Based on Copula-CoVaR model of risk spillover effect of oil markets and other commodity markets

Abstract

Based on the Copula-CoVaR framework, the Beta-skew-t-EGARCH model is used to capture the leverage effect of financial assets, the thick tail distribution and the conditional skewness. Then, based on the standard deviation, this paper introduces the extreme value theory, combined with the Copula function and the CoVaR method which are used to measure the effect of China’s crude oil market on the risk spillover effects of domestic and international commodity markets, and reveals the impact of China’s crude oil market. The results show that, regardless of the domestic Crude oil market and international commodity market, domestic crude oil market and domestic commodity market, there is a two-way positive risk spillover effect, which shows asymmetric characteristics. The results provide some theoretical supports for the regulatory authorities to enhance the marketization level of China’s crude oil market, some suggestions for the investment institutions to assess the risk level of China’s crude oil market, and some reference value for investors to invest in China’s crude oil market. Finally, based on the conclusions of the study, this paper provides some targeted recommendations to the development of China’s crude oil market.

Keywords

Oil markets commodity markets risk spillover effect

1 Introduction

As the important industrial base product worldwide, crude oil is the leader of bulk commodity and the crude oil market is also related to other bulk commodity markets in various ways. First of all, development of liquid biofuels i.e. bioethanol and biodiesel have accelerated the price linkage between the crude oil market and the agricultural market [24]. For example, the soaring price of crude oil price might result in the driving effect of increased product cost and influence the market price of agricultural products [20]. Secondly, during the oil refining, the fluctuation of crude oil price might influence the metal price through the upstream and downstream supply chains [11]. Thirdly, the development of the global economic integration and the electronic trade system make the traders adopt the investment strategies to reduce the market risks; especially the commodity market financialization has accelerated this process and brought forth greater academic and industrial concerns to the connection between the crude oil market and other commodity markets. Since 2003, China has achieved huge development in various fields such as energy, agricultural product, metal and so on. The Chinese commodity market and the international commodity market are sharing an increasingly close information dissemination and mutual influence. However, there remains limited study on the relationship between the Chinese crude oil market and the international commodity market, and the actual effects of the information dissemination need to be further verified. Therefore, to study the information dissemination between the Chinese crude oil market and the international commodity market is vital to improve Chinese market system and understand the international economic rules.

2 Literature review

Studies on the relationship between the crude oil market and other bulk commodity markets have been a long-standing concern of the domestic and foreign scholars. Pindyck and Rotemberg [17], Sieczka and Holyst [15], Harri et al. [2], Nazlioglu [22], ReboredoandUgolini et al. [12-13] had explored the spillover effect between the crude oil price and prices of other bulk commodities from different aspects.

In addition, many researchers have analyzed the price of crude oil and agricultural commodities overflow and information transmission. Metthey adopted were mainly focused on the classical econometric method and the mathematical model, including the GARCH type model [19], panel cointegration [18], generalized forecast error variance decomposition [21] and wavelet method [31]. Most of the research results confirmed that the close relations between crude oil and other commodity markets.

However, there are few studies on the mutual interactions between the Chinese crude oil markets and the domestic and international commodity markets; the existing literature mainly explores the relationship between the domestic and the international crude oil markets. For instance, Jiao Jianling et al. [14], PanHuifeng et al. [8], Dong Xiulaing and Zhang Yishan [29], Pan Huifeng et al. [9], Ma Chaoqun et al. [7] inspected the spillover effect of the Chinese crude oil market with the GARCH models.

In conclusion, most of the current studies are limited to explore the relationship of a country or between two countries with the existing foreign models while few are involved the spillover effect of the crude oil market to the risks of the related markets. On the other hand, there are less studies on the risk spillover effect between the Chinese crude oil market and other commodity markets, especially the risk spillover effect studies under the extreme conditions of the crude oil market. In this article, it considers the financial features of crude oil market, and fits its fluctuation features through the Beta-Skew-t-EGARCH model and POT model; on this basis, the Copula model is introduced to describe the nonlinear dependency of the structure between the Chinese crude oil market and the domestic and international commodity markets. The construction is based on the measure model of the extreme risk spillover effect of the Copula-Beta-Skew-t-EGARCH-POT-CoVaR model to quantize the extreme risk spillover level of the crude oil market and the commodity market.

3 Model construction

3.1 CoVaR

Reviewing the definition of VaR to represents the financial institute i under the yield rate $r_{t}^{i}$ and confidence level 1 - q, ${VaR}_{q}^{i}$ can be expressed as:

$Pr (r_{t}^{i} \leq {VaR}_{q}^{i}) = q$ (1)

${VaR}_{q}^{i}$ is generally negative, but in actual application, it is expressed by the positive value.

