Volatility Transmissions and Hedging Between Petroleum and Equity Market Sectors: Insights from Petroleum Exporters and Importers

Abstract

This study investigates the return and volatility transmissions between petroleum prices and stock sector indices of 7 net petroleum-exporting and 19 net petroleum-importing countries over the period from January 2005 to September 2018. Given that indices representing sectors of most considered countries are not available, a unique approach is implemented to manually construct sector indices using daily data of 5768 stocks listed in 10 sectors. The VAR-GARCH model is applied that allows to capture bilateral volatility interactions. Furthermore, the estimates of the model are employed to analyse optimal portfolio holdings and hedge ratios. The findings reveal significant volatility transmissions between petroleum prices and stock sector indices of exporters and importers. However, the direction and magnitude of spillover effects are country- and sector-specific. The optimal portfolio weights and hedge ratios indicate that sector indices of Saudi Arabia (net exporter) and China (net importer) offer better opportunities with respect to hedging petroleum price risks.

1. Introduction

Petroleum remains the dominant source of energy, accounting for approximately 35 percent of global primary energy consumption in 2020 (British Petroleum, 2021), and is among the most actively traded commodities. The level of interconnection between stock markets and petroleum prices has substantially enhanced with the financialization of commodity markets. The extreme fluctuations in the price of petroleum over the past decades that were reflected with different magnitudes in stock markets have generated particular interest among policymakers and investors. Consequently, comprehending the volatility spillover effects between petroleum prices and stock markets is essential for the efficient management of risks and investment portfolios. Given that not all sectors in petroleum-exporting and -importing countries are equally sensitive to swings in petroleum prices, the utilisation of aggregate market indices in this matter would be disadvantageous. Thus, a sector-level analysis can provide investors with valuable insights into diversification opportunities and help regulators in formulating relevant frameworks addressed towards the prevention of contagion risks (Bouri et al., 2016).

The number of works in the energy finance literature that study the interrelationship between petroleum price and stock market volatility has been growing. These works are primarily focused on the aggregate market indices and document different findings depending on the employed methodologies and study periods. Malik and Hammoudeh (2007) carried out one of the early works in this strand of literature. Their empirical results indicate that volatility is transmitted from the petroleum market to equity markets of all Gulf countries, but a spillover in the reverse direction is detected only in the case of Saudi Arabia. In addition, they show the existence of volatility interactions between the second moments of the United States equity and petroleum markets. Other studies consider solely the United States or a group of developed markets. Mensi et al. (2013) showed evidence of the bidirectional shock and volatility transmissions between the S&P 500 index and the WTI grade, and the unidirectional spillovers of shock from the Brent sort to the S&P 500 index and volatility in the opposite course. Salisu and Oloko (2015) observed bilateral spillovers of shock between the petroleum market and United States stock market, and the unilateral volatility transmissions from the first to the latter before and after structural breaks. Furthermore, Ewing and Malik (2016), incorporating structural breaks in their study, also reported significant volatility transmissions between petroleum prices and the United States stock market. Khalfaoui et al. (2015) provided strong evidence of volatility linkages between petroleum and stock markets of G7 countries. On the contrary, Chang et al. (2013) found little evidence of volatility transmissions between crude petroleum and the NYSE, Dow Jones, FTSE 100, and S&P 500 indices.

Other studies conduct analyses in the context of emerging markets. The study of Arouri et al. (2011b) indicated the existence of significant transmissions of shock and volatility between petroleum and most of the Gulf Cooperation Council countries’ stock markets, particularly during the crisis period. Lin et al. (2014) focused on developing stock markets in the West African region, namely the GSE All-Share Index of Ghana and the NSE All-share index of Nigeria. They detected volatility spillovers between the petroleum and stock markets of both countries. Yousaf and Hassan (2019) studied the return and volatility transmissions between crude petroleum and nine Asian emerging stock markets. The authors showed that volatility linkages vary across the considered stock markets and study periods. Sarwar et al. (2020) investigated the volatility transmission between petroleum and stock market returns of China, Pakistan, and India. They provided evidence of bidirectional, unidirectional, and mixed-volatility spillovers for the stock markets of Pakistan, China, and India, respectively, and indicated that results differ across subperiods and data frequencies. Molina-Muñoz et al. (2025) focused on studying volatility spillovers between energy and financial stock indices in emerging markets over the period from 2001 to 2021, considering the sub-prime crisis and the COVID-19 pandemic. Their findings, based on the full sample, reveal the presence of bidirectional transmissions. Considering different subsamples, the authors reported that during the sub-prime crisis, volatility spillovers ran from the financial index to the energy index, whereas during the COVID-19 period, volatility was transmitted in the reverse direction.

A group of works distinguish between petroleum-exporting and -importing countries. For instance, Wang and Liu (2016) examined volatility spillovers between petroleum and stock markets of seven petroleum-exporting and nine petroleum-importing economies. Their results show the presence of volatility transmissions from petroleum to the stock markets of Canada, Norway, Russia, and Venezuela, while in the case of importers, the petroleum market volatility is affected by the stock market volatility of Germany, the United Kingdom, and the United States. Ashfaq et al. (2019) considered the stock markets of three petroleum-exporting countries (Iraq, Saudi Arabia, and the United Arab Emirates) and four petroleum-importing countries (China, India, Japan, and South Korea). They detected shock and volatility transmissions between the petroleum and stock markets of South Korea, Saudi Arabia, and Iraq, but did not report significant effects in all other cases. Belhassine and Karamti (2021) investigated the volatility interactions between petroleum and stock markets of the top three petroleum exporters, Canada, Russia, and Saudi Arabia, and three petroleum importers, China, India, and the United States. The authors documented the existence of significant price and volatility spillovers that are contingent on the studied countries and time scales. Sarwar et al. (2019) analysed the volatility transmission between the crude petroleum returns and stock market returns of three large Asian petroleum-importing countries, namely China, Japan, and India. Their findings depict that spillovers of shock and volatility are unidirectional from the stock market returns of India to petroleum returns and that the effects are bidirectional in the case of the Japanese stock market. However, they provide no evidence for the Chinese stock market.

Given that the sensitivity to the petroleum price volatility could vary across individual sectors, some studies focus on investigating the volatility interdependencies between petroleum and stock sectors. Malik and Ewing (2009) examined the volatility and shock transmission mechanisms between petroleum prices and five stock sectors of the United States. Their results show significant transmissions between petroleum prices and some of the studied sectors. Arouri et al. (2011a) analysed volatility interactions between petroleum and stock sectors of the United States and Europe. The authors provided evidence of volatility transmissions, which are generally unidirectional from petroleum to global European sectors and bidirectional for sectors of the United States. These findings are corroborated by the study of Arouri et al. (2012), which investigated volatility spillovers between the Brent grade of petroleum and European sector indices. Belhassine (2020) examined the transmission of volatility between petroleum prices and nineteen Eurozone sector indices. The findings point to the heterogeneous and time-varying nature of volatility spillovers between petroleum and most Eurozone sector indices. Bouri et al. (2016) studied the association between petroleum and the financial, industrial, and service sectors in Jordan. Their results provide evidence of long-term effects from petroleum to the industrial and service sectors during the period preceding the Arab uprisings, and both short- and long-term transmissions from petroleum to the industrial sector for the period that followed the Arab uprisings.

While the existing studies have been mostly devoted to the US and aggregate European sectors when assessing volatility interactions between petroleum prices and equity markets, evidence drawn from other asset classes further supports the relevance of modelling the cross-market volatility transmission mechanisms (Mensi et al., 2021; Yadav et al., 2022; Koulis & Kyriakopoulos, 2023; among others). Taking this into account, the present paper, by distinguishing between net petroleum-exporting and -importing economies, strives to shed new light on volatility interdependencies between the petroleum and stock sectors. As major aggregate market indices generally exhibit favouritism towards certain industries, the weights of which vary depending on their importance across countries (Mateus et al., 2017), a sector-level investigation is essential. Unlike the closely related study of Bagirov and Mateus (2022), which focuses on Mexico and the United Kingdom, this paper considers sectors of a broad sample of petroleum exporters and importers, which includes countries with different levels of economic development and degrees of petroleum dependency, thereby providing a comprehensive picture on spillover effects. Global investors typically select stocks which have both large market capitalisation and high liquidity within a particular or group of sectors for their portfolios. However, the analysis of the heterogeneity of stock sectors’ reactions is hampered by the absence of sector indices that represent components of the aggregate market indices utilised in the present study. To address this gap, following the works of Mateus et al. (2017) and Bagirov and Mateus (2022), a unique approach for the manual construction of stock sector indices is adopted. In the view of the fact that stock markets in each country have their own regulatory and construction procedures, this approach enables the application of the same construction methods across various markets. This study possesses two main objectives. First, the investigation of the return and volatility transmissions between petroleum prices and self-constructed stock sector indices of seven net petroleum-exporting and nineteen net petroleum-importing countries utilising daily firm-level data. For this purpose, the VAR-GARCH process, where the variance equation takes the form proposed by Ling and McAleer (2003), is applied. The main advantages of this specification are that it permits studying bidirectional volatility spillovers and is not excessive in parameters. Second, using estimates of the model, the quantification of optimal weights and hedge ratios for portfolios comprising stock sector index and petroleum assets is achieved, based on the methodologies of Kroner and Ng (1998) and Kroner and Sultan (1993), respectively.

