^{1}

^{1}

This paper proposes a Markov-switching copula model to examine the presence of regime change in the time-varying dependence structure between oil price changes and stock market returns in six GCC countries. The marginal distributions are assumed to follow a long-memory model while the copula parameters are supposed to evolve according to the Markov-switching process. Furthermore, we estimate the Value-at-Risk (VaR) based on the proposed approach. The empirical results provide evidence of three regime changes, representing precrisis, financial crisis and post-crisis, in the dependence structure between energy and GCC stock markets. In particular, in the pre- and post-crisis regimes, there is no dependence, while in the crisis regime, there is significant tail dependence. For OPEC countries, we find lower tail dependence whereas in non-OPEC countries, we see upper tail dependence. VaR experiments show that the Markov-switching time- varying copula model performs better than the time-varying copula model.

It is widely recognized that the energy and stock markets are very closely tied. Theoretically, changes in the oil price are the most significant factor influencing the returns of stock market indices, either directly by affecting the future cash flows or indirectly through impacting the interest rate considered to discount the future cash flows.

Regarding the Gulf Cooperation Council (GCC) countries^{1}, numerous empirical studies have been developed to examine the linkages between oil price changes and stock market returns using various econometric approa- ches. Previous studies rely on linear times series models like VAR and VAR-GARCH to study short-term dynamics [

An important precondition for the validation of the linear models is the stability of the models and the invariability of the parameters over time. In practice, this assumption is far from being satisfied due to the pre- sence of structural breaks [

The evidence of non-linearity of the relationship between oil price changes and GCC stock market returns has been provided by [

It is well-known that these models are limited because they do not allow for the asymmetric effect of increases and decreases in oil prices on stock markets returns. In this sense, some studies show that the stock markets are more sensitive to negative oil shocks than to positive oil shocks [^{2} in the linear model and find that the decreases in oil prices have a significant negative impact on stock market returns, whereas the increases in oil prices present a strong positive effect on the stock market returns in Saudi Arabia and the UAE only. [

Although the DCC-GARCH model allows for the time-varying conditional correlation, it fails to reproduce the non-linear dependence that may exist between the variables and does not provide information about the tail dependence. The tail dependence corresponds to the possibility of joint events such as low or high extreme event occurrence. To do so, an alternative approach based on copula functions has been adopted. The main advantage of the copulas lies in separating the dependence structure from the marginals without making any assumptions about the distribution. Using several copula functions, [

The main insufficiency of these copula functions is that the dependence structure is supposed to be constant over time. To allow for variability in the dependence structure, [

More recent studies show that the financial crisis has a considerable impact on the dependence structure between oil price changes and stock market returns. For instance, [

To test for the presence of change in the dependence structure between oil price changes and GCC stock market returns, [

Although the two later studies provide interesting findings about the existence of structural change in the dependence structure and the instability of the copula parameter, they do not give information about the existence of regime change in the dependence. In this paper, we propose a novel regime switching copula model that allows for regime change in the copula parameter in order to identify the financial crisis regime through the time-varying dependence structure between oil price changes and six GCC stock market returns. Interestingly, we employ Markov-switching copula functions that permit the copula parameter to evolve according to three regimes (pre-crisis, during crisis and post-crisis) depending on the state of an unobserved Markov chain with corresponding transition probabilities as suggested by [

The main advantage of this model is that it does not require an ad hoc determination of change point in the dependence structure. Prior studies like [

The rest of this paper is organized as follows. Section 2 describes the econometric methodology. Section 3 presents the data, gives the empirical results and discusses the policy implications. Section 4 concludes.

This section introduces the econometric methodology that we adopt to reproduce the presence of regime change in the dynamic dependence structure between oil prices and stock markets. We firstly recall the bivariate copulas. Secondly, we discuss the Markov-switching time-varying copula functions. Finally, we present the method considered to estimate the copula parameter.

A copula is a function that allows to join different univariate distributions to form a valid multivariate dis- tribution without losing any information from the original multivariate distribution^{3}. According to theorem of [

Formally, let

The theorem also states that if

where the copula density c is obtained by differentiating (1).