VaR is to make risk estimate on the single financial asset, but it is not to reflect the spillover level of the risks among financial assets. Therefore, Adrian and Brunnermeier [23] proposed the concept of CoVaR on the basis of VaR, which refers to the risk value faced by the financial asset i when the financial asset j is at the risk level. ${CoVaR}_{q}^{ij}$ is a kind of conditional risk reflecting the financial asset i to financial asset j that can be expressed by the formula as follows:

$Pr (r_{t}^{i} \leq {CoVaR}_{q}^{ij} | r_{t}^{i} = {VaR}_{q}^{j}) = q$ (2)

According to Equation (2), ${CoVaR}_{q}^{ij}$ is actually a conditional VaR to measure the total degree of risk faced by the financial asset i, specifically including the risk value of i and the risk spillover effect of j. CoVaR reflects the conditional risk and infection risk; it is to mainly measure extreme event risks under the tail extreme probability as well as the conditional concepts to capture the effect of the risk infection. $Δ {CoVaR}_{q}^{ij}$ is defined to estimate the risk spillover effect of the financial asset j and financial asset i as follows:

$Δ {CoVaR}_{q}^{ij} = {CoVaR}_{q}^{ij}$ (3)

Considering the larger differences of VaR for the different financial assets, $Δ {CoVaR}_{q}^{ij}$ can only express the selection and estimation of marginal distribution of the risk spillover effect.

$% Δ {CoVaR}_{q}^{ij} = \frac{Δ {CoVaR}_{q}^{ij}}{{CoVaR}_{q}^{i}} \times 100 %$ (4)

3.2 Selection and estimation of marginal distribution

3.2.1 Construction and estimation of Beta-skew-t-EGARCH

To better reflect the features i.e. lever effect, tail distribution and conditional skewness of financial assets, Harvey and Chakravarty [5] proposed the Beta-skew-t-EGARCH model on the basis of EGARCH model:

${\begin{matrix} r_{t} = exp λ_{t} ɛ_{t} = ɛ_{t} σ_{t} \\ ɛ_{t} \sim St 0, σ_{ɛ}^{2}, ν, γ, ν > 2, γ \in 0, \infty \\ λ_{t} = ω + λ_{t}^{+} \\ λ_{t}^{+} = φ λ_{t - 1}^{+} + κ_{1} u_{t - 1} + κ^{*} sgn - r_{t - 1}, | φ | < 1 \\ ɛ_{t} = ɛ_{t}^{*} - μ_{ɛ} \end{matrix}$ (5)

In which, r_t refers to the yield rate of financial asset, λ_t refers to the logarithmic scale as the linear combination of the past values of variable, ɛ_t refers to the conditional error, σ_t is the conditional variance, $σ_{ɛ}^{2}$ is the variance of variable ɛ_t (which indicates that ɛ_t is not the standard variance), ν refers to the degree of freedom, γ refers to the deviation parameter, ω refers to the constant term fluctuating in long term, φ refers to the persistent parameter (the larger the value, the stronger the agglutination), κ₁ refers to ARCH parameter (the larger the absolute value, the larger the reflection to the impact), κ^* refers to the lever parameter, u_t refers to the conditional score item $u_{t} = β (0, \frac{2 ν}{ν + 3})$ .

Assuming ɛ^* is the variable ordinarily in line with t distribution, f (ɛ^*) is the density function, borrowing the method of Fernandez and Steel [6], the distribution function of the deviation t distribution shall be:

$f (ɛ^{*} | γ) = \frac{2}{γ + γ^{- 1}} f (\frac{ɛ^{*}}{γ^{sgn} (ɛ^{*})})$ (6)

The conditional score u_t of martingale difference of the Beta-skew-t-EGARCH model shall be:

$\begin{matrix} \frac{\partial ln (r_{t})}{\partial λ_{t}} = u_{t} \\ = \frac{(ν + 1) [r_{t}^{2} + r_{t} exp (λ_{t})]}{ν exp (2 λ_{t}) γ^{2 sgn (r_{t} + μ_{ɛ} exp (λ_{t})}) + (r_{t} + μ_{ɛ} exp (λ_{t}^{2}))} - 1 \\ u_{t} = \frac{(ν + 1) (ɛ_{t}^{* 2} - μ_{ɛ} ɛ_{t}^{*})}{ν γ^{2 sgn (ɛ^{*})} + ɛ_{t}^{* 2}} - 1 \end{matrix}$ (7)

Equation (7) can be further simplified as:

$u_{t} = \frac{(v + 1) [ɛ_{t}^{* 2} - u_{ɛ} ɛ_{t}^{*}]}{v γ^{2 sgn (ɛ^{*})} + ɛ_{t}^{* 2}} - 1$ (8)

Parameters of Equations (5) and (8) can be obtained through the maximum likelihood.