The present paper makes several important contributions to the growing literature on the volatility interconnection between petroleum prices and stock sectors. First, it demonstrates that stock sector indices in petroleum-exporting and -importing countries exhibit heterogeneous sensitivities to petroleum price volatility. These nuanced interconnections are often concealed when using aggregate market indices. By highlighting such differences, this study provides valuable insights for international investors and policymakers, enabling more informed decisions regarding cross-market asset allocations, hedging of risks, and regulatory frameworks. Second, in contrast to prior sector-based studies that are typically region- or country-specific and predominantly focused on developed economies, this paper, to our knowledge, is the first to examine both the intensity and direction of volatility spillovers between petroleum prices and manually constructed stock sector indices. This analysis encompasses a broad set of net petroleum exporters and importer with developed, emerging, and frontier markets, thereby offering a more global and inclusive perspective. Third, it introduces a novel methodology for constructing sector indices based on daily firm-level data. This approach facilitates a more precise examination of intermarket volatility transmission mechanisms.

Employing daily data of 5768 unique stocks listed from 3 January 2005 to 28 September 2018 in ten sectors, namely Basic Materials, Consumer Cyclicals, Consumer Non-Cyclicals, Energy, Financials, Healthcare, Industrials, Technology, Telecommunications Services, and Utilities, of seven petroleum-exporting and nineteen petroleum-importing countries, this study yields interesting empirical findings. The evidence of significant shock or innovation and volatility spillovers between stock sector indices and petroleum is documented. The direction and magnitude of transmissions vary across markets and sectors. The assessment of optimal portfolio holdings and hedge ratios suggests that stock sector indices of Saudi Arabia (net exporter) and China (net importer) provide better opportunities for hedging petroleum price risks. Thus, the conducted investigation ascertains the importance of distinguishing between the net petroleum exporter and importer along with the sector-level analysis in understanding the heterogeneity of spillover effects.

The remaining parts of the work are organised as follows. The next section describes data sources, sample selection, and sector indices’ construction procedures. The employed econometric model is presented in Section 3. Section 4 discusses the empirical findings. Section 5 analyses portfolio weights and hedge ratios. The last section finalises the work.

2. Data and Construction of Sector Indices

2.1. Sample Selection

To set the sample for the investigation of volatility spillovers between petroleum prices and stock sector indices of petroleum exporters and importers, this study follows several selection procedures. First, petroleum-exporting and -importing countries are selected based on the annual crude petroleum production and consumption data provided by the British Petroleum Statistical Review of World Energy. The group of petroleum exporters includes countries where the level of petroleum production exceeds petroleum consumption. The group of petroleum importers comprises countries where the level of petroleum consumption surpasses petroleum production. Since this study covers the period from 2005 to 2018, the additional condition, which follows the empirical work of Ramos and Veiga (2013), is that the levels of petroleum production and consumption should remain unchanged during the entire investigation period; that is, petroleum-exporting and -importing countries should belong to the same group. The applied criteria contributed to the identification of countries, such as Argentina and Denmark, that fell into both petroleum exporter and importer categories during the estimation period. Given that the level of petroleum production exceeded petroleum consumption during more than 60% of the considered period, it was decided to include Argentina and Denmark in the group of petroleum-exporting countries. Second, the final sample of countries was conditional on the maturity of stock markets and the availability of data. Thus, in order to avoid liquidity-related issues with stock prices, the major stock market indices were chosen. For each of the considered stock market indices, the quarterly component lists were retrieved from the Eikon and Datastream databases, including official sources of indices, to select sectors and stocks based on the following criteria: (i) sectors of stock market indices should comprise minimum five stocks in each quarter during the entire period of analysis; (ii) stocks should be liquid, particularly in countries with less developed stock markets; and (iii) stock markets should have at least three sectors with five or more stocks. The applied conditions permit not only to reach sectoral-level conclusions, but also to contrast findings among various sectors within one or more countries.

The final sample comprises 7 petroleum-exporting countries, namely Argentina, Canada, Denmark, Mexico, Norway, Russia, and Saudi Arabia, and 19 petroleum-importing countries, namely Australia, Brazil, Chile, China, the Eurozone, India, Indonesia, Israel, Japan, New Zealand, Pakistan, South Africa, South Korea, Sweden, Taiwan, Thailand, Turkey, the United Kingdom, and the United States.1 The Eurozone consists of 11 member countries, Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal, and Spain, that started using the euro as their official currency prior to the investigation period and have unified monetary policies. Since the total number of stocks representing each sector in individual countries did not meet the set requirements, it was practical and methodologically sound to combine them in a single group. This approach not only ensures the inclusion of major petroleum-importing economies, but also enables a more comprehensive assessment of how equity sectors across the broader European region are exposed to petroleum price risks.

The total number of considered stocks by market and sector is presented in

Table 1. Total number of stocks by market and sector listed from January 2005 to September 2018.

Country	Index	Basic Materials	Consumer Cyclicals	Consumer Non-Cyclicals	Energy	Financials	Healthcare	Industrials	Technology	Telecommunications Services	Utilities	Total
Panel A: Petroleum Exporters
Argentina	S&P BYMA	17	-	18	11	14	-	-	-	-	10	70
Canada	S&P TSX	151	44	20	129	74	23	42	28	10	22	543
Denmark	OMXC BENCHMARK	-	18	-	-	46	10	22	-	-	-	96
Mexico	S&P BMV IPC	12	12	14	-	14	-	14	-	-	-	66
Norway	OSE BENCHMARK	-	14	13	36	20	-	29	-	-	-	112
Russia	MOEX	19	-	-	14	-	-	-	-	-	10	43
Saudi Arabia	TADAWUL ALL SHARE	42	18	20	6	67	-	22	-	-	-	175
Total		241	106	85	196	235	33	129	28	10	42	1105
Panel B: Petroleum Importers
Australia	S&P ASX 200	102	65	34	51	89	29	59	-	13	11	453
Brazil	IBX	31	21	23	13	53	-	30	-	-	27	198
Chile	S&P IPSA CLP	-	-	13	-	16	-	9	-	-	15	53
China	CSI 300	85	82	38	37	98	37	124	61	-	-	562
Eurozone	MULTIPLE INDICES	75	94	44	28	119	30	105	32	24	34	585
India	S&P BSE 100	23	22	14	15	37	15	18	10	-	9	163
Indonesia	JSX LQ 45	10	-	15	18	20	-	-	-	-	-	63
Israel	TA 125	-	24	11	18	69	-	21	32	-	-	175
Japan	NIKKEI 225	38	38	19	-	30	11	69	26	-	5	236
New Zealand	S&P NZX 50	-	17	-	-	21	-	13	-	-	-	51
Pakistan	KSE 100	33	32	-	12	48	-	-	-	-	12	137
South Africa	FTSE JSE ALL SHARE	53	45	27	-	112	-	34	14	7	-	292
South Korea	KOSPI 200	69	66	43	9	27	32	65	36	-	-	347
Sweden	OMXS 30	-	6	-	-	9	-	10	-	-	-	25
Taiwan	FTSE TW 50	7	-	-	-	16	-	-	29	-	-	52
Thailand	SET 50	-	18	-	13	25	-	15	-	-	-	71
Turkey	BIST 100	34	47	21	-	50	-	30	-	-	-	182
United Kingdom	FTSE 100	27	40	17	12	42	9	33	-	-	12	192
United States	S&P 500	48	138	63	65	145	89	100	119	16	43	826
Total		635	755	382	291	1026	252	735	359	60	168	4663
Grand Total		876	861	467	487	1261	285	864	387	70	210	5768

Notes: The Thomson Reuters Business Classification standard is followed. The Eurozone comprises 11 member countries that use the euro as their official currency, namely Austria (ATX), Belgium (BEL 20), Finland (OMXH 25), France (CAC 40), Germany (DAX 30), Greece (ATHEX COMPOSITE), Ireland (ISEQ 20), Italy (FTSE MIB), the Netherlands (AEX), Portugal (PSI 20), and Spain (IBEX 35).

