# Journal of Energy Markets

**ISSN:**

1756-3607 (print)

1756-3615 (online)

**Editor-in-chief:** Derek W. Bunn

# Optimal weights and hedge ratio behavior in Brent oil and Islamic Gulf stock markets

####
Need to know

- The hedging performance between oil and Gulf region Islamic stock markets in investigated.
- Findings show shock effect and volatility transmission between all oil/stock market pairs except for oil/Jordan and oil/UAE pairs.
- The own and cross-weights volatility spillover between time-varying series is analyzed.
- Results show that investors have more information about the volatility of hedged portfolios they own in terms of time-varying weight and hedge ratio.

####
Abstract

This paper examines the dynamics and spillover behavior between time-varying optimal weights and hedge ratios in order to analyze optimal volatility allocation spillover and characteristic structure.We estimate the vector autoregression moving-average-Baba–Engle–Kraft–Kroner-generalized autoregressive conditional heteroscedasticity model of Ling and McAleer based on weights and hedge ratios following the 2017 work of Majdoub and Ben Sassi. Optimal dynamic portfolio compositions are obtained from Brent oil and Gulf region stock markets estimations based on Kroner and Ng’s approach from 1998. The results reveal that our proposed time-varying weights and hedge ratios are dynamic, show active time-varying behavior and have a different performance than the mean value used in all previous studies in order to hedge different portfolios. Our findings are interesting for local and international portfolio holders wishing to more precisely forecast hedged portfolio composition and structure within the framework of international portfolio diversification.

####
Introduction

## 1 Introduction

Both the scope and variety of empirical studies of financial and economic linkages between energy and financial markets have recently increased. A focus on market connectedness, volatility spillover and transmission of shocks provides suitable diversification guidance for portfolio owners. In this paper, we investigate time-varying weights and hedge ratios to show how active the time-varying behavior is and how different its performance is from the mean value used in all previous studies on hedging different portfolios. The optimal time-varying hedge ratio based on different economic issues has been studied by, for example, Conlon and Cotter (2012), Fan et al (2016), Lee et al (2006), Lien (2010), Liu et al (2014), Kim and Park (2016) and Park and Jei (2010). Although they used a variety of different econometric methods and data coverage, these studies showed a common result, which is the existence of return and volatility spillovers from energy markets to stock markets in developed and emerging economies.

Balli et al (2015) studied the volatility spillovers from developed markets to emerging markets over the period 2000–13. Their results show the existence of significant spillover effects, which can be explained in terms of economic factors such as bilateral trade. Awartani et al (2018) examined the linkage between oil and five Middle East and North Africa region countries over the period 2006–17 by using the dynamic conditional correlation-mixed data sampling (DCC-MIDAS) model. They show that high oil prices have a positive impact on stock markets. Indeed, the increase in oil price returns tends to reduce the long-term risk of the Saudi market in particular. The linkage increases during periods of excess volatility and distress.

The importance of the study of the linkage between the oil market and stock markets stems from the informational content that may affect equity values. Indeed, the fluctuation of oil prices affects firms’ dynamics in terms of growth, survival, profit generation and business prospects. Households may also see their net worth affected, depending on whether they live in an oil-producing or an oil-consuming economy. In addition, an economy’s fundamentals can be influenced by positive or negative oil shocks that affect monetary, budgetary and fiscal policies. For all these reasons there exists an interrelationship between stock markets and oil price. Various empirical studies have explored the different forms of such interrelationships, which are documented as either positive (see, for example, Basher and Sadorsky 2006; Choi and Hammoudeh 2010) or negative (see, for example, Huang et al 1996). Most of these studies link their results to aspects of investors’ behavior in terms of diversification of their portfolio positions.

The focus on the study of Islamic marketplaces stems from their specificities in terms of financial transactions, business practices and ethical conduct. Indeed, Islamic finance practices are based on the principles of nonuse of interest as an incentive mechanism in contracting, the necessity of using the asset-backed principle and the prohibition of all transactions that lead to zero-sum games. Various studies have explored whether such specificities can render Islamic marketplaces more resilient during periods of distress and economic troubles. There are two main streams of research findings. The first supports the decoupling hypothesis and shows that the transmission of shocks and the volatility spillover between conventional and Islamic markets are both alleviated. The second supports the opposite result by documenting that Islamic marketplaces are strongly connected with their conventional counterparts.^{1}^{1} 1 See Majdoub and Mansour (2014) and Majdoub et al (2016) for a review of empirical results, and Mansour et al (2015) and Mansour and Bhatti (2018) for a review of Islamic finance’s specificities.

Majdoub et al (2018, p. 27) examined the volatility spillover between Islamic equity markets from the Gulf region and oil prices using a sample of five countries from the region. The results show that there is a reduction in the volatility spillover, particularly for the Saudi market. This can be interpreted, in our opinion, in terms of the distinguishing features of the Islamic financial intermediation model, which is better able to alleviate the transmission of shocks to domestic markets and ensure a better stability than non-Islamic financial markets. The new structure of volatility spillover has important implications for international investors with respect to portfolio diversification benefits and for financial policy makers regarding contagion risks and portfolio allocation policies. Other studies focus on different regions of the world, such as developed versus developing countries. For instance, Bhatia and Mitra (2018), Yang et al (2018), Bhuyan et al (2016), Hammoudeh et al (2016), Mensi et al (2016a,b), Yarovay and Lau (2016) and Zhang et al (2013) studied Brazil, Russia, India, China and South Africa (BRICS) and developed stock markets. Bhar and Nikolova (2009) examined the integration and dynamic relationship between BRICS countries and the rest of the world and found that India shows the highest regional and global integration, followed by Brazil and Russia and lastly by China. Mensi et al (2014) investigated the dynamics of the dependence structure between the emerging stock markets of the BRICS countries and global factors from September 1997 to September 2013 based on the quantile regression approach. They found that the BRICS stock markets exhibited dependence on the global stock and commodity markets (Standard & Poor’s index, oil and gold). Andrikopoulos et al (2014) found that international stock markets are characterized by persistent illiquidity and significant correlation shocks across markets. Their results show Granger causality between risk, return and illiquidity both across Group of Seven (G7) stock markets and within each stock market. Golosnoy et al (2015) investigated intraday volatility spillovers within and across the US, German and Japanese stock markets before and during the subprime crisis using a conditional autoregressive model framework for realized variance and covariance. They found significant short-term spillovers from one stock market to another. Antonakakis and Badinger (2016) examined the linkages between growth and volatility in the G7 countries over a long period (1958–2013) using a vector autoregression approach. They found a high volatility spillover between the US and G7 markets during the global financial crisis. The United States has been found to be the largest transmitter of volatility shocks.