An important property of a copula is that it can capture the tail dependence: the upper tail dependence

where

The time-varying copulas have been introduced by [^{4}. They constitute an extension of Sklar’s theorem, which shows that any joint distribution function may be decomposed into its marginal distributions and a copula that describes the dependence between the variables, for conditional case. In what follows, we give a general definition of the conditional copula and we present the time-varying copula functions used to examine the dependence between the series over time. We consider several time-varying copulas that capture different patterns of dependence, namely, time-varying Normal, time- varying Student, time-varying Gumbel, time-varying Clayton and time-varying Symmetrized Joe-Clayton copulas. The time-varying Gaussian and Student are characterized by symmetric dependence while the time- varying Gumbel and Clayton are used to capture the right and the left dependences respectively. The SJC copula is more general because it allows the tail dependences to be either symmetric or asymmetric.

Definition The conditional copula C is the joint distribution function of

Theorem extension of Sklar’s ( [

Let

To model the joint conditional distribution the evolution of the conditional copula C has to be specified and the functional form of C is fixed (see [

In this paper, we assume that the dependence parameter is allowed to vary over time follows a restricted ARMA(1,10) process where the intercept term switches according to some homogeneous Markov process. However, we consider ^{5}, i.e., P is a

where

regime j at time

The Normal copula is the copula of the multivariate normal distribution and is given by:

where

In order to allow for time-varying dependence, we assume the parameter

where

Equation (8) reveals that the copula parameter follows an ARMA(1,10) type process in which the auto- regressive term

The Student copula proposed is defined as:

where

symmetric non-zero tail dependence

Similar to the Normal copula, we assume that the dynamics of

The Gumbel copula introduced by [

where

For the non-Gaussian case, we consider that the dependence parameter varies over time. More precisely, we consider that the Kendall’s tau

where

Equation (12) shows that the Kendall’s tau follows an ARMA(1,10) type process in which the autoregressive term

The Clayton copula proposed by [

where

To allow for time-varying dependence, we assume that

where

The SJC copula is [

where

With

In contrast to the Clayton and the Gumbel copulas, the SJC copula considers both the lower and the upper tail dependence. If

where

at all times.

Equation (17) and Equation (18) show that the upper and lower tail dependence parameters follow an ARMA(1,10) type process in which the autoregressive terms

To estimate the vector with all model parameters of the time-varying copula

Considering

In this paper, the estimation process is performed in two steps adopting the IFM method. This method consists of estimating the parameters of the univariate marginal distributions in a first step and then using these estimates to estimate the dependence parameters in a second step.

In a first step, the each marginal distributions of dataset ^{6} presented in section 3.2. After fitting marginal distributions, the filtered (standardized) residuals are used to specified the copula parameters.

The approximate log-likelihood function is given by the following equation:

where

Thus, the approximate log-likelihood copula function is obtained via:

However, the dependence parameter estimation through copula in our case depends on a non-observable discrete variable

where the set

To calculate the conditional probabilities

and

where

end j.

However, the smoothed probabilities regarding

In a second step, the time-varying dependence parameter

Under certain regularity conditions (for more details, see [^{7}. After estimating the parameters of the copula, a typical problem that arises is how to choose the best copula, i.e., the copula that provides the best fit with the data set at hand. To this purpose, we consider the log likelihood (LL), the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

This section presents the data, gives the empirical results and discusses some policy implications.

Our data set consists of daily oil prices and stock market indices in six GCC countries, namely, Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and the United Arab Emirates (UAE) over the period May 25, 2005 until March 31, 2015. The chosen period allows us to take into account the effect of the recent global financial crisis of 2007- 2009. We obtain a total of 2555 observations. These countries may be divided into two groups: 1) OPEC (Organization of Petroleum Exporting Countries) including Kuwait, Qatar, Saudi Arabia and the UAE and 2) non-OPEC including Bahrain and Oman.

As a proxy for stock markets, we use the major stock market index for each country extracted from MSCI (Morgan Stanley Capital International). To represent the world oil price, we use the Brent crude oil price collected from the US Energy Information Administration (EIA) website. We consider the Brent crude oil price rather than the West Texas Intermediate (WTI) crude oil price to represent the international oil market because the Brent crude oil price is widely used as the benchmark for oil-pricing. In addition, the Brent crude oil price is closely related to other crude oils such as WTI, Maya, Dubai (see [

These data are transformed into logarithm form and considered in first difference, so the series obtained correspond to stock market returns and oil price changes. More precisely, we consider the stock market returns (resp. oil price changes) ^{8}, which is a standard finding in the literature for such series.