. The selected market indices do not follow the same industry classification standards. Therefore, this study follows the Thomson Reuters Business Classification (TRBC), which is regarded as a market-based classification scheme, with the purpose of making the standards uniform. There were 1105 and 4663 unique stocks listed from January 2005 to September 2018 in 10 sectors, namely Basic Materials, Consumer Cyclicals, Consumer Non-Cyclicals, Energy, Financials, Healthcare, Industrials, Technology, Telecommunications Services, and Utilities, of petroleum-exporting and -importing countries, respectively. The sectors that make up the largest part of the sample are Basic Materials, Energy, and Financials in the case of exporters and Basic Materials, Consumer Cyclicals, Financials, and Industrials in the case of importers. In total, their shares are approximately 61 percent and 68 percent, respectively. At the individual level, certain markets within the petroleum exporters and importers groups are more focused on some sectors. For instance, the Basic Materials and Energy sectors stand out with the largest number of stocks in Canada, while in the United States the Consumer Cyclicals, Financials, Industrials and Technology sectors dominate. This suggests that the examination of volatility spillover effects utilising aggregate market indices could be biased towards specific sectors since, depending on the importance of sectors, their weights vary in petroleum-exporting and -importing countries. It would therefore be rational to conduct the sector-level analysis.

2.2. Data Sources

This study makes use of daily data covering the period from 3 January 2005 to 28 September 2018. The sample range is guided by the goal of prioritising the recent period in order to encompass different global events, namely the global financial crisis, Arab uprisings, and geopolitical tensions, that resulted in petroleum market imbalances, and the availability of stock market data for important petroleum-exporting and -importing countries. Given the fact that crude petroleum grades are priced and traded in US dollars, this currency is chosen as the main currency in this study.

The petroleum market is represented by the Brent grade that is extracted from petroleum fields in the North Sea and easily transported to distant locations. It acts as a price benchmark and is classified as sweet and light crude because of its low sulphur content and density, which does not require additional refinery costs. Batten et al. (2021) emphasise the appropriateness of the Brent grade for the analysis of hedge ratios between stocks and petroleum, as the other widespread benchmark, West Texas Intermediate (WTI), comprises a local price spread related to petroleum movements at the Cushing (Oklahoma) crude petroleum storage and transportation hub. Their findings suggest that on average the hedging effectiveness is greater when the Brent grade is used compared to the WTI grade. The Brent crude petroleum spot prices quoted in US dollars and the adjusted closing prices of 5768 selected stocks expressed in local currencies, including exchange rates used for conversion into US dollars, were extracted from the Datastream database.

2.3. Construction of Sector Indices and Preliminary Analysis

Considering that stock sectors in petroleum-exporting and -importing countries may experience different levels of susceptibility to the petroleum price volatility, conducting the aggregate level analysis would be rather general. Thus, the present work, following the studies of Mateus et al. (2017) and Bagirov and Mateus (2022), applies a unique approach to self-constructed sector indices employing daily data on individual stocks in order to overcome the limitations associated with the non-availability of sector indices constituting the largest stocks in most countries of interest. First, the major stock market indices presented in Table 1 were considered in the process of choosing sectors and individual stocks. Since the aggregate market indices pursue dissimilar industry classification schemes, the TRBC is adopted to standardise the schemes. Second, the constituent lists of the aggregate stock market indices were obtained for each quarter from January 2005 to September 2018. Third, multiple criteria described in the preceding subsections were applied to finalise the list of sectors and stocks for the analysis. Fourth, an equal-weighted approach, which is less biased towards high price and large cap stocks and hence results in greater diversification, was employed to manually build sector indices for each of the markets, $I_{i,t}=I_{i,t-1}\times \left(1+r_{i,t}\right)$, where the base value of sector indices is set as 100, $I_{i,t-1}$ refers to the value of the ith sector index at time t − 1, the value of the ith sector index at time t is signified by $I_{i,t}$, and $r_{i,t}$ denotes the average of daily logarithmic returns2 of all stocks listed at time t. In order to account for each joining and leaving constituent and sustain the continuity of the stock sector indices, they were rebalanced every quarter throughout the construction process. The applied methodology enables the utilization of the same approach for the formation of sector indices across 26 studied markets.

It is worth mentioning some exceptions made in relation to certain markets during the construction process. The number of stocks listed in the Utilities sector of India until the 4th quarter of 2006, in the Basic Materials and Utilities sectors of Russia until the 1st quarters of 2006 and 2007, respectively, and in the Consumer Cyclicals, Consumer Non-Cyclicals, and Energy sectors of Saudi Arabia until the 3rd and 1st quarters of 2005 and 2006, respectively, were below the set criteria. Given the importance of these large petroleum-importing and -exporting countries, the sector indices were built utilising the available stocks to ensure that the starting period of the sample is the same across all markets. In the case of Argentina, all stock sector indices were smoothed on 17 December 2015 due to a devaluation of the local currency by approximately 30 percent3 associated with the government’s decision rather than market factors. In addition, some countries’ sectors had stocks with missing values despite the markets being open, which were replaced using linear interpolation.

Table S3 in Supplementary Materials reports the results of ARCH and unit root tests.4 The statistically significant values of the Engle’s test for autoregressive conditional heteroscedasticity of order five, which is applied as a preliminary test for the presence of ARCH effects in residuals, shows the presence of the ARCH effects in all series, thereby pointing to the suitability of GARCH family models for the investigation of volatility transmissions. The applied Augmented Dickey–Fuller unit root test, where the number of lags was selected using the Schwarz Bayesian Criterion, shows that return series are stationary in all cases.

3. Econometric Model

The utilization of GARCH-type models in analysing financial volatility has become widespread among academics and practitioners over the past decades. Studies in the finance literature have shown that the multivariate versions of GARCH models, such as DVECH, BEKK, CCC, and DCC, among others, are more appropriate and effective in examining the volatility interactions between different financial variables compared to their univariate counterparts. However, these multivariate models encounter some limitations: the excessive number of parameters that are not easily interpretable; achieving convergence during the estimation process is difficult, especially with the introduction of additional variables; and the lack of ability in measuring the extent of volatility interdependence that is important given the increased level of integration between markets (Hammoudeh et al., 2009; Arouri et al., 2011b, 2015; Sadorsky, 2012). Thus, the present work applies a multivariate GARCH approach comprising two elements, the vector autoregressive (VAR) model and the specification of the GARCH model proposed by Ling and McAleer (2003), in order to examine the transmission of volatility between petroleum prices and manually constructed stock sector indices of petroleum-exporting and -importing countries. The model encompasses the constant conditional correlation (CCC) GARCH process of Bollerslev (1990), where the conditional correlations between variables are constrained to be constant over the investigation period. The empirical studies of Hammoudeh et al. (2009, 2010), Chang et al. (2010, 2013), Arouri et al. (2011a, 2012, 2015), Sadorsky (2012), Salisu and Mobolaji (2013), Lin et al. (2014), and Bagirov and Mateus (2022) have demonstrated the efficacy of the VAR-GARCH approach in modelling volatility transmissions between various markets.

The VAR model is employed to estimate the conditional mean equation for each pair of stock sector and petroleum return series. The optimal number of lags was selected based on the Schwarz Bayesian Criterion (SBC). Although the SBC criterion suggested different lags for each sector and country, in the interest of economy, the model with one lag was selected, since the inclusion of longer lags showed negligible difference based on the preliminary analysis (see Supplementary Materials). Consequently, the conditional mean equation is defined as follows:

$$R_{i,t}={\mu }_i+{\displaystyle \sum }_{j=1}^n{\varphi }_{ij}{R}_{i,t-1}+{\epsilon }_{i,t}$$

(1)

$${\epsilon }_{i,t}=h_{i,t}^{1/2}{\eta }_{i,t}$$

where the logarithmic return of the ith stock sector index and Brent crude petroleum prices at time t is denoted by $R_{i,t}$, ${\epsilon }_{i,t}$ signifies error terms, ${\eta }_{i,t}$ refers to a sequence of i.i.d. random errors, and the conditional variance of the ith stock sector index and Brent petroleum price returns at time t is represented by $h_{i,t}$.

Chang et al. (2011) suggest that practically multivariate GARCH models with a longer number of lags can cause difficulties during the estimation process. Table S2 in Supplementary Materials confirms this and illustrates that in all cases the GARCH(1,1) process fits better based on the SBC criterion. Thus, the conditional variance equation, which is represented by the GARCH(1,1) specification of Ling and McAleer (2003), where one variable’s past shocks and volatilities are allowed to impact volatilities of other variable, takes the following form:

$$h_{i,t}=c_{ii}+{\displaystyle \sum }_{j=1}^n{\alpha }_{ij}{\epsilon }_{j,t-1}^2+{\displaystyle \sum }_{j=1}^n{\beta }_{ij}{h}_{j,t-1}$$

(2)

where the ARCH term, which quantifies the effects of innovations or shocks, is denoted by ${\displaystyle \sum }{\alpha }_{ij}{\epsilon }_{j,t-1}^2$ and the GARCH term, which measures the effects of volatilities, is signified by ${\displaystyle \sum }{\beta }_{ij}{h}_{j,t-1}$. It can be seen from the second equation that the conditional variance of the ith stock sector index (or Brent crude petroleum prices) is affected by both own past innovations or shocks, as well as by those of Brent crude petroleum prices (or the ith stock sector index).