The aim of this paper is to explore time-varying optimal weights and the hedge ratio spillover structure as well as to investigate the dynamics of the relationship between the time-varying cross-weights and hedge ratios spillover in order to better understand a portfolio holder’s decision-making behavior. It contributes to the literature by proposing a new framework to model the dynamic relationship between return, weight and hedge ratios based on the concepts of volatility spillover. In order to do this, we use daily data on oil price and Islamic market returns in the Gulf region spanning the period from February 21, 2011 to February 16, 2016. This paper is organized as follows. Section 2 provides the methodology. Section 3 describes the data and preliminary analysis. Section 4 presents the interpretation of results and policy implications. Section 5 gives our conclusions.

## 2 Methodology

Multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) models seem to be very useful in studying volatility spillover effects between equity markets (see, for example, Golab et al 2014; Jin 2015; Malik and Hammoudeh 2007). In this paper, we apply the vector autoregression moving-average-Baba–Engle–Kraft–Kroner-GARCH (VARMA(1)-BEKK-GARCH(1,1)) model developed by Ling and McAleer (2003) to investigate the volatility spillover between optimal time-varying weights and hedge ratios series following Kroner and Ng (1998). Hammoudeh et al (2009), Rahman and Serletis (2012), Sadorsky (2014), Arouri et al (2015) and Majdoub and Ben Sassi (2017), among others, have applied symmetric and asymmetric versions of this model to various economic issues and find it to be the best fitting model to investigate the volatility spillover between selected markets.

We first estimate five models between crude oil prices and Morgan Stanley Capital International (MSCI) equity market indexes from the Gulf region. We initially specify a bivariate VARMA(1)-BEKK-GARCH(1,1) model under separate headings for the conditional mean and conditional variance equations. Then, following Kroner and Ng (1998), we compute the optimal weight and hedge ratio series for crude oil and each stock market portfolio. Finally, we estimate all models using time-varying optimal weights to assess the dynamics between them, based on the multivariate VARMA-BEKK-GARCH model, in order to analyze their spillover behavior.

### 2.1 Return and volatility spillover index methodology

#### 2.1.1 The bivariate VARMA(1)-BEKK-GARCH(1,1) model

The conditional mean equation and conditional variance equation are expressed as follows:

$$\begin{array}{cc}\hfill {R}_{t}& =\mu +\mathrm{\Omega}{R}_{t-1}+{\epsilon}_{t}+\mathrm{\Phi}{\epsilon}_{t-1},\hfill \\ \hfill {\epsilon}_{t}& ={D}_{t}{\eta}_{t},\hfill \end{array}\}$$ | (2.1) |

where we have the following: ${R}_{t}=({r}_{t}^{\mathrm{o}},{r}_{t}^{i})$, with ${r}_{t}^{\mathrm{o}}$ and ${r}_{t}^{i}$ the returns on crude oil price and each stock market MSCI index return, respectively, for $i=(1,2,3,4,5)\equiv (\text{Jordan},\text{Kuwait},\text{Oman},\text{Qatar},\text{UAE})$; $\mu $ is a $(2\times 1)$ vector of constant terms of the form $({\mu}^{\mathrm{o}},{\mu}^{i})$; $\mathrm{\Omega}$ is a $(2\times 2)$ matrix of coefficients of the form

$$\mathrm{\Omega}=\left(\begin{array}{cc}\hfill {\omega}_{11}\hfill & \hfill {\omega}_{12}\hfill \\ \hfill {\omega}_{21}\hfill & \hfill {\omega}_{22}\hfill \end{array}\right);$$ |

${\epsilon}_{t}=({\epsilon}_{t}^{\mathrm{o}},{\epsilon}_{t}^{i})$, where ${\epsilon}_{t}^{\mathrm{o}}$ and ${\epsilon}_{t}^{i}$ are the residual terms of the mean equations for the oil price and each stock market return of the Gulf region under consideration, respectively; $\mathrm{\Phi}$ is a $(2\times 2)$ matrix of the coefficients of lagged terms of residuals in the form

$$\mathrm{\Phi}=\left(\begin{array}{cc}\hfill {\phi}_{ii}\hfill & \hfill {\phi}_{ij}\hfill \\ \hfill {\phi}_{ji}\hfill & \hfill {\phi}_{jj}\hfill \end{array}\right),$$ |

indicating the shock spillovers between the crude oil return $(i)$ and each stock market return $(j)$ from the Gulf region, respectively; ${\eta}_{t}=({\eta}_{t}^{\mathrm{o}},{\eta}_{t}^{i})$ refers to a sequence of independently and identically distributed (iid) random vectors; and ${D}_{t}=\mathrm{diag}(\sqrt{{h}_{t}^{\mathrm{o}}},\sqrt{{h}_{t}^{i}})$, where ${h}_{t}^{\mathrm{o}}$ and ${h}_{t}^{i}$ are, respectively, the conditional variances of ${r}_{t}^{\mathrm{o}}$ and ${r}_{t}^{i}$.

#### 2.1.2 The conditional variance equation

The conditional variance is defined by the following equation:

$${H}_{t}={\mathrm{\Theta}}^{\prime}\mathrm{\Theta}+{A}^{\prime}{\epsilon}_{t-1}{\epsilon}_{t-1}^{\prime}A+{B}^{\prime}{H}_{t-1}B,$$ | (2.2) |

where $A$ and $B$ are square matrixes and $\mathrm{\Theta}$ is a lower triangular matrix, defined as

$$A=\left(\begin{array}{cc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill \end{array}\right),B=\left(\begin{array}{cc}\hfill {b}_{11}\hfill & \hfill {b}_{12}\hfill \\ \hfill {b}_{21}\hfill & \hfill {b}_{22}\hfill \end{array}\right),\mathrm{\Theta}=\left(\begin{array}{cc}\hfill {\theta}_{11}\hfill & \hfill 0\hfill \\ \hfill {\theta}_{21}\hfill & \hfill {\theta}_{22}\hfill \end{array}\right).$$ |