We see that the average stock market returns are negative for all GCC countries while the average oil price changes are positive. Moreover, we observe that the UAE shows the highest risk degree as measured by the standard deviation (2.095%) followed by Saudi Arabia (1.840%) and Qatar (1.677%), while Bahrain experiences the lowest risk (1.335%) followed by Oman (1.396%), indicating that the OPEC stock markets are more risky than the non-OPEC stock markets. The oil price changes show a higher average return (0.044%) and a higher standard deviation (2.226%) than those of stock markets since oil prices doubled during the study period from $50.46 to $118.29. All series exhibit negative skewness and show excess kurtosis. The Jarque-Bera test strongly rejects the null hypothesis of normality for all series, which justifies the choice of copula theory.The Ljung-Box test shows significant evidence of serial correlation for all series and the ARCH-LM test indicates presence of heteroskedasticity in all series.

Prior studies have documented that stock market returns and oil price changes exhibit some common charac- teristics such as fat-tails, conditional heteroskedasticity and long-memory behavior. The most popular approach used is the ARFIMA-FIGARCH model introduced by [

Countries | Kuwait | Qatar | Saudi | UAE | Bahrain | Oman | Brent |
---|---|---|---|---|---|---|---|

Mean (%) | −0.012 | −0.001 | −0.027 | −0.048 | −0.100 | −0.020 | 0.044 |

Std Dev (%) | 1.533 | 1.677 | 1.840 | 2.095 | 1.335 | 1.396 | 2.226 |

Skewness | −1.288 | −1.018 | −2.108 | −0.908 | −3.359 | −1.612 | −0.010 |

Kurtosis | 17.317 | 16.958 | 29.805 | 15.728 | 42.223 | 29.720 | 9.051 |

JB × 10^{−4} | 1.709^{***} | 1.607^{***} | 5.945^{***} | 1.335^{***} | 28.686^{***} | 5.849^{***} | 2.957^{***} |

Q(10) | 25.585^{***} | 24.597^{***} | 26.548^{***} | 27.645^{***} | 25.459^{***} | 28.692^{***} | 24.734^{***} |

ARCH(10) | 29.556^{***} | 21.127^{***} | 13.238^{***} | 21.095^{***} | 12.498^{***} | 15.415^{***} | 16.333^{***} |

Notes: JB is the statistic of Jarque and Bera test for normality, Q(10) is the statistic of Ljung-Box test for serial correlation, corrected for heteroskedasticity, computed with 10 lags and ARCH(10) is the statistic of ARCH test for heteroskedasticity for order 10. ^{***}indicates a rejection of the null hypothesis at the 1% level.

In this paper, we use the ARFIMA-FIAPARCH (Autoregressive Fractionally Integrated Moving Average- Fractionally Intergrated Asymmetric Power AutoRegressive Conditionally Heteroskedastic) model proposed by [^{9}.

Let

Equation (27) represents the mean equation. ^{10}.

Equation (28) defines the residual terms of the mean equation

Equation (29) assumes that the standardized residuals

Equation (30) corresponds to the FIAPARCH

We see that the fractional integration parameter in mean

It should be stressed that, within the FIAPARCH model, we can test for the restrictions embodied in the FIGARCH model, i.e.,

Kuwait | Qatar | Saudi | UAE | Bahrain | Oman | Brent | |
---|---|---|---|---|---|---|---|

0.796^{*} | |||||||

(1.686) | |||||||

0.164^{***} | 0.045^{***} | 0.157^{***} | 0.062^{***} | 0.038^{***} | 0.029^{**} | 0.043^{***} | |

(2.979) | (3.058) | (4.873) | (3.170) | (2.672) | (2.073) | (3.735) | |

0.802^{***} | −0.756^{***} | 0.239^{**} | 0.836^{***} | −0.206^{**} | 0.327^{***} | ||