The conditional covariance between the ith stock sector index and Brent crude petroleum return series is given as follows:

$$h_{ij,t}={\rho }_{ij}{h}_{ii,t}^{1/2} h_{jj,t}^{1/2}$$

(3)

where ${\rho }_{ij}$ refers to the constant conditional correlation. The considered VAR-GARCH model is estimated by using the quasi-maximum likelihood estimation (QMLE) method, since the normality condition does not often hold in the case of financial time series. Ling and McAleer (2003) demonstrated in their study, which provides a detailed explanation of the model’s necessary conditions, that for consistency of QMLE, the existence of the second moment is sufficient. The estimates of the VAR-GARCH model for each pair are subsequently utilised to build optimal portfolio weights and hedge ratios.

4. Empirical Results

The discussion of the VAR-GARCH model estimates is partitioned into several subsections. The results of the return spillovers are presented first, followed by the findings of own shock and volatility dependencies as well as interdependencies, and concluding with the outcomes of constant conditional correlations and diagnostic tests.

4.1. Return Spillovers

Table S4 (panels A to G) and Table S5 (panels A to S) in Supplementary Materials report detailed empirical results of the VAR-GARCH model for petroleum and stock sector index pairs of petroleum-exporting and -importing countries, respectively. The estimated parameters of the mean equation show that the interdependence between returns of stock sector indices and petroleum are contingent on the considered sector and market. The current period returns of the majority stock sector indices in petroleum-exporting countries, namely Norway and Saudi Arabia, and petroleum-importing countries, namely Australia, Indonesia, Japan, New Zealand, South Korea, and Taiwan, are affected by one-period lagged petroleum returns, as indicated by the significance of corresponding coefficients (${\varphi }_{1,2}$) at conventional levels, which implies that past petroleum returns can be employed to forecast the direction of sector indices. The statistically significant effects in the reverse course (${\varphi }_{2,1}$) are more evident for stock sector indices of petroleum exporters, such as Mexico and Canada, and petroleum importers, such as Brazil, South Africa, the United Kingdom, and the United States. This finding points to the possibility of utilising returns of sector indices to predict petroleum returns. The estimated own autoregressive parameters of stock sector indices represented by ${\varphi }_{1,1}$ are statistically significant for all markets, except Norway and the United Kingdom, and thus suggest that past returns of stock sector indices can be efficient in predicting future movements. On the contrary, the results suggest that the current period petroleum returns are not impacted by past values in most cases, as shown by the insignificance of ${\varphi }_{2,2}$ coefficients, thereby indicating their weak form of efficiency in forecasting future performance.

4.2. Shock and Volatility Spillovers

Turning to the variance equations, the parameters ${\alpha }_{1,1}$ and ${\alpha }_{2,2}$ represent the past own conditional ARCH effects, which estimate the sensitivity to shocks, or short-term persistence, of stock sector indices and petroleum, respectively (see Supplementary Materials). The results display that stock sector indices and petroleum are sensitive to past own shocks or innovations. The estimated coefficients are highly significant in all cases, with the exception of New Zealand and Thailand, where the significance levels vary from 1% to 10%. Among both groups of petroleum-exporting and -importing countries, the stock sector indices of Argentina have the greatest degree of sensitivity. The past own conditional GARCH effects, which estimate the volatility sensitivity or long-term persistence, of stock sector indices and petroleum are represented by parameters ${\beta }_{1,1}$ and ${\beta }_{2,2}$, respectively (see Supplementary Materials). The conditional volatility of all stock sector indices and petroleum are positively influenced by past own volatility, as shown by the significance of coefficients at the 1% level. The magnitude of sensitivity differs across sectors and countries, but appears to be higher for stock sector indices of China. The estimated coefficients of own ARCH terms are relatively smaller in size compared to the coefficients of own GARCH terms for all studied series. This finding suggests that stock sector indices and petroleum are generally more impacted by fundamental factors rather than news and the conditional volatility exhibits more gradual swings over time rather than rapid responses to shocks or innovations.

Table 2. Summary of shock and volatility transmissions between petroleum and stock sector indices of petroleum-exporting countries.

Panel A	Argentina		Canada		Denmark		Mexico
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	0	0	S → P	0	-	-	S → P	S → P
Consumer Cyclicals	-	-	0	0	0	0	S → P	S ← P
Consumer Non-Cyclicals	0	0	S → P	0	-	-	S → P	S ↔ P
Energy	0	0	S → P	0	-	-	-	-
Financials	0	0	S → P	S ↔ P	0	0	S → P	S → P
Healthcare	-	-	S → P	S ← P	0	0	-	-
Industrials	-	-	S → P	S ↔ P	0	0	S → P	0
Technology	-	-	S → P	S → P	-	-	-	-
Telecommunications Services	-	-	S → P	0	-	-	-	-
Utilities	S ← P	S ← P	S ↔ P	0	-	-	-	-
Panel B	Norway		Russia		Saudi Arabia
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	-	-	S → P	S → P	0	0
Consumer Cyclicals	S ← P	S ↔ P	-	-	0	0
Consumer Non-Cyclicals	0	0	-	-	0	0
Energy	0	0	S → P	0	0	0
Financials	S ← P	S ← P	-	-	0	0
Healthcare	-	-	-	-	-	-
Industrials	0	0	-	-	0	0
Technology	-	-	-	-	-	-
Telecommunications Services	-	-	-	-	-	-
Utilities	-	-	0	0	-	-

Notes: S and P denote stock sector indices and petroleum, respectively. The ARCH term represents innovation or shock spillovers and the GARCH term represents volatility spillovers. The arrows → (←) indicate the direction of innovation or shock and volatility spillovers from stock sector indices to petroleum (from petroleum to stock sector indices), ↔ captures bidirectional spillovers, and 0 refers to the absence of innovation or shock and volatility spillovers between the studied pairs.

(panels A and B) summarises shock or innovation and volatility transmissions between petroleum and stock sector indices of seven petroleum-exporting countries. The empirical results provide no evidence of shock and volatility interdependencies between petroleum and all stock sector indices of Denmark and, unexpectedly, Saudi Arabia. Wang and Liu (2016) also do not report volatility spillovers between the petroleum market and the stock market of Saudi Arabia, although the authors consider the aggregate market index. The absence of cross effects for Saudi Arabia could be elucidated by several reasons, such as local regulations related to the ownership of listed companies by investors from the Gulf Cooperation Council countries up to 25% and by foreign investors through legal entities only (Arouri et al., 2011b), which were eased after 2019, and non-market factors, such as the geopolitical events that tend to frequently influence markets in the Middle East (Wang & Liu, 2016). In the case of Denmark, the country’s lower level of petroleum production compared to other large exporters and the increased production, including consumption, of renewable energy over the past decades5 could provide a plausible explanation for the obtained results.

It is interesting to note that past shocks or innovations emanating from all stock sector indices of Mexico and Canada, except the Consumer Cyclicals sector, and the Basic Materials and Energy sectors of Russia significantly affect the conditional volatility of petroleum, thereby emphasising the importance of news arising from the sectors of these largest petroleum-exporting countries for the petroleum market. The effects in the reverse direction were detected only for the Utilities sectors of Argentina and Canada and the Consumer Cyclicals and Financials sectors of Norway. The findings provide evidence of volatility transmissions from petroleum to stock sector indices. More specifically, the past volatility of petroleum spills over to the Utilities sector of Argentina, the Financials, Healthcare, and Industrials sector of Canada, the Consumer Cyclicals and Consumer Non-Cyclicals sectors of Mexico, and the Consumer Cyclicals and Financials sectors of Norway. Inversely, it was observed that the past volatility of the Financials, Industrials, and Technology sectors of Canada, the Basic Materials, Consumer Non-Cyclicals, and Financials sectors of Mexico, the Consumer Cyclicals sector of Norway, and the Basic Materials sector of Russia impact the conditional volatility of petroleum, but negatively, as indicated by signs of corresponding coefficients (see Supplementary Materials). This result suggests that sector indices’ volatility cools off petroleum volatility in the long run, possibly because of the efficient performance of companies within these sectors, which potentially might signal less stagnant periods in the economies of large petroleum-exporting countries, such as Canada, Mexico, Norway, and Russia. Some empirical studies obtained a similar finding for the large petroleum-exporting countries; that is, significant volatility transmissions from the aggregate stock markets to the petroleum market were documented (Malik & Hammoudeh, 2007; Lin et al., 2014; Belhassine & Karamti, 2021; among others). The negative conditional volatility interdependency is present only between the Consumer Cyclicals sector of Norway and petroleum, implying that their past volatility offsets the current volatility of each other.

Table 3. Summary of shock and volatility transmissions between petroleum and stock sector indices of petroleum-importing countries.