As indicated by (2.2), ${H}_{t}$ is the conditional variance–covariance matrix, which defines market volatility. The elements of matrix $A$ are the coefficients of ARCH terms that reflect the effect of shock in the own market and shock spillover from other markets on the conditional volatility of a selected market. The elements of matrix $B$ are the coefficients of GARCH terms that reflect the effect of past volatility in the own market and past volatility spillover from the other market on the conditional volatility of a selected market. The ARCH terms represent the short-term persistence volatility, while the GARCH terms represent the long-term persistence volatility, given the autoregressive nature of conditional volatility. The sum of the ARCH and GARCH terms is expected to be positive and less than unity in order to satisfy the mean-reverting condition. In addition, the magnitude of the sum of the ARCH and GARCH terms for a particular market determines the speed of convergence of the conditional volatility in a given market to its long-run equilibrium. The structural and statistical properties of the model, including the necessary and sufficient conditions for stationarity and ergodicity of VARMA-GARCH, are explained in detail in Ling and McAleer (2003).

The resulting variance and covariance equations for the estimates of bivariate VARMA-BEKK-GARCH can be expressed as follows:

${h}_{11,t}$ | $={\theta}_{11}^{2}+{a}_{11}^{2}{\epsilon}_{1,t-1}^{2}+{a}_{21}^{2}{\epsilon}_{2,t-1}^{2}+2{a}_{11}{a}_{21}{\epsilon}_{1,t-1}{\epsilon}_{2,t-1}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{b}_{11}^{2}{h}_{11,t-1}+{b}_{21}^{2}{h}_{22,t-1}+2{b}_{11}{b}_{21}{h}_{21,t-1},$ | (2.3) | |||

${h}_{22,t}$ | $={\theta}_{22}^{2}+{a}_{12}^{2}{\epsilon}_{1,t-1}^{2}+{a}_{22}^{2}{\epsilon}_{2,t-1}^{2}+2{a}_{12}{a}_{21}{\epsilon}_{1,t-1}{\epsilon}_{2,t-1}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{b}_{22}^{2}{h}_{22,t-1}+{b}_{12}^{2}{h}_{11,t-1}+2{b}_{22}{b}_{21}{h}_{21,t-1},$ | (2.4) | |||

${h}_{21,t}$ | $={\theta}_{21}{\theta}_{22}+{a}_{11}{a}_{22}{\epsilon}_{1,t-1}^{2}+{a}_{21}{a}_{22}{\epsilon}_{2,t-1}^{2}+({a}_{21}{a}_{12}+{a}_{11}{a}_{22}){\epsilon}_{1,t-1}{\epsilon}_{2,t-1}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{b}_{11}{b}_{22}{h}_{11,t-1}+{b}_{21}{b}_{22}{h}_{22,t-1}+({b}_{21}{b}_{12}+{b}_{11}{b}_{22}){h}_{21,t-1}.$ | (2.5) |

### 2.2 Time-varying weight estimation methodology

We estimate five models based on the multivariate VARMA-BEKK-GARCH to assess the dynamics between the optimal daily time-varying weights and the hedge ratio series for oil and each of the stock market portfolios, in order to analyze the own and cross-weights volatility spillover.

Following Kroner and Ng (1998), the portfolio optimal weight of the oil and stock markets is given by

$$ | (2.6) |

where ${w}_{t}^{\mathrm{o}i}$ refers to the weight of crude oil per USD1 of the two assets defined at time $t$; ${h}_{t}^{i}$ and ${h}_{t}^{\mathrm{o}}$ are the conditional variances of the MSCI Islamic stock market index for the $i$th market and crude oil, respectively; and ${h}_{t}^{\mathrm{o}i}$ is the conditional covariance between crude oil and each stock market at time $t$.

The weights of the stock market MSCI index in the portfolios we consider are given by $(1-{w}_{t}^{\mathrm{o}i})$. The investor’s objective is to optimally hedge the risk of their investment in oil; they should take an appropriate position on the stock market so that it minimizes the risk of the hedged portfolio. Concretely, a long position (buying) of USD1 on oil must be hedged by a short position (selling) of ${\beta}_{t}^{\mathrm{o}i}$ US dollars on stock market $i$.

Following Kroner and Sultan (1993), the optimal hedge ratio ${\beta}_{t}^{\mathrm{o}i}$ can be expressed as

$${\beta}_{t}^{\mathrm{o}i}=\frac{{h}_{t}^{\mathrm{o}i}}{{h}_{t}^{\mathrm{o}}}.$$ | (2.7) |

#### 2.2.1 The conditional mean equation

The conditional mean equation can be written as

$$\begin{array}{cc}\hfill {W}_{t}& =\mu +\mathrm{\Omega}{w}_{t-1}+{\epsilon}_{t}+\mathrm{\Phi}{\epsilon}_{t-1},\hfill \\ \hfill {\epsilon}_{t}& ={D}_{t}{\eta}_{t},\hfill \end{array}\}$$ | (2.8) |

where we have the following: ${W}_{t}={({w}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$, ${w}_{t}^{\mathrm{o}i}$ are the optimal weights on oil and on each stock market $i$ obtained from the first step estimation following Kroner and Ng (1998); $i=(1,2,3,4,5)\equiv (\text{Jordan},\text{Kuwait},\text{Oman},\text{Qatar},\text{UAE})$; $\mu $ is a $(5\times 1)$ vector of constant terms of the form ${\mu}_{t}={({\mu}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$; $\mathrm{\Omega}$ is a $(5\times 5)$ matrix of coefficients of the form

$$\mathrm{\Omega}=\left(\begin{array}{ccc}\hfill {\omega}_{11}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\omega}_{1i}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {\omega}_{j1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\omega}_{ji}\hfill \end{array}\right);$$ |

${\epsilon}_{t}={({\epsilon}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$, where ${\epsilon}_{t}^{\mathrm{o}i}$ are the residual terms of the mean equation for ${({w}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$ optimal weights; $\mathrm{\Phi}$ is a $(5\times 5)$ matrix of the coefficients of lagged terms of residuals in the form

$$\mathrm{\Phi}=\left(\begin{array}{ccc}\hfill {\phi}_{11}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\phi}_{1i}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {\phi}_{j1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\phi}_{ji}\hfill \end{array}\right),$$ |

which explains the shock spillovers between each ${({w}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$; ${\eta}_{t}={({\eta}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$ denotes a sequence of iid random vectors; and

$${D}_{t}=\mathrm{diag}{(\sqrt{{h}_{t}^{{W}^{\mathrm{o}i}}})}_{i=1,\mathrm{\dots},5},$$ |

where ${h}_{t}^{{W}^{\mathrm{o}i}}$ denotes the conditional variances of ${({w}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$.