(2.608) | (−6.588) | (2.312) | (8.345) | (−2.378) | (3.132) | ||

0.172^{***} | 0.346^{***} | 0.268^{**} | −0.242^{***} | ||||

(5.837) | (7.489) | (2.017) | (−2.981) | ||||

0.035^{***} | |||||||

(2.635) | |||||||

0.398^{***} | 0.478^{***} | 0.531^{***} | 0.446^{***} | 0.423^{***} | 0.563^{***} | 0.267^{***} | |

(7.747) | (4.711) | (5.98) | (4.894) | (5.312) | (5.138) | (4.863) | |

0.285^{**} | 0.468^{***} | 0.268^{***} | 0.327^{***} | 0.168^{***} | 0.192^{***} | 0.639^{***} | |

(2.079) | (2.665) | (2.896) | (2.874) | (2.973) | (2.679) | (3.248) | |

1.685^{***} | 2.830^{***} | 2.396^{***} | 2.618^{***} | 2.887^{***} | 3.332^{***} | 1.733^{***} | |

(8.347) | (4.684) | (5.885) | (10.541) | (9.752) | (7.530) | (6.292) | |

0.435^{***} | 0.825^{***} | 0.732^{***} | 0.601^{***} | 0.467^{***} | 0.752^{***} | 0.563^{***} | |

(6.234) | (6.672) | (7.894) | (6.637) | (3.744) | (12.703) | (4.729) | |

0.259^{***} | 0.269^{***} | 0.179^{***} | 0.331^{***} | 0.185^{***} | 0.156^{***} | 0.321^{***} | |

(8.572) | (4.194) | (6.783) | (2.968) | (3.307) | (3.679) | (3.681) | |

−0.029^{***} | −0.008^{***} | 0.209^{***} | −0.007^{**} | −0.057^{***} | 0.074^{***} | −0.068^{***} | |

(−2.871) | (−2.689) | (3.761) | (−1.968) | (−2.689) | (2.791) | (−2.847) | |

2.704^{***} | 2.295^{***} | 4.985^{***} | 2.413^{***} | 2.217^{***} | 2.259^{***} | 2.564^{***} | |

(13.735) | (15.886) | (10.643) | (15.891) | (23.561) | (24.358) | (14.576) | |

Skw | −0.783^{***} | −0.938^{***} | −0.857^{***} | −0.957^{***} | 0.768^{***} | −0.921^{***} | −0.162^{***} |

Ex. Kurt | 3.479^{***} | 4.367^{***} | 4.452^{***} | 3.694^{***} | 2.637^{***} | 4.995^{***} | 1.987^{***} |

20.436 | 22.768 | 19.639 | 17.652 | 14.369 | 18.437 | 14.768 | |

17.847 | 16.453 | 14.573 | 12.758 | 13.739 | 16.752 | 12.678 |

Notes: The values in parenthesis are the t-Student. Skw is Skewness. Ex. Kurt is Excess of Kurtosis. ^{*}, ^{**} and ^{***} denote significance at the 10%, 5% and 1% levels respectively.

statistical test used to compare the in-sample performance of nested models. The statistic test is asymptotically Chi-squared distributed with a degree of freedom equal to the number of restrictions being tested. Let ^{11}.

For all series, we find that the statistic of test exhibits higher values than 9.210^{12}, and we find that the statistic test clearly rejects the constraint implied by the FIGARCH-type specification at the 1% significance level. Hence, we can conclude that the FIAPACH adaptation appears to be the most satisfactory representation to describe the long-memory behavior in the second conditional moment.

For all series, the leverage coefficient

For all series, the kurtosis parameter

Now, we focus on the regime change dynamic dependence between oil price changes and stock market returns in six GCC countries. For each country, we estimate the Markov-switching time-varying copula functions presented in section 2.2 (Equations (8), (10), (12), (14), (17) and (18). The obtained results are displayed in Tables 3-7. To determine the number of regimes of the appropriate specification for Markov-switching time- varying copula, we consider null hypothesis of

Normal | Kuwait | Qatar | Saudi | UAE | Bahrain | Oman |
---|---|---|---|---|---|---|