Panel A	Brazil		China		United Kingdom		United States
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	S → P	0	0	0	0	0	S ↔ P	S → P
Consumer Cyclicals	S → P	0	0	S ← P	0	0	S → P	S → P
Consumer Non-Cyclicals	S ↔ P	0	0	0	0	0	S → P	S → P
Energy	S → P	S → P	0	0	S → P	0	S → P	0
Financials	S → P	0	0	S ← P	0	0	S → P	S → P
Healthcare	-	-	0	0	0	0	S → P	S → P
Industrials	S → P	0	0	0	0	0	S ↔ P	S → P
Technology	-	-	0	S ← P	-	-	S ↔ P	S → P
Telecommunications Services	-	-	-	-	-	-	0	0
Utilities	S → P	S → P	-	-	S → P	S ← P	S → P	0
Panel B	Australia		Chile		Eurozone		India
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	0	0	-	-	0	0	0	0
Consumer Cyclicals	S ← P	0	-	-	0	S ← P	0	0
Consumer Non-Cyclicals	S ← P	S ← P	0	0	0	0	0	0
Energy	S ↔ P	0	-	-	0	0	0	0
Financials	S ← P	0	0	0	0	0	0	0
Healthcare	S ← P	0	-	-	0	0	0	0
Industrials	S ← P	S ← P	0	0	S ← P	S ← P	0	0
Technology	-	-	-	-	S ← P	0	0	0
Telecommunications Services	0	0	-	-	S → P	S → P	-	-
Utilities	0	0	0	0	0	0	0	0
Panel C	Indonesia		Israel		Japan		New Zealand
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	0	0	-	-	S ← P	0	-	-
Consumer Cyclicals	-	-	S → P	0	S ← P	0	0	0
Consumer Non-Cyclicals	0	0	S → P	S → P	S → P	0	-	-
Energy	0	0	S → P	S → P	-	-	-	-
Financials	0	0	S → P	S → P	0	0	0	0
Healthcare	-	-	-	-	S ← P	0	-	-
Industrials	-	-	S ↔ P	S → P	S ← P	0	0	0
Technology	-	-	0	0	S ← P	0	-	-
Telecommunications Services	-	-	-	-	-	-	-	-
Utilities	-	-	-	-	S ← P	S ← P	-	-
Panel D	Pakistan		South Africa		South Korea		Sweden
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	S ↔ P	S ↔ P	0	0	S ← P	0	-	-
Consumer Cyclicals	S ↔ P	S ↔ P	0	0	0	0	0	0
Consumer Non-Cyclicals	-	-	0	0	S ← P	0	-	-
Energy	S ← P	S ← P	-	-	0	0	-	-
Financials	S ← P	S ← P	0	0	S → P	0	0	0
Healthcare	-	-	-	-	S ↔ P	0	-	-
Industrials	-	-	0	0	S → P	0	0	0
Technology	-	-	0	S ← P	S ↔ P	0	-	-
Telecommunications Services	-	-	0	0	-	-	-	-
Utilities	0	0	-	-	-	-	-	-
Panel E	Taiwan		Thailand		Turkey
	ARCH	GARCH	ARCH	GARCH	ARCH	GARCH
Basic Materials	S → P	0	-	-	0	0
Consumer Cyclicals	-	-	0	0	0	0
Consumer Non-Cyclicals	-	-	-	-	S ← P	0
Energy	-	-	0	0	-	-
Financials	0	0	0	0	0	0
Healthcare	-	-	-	-	-	-
Industrials	-	-	0	0	S ← P	0
Technology	S → P	0	-	-	-	-
Telecommunications Services	-	-	-	-	-	-
Utilities	-	-	-	-	-	-

Notes: S and P denote stock sector indices and petroleum, respectively. The ARCH term represents innovation or shock spillovers and the GARCH term represents volatility spillovers. The arrows → (←) indicate the direction of innovation or shock and volatility spillovers from stock sector indices to petroleum (from petroleum to stock sector indices), ↔ captures bidirectional spillovers, and 0 refers to the absence of innovation or shock and volatility spillovers between the studied pairs.

(panels A to E) provides a summary of shock or innovation and volatility spillovers between the petroleum and stock sector indices of nineteen petroleum-importing nations. Particular attention should be given to important petroleum-importing countries presented in panel A that are among the largest producers of petroleum, but whose production levels do not allow to meet local needs. Starting with China and the United Kingdom, no evidence of bilateral transmissions of shocks or innovations and volatility was found between petroleum and the majority of stock sector indices. While such a result is foreseeable for some sectors of the United Kingdom, the absence of interdependencies for the Basic Materials, Industrials, and Financials sectors is to a certain degree unexpected and could be attributed to the effectiveness of strategies employed by companies within these sectors to mitigate risks originating from petroleum prices. For the Energy and Utilities sectors, it was detected that past shocks or innovations of sector indices impact the conditional volatility of petroleum. However, in the opposite direction, volatility from petroleum only transmits to the Utilities sector. The shock or innovation and volatility interactions between petroleum and the stock sector indices of China, the second largest petroleum consumer in the world, are rather limited. This outcome is partially consistent with the works of Yousaf and Hassan (2019), Ashfaq et al. (2019), and Sarwar et al. (2019), which do not report significant volatility spillovers between the Chinese stock market and petroleum. The latter two studies suggest that regulations imposed by the government with respect to the ownership caps for foreign investors may potentially act as a shield from petroleum price shocks. It is worth noting that the conditional volatility of the Consumer Cyclicals, Financials, and Technology sectors is affected by the past volatility of petroleum, although negatively, but with low magnitude, as shown by the coefficients’ signs (see Supplementary Materials). This observation appears to indicate that periods of relative stability in the petroleum market are associated with a decline of volatility in the three sectors.

The empirical findings depict that volatility spillover effects are more evident between petroleum and the stock sector indices of Brazil and the United States. Shocks or innovations that originated from all considered sectors of both countries, except the Telecommunications Services sector of the United States, significantly affected the conditional volatility of petroleum, thus pointing to the capability of unforeseen events and news in these markets to cause increased volatility in the petroleum market. The transmissions from petroleum to stock sector indices are present only for the Consumer Non-Cyclicals sector of Brazil and the Basic Materials, Industrials, and Technology sectors of the United States. In terms of cross volatility spillovers, it was observed that the past volatility of only the Energy and Utilities sectors of Brazil influenced the conditional volatility of petroleum. For the United States, the past volatility of most sector indices is important in explaining the conditional volatility of petroleum. However, one can note that the corresponding coefficients have negative signs, as in the case of petroleum-exporting countries (see Supplementary Materials). Hence, the rationale provided in the earlier paragraph also holds for the stock sector indices of Brazil and the United States. Interestingly, effects in the contrary direction were not observed. The documented results are in line, albeit partially, with the studies of Malik and Ewing (2009), Arouri et al. (2011a), and Salisu and Oloko (2015) that focus on the United States.

Looking across other petroleum-importing countries presented in panels B to E of Table 3, the volatility spillover effects are country- and sector-specific. There is no evidence indicating the presence of shock or innovation and volatility transmissions between petroleum and the stock sector indices of developing markets such as Chile, India, Indonesia, and Thailand. Wang and Liu (2016) and Ashfaq et al. (2019) also did not observe statistically significant effects for India. On the other hand, Yousaf and Hassan (2019) found some evidence of volatility interactions between petroleum and the aggregate market indices of India and Indonesia, but not of Thailand, during the full sample period. The obtained findings suggest that the considered stock sectors are not sensitive at the individual level, possibly due to their low integration with the world’s petroleum market. In the case of New Zealand and Sweden, the absence of volatility spillovers between petroleum and the stock sector indices of developed markets could be explained by a relatively large consumption of energy from renewable sources, which appears to make them immune to the petroleum price volatility. Some evidence was obtained for South Africa, Taiwan, and Turkey. Specifically, shocks or innovations spill over from the Basic Materials and Technology sectors of Taiwan to petroleum, while the opposite effects from petroleum to the Consumer Non-Cyclicals and Industrials sectors of Turkey were detected. In the case of South Africa, the past volatility of petroleum only impacted the conditional volatility of the Technology sector. Yousaf and Hassan (2019) reported statistically significant results for Taiwan and Vardar et al. (2018) for Turkey and South Africa.

