpha_{g} + \beta_{gc} (\alpha_{c} + \beta_{cx} r_{x,t} + \varepsilon_{c,t} ) + \varepsilon_{g,t} ,$$

(16)

Take the expected second moment of Eqs. (15) and (16).

$$\begin{aligned} Er_{c,t}^{2} & = E\alpha_{c}^{2} + 2\alpha_{c} \beta_{cx} Er_{x,t} + 2\alpha_{c} E\varepsilon_{c,t} + \beta_{cx}^{2} Er_{x,t}^{2} + 2\beta_{cx} E\varepsilon_{c,t} r_{x,t} + E\varepsilon_{ct}^{2} \\ h_{c,t} & = \beta_{cx,t}^{2} \sigma_{x,t}^{2} + \sigma_{c\varepsilon ,t}^{2} , \\ \end{aligned}$$

(17)

$$\begin{aligned} Er_{g,t}^{2} & = E\alpha_{c}^{2} \beta_{cg,t}^{2} + 2\alpha_{c} \alpha_{g} \beta_{cg,t} + 2\alpha_{c} \beta_{cx,t} \beta_{cg,t}^{2} Er_{x,t} + 2\alpha_{c} \beta_{cg,t}^{2} E\varepsilon_{ct} + 2\alpha_{c} \beta_{cg,t} E\varepsilon_{g,t} \\ & \quad + E\alpha_{c}^{2} + 2\alpha_{g} \beta_{cg,t} \beta_{cx,t} Er_{x,t} + 2\alpha_{g} \beta_{cg,t} E\varepsilon_{c,t} + 2\alpha_{g} E\varepsilon_{g,t} + \beta_{cg,t}^{2} \beta_{cx,t}^{2} Er_{x,t}^{2} \\ & \quad + 2\beta_{cx,t} \beta_{cg,t}^{2} E\varepsilon_{c,t} r_{x,t} + \beta_{cg,t}^{2} E\varepsilon_{c,t}^{2} + 2\beta_{cg,t} \beta_{cx,t} E\varepsilon_{g,t} r_{x,t} + 2\beta_{cg,t} E\varepsilon_{c,t} \varepsilon_{g,t} + E\varepsilon_{g,t}^{2} , \\ \end{aligned}$$

(18)

$$\begin{aligned} Er_{g,t}^{2} & = \beta_{cg,t}^{2} \beta_{cx,t}^{2} Er_{x,t}^{2} + \beta_{cg,t}^{2} E\varepsilon_{c,t}^{2} + 2\beta_{cg,t} E\varepsilon_{c,t} \varepsilon_{g,t} + E\varepsilon_{g,t}^{2} , \\ h_{g,t} & = \beta_{cg,t}^{2} \beta_{cx,t}^{2} \sigma_{x,t}^{2} + \beta_{cg,t}^{2} \sigma_{c\varepsilon ,t}^{2} + 2\beta_{cg,t} \sigma_{cg,t} + \sigma_{g\varepsilon ,t}^{2} \\ \end{aligned}$$

(19)

$$\begin{aligned} Er_{c,t} r_{g,t} & = \beta_{cg,t} E\varepsilon_{c,t}^{2} + E\varepsilon_{c,t} \varepsilon_{g,t} + E\alpha_{c} \alpha_{g} + E\alpha_{c}^{2} \beta_{cg,t} + \alpha_{c} E\varepsilon_{g,t} + \beta_{cg,t} \beta_{cx,t}^{2} Er_{x,t}^{2} \\ & \quad + \alpha_{g} \beta_{cx,t} Er_{x,t} + 2\alpha_{c} \beta_{cg,t} E\varepsilon_{c,t} + \beta_{cx,t} E\varepsilon_{g,t} r_{x,t} + 2\alpha_{c} \beta_{cg,t} \beta_{cx,t} Er_{x,t} + 2\beta_{cg,t} \beta_{cx.t} E\varepsilon_{c,t} r_{x,t} \\ h_{cg,t} & = \beta_{cg,t} \sigma_{c\varepsilon ,t}^{2} + \sigma_{cg,t} + \beta_{cg,t} \beta_{cx,t}^{2} \sigma_{x,t}^{2} , \\ \end{aligned}$$

(20)

Express the betas of Eqs. (17–20) in relative terms as

$$\begin{aligned} \beta_{cx,t} & = \rho_{cx,t} \frac{{\sigma_{c,t} }}{{\sigma_{x,t} }} \\ \beta_{gc,t} & = \rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}, \\ \end{aligned}$$

(21)

Standardize the covariance in terms of crude oil variance and simplify thereafter to obtain,

$$\begin{aligned} \frac{{h_{cg,t} }}{{h_{c,t} }} & = \frac{{\rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}\sigma_{c\varepsilon ,t}^{2} + \sigma_{cg,t} + \rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}\left( {\rho_{cx,t} \frac{{\sigma_{c,t} }}{{\sigma_{x,t} }}} \right)^{2} \sigma_{x,t}^{2} }}{{\left( {\rho_{cx,t} \frac{{\sigma_{c,t} }}{{\sigma_{x,t} }}} \right)^{2} \sigma_{x,t}^{2} + \sigma_{c\varepsilon ,t}^{2} }}, \\ & = \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }} \times \rho_{gc,t} \times \frac{{\left( {2 + \rho_{cx,t}^{2} } \right)}}{{\left( {1 + \rho_{cx,t}^{2} } \right)}} \\ \end{aligned}$$

(22)

Standardize the covariance in terms of gasoline variance and simplify thereafter to obtain,

$$\begin{aligned} \frac{{h_{cg,t} }}{{h_{g,t} }} & = \frac{{\rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}\sigma_{c\varepsilon ,t}^{2} + \sigma_{cg,t} + \rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}\left( {\rho_{cx,t} \frac{{\sigma_{c,t} }}{{\sigma_{x,t} }}} \right)^{2} \sigma_{x,t}^{2} }}{{\left( {\rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}} \right)^{2} \left( {\rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}} \right)^{2} \left( {\rho_{cx,t} \frac{{\sigma_{c,t} }}{{\sigma_{x,t} }}} \right)^{2} \sigma_{x,t}^{2} + \left( {\rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}} \right)^{2} \sigma_{c\varepsilon ,t}^{2} + 2\rho_{gc,t} \frac{{\sigma_{g,t} }}{{\sigma_{c,t} }}\sigma_{cg,t} + \sigma_{g\varepsilon ,t}^{2} }}, \\ & = \frac{{\sigma_{c,t} }}{{\sigma_{g,t} }} \times \rho_{gc,t} \times \frac{{\left( {2 + \rho_{cx,t}^{2} } \right)}}{{\left( {1 + \rho_{gc,t}^{2} \rho_{cx,t}^{2} + 3\rho_{cx,t}^{2} } \right)}} \\ \end{aligned}$$

(23)