The dependence and risk spillover between economic uncertainties and the crude oil market: new evidence from a Copula-CoVaR approach incorporating the decomposition technique

Abstract

Understanding the risk spillover of the oil market in economic uncertainty is of great importance. However, it is difficult to take on a traditional single perspective in describing the risk spillover law of economic uncertainty in the crude oil market on different timescales. In order to fill the research gap resulting from such difficulty, this paper incorporates empirical mode decomposition into the time-varying Copula-CoVaR model, and for the first time explores the risk spillover path of economic uncertainty on the two international crude oil pricing benchmarks—Brent and West Texas Intermediate crude oil prices—using different timescales. The empirical results not only verify the necessity of research from the perspective of different timescales, but also reveal the heterogeneity of the risk spillover paths of different types of economic uncertainty on crude oil prices. The research in this paper provides a multi-perspective interpretation for understanding the complex risk spillovers between various economic uncertainties and the crude oil market, as well as providing meaningful information to support stakeholders in making rational decisions.

Appendix

This section shows the specific forms of five Copula functions: Gaussian-Copula, Clayton-Copula, Frank-Copula, Gumbel-Copula and t-Copula. Different types of Copula functions have their own emphasis when describing the tail dependencies between variables. Gaussian-Copula function is the most commonly used copula function. Due to its obvious symmetry, the Gaussian-Copula function can better describe the symmetric relationship and linear correlation between variables, but it cannot describe the asymmetric relationship between variables. The t-Copula function takes into account the skew and peak of the data, and is more sensitive to the change of the tail correlation coefficient, so the correlation structure of the tail can be described more accurately. The tail of the Clayton copula function is asymmetric, which can more sensitively capture the change of the lower tail’s correlation coefficient. Frank copula function has been widely used because of its relatively simple and effective structure.

• 1. Gaussian-Copula

Gaussian-Copula function, one of the most widely used Copula functions, has good upper and lower tail symmetry. For random variables (u, v), the expressions of the distribution function and density function of the Gaussian-Copula function are as follows:

$$C\left(u,v;\rho \right)={\int}_{-\infty}^{\varnothing^{-1}(u)}{\int}_{-\infty}^{\varnothing^{-1}(v)}\frac{1}{2\pi \sqrt{1-{\rho}^2}}\exp \left(\frac{-\left({r}^2+{s}^2-2\rho rs\right)}{2\left(1-{\rho}^2\right)}\right) drds$$

(1)

$$c\left(u,v;\rho \right)=\frac{1}{\sqrt{1-{\rho}^2}}\exp \left(-\frac{\varnothing^{-1}{(u)}^2+{\varnothing}^{-1}{(v)}^2-2\rho {\varnothing}^{-1}(u){\varnothing}^{-1}(v)}{2\left(1-{\rho}^2\right)}\right)\exp \left(-\frac{\varnothing^{-1}{(u)}^2{\varnothing}^{-1}{(v)}^2}{2}\right)$$

(2)

• 2. Clayton-Copula

The binary Clayton-Copula function describes the asymmetric binary correlation. Unlike the binary Gumbel-Copula function, the binary Clayton-Copula function can capture the lower tail correlation square more accurately, and its distribution function expression is as follows:

$$C\left(u,v;\theta \right)={\left({u}^{-\theta }+{v}^{-\theta }-1\right)}^{-\frac{1}{\theta }}$$

(3)

• 3. Frank-Copula

Although Frank copula is also suitable for describing symmetric data, it has the special advantage of being able to describe the negative correlation between variables, unlike the Clayton copula and Gumbel copula. The binary Frank copula distribution function is expressed as follows:

$$C\left(u,v;\lambda \right)=-\frac{1}{\lambda}\ln \left(1+\frac{\left({e}^{-\lambda u}-1\right)\Big(\left({e}^{-\lambda v}-1\right)}{e^{-\lambda }-1}\right)$$

(4)

• 4. Gumbel-Copula

Compared with the Gaussian-Copula and t-Copula functions, the Gumbel-Copula function is asymmetric, with the upper tail being higher and the lower tail being lower, making it more sensitive and accurate in catching the relationship between the upper tail and the lower tail. The distribution function expression of the Gumbel-Copula is as follows:

$$C\left(u,v;\alpha \right)=\exp \left(-{\left[{\left(-\ln u\right)}^{\frac{1}{\alpha }}+{\left(-\ln v\right)}^{\frac{1}{\alpha }}\right]}^{\alpha}\right)$$

(5)

• 5. t-Copula

Compared with the binary Gaussian-Copula function, the binary t-Copula function can not only describe the binary phase-dependent structure with symmetric distribution, but also can describe the degree of tail correlation between variables in a more detailed way. The specific expression of its function is as follows:

$$C\left(u,v;\rho, v\right)={\int}_{-\infty}^{T_v^{-1}(u)}{\int}_{-\infty}^{T_v^{-1}(v)}\frac{1}{2\pi \sqrt{1-{\rho}^2}}\left[1+\frac{t^2+{s}^2-2\rho st}{v\left(1-{\rho}^2\right)}\right] dsdt$$

(6)