The shocks or innovations originating from petroleum were found to significantly influence the conditional volatility of the Consumer Cyclicals, Consumer Non-Cyclicals, Financials, Healthcare, and Industrials sectors of Australia, the Industrials and Technology sectors of the Eurozone, the Basic Materials, Consumer Cyclicals, Healthcare, Industrials, Technology, and Utilities sectors of Japan, the Energy and Financial sectors of Pakistan, and the Basic Materials and Consumer-Cyclicals sectors of South Korea. The spillover effects in the inverse route were detected for the Telecommunications Services sector of the Eurozone, the Consumer Cyclicals, Consumer Non-Cyclicals, Energy, and Financials sectors of Israel, the Consumer Non-Cyclicals sector of Japan, and the Financials and Industrials sectors of South Korea. In addition, bidirectional transmissions of shocks or innovations between petroleum and stock sector indices were documented for the Energy sector of Australia, the Industrials sector of Israel, the Basic Materials and Consumer Cyclicals sectors of Pakistan, and the Healthcare and Technology sectors of South Korea. The findings provide less evidence with regard to conditional volatility interdependencies. The past volatility of petroleum spilled over to the Consumer Non-Cyclicals and Industrials sectors of Australia, the Consumer Cyclicals and Industrials sectors of the Eurozone, the Energy and Financials sectors of Pakistan, and the Utilities sector of Japan, whilst the past volatility of the Telecommunications sector of the Eurozone and the Consumer Non-Cyclicals, Energy, Financials, and Industrials sectors of Israel impacted the conditional volatility of petroleum. The two-way volatility transmissions were observed only between petroleum and the Basic Materials and Consumer Cyclicals sectors of Pakistan, where the level of petroleum dependency remains high. However, it is necessary to underline that all coefficients of volatility interactions have negative signs (see Supplementary Materials). The results are moderately consistent with the research of Arouri et al. (2012), Sarwar et al. (2019), Sarwar et al. (2020), Yousaf and Hassan (2019), and Belhassine (2020). In the case of Israel, as one can note, shock and volatility spillovers are surprisingly unidirectional from most sector indices to petroleum. The stock market of Israel is strongly integrated with the world’s financial and commodity markets. The large share of petroleum in primary energy consumption (British Petroleum, 2018), and location in the petroleum-rich region with frequent instabilities that the country’s stock market might be exposed to could elucidate the obtained findings.

To summarise, the extensive evidence of significant shock and volatility transmissions between petroleum and the stock sector indices of petroleum-exporting and -importing countries were obtained. The direction and magnitude of spillover effects depend on the considered markets and sectors. This result is in line with the study of Bagirov and Mateus (2022), in the sense that the reactions differ across net exporters and net importers, thereby reflecting structural distinctions between these groups. It is interesting to observe that transmissions generally occur from stock sector indices to petroleum in the case of petroleum exporters and from petroleum to stock sector indices of petroleum importers, excluding countries that have large production levels. In the context of petroleum-exporting countries, this finding points to the forward-looking nature of stock markets, which tend to swiftly incorporate expectations regarding future economic fundamentals ahead of their full transmission into commodity prices. Conversely, in the case of petroleum-importing counties, this underscores the role of petroleum prices as exogenous drivers of input costs, which, as vital intermediate inputs in the industrial processes, impact corporate cash flows and shape monetary policy expectations, ultimately being reflected in equity valuations. The comparatively weaker or non-existent spillover effects identified among certain petroleum-exporting and -importing countries suggest that energy diversification, robust regulatory frameworks, effective risk hedging strategies, and government interventions play a significant role in dampening the transmission of shocks from the petroleum market.

In the group of petroleum-exporting countries, the volatility cross effects are more apparent for the sector indices of Canada and Mexico. Among petroleum-importing countries, volatility interactions were detected more for sector indices of Brazil, the United States, Australia, Israel, Japan, Pakistan, and South Korea.

Table 4. Total number of shock and volatility transmissions, both unidirectional and bidirectional, between petroleum and stock sector indices of petroleum-exporting and -importing countries.

Sectors	Exporters			Importers
	#Countries	#ARCH	#GARCH	#Countries	#ARCH	#GARCH
Basic Materials	5	3	2	14	6	2
Consumer Cyclicals	5	2	2	16	6	4
Consumer Non-Cyclicals	5	2	1	14	7	3
Energy	5	2	0	12	6	3
Financials	6	3	3	19	6	4
Healthcare	2	1	1	8	4	1
Industrials	5	2	1	16	8	4
Technology	1	1	1	9	5	3
Telecommunications Services	1	1	0	4	1	1
Utilities	3	2	1	9	4	3

Notes: The ARCH and GARCH terms represent innovation or shock and volatility spillovers, respectively.

reports the total number of shock and volatility transmissions, both bilateral and unilateral, which considerably vary across sectors of petroleum exporters and importers. The observed shock and volatility spillovers are greater in the Basic Materials and Financials sectors of petroleum exporters and for most sectors of petroleum importers. By contrast, the number of documented interdependencies is small in the Telecommunications Services sector of both country groups and the Healthcare sector of petroleum importers. Overall, the findings reveal that past shocks or innovations arising from sector indices (or petroleum) compared to past volatilities play a more prominent role in comprehending the conditional volatility of petroleum (or sector indices).

4.3. Constant Conditional Correlations

The estimated constant conditional correlations, represented by (${\rho }_{2,1}$), between petroleum and the stock sector indices of petroleum-exporting and -importing countries are positive and statistically significant at conventional levels in all cases, except the Utilities sector of Japan (see Supplementary Materials). In the case of petroleum exporters, as expected, the strongest conditional correlations were observed between petroleum and the Energy sectors of Russia (0.4360), Norway (0.4945), and Canada (0.4972), while the lowest were found between petroleum and the Consumer Non-Cyclicals (0.1023), Consumer Cyclicals (0.1047), and Financials (0.1051) sectors of Saudi Arabia. In the group of petroleum importers, the Energy sectors of the United States (0.4304), the Eurozone (0.4738), and the United Kingdom (0.4907) also have high conditional correlations with petroleum, whereas the Consumer Non-Cyclicals (0.0691) and Healthcare (0.0510) sectors of China, the Basic Materials (0.0320), Financials (0.0343), and Utilities (0.0390) sectors of Pakistan, and Consumer Non-Cyclicals (0.0584) and Healthcare (0.0577) sectors of Japan have the smallest conditional correlations with petroleum. The sector indices of petroleum exporters on average have stronger correlations than those of petroleum importers. The weak constant conditional correlations may be a tempting indicator for potential gains from holding both stock sector index and petroleum assets in one portfolio. However, Arouri et al. (2012) suggest that shock and volatility transmissions should be considered by portfolio managers when identifying optimal portfolio weights and hedging ratios in order to properly manage petroleum risks.

4.4. Diagnostics

The necessary and sufficient condition for consistency of the QMLE for the considered GARCH model is the existence of the second moment, where $\alpha +\beta <1$.6 Tables S4 and S5 in Supplementary Materials show that second moment conditions for stock sector index and petroleum pairs, presented as ${\alpha }_{1,1}+{\beta }_{1,1}$ and ${\alpha }_{2,2}+{\beta }_{2,2}$, respectively, are less than one in all cases. In addition, a series of diagnostic tests were applied for standardised residuals and squared standardised residuals in order to check the adequacy of VAR-GARCH models for all pairs. Overall, the results indicate that serial correlations are mostly absent at the 1% significance level (see Supplementary Materials). However, it is worth mentioning that diagnostic tests in certain cases show some signs of autocorrelations, albeit the derived values are trivial. Thus, one can conclude that the VAR-GARCH models estimated for each pair were properly defined and covered all properties.

5. Optimal Portfolio Weights and Hedge Ratios

Given the considerable shock and volatility spillovers between petroleum and stock sector indices of petroleum exporters and importers, the present study demonstrates the process of managing efficiently risks associated with petroleum price swings by quantifying optimal portfolio weights and hedge ratios based on the conditional volatility estimates of the VAR-GARCH models for each stock sector index and petroleum pairs.

First, applying the methodology of Kroner and Ng (1998), optimal weights of possessing a stock sector index of a petroleum-exporting or -importing country and petroleum in an investment portfolio, where no short-selling strategy is permitted, that seeks to diminish risks while upholding expected returns at the same level are constructed as follows:

$$w_{ij,t}={\displaystyle \frac{h_{jj,t}-h_{ij,t}}{h_{ii,t}-2h_{ij,t}+h_{jj,t}}}$$

(4)

and

$$w_{ij,t}=\{\begin{array}{c}0, if w_{ij,t}<0\\ w_{ij,t}, if 0\le w_{ij,t}\le 1 \\ 1, if w_{ij,t}>1\end{array}$$

(5)

where $w_{ij,t}$ represents the first asset’s weight in a one-dollar portfolio consisting of a stock sector index and petroleum at time t, $h_{ii,t}$ and $h_{jj,t}$ are the conditional variances of assets i and j at time t, respectively, and $h_{ij,t}$ refers to the conditional covariance between two considered assets at time t. Hence, the second asset’s weight in a portfolio is derived as $1-w_{ij,t}$.