0.1129 | 0.2712^{*} | 0.1953 | 0.0282 | −0.0068 | 0.3196^{***} | |

(1.476) | (2.548) | (0.934) | (1.094) | (−0.021) | (13.039) | |

0.0123 | −0.0811^{***} | 0.0037 | 0.0547 | 0.0003 | 0.0973^{***} | |

(0.865) | (7.129) | (0.276) | (1.009) | (1.108) | (3.078) | |

0.0043 | 0.0890^{***} | −0.0001 | 0.0031 | −0.0006 | 0.0674^{***} | |

(1.134) | (4.956) | (−0.127) | (0.027) | (−0.861) | (21.967) | |

0.0001 | −1.5146^{***} | −0.3148 | 1.6880 | 1.2992 | −1.1618^{**} | |

(1.153) | (−3.976) | (−1.423) | (1.438) | (0.923) | (−2.310) | |

0.0111 | −0.1540^{***} | −0.0872^{*} | 0.0492 | 0.0989 | 0.3187 | |

(1.568) | (−1.726) | (−1.753) | (1.545) | (1.368) | (1.243) | |

(0) | (0) | (0) | (0) | |||

−2.9951 | −9.3970 | −13.9373 | −13.2904 | −3.7923 | −10.8337 | |

−5.9892 | −18.7908 | −27.8716 | −26.5777 | −6.5816 | −21.6644 | |

−5.9863 | −18.7822 | −27.8630 | −26.5691 | −6.5730 | −21.6557 |

Note: The numbers in parentheses are t-student. ^{***}, ^{**} and ^{*} indicate statistical significance at 1%, 5% and 10% levels respectively. The symbols ^{−5} (>10^{−5}).

Student | Kuwait | Qatar | Saudi | UAE | Bahrain | Oman |
---|---|---|---|---|---|---|

0.1123 | 0.1907^{**} | 0.1265^{***} | 0.3258^{***} | −0.0136 | 0.0327^{***} | |

(1.012) | (1.966) | (3.368) | (3.255) | (−1.338) | (3.352) | |

0.0130 | 0.0042 | −0.1181 | 0.1097 | −0.0001 | 0.0211 | |

(0.963) | (1.002) | (0.452) | (0.575) | (−0.088) | (1.265) | |

−0.0015 | 0.0138 | 0.1019^{***} | 0.0644 | 0.0001 | 0.0013 | |

(−0.002) | (1.351) | (6.8323) | (1.216) | (0.005) | (0.085) | |

−0.0113 | −0.1794^{**} | −1.4610^{***} | −1.2194^{***} | 1.3090 | 1.6395^{*} | |

(−0.350) | (−2.023) | (−3.679) | (−3.281) | (0.976) | (1.938) | |

−0.0140 | −0.0774 | −0.1312^{***} | 0.2392^{***} | 0.0779 | 0.0394 | |

(−0.388) | (−1.452) | (−4.256) | (3.674) | (1.027) | (1.413) | |

(0) | (0) | (0) | (0) | |||

−2.7906 | −8.6331 | −17.8127 | −16.0150 | −4.1188 | −7.9173 | |

−5.5750 | −17.2631 | −35.6191 | −32.0269 | −8.2364 | −15.8284 | |

−5.5578 | −17.2544 | −35.6019 | −32.0183 | −8.2295 | −15.8112 |

Note: see note of

Gumbel | Kuwait | Qatar | Saudi | UAE | Bahrain | Oman |
---|---|---|---|---|---|---|

−1.0262 | −0.9277 | 1.1383 | 1.4772 | 1.0115^{***} | 1.0475^{***} | |

(−1.222) | (−1.186) | (1.462) | (1.433) | (3.751) | (4.132) | |

0.0172 | 0.2331 | 1.2331 | −0.0032 | −1.0565^{***} | −1.0595^{***} | |

(1.006) | (1.2560) | (1.012) | (−0.742) | (−5.573) | (−5.342) | |

0.0022 | −0.1347 | 0.0033 | 0.0081 | 1.0734^{***} | 1.0778^{***} | |

(0.876) | (−1.001) | (0.544) | (0.008) | (3.259) | (3.776) | |

1.0270^{**} | 1.1706^{***} | −1.5095 | −1.2530^{**} | 0.7296^{***} | 0.5892^{***} | |

(2.325) | (3.431) | (−1.509) | (−2.220) | (3.956) | (3.498) | |

−0.5848 | −0.2424 | 0.3808^{*} | −0.8928^{***} | 1.7593^{***} | −0.3538^{***} | |