#### 2.2.2 The conditional variance equation

The conditional variance equation can be written as

$${H}_{t}={\mathrm{\Theta}}^{\prime}\mathrm{\Theta}+{A}^{\prime}{\epsilon}_{t-1}{\epsilon}_{t-1}^{\prime}A+{B}^{\prime}{H}_{t-1}B,$$ | (2.9) |

where $A$ and $B$ are square matrixes and $\mathrm{\Theta}$ is a lower diagonal matrix, defined as

$$A=\left(\begin{array}{ccc}\hfill {a}_{11}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{1i}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{j1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{ji}\hfill \end{array}\right),B=\left(\begin{array}{ccc}\hfill {b}_{11}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {b}_{1i}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {b}_{j1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {b}_{ji}\hfill \end{array}\right),\mathrm{\Theta}=\left(\begin{array}{ccc}\hfill {\theta}_{11}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {\theta}_{j1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\theta}_{ji}\hfill \end{array}\right).$$ |

The resulting variance and covariance equations for the estimates of multivariate VARMA-BEKK-GARCH measure the own volatility for ${({w}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$ and the volatility spillover between all ${({w}_{t}^{\mathrm{o}i})}_{i=1,\mathrm{\dots},5}$ for all weight allocations.

## 3 Data description and preliminary analysis

We consider the Islamic MSCI index for five countries in the Gulf region (Jordan, Kuwait, Oman, Qatar and the United Arab Emirates (UAE)) and Brent crude oil stock over the period from February 7, 2011 to February 5, 2016 (thus yielding 1305 daily observations). The data is obtained from the MSCI database and expressed in US dollars to preserve homogeneity across equity markets and to avoid currency risk effects. We do not use additional historical data for series to exclude the effect of the 2007–8 global financial crisis in our study. Some empirical results are not reported in our paper due to space constraints but are available upon request from the corresponding author.

Tables 1 and 2 present the descriptive statistics and preliminary results of our data sample. All return series show similar sample means that are close to zero for all stock indexes. Moreover, the distributions of stock indexes are negatively skewed for Brent oil, Jordan and Kuwait, and positively skewed for the rest. All series are leptokurtic because of the excess of kurtosis coefficients beyond 2. The deviation compared with the normal distribution is confirmed by the Jarque–Bera (JB) test. The ARCH Lagrange multiplier (LM) tests show strong evidence of ARCH effects for one, five and ten lags. On the other hand, the Ljung–Box test shows a significant serial autocorrelation for stock indexes for all series at a 1% level of significance up to ten lags. The Ljung–Box $Q$-statistic test implies there is a statistically significant autocorrelation in return series up to ten lags for all countries except Qatar. Figures 1 and 2 show the dynamics of stock indexes and returns, respectively.

Brent | Jordan | Kuwait | Oman | Qatar | UAE | |

Observations | 1 302 | 1 302 | 1 302 | 1 302 | 1 302 | 1 302 |

Mean | 94.847 | 390.729 | 346.669 | 931.972 | 1 546.743 | 333.445 |

Median | 108.025 | 365.463 | 357.055 | 932.366 | 1 391.085 | 329.772 |

Maximum | 128.140 | 579.703 | 490.542 | 1 086.888 | 2 413.405 | 559.395 |

Minimum | 26.010 | 205.881 | 200.542 | 788.624 | 1 155.235 | 164.432 |

SD | 26.522 | 119.158 | 52.080 | 69.731 | 297.813 | 105.684 |

CV | 0.280 | 0.305 | 0.150 | 0.075 | 0.193 | 0.317 |

Skewness | $-$1.097 | $-$0.057 | $-$0.412 | 0.128 | 0.971 | 0.276 |

Kurtosis | $-$0.298 | $-$1.670 | 0.576 | $-$1.059 | $-$0.280 | $-$1.205 |

JB | 266.024${}^{***}$ | 151.945${}^{***}$ | 54.796${}^{***}$ | 64.331${}^{***}$ | 208.726${}^{***}$ | 95.283${}^{***}$ |

LB-$Q$(1) | 1 295.361${}^{***}$ | 1 297.228${}^{***}$ | 1 287.394${}^{***}$ | 1 286.418${}^{***}$ | 1 297.190${}^{***}$ | 1 299.647${}^{***}$ |

LB-$Q$(5) | 6 385.983${}^{***}$ | 6 422.899${}^{***}$ | 6 267.024${}^{***}$ | 6 232.752${}^{***}$ | 6 409.017${}^{***}$ | 6 450.228${}^{***}$ |

LB-$Q$(10) | 12 542.223${}^{***}$ | 12 706.124${}^{***}$ | 12 140.820${}^{***}$ | 11 911.408${}^{***}$ | 12 626.529${}^{***}$ | 12 772.452${}^{***}$ |

ARCH LM test(1) | 1 293.188${}^{***}$ | 1 266.726${}^{***}$ | 1 283.458${}^{***}$ | 1 248.143${}^{***}$ | 1 267.644${}^{***}$ | 1 269.035${}^{***}$ |

ARCH LM test(5) | 1 289.310${}^{***}$ | 1 263.291${}^{***}$ | 1 279.874${}^{***}$ | 1 244.689${}^{***}$ | 1 264.603${}^{***}$ | 1 265.413${}^{***}$ |

ARCH LM test(10) | 1 284.416${}^{***}$ | 1 259.809${}^{***}$ | 1 277.455${}^{***}$ | 1 241.043${}^{***}$ | 1 260.448${}^{***}$ | 1 262.272${}^{***}$ |

Brent | Jordan | Kuwait | Oman | Qatar | UAE | |

Observations | 1 301 | 1 301 | 1 301 | 1 301 | 1 301 | 1 301 |

Mean | $-$0.091 | $-$0.076 | $-$0.056 | $-$0.009 | 0.002 | 0.015 |

Median | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Maximum | 9.896 | 9.594 | 10.267 | 8.170 | 11.606 | 11.319 |