3.2.2 Construction and estimation of extreme model

On the basis of modelling the Beta-Skew-t-EGARCH model in the last section, to further describe the tail features of the financial time sequence, the POT model of the extreme theory is introduced. The marginal distribution model based on the Beta-Skew-t-EGARCH-POT is finally constructed, and the specific process is as follows:

First of all, the Beta-Skew-t-EGARCH is adopted to separate the residual sequence r_t of the financial asset to obtain he standard residual sequence Z_t after standardizing it $(Z_{t - n - + 1}, \dots Z_{t}) = (\frac{r_{t - n + 1} - μ_{t - n + 1}}{σ_{t - n + 1}}, \dots, \frac{r_{t} - μ_{t}}{σ_{t}})$ (9)

In which, μ_t is the conditional mean of the residual sequence, and σ_t is the conditional variance of the residual sequence. The key pointed to model POT model is to determine the threshold u in the general methods such as excess expectation function plot, Hill graph and Du Mouchel 10% principle [28]. As the former two are too subjective, the last one isn’t robust enough, and the three methods are integrated in this paper to determine the threshold. This paper assumes the distribution function of standard residual sequence Z_t as F_u (y), so that:

$\begin{matrix} F_{u} (y) = P (r - u \leq y | r > u) \\ = \frac{F (u + y) - F (u)}{1 - F (u)} \end{matrix}$ (10)

When the threshold is large enough, according to the extreme theory [1], there will be F_u (y) ≈ G_ξ,β (y):

$F_{u} (y) \approx G_{ξ, β} (y) = {\begin{matrix} 1 - {(1 - \frac{y ξ}{β})}^{- \frac{1}{ξ}} ξ \neq 0 \\ 1 - exp - (\frac{y}{β}) ξ = 0 \end{matrix}$ (11)

G_ξ,β (y) is in GPD distribution, in which β and ξ are respectively the scale parameter and shape parameter. Then the larger ξ is, the thicker the tail is. Values of β and ξ can be obtained through the maximum likelihood estimation. Make x = y - u, according to Equations (10) and (11) there will be:

$F (x) = F (u) + G_{ξ, β} (y) [1 - F (u)]$ (12)

To obtain the upper and lower tail distribution of the sample yield rate, N represents the total number of samples, n represents the number of super-thresholds, the threshold distribution function F (u) can be expressed as (N - n)/N, so that the marginal distribution function of the sample yield rate shall be:

$F (x) = {\begin{matrix} 1 - \frac{n_{nR}}{N} {(1 + ξ \frac{x - u}{β})}^{- \frac{1}{ξ}} & x > u_{R} \\ Ecdf (x) & u_{L} < x < u_{R} \\ \frac{n_{nR}}{N} {(1 - ξ \frac{x - u}{β})}^{- \frac{1}{ξ}} & x < u_{L} \end{matrix}$ (13)

In which, u_R and u_L are respectively the thresholds of lower tail and upper tail, Ecdfx is the empirical distribution function distributed in the center of Ecdfx, to substitute Equations (11), (12) and (13) into (1), there will be:

${VaR}_{q}^{z} = u - \frac{β}{ξ} {{{1 - [\frac{N}{n} (1 - q)]}^{- ξ}}$ (14)

3.2.3 Selection of copula function

Copula function is a kind of contiguous functions proposed by Sklar (1959) [3]. It is essentially another conversion form of the joint distribution and according to the Sklar theorem [4], when the marginal distribution F₁, ⋯ F_N is continuous, there will be the only Copula function:

$F (x_{1}, x_{2}, \dots x_{N}) = C (F_{1} (x_{1}), \dots F_{N} (x_{N}))$ (15)

Then, according to the Sklar theorem, the density function of the joint distribution function can be obtained according to Equation (15):

$\begin{matrix} f (x_{1}, \dots x_{n}, \dots x_{N}) \\ = c (F_{1} (x_{1}), \dots F_{n} (x_{n}), \dots F_{N} (x_{N})) \prod_{n = 1}^{N} f_{n} (x_{n}) \end{matrix}$ (16)

In which, c (F₁ (x₁) , ⋯ F_n (x_n) , ⋯ F_N (x_N)) and f_n (x_n) are respectively the density functions of the Copula function and the marginal distribution function. Considering it is the spillover effect that is studied in this paper, the binary Copula function with lower data dimension is mainly applied; therefore, the maximum likelihood estimation method can be adopted to estimate the Copula function [27, 30]. In addition, Copula function family has a great variety generally divided into 16 kinds in 4 categories namely the oval Copula family, Archimedes Copula family, extreme Copula family and Archimax Copula. In this paper, the maximum likelihood value and AIC value, BIC value and HQ value minimum principle are adopted to select the optimal Copula function.