Second, in order to reduce the risks of an investment portfolio comprising a stock sector index of a petroleum-exporting or -importing country and petroleum, the hedge ratios between assets i and j are computed following the approach of Kroner and Sultan (1993), which is one of the most widely utilised methods, in the bellow manner:

$${\beta }_{ij,t}= {\displaystyle \frac{h_{ij,t}}{h_{jj,t}}}$$

(6)

In the above equation, a risk-minimising hedge ratio at time t is denoted as ${\beta }_{ij,t}$, that is, a one-dollar ($1) long position taken in the first asset should be hedged by a short position comparable to $${\beta }_{ij,t}$ in the second asset at time t. Thus, it is important to note that hedging is considered effective if the obtained hedge ratios are low or inexpensive (Hammoudeh et al., 2010).

The average values of optimal portfolio weights and hedge ratios associated with petroleum-exporting and -importing countries are conveyed in

Table 5. Average optimal portfolio weights and hedge ratios for petroleum-exporting countries.

Panel A	Argentina		Canada		Denmark		Mexico
	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$
Basic Materials/Petroleum	0.623	0.132	0.470	0.361	-	-	0.577	0.257
Petroleum/Basic Materials	0.377	0.209	0.530	0.326	-	-	0.423	0.327
Consumer Cyclicals/Petroleum	-	-	0.845	0.178	0.740	0.151	0.654	0.185
Petroleum/Consumer Cyclicals	-	-	0.155	0.590	0.260	0.363	0.346	0.313
Consumer Non-Cyclicals/Petroleum	0.671	0.132	0.872	0.142	-	-	0.705	0.167
Petroleum/Consumer Non-Cyclicals	0.329	0.258	0.128	0.570	-	-	0.295	0.345
Energy/Petroleum	0.575	0.267	0.583	0.462	-	-	-	-
Petroleum/Energy	0.425	0.332	0.417	0.550	-	-	-	-
Financials/Petroleum	0.532	0.145	0.883	0.182	0.756	0.174	0.607	0.203
Petroleum/Financials	0.468	0.158	0.117	0.709	0.244	0.461	0.393	0.292
Healthcare/Petroleum	-	-	0.650	0.175	0.698	0.169	-	-
Petroleum/Healthcare	-	-	0.350	0.295	0.302	0.343	-	-
Industrials/Petroleum	-	-	0.840	0.199	0.666	0.237	0.618	0.196
Petroleum/Industrials	-	-	0.160	0.610	0.334	0.424	0.382	0.290
Technology/Petroleum	-	-	0.744	0.167	-	-	-	-
Petroleum/Technology	-	-	0.256	0.391	-	-	-	-
Telecommunications Services/Petroleum	-	-	0.840	0.149	-	-	-	-
Petroleum/Telecommunications Services	-	-	0.160	0.514	-	-	-	-
Utilities/Petroleum	0.523	0.136	0.858	0.196	-	-	-	-
Petroleum/Utilities	0.477	0.136	0.142	0.642	-	-	-	-
Panel B	Norway		Russia		Saudi Arabia
	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$
Basic Materials/Petroleum	-	-	0.496	0.393	0.671	0.126
Petroleum/Basic Materials	-	-	0.504	0.392	0.329	0.242
Consumer Cyclicals/Petroleum	0.627	0.268	-	-	0.636	0.089
Petroleum/Consumer Cyclicals	0.373	0.408	-	-	0.364	0.151
Consumer Non-Cyclicals/Petroleum	0.552	0.283	-	-	0.569	0.103
Petroleum/Consumer Non-Cyclicals	0.448	0.331	-	-	0.431	0.130
Energy/Petroleum	0.367	0.588	0.510	0.452	0.577	0.129
Petroleum/Energy	0.633	0.434	0.490	0.461	0.423	0.171
Financials/Petroleum	0.616	0.315	-	-	0.618	0.088
Petroleum/Financials	0.384	0.452	-	-	0.382	0.139
Healthcare/Petroleum	-	-	-	-	-	-
Petroleum/Healthcare	-	-	-	-	-	-
Industrials/Petroleum	0.647	0.306	-	-	0.585	0.119
Petroleum/Industrials	0.353	0.480	-	-	0.415	0.160
Technology/Petroleum	-	-	-	-	-	-
Petroleum/Technology	-	-	-	-	-	-
Telecommunications Services/Petroleum	-	-	-	-	-	-
Petroleum/Telecommunications Services	-	-	-	-	-	-
Utilities/Petroleum	-	-	0.411	0.384	-	-
Petroleum/Utilities	-	-	0.589	0.273	-	-

Notes: Petroleum is represented by the Brent grade. w and β denote average weights and hedge ratios (long/short), respectively, of assets in the portfolio comprising a stock sector index and petroleum.

(panels A and B) and

Table 6. Average optimal portfolio weights and hedge ratios for petroleum-importing countries.