(−1.388) | (−0.989) | (−1.876) | (−3.330) | (3.567) | (−3.643) | |

(0) | (0) | (0) | (0) | |||

−2.4893 | −6.5129 | −11.6831 | −15.3473 | −17.1689 | −14.0013 | |

−4.9755 | −13.0228 | −23.3631 | −30.6914 | −34.3369 | −27.9994 | |

−4.9669 | −13.0142 | −23.3544 | −30.6828 | −34.3299 | −27.9887 |

Note: see note of

Clayton | Kuwait | Qatar | Saudi | UAE | Bahrain | Oman |
---|---|---|---|---|---|---|

1.0238^{***} | −2.9515^{**} | −2.6441 | −1.5003 | −2.4787^{***} | −0.7919 | |

(3.585) | (−2.158) | (−1.393) | (−1.419) | (−3.292) | (−0.787) | |

−1.0265^{***} | −2.5471 | 0.6510 | 0.7773^{***} | −0.7541 | ||

(−4.998) | (−1.009) | (1.001) | (4.003) | (−0.135) | ||

1.3089 | 0.0049 | −0.0014 | −0.7420^{***} | 0.0022 | ||

(4.845) | (0.383) | (0.9361) | (−3.576) | (0.005) | ||

−0.4967^{***} | −0.7113^{*} | −0.3183 | 0.4088 | −0.6993^{*} | 0.3341 | |

(3.698) | (−1.667) | (0.417) | (1.165) | (−1.876) | (0.604) | |

1.4058^{***} | −1.0138 | −0.4748 | −0.6240 | 0.8314 | −0.4723 | |

(4.132) | (−0.941) | (−0.268) | (−0.515) | (1.118) | (−0.615) | |

(0) | (0) | (0) | (0) | |||

−12.4932 | −4.7341 | −5.4993 | − 6.1496 | −6.2265 | −10.1203 | |

−25.9086 | −10.4693 | −11.7118 | −12.2642 | −12.4468 | −20.2402 | |

−25.8854 | −10.3838 | −11.6829 | −12.2537 | −12.4295 | −22.2396 |

Note: see note of

copula exhibits three possible state space^{13}.

We find that the Markov-switching dynamic SJC copula gives a better fit for Qatar, Saudi Arabia and UAE, since it exhibits the smallest LL^{14}, AIC and BIC. The Markov-switching dynamic Clayton copula gives a better fit for Kuwait. Bahrain and Oman show the same dependence structure as described by the Markov-switching dynamic Gumbel copula.

For all countries, we see that the estimates of

This section shows how the proposed copula model with Markov-switching dynamic dependence can improve the accuracy of market risk forecasts for an equally weighted energy and stock markets in GCC countries portfolio^{15}. We indeed consider the Value-at-Risk (VaR) as the portfolio’s market risk measure and estimate it using Monte Carlo simulations, instead of the analytical method that is only valid for Gaussian copula models. It is worth noting that when copula functions are used to gauge the dependence structure between two variables, it

SJC | Kuwait | Qatar | Saudi | UAE | Bahrain | Oman |
---|---|---|---|---|---|---|

−1.6768^{***} | 0.0142^{***} | −0.0451^{***} | 0.0106^{***} | −0.5787 | −2.1245 | |

(−7.883) | (17.169) | (−10.422) | (3.890) | (−0.712) | (−0.414) | |

−2.1736 | 0.0061^{***} | −0.0081 | 0.0834^{***} | −0.0766 | −1.9755 | |

(−0.484) | (27.025) | (−0.251) | (5.578) | (−0.187) | (=0.132) | |

0.1308 | 0.0557^{***} | −0.0032 | 0.0341^{***} | 0.0077 | −0.1867 | |

(0.653) | (15.098) | (−0.132) | (3.983) | (0.942) | (−0.883) | |

0.3695 | 0.3278^{***} | 0.4005^{***} | 0.3274^{**} | −1.4722 | 0.4537 | |

(0.953) | (8.123) | (6.257) | (2.3171) | (−1.499) | (0.981) | |

−7.5930^{**} | −4.5970^{***} | −2.9999^{***} | −2.9951^{***} | 5.9937 | −8.7337 | |