3.3 CoVaR computation based on the Copula-Beta-skew-t-EGARCH-POT model

After identifying the marginal distribution and selecting the optimal Copula function, the spillover effect of the financial asset shall be computed according to the definition of CoVaR. The specific process is as follows:

First of all, there is (U,V)∼C; C is the distribution function of the Copula function F_i,j (xⁱ, x^j) representing the joint distribution, U refers to the marginal distribution function F_i (xⁱ) of the financial asset Xⁱ, Vrefers to the marginal distribution function F_j (x^j) of the financial asset X^j. The corresponding density functions shall be respectively as f_i,j (xⁱ, x^j), f_i (xⁱ) and f_j (x^j); according to the definition in document of Gropp [16], the form of the conditional distribution density function of financial asset Xⁱ shall be as follows: $f_{i | j} (x^{i} | x^{j}) = \frac{f_{i, j} (x^{i}, x^{j})}{f_{j} (x_{j})}$ (17)

According to Equations (16) and (17), there will be the reduction as: $f_{i | j} (x^{i} | x^{j}) = c (F_{i} (x^{i}), F_{j} (x_{j})) f_{i} (x^{i})$ (18)

The distribution function of Equation (18) can be: $F_{i | j} (x^{i} | x^{j}) = \int_{- \infty}^{x_{i}} c ((F_{i} (x^{i}), F_{j} (x_{j})) f_{i} (x^{i}) {dx}^{i}$ (19)

The marginal distribution function F_i (xⁱ), F_j (x^j) in Equation (19) is obtained through the Beta-skew-t-EGARCH-POT in the above; the density function f_i (xⁱ) is the derivative of the marginal distribution function F_i (xⁱ), and c (·) is the density function of optimal Copula function selected in the above, so that according to the method of Mainik and Schaanning [10], ${CoVaR}_{q}^{ij}$ can be expressed as follows:

${CoVaR}_{q}^{ij} = F_{i | j}^{- 1} (q | {VaR}_{q}^{j})$ (20)

In Equation (19), $F_{i | j}^{- 1} (\cdot)$ is the inverse function of F_i|j (·), its analysis formula is difficult to get. Generally, Equation (19) can be transformed into the following expression for solution: $\int_{- \infty}^{x_{i}} c (F_{i} (x^{i}), F_{j} (x_{j})) f_{i} (x^{i}) {dx}^{i} = q$ (21)

The solution xⁱ of Equations (20) can be ${CoVaR}_{q}^{ij}$ .

4 Empirical study

4.1 Data source and descriptive statistics

The representative indexes on the Chinese and international forward markets are selected for analysis. Considering the data availability, as the domestic crude oil futures are in the preparatory phase. In this paper, the spot price of Daqing crude oil (CCO) is selected for substitution. The Nanhua Futures price index (NFI) is selected on the domestic market and the CRB futures index (RJ/CRB) on the international commodity market [31]. The entire data are the daily closing prices from June 1, 2016 to July 20, 2017. The proposed data are from the WIND database, and the defined logarithmic yield is r_t = 100 × log(P_t/P_t-1).

According to Table 1, the sample sequences are of the left-skewed pattern (skewness < 0) with leptokurtosis and fat-tail features (kurtosis > 3). Also, there is J-B inspection that under the 1% significance level, and all sample sequences are significantly different from the normal distribution. To further inspect the distribution features of the sample sequences, the QQ charts of sample sequences are described in this paper (Fig. 1). Take the CRB on the up most as an example, the upper and lower tails of the CRB sequence deviate significantly from the normal distribution, presenting the heavy tail feature; therefore, there can be a conclusion that the sample sequences have a typical feature of “leptokurtosis and fat-tail”. According to Fig. 2, the fluctuation trends of the three sample sequences are basically consistent, which presents the asymmetrical distribution and shows certain explosion and aggregation features. According to Ljung-Box inspection, under the 1% significance level, there are no serial correlations for the yield rates of the three sequences; the ARCH inspection indicates that there are ARCH effects on three yield rates. According to the ADF inspection, the three yield rates are stable, so that the Beta-Skew-t-EGARCH-POT can be adopted for modelling.