Minimum | $-$8.245 | $-$15.475 | $-$7.076 | $-$7.700 | $-$8.347 | $-$11.183 |

SD | 1.770 | 1.554 | 1.098 | 0.860 | 1.256 | 1.780 |

CV | $-$19.347 | $-$20.357 | $-$19.559 | $-$95.303 | 507.970 | 118.466 |

Skewness | 0.166 | $-$0.259 | 0.043 | $-$1.098 | 0.164 | $-$0.128 |

Kurtosis | 4.102 | 13.003 | 9.998 | 25.693 | 14.298 | 7.292 |

JB | 917.939${}^{***}$ | 9 179.978${}^{***}$ | 5 418.868${}^{***}$ | 36 045.516${}^{***}$ | 11 087.363${}^{***}$ | 2 886.051${}^{***}$ |

LB-$Q$(1) | 7.532${}^{***}$ | 0.165 | 3.564${}^{*}$ | 0.018 | 0.000 | 0.350 |

LB-$Q$(5) | 12.767${}^{**}$ | 7.754 | 6.922 | 26.584${}^{***}$ | 5.660 | 6.054 |

LB-$Q$(10) | 17.817${}^{*}$ | 21.329${}^{**}$ | 15.077 | 39.863${}^{***}$ | 10.415 | 16.454${}^{*}$ |

ARCH LM test(1) | 36.127${}^{***}$ | 9.287${}^{***}$ | 2.786${}^{*}$ | 14.533${}^{***}$ | 18.425${}^{***}$ | 8.322${}^{***}$ |

ARCH LM test(5) | 79.396${}^{***}$ | 16.211${}^{***}$ | 101.758${}^{***}$ | 99.414${}^{***}$ | 51.551${}^{***}$ | 165.291${}^{***}$ |

ARCH LM test(10) | 133.573${}^{***}$ | 23.016${}^{***}$ | 109.565${}^{***}$ | 123.246${}^{***}$ | 68.999${}^{***}$ | 186.300${}^{***}$ |

Table 3 presents the descriptive statistics of the weight series. All series show a similar sample mean for all weight series in the range 0.4–0.8. The empirical distributions of all weights are negatively skewed, except for Qatar. The Kuwait series is the only leptokurtic one, as the associated excess kurtosis coefficient is greater than 2. The deviation compared with the normal distribution is confirmed by the Jarque–Bera test. The ARCH LM tests indicate strong evidence of ARCH effects for one, five and ten lags. Unit root tests, not reported, indicate that each weight series is stationary. On the other hand, all weight series reject the null hypothesis of no ARCH effects at a 1% level of significance until ten lags. In addition, the Ljung–Box $Q$-statistic test indicates that there is statistically significant autocorrelation in weight series up to ten lags for all countries. Tables 4 and 5 show the empirical unconditional correlations between the Gulf countries and Brent oil and the empirical unconditional correlations between weights, respectively.

Brent | Jordan | Kuwait | Oman | Qatar | |

Observations | 1 300 | 1 300 | 1 300 | 1 300 | 1 300 |

Mean | 0.537 | 0.691 | 0.788 | 0.705 | 0.486 |

Median | 0.560 | 0.709 | 0.817 | 0.789 | 0.489 |

Maximum | 0.983 | 0.989 | 1.000 | 1.000 | 0.962 |

Minimum | 0.017 | 0.164 | 0.000 | 0.008 | 0.021 |

SD | 0.244 | 0.165 | 0.149 | 0.253 | 0.205 |

CV | 0.454 | 0.239 | 0.189 | 0.359 | 0.422 |

Skewness | $-$0.316 | $-$0.471 | $-$1.581 | $-$0.941 | 0.062 |

Kurtosis | $-$0.950 | $-$0.537 | 3.881 | $-$0.072 | $-$0.747 |

JB | 70.480${}^{***}$ | 63.606${}^{***}$ | 1 357.294${}^{***}$ | 192.015${}^{***}$ | 31.042${}^{***}$ |

LB-$Q$(1) | 1 278.428${}^{***}$ | 1 210.835${}^{***}$ | 1 049.757${}^{***}$ | 1 276.800${}^{***}$ | 1 253.465${}^{***}$ |

LB-$Q$(5) | 6 145.501${}^{***}$ | 5 317.641${}^{***}$ | 3 681.941${}^{***}$ | 6 138.919${}^{***}$ | 5 825.851${}^{***}$ |

LB-$Q$(10) | 11 717.167${}^{***}$ | 9 354.801${}^{***}$ | 5 237.838${}^{***}$ | 11 645.258${}^{***}$ | 10 718.192${}^{***}$ |

ARCH LM test(1) | 1 242.117${}^{***}$ | 1 015.879${}^{***}$ | 776.670${}^{***}$ | 1 262.172${}^{***}$ | 1 208.355${}^{***}$ |

ARCH LM test(5) | 1 239.082${}^{***}$ | 1 014.036${}^{***}$ | 797.061${}^{***}$ | 1 258.512${}^{***}$ | 1 206.944${}^{***}$ |

ARCH LM test(10) | 1 234.789${}^{***}$ | 1 020.512${}^{***}$ | 766.858${}^{***}$ | 1 254.500${}^{***}$ | 1 202.607${}^{***}$ |