(−2.402) | (−8.734) | (−2.7959) | (−4.356) | (0.2523) | (−0.475) | |

−2.6017 | 0.0092^{***} | 0.0312^{***} | 0.0068^{***} | −3.4327 | −3.3229 | |

(−1.059) | (13.618) | (25.629) | (4.051) | (−0.284) | (−0.457) | |

−2.3662 | 0.7902^{***} | 0.0558^{***} | 0.0139^{***} | −2.3662 | −4.3667 | |

(−0.887) | (17.158) | (30.182) | (3.576) | (−0.887) | (−1.002) | |

−1.1647 | 1.0313^{***} | 0.1903^{***} | 0.0055^{***} | −1.1647 | −1.2539 | |

(−1.113) | (15.317) | (32.485) | (3.983) | (−1.113) | (−0.993) | |

7.9710^{*} | 0.7118^{***} | 0.9946^{***} | 0.8998^{***} | 2.6279 | 0.2128 | |

(2.211) | (12.603) | (5.094) | (3.683) | (1.528) | (0.729) | |

−6.7355 | −4.4958^{***} | −2.2832^{***} | −2.9875^{***} | 8.8226 | 9.4365 | |

(−1.349) | (−3.033) | (−3.426) | (−3.322) | (0.111) | (0.049) | |

(0) | (0) | (0) | ||||

−12.1812 | −10.8822 | −18.0976 | −20.2465 | −14.7464 | −11.0141 | |

−24.7622 | −21.6573 | −36.1921 | −40.4867 | −29.4925 | −22.0251 | |

−24.7467 | −21.6414 | −36.1835 | −40.4695 | −29.4879 | −22.0164 |

Note: see note of

is relatively easy to construct and simulate random scenarios from their joint distribution, based on any choice of marginals and any type of dependence structure.

The VaR is a forecast of a given percentile, usually in the lower tail, of the distribution of returns on a portfolio over a given time period. At time t, the VaR of a portfolio, with confidence level

as a result

Our method for computing the VaR requires the following steps. First, we simulate dependent uniform variates from the fitting copula model and transform them into standardized residuals by inverting the semi-parametric marginal Cumulative Distribution Function (CDF) of each index. We then consider the simulated standardized residuals and calculate the returns by reintroducing the FIAPARCH volatility and the ARFIMA parameters observed in the original return series. Finally, given the simulated return series

In order to asses the accuracy of the VaR estimates we backtest the method at 99% and 99.5% confidence levels by the following procedure. We start by estimating the model using the first 1655 observations; then, we simulate 2000 values of the standardized residuals, estimate the VaR and count the number of losses that exceeds the estimated VaR values. This procedure can be repeated until the last observation and we compare the estimated VaR with the actual next-day value change in the portfolio. The whole process is repeated only once in every 75 observations owing to the computational cost of this procedure.

Turning to tail dependence, for OPEC countries, there is evidence of significant low tail dependence between oil price changes and stock market returns^{16}, whereas the non-OPEC stock market returns and oil price changes exhibit upper tail dependence^{17}. The tail dependence indicates extreme co-movements and means that oil price changes and stock market returns crash together in OPEC countries, but boom together in non-OPEC countries. This could possibly be explained by the volatility of the stock market as measured by the standard deviation, since OPEC stock markets present a higher risk degree compared with non-OPEC stock markets (see

Figures 1-6 plot the copula parameters and the probabilities of being in regime 1, 2 and 3 for each pair of oil

Backtest | Proportion | Number | Proportion | Number | |

Kuwait | M-s conditional Clayton | 0.103 | (93) | 0.098 | (88) |

conditional Clayton | 0.119 | (107) | 0.108 | (97) | |

Qatar | M-s conditional SJC | 0.087 | (78) | 0.072 | (65) |

conditional SJC | 0.106 | (95) | 0.098 | (88) | |

Saudi | M-s conditional SJC | 0.113 | (102) | 0.104 | (94) |

conditional SJC | 0.126 | (113) | 0.118 | (106) | |

UAE | M-s conditional SJC | 0.119 | (107) | 0.099 | (89) |

conditional SJC | 0.126 | (113) | 0.109 | (98) | |

Bahrain | M-s conditional Gumbel | 0.072 | (65) | 0.051 | (46) |

conditional Gumbel | 0.079 | (71) | 0.059 | (53) | |

Oman | M-s conditional Gumbel | 0.076 | (68) | 0.054 | (49) |

conditional Gumbel | 0.087 | (78) | 0.062 | (56) |

Notes: This table reports the VaR backtesting results with the number of exceedances is given in brackets.