Table 1
Sample descriptive statistics

Min Max Mean SD Skewness kurtosis JB Test Q(12) Q²(12) ARCH ADF

CRB –16.3115 5.7462 –0.0153 1.2080 –0.9218 13.1294 22257* 23.382 299.88* 199.13* –39.0408*

NFI –4.5086 3.6855 0.0100 0.9694 –0.2361 3.8634 468* 47.236* 928.08* 363.54* –37.8271*

CCO –17.4061 20.4234 0.0047 2.4173 –0.0662 5.7923 4252* 25.956 873.37* 366.63* –37.6852*

	Min	Max	Mean	SD	Skewness	kurtosis	JB Test	Q(12)	Q²(12)	ARCH	ADF
CRB	–16.3115	5.7462	–0.0153	1.2080	–0.9218	13.1294	22257*	23.382	299.88*	199.13*	–39.0408*
NFI	–4.5086	3.6855	0.0100	0.9694	–0.2361	3.8634	468*	47.236*	928.08*	363.54*	–37.8271*
CCO	–17.4061	20.4234	0.0047	2.4173	–0.0662	5.7923	4252*	25.956	873.37*	366.63*	–37.6852*

Note:* Represent 1% significant levels.

Fig. 1

QQ diagram of sample sequence.

Fig. 2

The yield curve of the sample.

4.2 Copula-Beta-Skew-t-EGARCH-POT model estimation

4.2.1 Beta-Skew-t-EGARCH model estimation

Considering the excellent features of Beta-Skew-t-EGARCH model, in this paper, the Beta-Skew-t-EGARCH in the above is adopted to fit three sample yield rate sequences, the specific result is as shown in Table 2.

Table 2
Parameter estimation results of Beta-Skew-t-EGARCH model

Index ω φ κ ₁ κ ^* ν γ

CRB 0.0027 0.9338 0.1825 0.9629 0.9445 7.7709

(0.0626) (0.0000) (0.0137) (0.0000) (0.0000) (0.0000)

NFI 0.0122 0.9731 –0.0651 0.9160 0.9479 7.7598

(0.0022) (0.0000) (0.1393) (0.0000) (0.0000) (0.0000)

CCO 0.0128 0.9464 0.2839 0.9500 0.9734 6.2508

(0.0653) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)

Index	ω	φ	κ ₁	κ ^*	ν	γ
CRB	0.0027	0.9338	0.1825	0.9629	0.9445	7.7709
	(0.0626)	(0.0000)	(0.0137)	(0.0000)	(0.0000)	(0.0000)
NFI	0.0122	0.9731	–0.0651	0.9160	0.9479	7.7598
	(0.0022)	(0.0000)	(0.1393)	(0.0000)	(0.0000)	(0.0000)
CCO	0.0128	0.9464	0.2839	0.9500	0.9734	6.2508
	(0.0653)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)

According to Table 2, the persistent parameter φ of the three sample markets shall be 0.9338, 0.9771 and 0.9464 respectively, which means the fluctuation aggregation is obvious. The ARCH parameter κ₁ means that the market has larger responses to the fluctuation impacts, in which CCO yield rate sequence is the largest and the lever parameter κ^* is larger than 0. The market samples have significant lever effect while CRD yield rate sequence has the more obvious lever effect. The skewness parameter γ is larger than 0, which means that the distributions of sample market yield rate sequences are significantly deflective.

4.2.2 POT Model estimation

Based on the Beta-Skew-t-EGARCH mode estimation, the residual sequences of three sequences are obtained in this paper. After standardizing the residual sequences, according to the inspection, the three standard residuals are stable without autocorrelation; therefore, and the POT model can be used for modelling.

There are three stages including the model exploration stage, model fitting stage and model diagnosis stage to model the standard residual sequences. At the model exploration stage, the appropriate threshold selected can directly influence the model fitting effect. Despite the current popular methods for threshold selection namely have the excess expectation function plot, Hill graph and Du Mouchel 10% principle etc., there has no unified standard. In view of how important the threshold selection to the model estimation is, the three methods are integrated to determine the threshold finally.

Take CRB market as an example. First of all, the upper tail excess expectation function plot shall be provided (Fig. 3).

Fig. 3

CRB upper and excess expected function diagram.

And then there shall be the upper tail Hill graph (Fig. 4).

Fig. 4

CRB top and tail hill diagram.

There will be the exact thresholds from Figs. 3 and 4; however, the approximate range of value of the upper tail threshold can be estimated between 0 and 2; the threshold can be computed as 0.97 according to the Du Mouchel 10% method. To inspect the precision of the threshold selection, the interval of the range of the threshold can be determined according to the excess expectation function plot and Hill graph. The parameter values can be computed through the maximum likelihood estimation. If the parameter estimation is stable near the threshold taken threshold (which shall be an interval), the threshold selection will be relatively reasonable. The specific threshold can be computed according to the Du Mouchel 10% method; if the threshold is in the stable interval, the proposed threshold can be regarded as relatively accurate. According to Fig. 5, the CRB upper tail threshold can be determined as 0.97 and other thresholds can be computed in this method; the specific result is as shown in Table 3.