(a) Level series | ||||||
---|---|---|---|---|---|---|

Brent | Jordan | Kuwait | Oman | Qatar | UAE | |

Brent | 1 | 0.747 | 0.800 | $-$0.201 | $-$0.328 | $-$0.441 |

Jordan | 1 | 0.574 | $-$0.648 | $-$0.712 | $-$0.845 | |

Kuwait | 1 | 0.039 | $-$0.065 | $-$0.152 | ||

Oman | 1 | 0.810 | 0.835 | |||

Qatar | 1 | 0.910 | ||||

UAE | 1 | |||||

(b) Returns | ||||||

Brent | Jordan | Kuwait | Oman | Qatar | UAE | |

Brent | 1 | 0.0184 | 0.0598 | 0.0288 | 0.1123 | 0.1157 |

Jordan | 1 | 0.0138 | 0.0619 | 0.1036 | 0.1074 | |

Kuwait | 1 | 0.2192 | 0.2960 | 0.3291 | ||

Oman | 1 | 0.3149 | 0.3350 | |||

Qatar | 1 | 0.5342 | ||||

UAE | 1 |

Jordan | Kuwait | Oman | Qatar | UAE | |
---|---|---|---|---|---|

Jordan | 1 | 0.784 | 0.463 | 0.593 | 0.787 |

Kuwait | 1 | 0.579 | 0.627 | 0.780 | |

Oman | 1 | 0.493 | 0.527 | ||

Qatar | 1 | 0.635 | |||

UAE | 1 |

## 4 Empirical results and discussion

Table 6 reports the estimates of the VARMA-BEKK-GARCH model and presents the estimation results between Brent and each Gulf market.^{2}^{2} 2 The Akaike information criterion (AIC) suggests the same number of lags. We have also explored the robustness of our results by increasing the number of lags up to ten; our results remained qualitatively unchanged. These results are available from the authors upon request. Using the model selection criteria, we find that the VARMA-BEKK-GARCH model outperforms the constant conditional correlation and dynamic conditional correlation models. The mean equation provides the characteristics for the own series returns and spillovers returns. The variance equation gives the volatility effect for all market pairs.

(a) Mean equation | |||||

Brent | Brent | Brent | Brent | Brent | |

Jordan | Kuwait | Oman | Qatar | UAE | |

${\omega}_{\text{11}}$ | 0.047 | 0.046${}^{*}$ | 0.059${}^{**}$ | 0.035 | 0.041 |

${\omega}_{\text{12}}$ | $-$0.023 | $-$0.007 | $-$0.020 | $-$0.028 | $-$0.026 |

${\mu}_{\text{10}}$ | $-$0.067 | $-$0.051 | $-$0.060 | $-$0.054 | $-$0.028 |

${\phi}_{\text{11}}$ | 0.416${}^{***}$ | 0.034${}^{***}$ | 0.005${}^{***}$ | 0.057${}^{***}$ | 0.014${}^{***}$ |

${\phi}_{\text{12}}$ | 0.092${}^{***}$ | 0.019${}^{***}$ | $-$0.012${}^{***}$ | $-$0.067${}^{***}$ | $-$0.067${}^{***}$ |

${\omega}_{\text{21}}$ | 0.030 | 0.041${}^{***}$ | 0.047${}^{***}$ | 0.055${}^{***}$ | 0.122${}^{***}$ |

${\omega}_{\text{22}}$ | $-$0.027 | $-$0.102${}^{***}$ | 0.059 | 0.021 | 0.002 |

${\mu}_{\text{20}}$ | $-$0.058 | $-$0.057${}^{**}$ | $-$0.011 | 0.026 | 0.067 |

${\phi}_{\text{21}}$ | 0.003${}^{***}$ | $-$0.092${}^{***}$ | $-$0.046${}^{***}$ | 0.047${}^{***}$ | 0.005${}^{***}$ |

${\phi}_{\text{22}}$ | 0.298${}^{***}$ | $-$0.015${}^{***}$ | $-$0.013${}^{***}$ | $-$0.085${}^{***}$ | 0.131${}^{***}$ |

(b) Variance equation | |||||

Brent | Brent | Brent | Brent | Brent | |

Jordan | Kuwait | Oman | Qatar | UAE | |

${\theta}_{\text{11}}$ | 0.077${}^{***}$ | 0.131${}^{***}$ | 0.074${}^{***}$ | 0.050 | 0.079${}^{***}$ |

${\theta}_{\text{21}}$ | 0.008 | $-$0.348${}^{***}$ | 0.255${}^{***}$ | $-$0.048${}^{***}$ | $-$0.490${}^{***}$ |

${\theta}_{\text{22}}$ | 0.228${}^{***}$ | 0.000 | 0.000 | 0.000 | 0.000 |

${a}_{\text{11}}$ | 0.195${}^{***}$ | 0.208${}^{***}$ | 0.136${}^{***}$ | 0.174${}^{***}$ | 0.179${}^{***}$ |

${a}_{\text{12}}$ | $-$0.004 | $-$0.110${}^{***}$ | $-$0.104${}^{***}$ | $-$0.018${}^{*}$ | 0.035 |

${a}_{\text{21}}$ | 0.009 | 0.017 | 0.135${}^{***}$ | $-$0.032 | $-$0.012 |

${a}_{\text{22}}$ | 0.258${}^{***}$ | 0.322${}^{***}$ | 0.465${}^{***}$ | 0.313${}^{***}$ | 0.284${}^{***}$ |

${b}_{\text{11}}$ | 0.981${}^{***}$ | 0.976${}^{***}$ | 0.988${}^{***}$ | 0.985${}^{***}$ | 0.982${}^{***}$ |

${b}_{\text{12}}$ | 0.000 | 0.025${}^{***}$ | 0.016${}^{***}$ | 0.003 | 0.003 |

${b}_{\text{21}}$ | $-$0.002 | 0.035${}^{*}$ | $-$0.092${}^{***}$ | 0.011${}^{**}$ | 0.015${}^{**}$ |

${b}_{\text{22}}$ | 0.957${}^{***}$ | 0.873${}^{***}$ | 0.832${}^{***}$ | 0.958${}^{***}$ | 0.913${}^{***}$ |

(c) Model diagnostics | |||||

Brent | Brent | Brent | Brent | Brent | |

Jordan | Kuwait | Oman | Qatar | UAE | |

AIC | 7.262 | 6.577 | 5.964 | 6.440 | 7.525 |

SBC | 7.346 | 6.660 | 6.047 | 6.523 | 7.608 |

LOG-L | 7.262 | 6.577 | 5.964 | 6.440 | 7.525 |

Observations | 1300 | 1300 | 1300 | 1300 | 1300 |

### 4.1 Spillover returns effect

Summarizing the rich information given by the mean equation results shown in Table 6, we can identify that the Brent oil return does not spill over for all Gulf countries, suggesting there is no short-term predictability between Brent oil and all the Gulf markets under consideration. These features agree with those deduced from the dynamics of the Islamic market index and returns plotted in Figures 1 and 2. The coefficients ${\omega}_{12}$ are not significant for all models from Brent oil to stock market returns, but the coefficients ${\omega}_{21}$ are significant and positive for all models, except for Jordan because it is not an oil-exporting country. Our results show unidirectional shock spillover from the Gulf Islamic markets to Brent oil for all models. Our findings show that an increase in the Gulf Islamic markets’ returns positively affects the Brent oil return but not vice versa. This can be explained by the impact of the conditions of Islamic stock markets in the oil-producing countries on Brent oil returns and oil consumption for both producing and nonproducing countries. Such an insight in terms of Islamic stock markets differs from recent findings in the literature between developed and emerging conventional stock markets (Balli et al 2015; Boubaker and Jouini 2014; Hammoudeh et al 2016; Mensi et al 2016b). Our results are interesting from the perspective of international diversification for investors investing in Brent oil. Further, the UAE and Qatar Islamic market returns exhibit the highest shock transmission to Brent oil markets. Jordan exhibits the lowest and insignificant spillover. Our findings, which are partly supported by Mensi et al (2016a), show that crude oil responds positively to four of the five Gulf Islamic markets.