price changes and stock market returns. For Qatar, Saudi Arabia and the UAE (Figures 2-4), we can see a considerable change of dependence as the copula parameter varies over time. In regime 1 (pre-crisis regime), the tail dependence is relatively low, indicating no tail dependence. In regime 2 (crisis regime), the lower tail dependence drops significantly and positively with a biggest drop for Saudi Arabia. This could be attributed to the fact that Saudi Arabia is the largest oil producer and exporter. In regime 3, the lower tail dependence seems to be to the one found in the first regime. A similar behavior is observed in Kuwait (

These findings suggest some important implications. First, there is a higher dependence structure between oil price changes and all GCC stock market returns during the financial crisis period than during the calm ones (pre-crisis or post-crisis). This result means that the dependence structure between oil price changes and stock market returns is more intensified and suggests the presence of a contagion effect in sense of [

Second, during the financial crisis period, there is evidence of lower tail dependence in OPEC countries whereas in the non-OPEC countries, there is rather evidence of upper tail dependence. These findings have important implications for both investors who are interested in GCC stock markets, and for policymakers. During pre- and post-crisis periods, they can invest in all GCC stock markets to benefit from diversification and to reduce exposure to risk because the series are independent. During the financial crisis period, the investors who include oil as an asset in a diversified portfolio or energy risk managers who consider VaR (or other downside energy risk measures) should be particularly concerned about downside risk exposure and should emphasize the left side of the portfolio return distribution. Indeed, the risk diversification is less effective due to their stronger dependence and the investor must pay attention and the choice of the portfolio is related to whether the oil price is expected to increase or decrease. More precisely, if the oil price is expected to increase, a portfolio of OPEC stock market indices and oil can be better in terms of diversification because the series are not expected to boom together. In contrast, if the oil price is expected to decrease, a portfolio of non-OPEC stock market indices and oil can be preferred because the series are not expected to crash together.

This paper examines the presence of regime change in the dynamic dependence structure between oil price changes and stock market returns in six GCC countries during the period May 25, 2005 to March 31, 2015. In particular, we assume that there are three regime changes corresponding to low, high and crash volatility. The transition from one regime to other is conducted.

The econometric approach adopted is based on two steps. In a first step, we model the marginal distributions using an ARFIMA-FIAPARCH model with skewed-t distribution. We find evidence of dual long-range de- pendence and asymmetric reactions of the conditional variance to positive and negative shocks. In a second step, we focus on the dependence structure between filtered returns series using different Markov-switching time-varying copula functions.

For all countries, we find evidence of three-state Markov-switching regimes corresponding to pre-crisis, financial crisis and post-crisis regimes. More precisely, we see that in pre- and post-crisis regimes, there is no de- pendence. In contrast, in the financial crisis regime, there is a significant tail dependence. In particular, in OPEC countries (Qatar, Saudi Arabia, the UAE and Kuwait), we find lower tail dependence. In non-OPEC countries (Bahrain and Oman), we see upper tail dependence. The dependence structure seems to be related to oil production and consumption as well as to the importance of oil to its national economy. In particular, we find that Saudi Arabia, which is the largest oil producer and exporter and which presents the biggest market capitalization, shows the highest increase in lower tail dependence during the financial crisis period. Simulation results of VaR show that the proposed model outperforms the traditional time-varying copula model. Fur- thermore, these empirical findings are of great interest for investors in order to build profitable investment strategies. The fact that GCC stock market returns have different dependence structures to oil price changes implies valuable risk diversification opportunities across countries.

Heni Boubaker,Nadia Sghaier, (2016) Markov-Switching Time-Varying Copula Modeling of Dependence Structure between Oil and GCC Stock Markets. Open Journal of Statistics,06,565-589. doi: 10.4236/ojs.2016.64048