Fig. 5

CRB maximum likelihood estimation under different threshold conditions.

Table 3

POT model estimation results

	Upper tail		Lower tail
CRB	u	0.97	u	–1.11
	ξ	–0.10	ξ	–0.17
	β	0.55	β	0.59
NFI	u	1.14	u	–1.15
	ξ	–0.11	ξ	–0.04
	β	0.70	β	0.84
CCO	u	1.0	u	–1.08
	ξ	–0.10	ξ	–0.15
	β	0.67	β	0.45

The second one is the model fitting stage. Upon obtaining the threshold, the POT model is adopted to estimate the two standard residual sequences to obtain the scale parameter β and the shape parameter ξ of the POT model; the result is as shown in Table 3.

The last one is the model inspection stage. To inspect the POT model estimation effectiveness, the fitting diagnosis figure of the CRB standard residual sequence is made in this paper. As shown in Fig. 6, most of the points are directly dropping on or near the threshold distribution diagram and the tail distribution plot with only several are deviating but not influencing the fitting effect. The entire fitting effect of POT is satisfied. Other samples adopt the same method, which won’t be further explained here.

Fig. 6

The fitting diagnosis of CRB standard residual sequence.

4.2.3 Copula function estimation

After making POT modelling and establishing the marginal distribution on a standard residuals of the CCO samples, NFI samples and CRB samples, the Copula function is adopted in this paper to describe the dependency structure of the sample sequence. Considering that in this paper, it is the level of mutual risk spillover between the crude oil market and the commodity market that is mainly explored, and these samples are divided into two groups: CCO & NFI and CCO & CRB. Copula function is adopted for fitting; the following is the fitting result for the CCO and Copula function; and the details in Table 4.

Table 4
Copula function fitting results of CCO and NFI

Copula type Copula function loglike AIC BIC HQ

ElipseCopula Norm 162.2927 –322.5855 –316.5678 –320.4222

normal.mix 162.2927 – –318.5855 –300.5326 312.0956

Archimedescopula frank 161.8552 –321.7105 –315.6928 –319.5472

kimeldorf.sampson 142.4064 –282.8128 –276.7952 –280.6496

joe 117.9211 –233.8423 –227.8246 –231.6790

bb1 186.0930 –368.1859 –356.1506 –363.8593

bb2 142.3847 –280.7694 –268.7341 –276.4428

bb3 185.2633 –366.5266 –354.4913 –362.2000

bb6 117.9211 –231.8423 –219.8070 –227.5157

bb7 182.5755 –361.1510 –349.1157 –356.8244

EVT Copula Gumble 162.5007 –323.0013 –316.9837 –320.8380

galambos 148.0242 –294.0484 –288.0308 –291.8851

husler.reiss 135.9440 –269.8880 –263.8704 –267.7247

tawn 171.6576 –337.3152 –319.2623 –330.8253

bb5 162.5007 –321.0013 –308.9660 –316.6747

Archimax Copula bb4 175.5653 –347.1306 –335.0953 –342.8040

Copula type	Copula function	loglike	AIC	BIC	HQ
ElipseCopula	Norm	162.2927	–322.5855	–316.5678	–320.4222
	normal.mix	162.2927 –	–318.5855	–300.5326	312.0956
Archimedescopula	frank	161.8552	–321.7105	–315.6928	–319.5472
	kimeldorf.sampson	142.4064	–282.8128	–276.7952	–280.6496
	joe	117.9211	–233.8423	–227.8246	–231.6790
	bb1	186.0930	–368.1859	–356.1506	–363.8593
	bb2	142.3847	–280.7694	–268.7341	–276.4428
	bb3	185.2633	–366.5266	–354.4913	–362.2000
	bb6	117.9211	–231.8423	–219.8070	–227.5157
	bb7	182.5755	–361.1510	–349.1157	–356.8244
EVT Copula	Gumble	162.5007	–323.0013	–316.9837	–320.8380
	galambos	148.0242	–294.0484	–288.0308	–291.8851
	husler.reiss	135.9440	–269.8880	–263.8704	–267.7247
	tawn	171.6576	–337.3152	–319.2623	–330.8253
	bb5	162.5007	–321.0013	–308.9660	–316.6747
Archimax Copula	bb4	175.5653	–347.1306	–335.0953	–342.8040