### 4.2 Volatility spillover effect

Table 6 reports the shock and volatility spillover effects between Brent and the Gulf Islamic stock markets under consideration. Our results reveal ARCH effect shocks and volatility transmissions between all oil/market pairs except for Brent/Jordan and Brent/UAE. The coefficient ${a}_{12}$ is statistically significant and positive except for the Jordan and UAE Islamic stock markets. Such results can be explained by the fact that Jordan is a non oil-producing country and the diversification of the UAE economy is based more intensively on services and capital investment, which explains this behavior in comparison with the other Gulf region Islamic markets. Our results show a negative shock spillover effect on the conditional volatility from Brent oil to Kuwait, Qatar and UAE. In the same context, the coefficient ${a}_{21}$ is statistically significant only for the Oman Islamic stock market, which explains the conditional volatility spillover effect from Oman to Brent crude oil. Such results are interpreted in terms of a bidirectional volatility spillover between Oman and Brent oil.

The coefficients ${a}_{11}$ and ${a}_{22}$ are positive and significant for all models, indicating short-term volatility persistence (ARCH effect) in the Brent oil price and all Gulf Islamic markets. These insights can be further understood through the plots of conditional volatility of all markets in Figure 3. This current integration can be explained by financial causal links and comovement between Brent oil and Gulf Islamic markets. Our results regarding the volatility spillovers structure can help portfolio holders to design effective hedging strategies.

From the perspective of shock and volatility transmissions among Brent oil and Gulf region Islamic stock markets, the results in Table 6 reveal further interesting insights. Indeed, the GARCH coefficients ${b}_{11}$ and ${b}_{22}$ are high, positive and statistically significant at a level of 1% for all models, suggesting that the conditional volatility of all returns is highly sensitive to past own conditional volatility. The positivity of the GARCH coefficients points to volatility clustering and persistence in stock positive changes. Indeed, there is evidence of bidirectional positive shock spillovers between oil and Kuwait and Oman Islamic stock markets. Our results show additional unidirectional shock spillovers from Islamic stock markets to crude oil, except for Jordan. As for the volatility spillover effects, the results display only bidirectional links between oil and Kuwait and Oman stock markets. For the other Gulf region countries, our results show evidence only of volatility spillover from the stock market to Brent oil, except for Jordan. These results indicate no persistence in volatility spillover effects over the long term for the Islamic markets under consideration. These insights can be further understood through the plots of volatility spillovers of all pairs in Figure 4. There is also evidence of general low persistence in conditional volatility and weak mean reversion for long-run equilibrium for all Islamic stock markets, since the sum of the ARCH and GARCH coefficients in each model equation is less than unity. To sum up, the empirical findings obtained from the estimation of the VARMA-BEKK-GARCH model seem to be satisfactory, as they accurately capture the dynamics between oil and Gulf region Islamic stock markets.

### 4.3 Time-varying optimal weight and hedge ratio robustness analysis

The fact that the above empirical results evidence the new puzzle of a volatility spillover between oil and Gulf region Islamic stock markets motivated us to investigate the dynamics of such results on portfolio designs through the time-varying optimal weights and hedge ratios between oil and each Gulf region Islamic stock market. In this context, we investigate the dynamics of the relationship between optimal weights based on the VARMA-BEKK-GARCH model in order to assess the changes in portfolio composition over time and the degree of short- and long-term predictability of portfolio optimization weights and hedging strategies. Investors aim to minimize the risk of their portfolio of oil and other Gulf region Islamic indexes without reducing their expected returns in order to hedge their exposure to the movements in crude oil. As stated above, the validity of the estimated model allows us to ensure the conditional variance and covariance are precisely determined. Accordingly, the optimal weights and hedge ratios can be computed correctly. Such insight in terms of time-varying weights and hedge ratios does not support recent findings in the empirical literature, since previous studies do not investigate the time-varying aspect of weights and hedge ratios and proceed only with a mean weight value for the entire sample (see, for example, Kim and Park 2016; Brooks et al 2012; Miffre 2004).

(a) Mean equation | |||||

Brent | Brent | Brent | Brent | Brent | |

Jordan | Kuwait | Oman | Qatar | UAE | |

${\omega}_{i\text{1}}$ | 0.970${}^{***}$ | 0.029${}^{***}$ | $-$0.001 | 0.005 | 0.013${}^{*}$ |

${\omega}_{i\text{2}}$ | 0.016${}^{*}$ | 0.863${}^{***}$ | $-$0.019 | 0.018${}^{*}$ | 0.017 |

${\omega}_{i\text{3}}$ | 0.002 | 0.028${}^{***}$ | 0.851${}^{***}$ | 0.005 | 0.004 |

${\omega}_{i\text{4}}$ | $-$0.006 | 0.000 | 0.004 | 0.987${}^{***}$ | $-$0.001 |

${\omega}_{i\text{5}}$ | 0.027${}^{***}$ | 0.052${}^{***}$ | 0.075${}^{***}$ | $-$0.005 | 0.964${}^{***}$ |

${\mu}_{i}$ | $-$0.008 | 0.031${}^{***}$ | 0.089${}^{***}$ | $-$0.008 | $-$0.005 |

${\phi}_{i\text{1}}$ | $-$0.002${}^{***}$ | 0.002${}^{***}$ | 0.000${}^{***}$ | 0.002${}^{***}$ | 0.003${}^{***}$ |

${\phi}_{i\text{2}}$ | $-$0.004${}^{***}$ | $-$0.001${}^{***}$ | $-$0.001${}^{***}$ | $-$0.001${}^{***}$ | 0.022${}^{***}$ |

${\phi}_{i\text{3}}$ | $-$0.005${}^{***}$ | $-$0.015${}^{***}$ | $-$0.001${}^{***}$ | $-$0.001${}^{***}$ | 0.011${}^{***}$ |

${\phi}_{i\text{4}}$ | $-$0.004${}^{***}$ | $-$0.003${}^{***}$ | $-$0.001${}^{***}$ | 0.000${}^{***}$ | 0.004${}^{***}$ |

${\phi}_{i\text{5}}$ | $-$0.002${}^{***}$ | 0.000${}^{***}$ | 0.000${}^{***}$ | 0.000${}^{***}$ | 0.010${}^{***}$ |

(b) Variance equation | |||||

Brent | Brent | Brent | Brent | Brent | |

Jordan | Kuwait | Oman | Qatar | UAE | |

${\theta}_{i\text{1}}$ | 0.020${}^{***}$ | $-$0.010${}^{***}$ | 0.001 | 0.007${}^{***}$ | 0.010${}^{***}$ |

${\theta}_{i\text{2}}$ | — | $-$0.003 | 0.002 | $-$0.014${}^{***}$ | 0.013${}^{***}$ |

${\theta}_{i\text{3}}$ | — | — | 0.000 | $-$0.003 | 0.001 |

${\theta}_{i\text{4}}$ | — | — | — | $-$0.002 | 0.001 |

${\theta}_{i\text{5}}$ | — | — | — | — | 0.001 |

${a}_{i\text{1}}$ | 0.100${}^{***}$ | $-$0.011 | 0.017 | $-$0.015 | 0.049${}^{***}$ |

${a}_{i\text{2}}$ | 0.085${}^{***}$ | 0.012 | $-$0.016 | $-$0.023 | $-$0.002 |

${a}_{i\text{3}}$ | 0.229${}^{***}$ | 0.102${}^{***}$ | 0.204${}^{***}$ | 0.107${}^{***}$ | $-$0.031 |

${a}_{i\text{4}}$ | 0.272${}^{***}$ | 0.139${}^{***}$ | 0.156${}^{***}$ | 0.489${}^{***}$ | 0.184${}^{***}$ |

${a}_{i\text{5}}$ | 0.047${}^{**}$ | $-$0.015 | $-$0.036${}^{**}$ | 0.318${}^{***}$ | $-$0.038 |

${b}_{i\text{1}}$ | 0.740${}^{***}$ | $-$0.001 | $-$0.058${}^{***}$ | 0.207${}^{***}$ | $-$0.169${}^{***}$ |

${b}_{i\text{2}}$ | 0.366${}^{***}$ | 0.895${}^{***}$ | $-$0.056${}^{***}$ | $-$0.289${}^{***}$ | 0.233${}^{***}$ |

${b}_{i\text{3}}$ | 0.130${}^{***}$ | 0.224${}^{***}$ | 0.935${}^{***}$ | $-$0.061 | 0.033 |

${b}_{i\text{4}}$ | $-$0.231${}^{***}$ | 0.193${}^{***}$ | $-$0.047${}^{***}$ | 0.526${}^{***}$ | 0.138${}^{***}$ |

${b}_{i\text{5}}$ | $-$0.091${}^{**}$ | 0.004 | $-$0.047${}^{***}$ | 0.260${}^{***}$ | 0.732${}^{***}$ |

AIC | $-$18.496 | ||||

SBC | $-$18.018 | ||||

LOG-L | $-$18.495 | ||||

Obs. | 1299 |

The estimation results reported in Table 7 (as indicated by the $\omega $s) indicate that, for the mean equation, the weights ${w}_{\mathrm{oil}}^{\text{Jordan}}$, ${w}_{\mathrm{oil}}^{\text{Kuwait}}$, ${w}_{\mathrm{oil}}^{\text{Oman}}$, ${w}_{\mathrm{oil}}^{\text{Qatar}}$ and ${w}_{\mathrm{oil}}^{\text{UAE}}$ exhibit volatility spillover for most models, thus suggesting evidence of short-term predictability between all weights. These features agree with those inferred from the dynamics of time-varying optimal weights and ratios plotted in Figures 5 and 6. ARCH coefficients are significant for most models, except for Oman and Qatar. Such results reveal ARCH effect shock and volatility transmissions between most weights. Regarding the extent of own and cross-weight GARCH coefficients, the results highlight that the conditional volatility of weights fluctuates more rapidly over time under the own innovation impulsions and volatility than the cross-weights values in most cases. Such insights suggest that own values have more power to predict future conditional volatility than cross-weight values. These features confirm the dynamics of the time-varying optimal weights plotted in Figure 6. Our findings are beneficial for portfolio owners in order to optimize portfolio asset allocation and predict crude oil allocation changes based on the Gulf Islamic markets’ weights characteristics.

## 5 Conclusion

In this paper, we proposed a time-varying method to improve hedging performance between oil and Gulf region Islamic stock markets. Our results reveal ARCH and shock effects and volatility transmissions between all pairs except for Brent/Jordan and Brent/UAE and show no evidence of shock interdependence between oil/stock markets except for Brent/Kuwait and Brent/Oman. This fact has important implications for portfolio owners regarding international portfolio diversification and for policy makers regarding contagion risks. As regards the short- and long-term persistence of volatility, our results for oil and Gulf region Islamic stock markets evidence a significant return spillover effect from Brent to stock markets but not vice versa. Further, we propose time-varying optimal weights and hedge ratios following Kroner and Ng (1998) in order to assess the own and cross-weights volatility spillover between time-varying series based on the VARMA-BEKK-GARCH model. Time-varying series for weights and hedge ratios have some advantages in all the existing conditional hedge ratios, since the weight to the conditional hedge ratio framework is time varying. Our empirical results show that our proposed time-varying weight and hedge ratio are already different from the mean value and perform differently. Our proposed hedge ratio outperforms conditional and unconditional hedge strategies studies, since portfolio holders have more information about the volatility of a hedged portfolio in terms of time-varying weight and hedge ratios. This result is partly supported by Kim and Park (2016). This paper can be extended in various ways, such as to the investigation of marketplaces in different geographical zones.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

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