According to the likelihood maximum and AIC value, BIC value and HQ minimum principle, the optimal Copula function among the 4 categories of Copula functions are selected in this paper as the BB3 Copula function. On this basis, the τ correlation coefficient, ρ_s correlation coefficient and the upper and lower tail correlation coefficients (the lower tail is the main concern) of the two groups of samples are computed as shown in Table 4; therefore, the preliminary qualitative analysis can be made on the risk spillover strength and direction of the two groups of samples. The τ correlation coefficients are the two groups of samples are respectively 0.2048 and 0.1191; the ρ_s correlation coefficients are respectively –0.6145 and –0.5359. It means that there are strong correlations between the two groups of samples. The correlation of CCO-NFI is stronger than that of the CCO-CRB; λ_L larger than 0 indicates that the risk spillover effect among the samples is positive and the strength of CCO-NFI is obviously larger than that of CCO-CRB. An obvious reason is that both CCO and BFI belong to the domestic commodity market, the risk transference is rather rapid while CCO and CRB respectively belong to the domestic and foreign markets. Considering that the domestic capital market has been not completely open yet, the mutual risk spillover effects will be weakened greatly. On the plus side, this might play a role of firewall in the domestic capital market including the crude oil market; on the minus side, it has obstructed the free competition of the domestic capital markets.

To substitute the estimation parameter into the BB3 Copula function, there will be the BB3 Copula distribution function as follows: $\begin{matrix} \begin{matrix} C (u, v) = exp {- [δ^{- 1} ln (e^{δ (- ln (u)^{θ})} \\ {+ e^{δ (- ln (v)^{θ})} - 1)]}^{\frac{1}{θ}}} \\ θ > 1, δ > 0 \end{matrix} \end{matrix}$

Its density function can be obtained through the BB3 Copula distribution function to make the maximum likelihood estimation of the density function; the specific result is as shown in Table 5.

Table 5

Parameter estimation results of BB3 Copula function

Object	Parameter	estimation	SD	Tstatistic	τ	ρ _s	λ _U	λ _L
CCO-NFI	δ	0.1609	0.0296	5.4444	0.2084	–0.6145	0.1881	0.9571
	θ	1.1662	0.0208	56.0937
CCO-CRB	δ	0.1703	0.0251	6.7816	0.1191	–0.5359	0.1087	0.0170
	θ	1.0876	0.0186	58.4042

To verify the fitting effect of the BB3 Copula function, the empirical distribution of Copula function and the fitting effect diagram of CDF based on CCO-NFT are made. In comparison, the fitting effect diagram of BB7 Copula is made. According to Fig. 7, the fitting effect of BB3 Copula function satisfies the correlation structure of sample.

Fig. 7

Comparison between the empirical functions of BB7 Copula (left) and BB3 Copula (right) and the CDF fitting effect.

4.2.4 CoVaR computation and inspection of extreme risk spillover effect

For the further quantitative study on the risk spillover effect among the samples, the model established in the above is adopted to compute CoVaR, CoVaR, Δ CoVaR and % CoVaR under the 5% significance level after identifying the marginal distribution of sample sequences and determining the optimal Copula function. See the details in Table 6.

Table 6
Crude oil market and commodity market extreme risk spillover statistics

VaR CoVaR ΔCoVaR % CoVaR

CO→CRB 1.5403 2.1597 0.6194 0.4021

CCO→NFI 1.5403 2.1876 0.6473 0.4202

CRB→CCO 1.5052 2.3324 0.8272 0.5496

NFI→CCO 1.6046 2.4412 0.8366 0.5214

	VaR	CoVaR	ΔCoVaR	% CoVaR
CO→CRB	1.5403	2.1597	0.6194	0.4021
CCO→NFI	1.5403	2.1876	0.6473	0.4202
CRB→CCO	1.5052	2.3324	0.8272	0.5496
NFI→CCO	1.6046	2.4412	0.8366	0.5214

According to Table 6, under the given significance level, the domestic commodity market (NFI) has the largest risk VaR and the foreign commodity market (CRB) has the least riskVaR. From the perspective of risk spillover, the conditional risk CoVaR is larger than the risk VaR and the mutual risk spillover is larger than 0, there is positive spillover effect. From the perspective of risk spillover strength (% CoVaR), the mean value is 47%, in which, the lowest CCO→CRB is 0.4021 and the highest CRB→CCO is 0.5494, presenting an asymmetry. This is mainly because the pricing power of the crude oil is controlled by the international crude oil market while the domestic crude oil market started late and has been immature, hence there is no pricing power. Comparatively speaking, the domestic crude oil market is easier to be impacted by the international crude oil market, hence the asymmetric strength between the international and domestic crude oil markets. In the meantime, there are bidirectional positive risk spillover effects in the domestic commodity market (NFI) and the domestic crude oil market (CCO); the NFI→CCO risk spillover effect is larger than that of CCO→NFI, mainly as a result of the market maturity. Generally, there are strong bidirectional positive risk spillovers between the two groups of samples in addition to the asymmetry.

Therefore, in the specific risk management, the risks must be underestimated if only theVaR value shall be considered. Take the CCO→CRB as an example, the actual risk value CoVaR is obviously larger than the theoretical risk valueVaR, hence the risks will be underestimated seriously and there will be huge uncertainties in the Risk Management. On the other hand, Table 4 only shows the index values under the 5% significance level; to deepen the risk spillover effects under the different significance levels, Fig. 8 has provided the % CoVaR under the different significance levels. There is an inverse relation between the % CoVaR values and the significance levels, which means, the larger the risk value (inversely proportional to the significance level), the larger the corresponding % CoVaR value. The concerns on extreme risks are more important.

Fig. 8

% CoVaR different significance levels.

Method in the reference documentation [28] is adopted to inspect the effectiveness of the proposed model. To be specific: according to the marginal distribution model and the optimal Copula function determined in the above, the method of producing the random number is adopted to simulate and generate five thousand groups (Xⁱ, X^j) of values and then select the 200 values which are the closest to ${VaR}_{q}^{j}$ from X^j. The 200 Xⁱ (marked as X) corresponding to X^j can be obtained. The specific value between the number of X with values less than ${CoVaR}_{q}^{ij}$ and the total number is exactly the inspection value this paper wants to get. The specific operation result shall be: CCO→CRB(5.5%), CCO→NFI(4.5%), CRB→CCO(3%) and NFI→CCO(5%). Comprehensively speaking, only CRB→CCO(3%) has a weaker effect on overestimated risks; the rest is on or near the ideal levels. As a result, the model has been well fitted the risk features of the samples and made appropriate measures on the risk spillovers of the samples.

5 Conclusion

Based on the Copula-CoVaR framework, in this paper, the Beta-skew-t-EGARCH model is adopted to capture the features such as lever effect of the financial assets, fat tail distributions and conditional skewness etc. The extreme theory is introduced on the basis of standard residuals to construct the marginal distribution model on Beta-skew-t-EGARCH-POT. Upon selecting the optimal Copula function, the CoVaR method is adopted to measure the risk spillover effects on the Chinese oil market as well as the domestic and international commodity markets to reveal the influence of the Chinese oil market. There will be the following conclusions:

The persistent parameter φ of sample market shall be respectively 0.9338, 0.9771and 0.9464. It means that the fluctuation aggregation is an obvious aggregation. ARCH parameter κ₁ means that the sample market has larger responses to the fluctuation impacts. In which, the CCO yield sequence is the largest. The lever parameter κ^* is larger than 0, which indicates that there are obvious lever effects on the sample market, in which the lever effect of CRB yield sequence is more obvious; the skewness parameter γ is larger than 0, which indicates that there is obvious skewness in the distribution of the sample market yield sequences.

Regardless of the domestic crude oil market, international commodity market, or the domestic commodity market, there are positive bidirectional risk spillover effects and asymmetric features; the asymmetric level of the former is stronger than the later. In addition, according to the back-testing, the model based on Copula-CoVaR framework has been not only described the features i.e. leptokurtosis and fat-tail, leer effect and conditional skewness, but also made precise description on the nonlinear risk spillover features of the sample market.

Footnotes

Acknowledgments

Thanks for The National Social Science Fund of China (No.19BJY077and No.18BJY093), the Chongqing Normal University Fund (No.13XWB008 and No.16XYY268), Chongqing Social Science Planning Project (No.2018PY74), Science and Technology Project of Chongqing Education Commission (No.KJQN20180051). This paper are grateful to the reviewers for helpful comments.

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Based on Copula-CoVaR model of risk spillover effect of oil markets and other commodity markets

Abstract

Keywords

1 Introduction

2 Literature review

3 Model construction

3.1 CoVaR

3.2.1 Construction and estimation of Beta-skew-t-EGARCH

4.1 Data source and descriptive statistics

4.2.1 Beta-Skew-t-EGARCH model estimation

Table 6 Crude oil market and commodity market extreme risk spillover statistics VaR CoVaR ΔCoVaR % CoVaR CO→CRB 1.5403 2.1597 0.6194 0.4021 CCO→NFI 1.5403 2.1876 0.6473 0.4202 CRB→CCO 1.5052 2.3324 0.8272 0.5496 NFI→CCO 1.6046 2.4412 0.8366 0.5214

Footnotes

Acknowledgments

References

Table 6
Crude oil market and commodity market extreme risk spillover statistics

VaR CoVaR ΔCoVaR % CoVaR

CO→CRB 1.5403 2.1597 0.6194 0.4021

CCO→NFI 1.5403 2.1876 0.6473 0.4202

CRB→CCO 1.5052 2.3324 0.8272 0.5496

NFI→CCO 1.6046 2.4412 0.8366 0.5214