Panel A	Brazil		China		United Kingdom		United States
	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$	${\mathbf{w}}_{\mathit{t}}$	${\mathsf{\beta }}_{\mathit{t}}$
Basic Materials/Petroleum	0.420	0.300	0.496	0.111	0.502	0.408	0.717	0.165
Petroleum/Basic Materials	0.580	0.231	0.504	0.110	0.498	0.411	0.283	0.365
Consumer Cyclicals/Petroleum	0.447	0.263	0.542	0.080	0.704	0.172	0.745	0.081
Petroleum/Consumer Cyclicals	0.553	0.219	0.458	0.097	0.296	0.369	0.255	0.232
Consumer Non-Cyclicals/Petroleum	0.523	0.230	0.562	0.065	0.808	0.131	0.864	0.039
Petroleum/Consumer Non-Cyclicals	0.477	0.247	0.438	0.085	0.192	0.428	0.136	0.239
Energy/Petroleum	0.387	0.376	0.471	0.135	0.688	0.405	0.574	0.401
Petroleum/Energy	0.613	0.260	0.529	0.122	0.312	0.632	0.426	0.479
Financials/Petroleum	0.430	0.260	0.519	0.074	0.656	0.223	0.713	0.110
Petroleum/Financials	0.570	0.204	0.481	0.082	0.344	0.391	0.287	0.282
Healthcare/Petroleum	-	-	0.559	0.048	0.773	0.131	0.815	0.049
Petroleum/Healthcare	-	-	0.441	0.064	0.227	0.361	0.185	0.207
Industrials/Petroleum	0.507	0.227	0.546	0.078	0.737	0.177	0.776	0.106
Petroleum/Industrials	0.493	0.230	0.454	0.095	0.263	0.413	0.224	0.331
Technology/Petroleum	-	-	0.497	0.078	-	-	0.734	0.096
Petroleum/Technology	-	-	0.503	0.078	-	-	0.266	0.248
Telecommunications Services/Petroleum	-	-	-	-	-	-	0.697	0.094
Petroleum/Telecommunications Services	-	-	-	-	-	-	0.303	0.211
Utilities/Petroleum	0.503	0.209	-	-	0.762	0.140	0.811	0.059
Petroleum/Utilities	0.497	0.211	-	-	0.238	0.358	0.189	0.235
Panel B	Australia		Chile		Eurozone		India		Indonesia
	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$
Basic Materials/Petroleum	0.535	0.259	-	-	0.709	0.241	0.560	0.165	0.499	0.162
Petroleum/Basic Materials	0.465	0.287	-	-	0.291	0.498	0.440	0.205	0.501	0.159
Consumer Cyclicals/Petroleum	0.716	0.182	-	-	0.725	0.202	0.622	0.125	-	-
Petroleum/Consumer Cyclicals	0.284	0.387	-	-	0.275	0.470	0.378	0.200	-	-
Consumer Non-Cyclicals/Petroleum	0.717	0.168	0.737	0.151	0.812	0.150	0.671	0.120	0.575	0.148
Petroleum/Consumer Non-Cyclicals	0.283	0.360	0.263	0.373	0.188	0.511	0.329	0.234	0.425	0.194
Energy/Petroleum	0.498	0.319	-	-	0.746	0.360	0.560	0.119	0.463	0.209
Petroleum/Energy	0.502	0.315	-	-	0.254	0.683	0.440	0.148	0.537	0.179
Financials/Petroleum	0.710	0.176	0.787	0.148	0.622	0.227	0.486	0.166	0.553	0.138
Petroleum/Financials	0.290	0.376	0.213	0.454	0.378	0.362	0.514	0.153	0.447	0.165
Healthcare/Petroleum	0.711	0.166	-	-	0.756	0.158	0.663	0.119	-	-
Petroleum/Healthcare	0.289	0.346	-	-	0.244	0.408	0.337	0.223	-	-
Industrials/Petroleum	0.663	0.203	0.678	0.161	0.711	0.208	0.515	0.161	-	-
Petroleum/Industrials	0.337	0.357	0.322	0.311	0.289	0.455	0.485	0.165	-	-
Technology/Petroleum	-	-	-	-	0.682	0.196	0.606	0.120	-	-
Petroleum/Technology	-	-	-	-	0.318	0.381	0.394	0.183	-	-
Telecommunications Services/Petroleum	0.687	0.163	-	-	0.749	0.176	-	-	-	-
Petroleum/Telecommunications Services	0.313	0.314	-	-	0.251	0.439	-	-	-	-
Utilities/Petroleum	0.696	0.187	0.786	0.132	0.740	0.185	0.541	0.145	-	-
Petroleum/Utilities	0.304	0.368	0.214	0.404	0.260	0.445	0.459	0.166	-	-
Panel C	Israel		Japan		New Zealand		Pakistan		South Africa
	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$
Basic Materials/Petroleum	-	-	0.648	0.101	-	-	0.672	0.023	0.596	0.319
Petroleum/Basic Materials	-	-	0.352	0.179	-	-	0.328	0.052	0.404	0.415
Consumer Cyclicals/Petroleum	0.684	0.103	0.694	0.072	0.768	0.130	0.695	0.039	0.676	0.204
Petroleum/Consumer Cyclicals	0.316	0.215	0.306	0.158	0.232	0.352	0.305	0.098	0.324	0.363
Consumer Non-Cyclicals/Petroleum	0.706	0.119	0.746	0.035	-	-	-	-	0.696	0.198
Petroleum/Consumer Non-Cyclicals	0.294	0.272	0.254	0.103	-	-	-	-	0.304	0.374
Energy/Petroleum	0.630	0.120	-	-	-	-	0.658	0.054	-	-
Petroleum/Energy	0.370	0.199	-	-	-	-	0.342	0.113	-	-
Financials/Petroleum	0.690	0.133	0.603	0.089	0.824	0.124	0.686	0.024	0.691	0.196
Petroleum/Financials	0.310	0.286	0.397	0.132	0.176	0.434	0.314	0.061	0.309	0.366
Healthcare/Petroleum	-	-	0.713	0.038	-	-	-	-	-	-
Petroleum/Healthcare	-	-	0.287	0.095	-	-	-	-	-	-
Industrials/Petroleum	0.680	0.107	0.683	0.098	0.770	0.121	-	-	0.669	0.214
Petroleum/Industrials	0.320	0.222	0.317	0.202	0.230	0.336	-	-	0.331	0.365
Technology/Petroleum	0.687	0.123	0.663	0.087	-	-	-	-	0.631	0.210
Petroleum/Technology	0.313	0.258	0.337	0.167	-	-	-	-	0.369	0.322
Telecommunications Services/Petroleum	-	-	-	-	-	-	-	-	0.585	0.215
Petroleum/Telecommunications Services	-	-	-	-	-	-	-	-	0.415	0.282
Utilities/Petroleum	-	-	0.636	0.013	-	-	0.677	0.028	-	-
Petroleum/Utilities	-	-	0.364	0.024	-	-	0.323	0.063	-	-
Panel D	South Korea		Sweden		Taiwan		Thailand		Turkey
	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$	$w_{\mathit{t}}$	${\beta }_{\mathit{t}}$
Basic Materials/Petroleum	0.624	0.143	-	-	0.715	0.110	-	-	0.457	0.225
Petroleum/Basic Materials	0.376	0.225	-	-	0.285	0.252	-	-	0.543	0.187
Consumer Cyclicals/Petroleum	0.665	0.117	0.617	0.245	-	-	0.656	0.107	0.495	0.205
Petroleum/Consumer Cyclicals	0.335	0.221	0.383	0.364	-	-	0.344	0.203	0.505	0.196
Consumer Non-Cyclicals/Petroleum	0.678	0.092	-	-	-	-	-	-	0.509	0.198
Petroleum/Consumer Non-Cyclicals	0.322	0.183	-	-	-	-	-	-	0.491	0.200
Energy/Petroleum	0.552	0.178	-	-	-	-	0.588	0.211	-	-
Petroleum/Energy	0.448	0.214	-	-	-	-	0.412	0.283	-	-
Financials/Petroleum	0.541	0.145	0.569	0.266	0.683	0.103	0.640	0.118	0.431	0.233
Petroleum/Financials	0.459	0.170	0.431	0.342	0.317	0.218	0.360	0.208	0.569	0.178
Healthcare/Petroleum	0.540	0.086	-	-	-	-	-	-	-	-
Petroleum/Healthcare	0.460	0.101	-	-	-	-	-	-	-	-
Industrials/Petroleum	0.541	0.163	0.565	0.290	-	-	0.648	0.106	0.483	0.205
Petroleum/Industrials	0.459	0.189	0.435	0.360	-	-	0.352	0.190	0.517	0.189
Technology/Petroleum	0.603	0.119	-	-	0.659	0.108	-	-	-	-
Petroleum/Technology	0.397	0.176	-	-	0.341	0.202	-	-	-	-
Telecommunications Services/Petroleum	-	-	-	-	-	-	-	-	-	-
Petroleum/Telecommunications Services	-	-	-	-	-	-	-	-	-	-
Utilities/Petroleum	-	-	-	-	-	-	-	-	-	-
Petroleum/Utilities	-	-	-	-	-	-	-	-	-	-

Notes: Petroleum is represented by the Brent grade. w and β denote average weights and hedge ratios (long/short), respectively, of assets in the portfolio comprising a stock sector index and petroleum.

(panels A to D). The findings indicate that optimal portfolio weights considerably vary among markets and sectors. Turning out to petroleum exporters first, for instance, the average weight for the Financials/Petroleum portfolio of Canada is 0.883, which is the highest value compared to other markets, suggesting that in a $1 portfolio, 88.3 cents should be invested in the Financials sector index and the remaining amount of 11.7 cents in petroleum. The smallest weight is detected for Argentina, where the optimal holding of the Financials sector index in a $1 portfolio is 53.2 cents, with the outstanding amount of 46.8 cents to be invested in petroleum. It can be noted that weights of petroleum exceed those of sector indices in the Basic Materials/Petroleum and Utilities/Petroleum portfolios of Russia, and the Basic Materials/Petroleum portfolio of Canada. On average, optimal holdings of petroleum in portfolios are lower for Canada and higher for Russia, implying that risks associated with petroleum prices are greater for the former than for the latter petroleum-exporting country.

In the case of petroleum-importing countries, average weights for the Financials/Petroleum portfolios range from 0.430 for Brazil to 0.824 for New Zealand. These figures suggest that in the case of Brazil, the amount of funds allocated in a $1 portfolio to the Financials sector index and petroleum should be 43.0 cents and 57.0 cents, respectively. For New Zealand, in a $1 portfolio, the optimal investment in the Financials sector index is 82.4 cents, while the remainder of 17.6 cents should be assigned to petroleum. The highest weight of a sector index was found in the Consumer Non-Cyclicals/Petroleum portfolio of the United States, which equals to 0.864. On the other hand, investors should own more petroleum than a stock sector index in the Basic Materials/Petroleum, Consumer Cyclicals/Petroleum, Energy/Petroleum, and Financials/Petroleum portfolios of Brazil, the Basic Materials/Petroleum, Consumer Cyclicals/Petroleum, Financials/Petroleum, and Industrials/Petroleum portfolios of Turkey, the Basic Materials/Petroleum, Energy/Petroleum, and Technology/Petroleum portfolios of China, the Basic Materials/Petroleum, and Energy/Petroleum portfolios of Indonesia, the Energy/Petroleum portfolio of Australia, and the Financials/Petroleum portfolio of India. Interestingly, the results indicate that the average optimal holdings of petroleum are greater in portfolios that include sector indices of developing markets, namely Brazil, China, India, Indonesia, and Turkey, which implies that risks arising from petroleum prices are lower for these markets than for their developed counterparts. Overall, one can observe that portfolios should consist of more sector indices and less petroleum, excluding some sectors of petroleum exporters and importers where the opposite trends are recorded, in order to minimise risks without reducing expected returns.

The average figures of hedge ratios exhibit considerable variability across different sectors. Starting with petroleum-exporting countries, for example, the average value between petroleum and the Financials sector index of Canada is 0.709, implying that a long position of $1 in a petroleum asset can be hedged with a short position of 70.9 cents in the Financials sector index. However, the average figure of the optimal hedge ratio between the same assets is much lower for Saudi Arabia, suggesting that a $l long position in petroleum should be shorted by approximately 13.9 cents of the Financials sector index. It can be noted that the most efficient strategy to hedge petroleum risks is to take a short position in the Consumer Non-Cyclicals sector index of Saudi Arabia and the least efficient is to use the Financials sector index of Canada. In general, sector indices of Saudi Arabia provide lower costs of hedging exposure to petroleum risks, while the hedging costs are the most expensive in the case of Canada. This outcome is in line with the study of Arouri et al. (2011b) that reports low hedge ratios for Saudi Arabia. Among sector indices of all petroleum-exporting countries, the smallest hedge ratios are between the Consumer Cyclicals and Financials sector indices of Saudi Arabia and petroleum, where long positions of $1 in sector indices can be hedged by short positions of less than 10 cents in the latter.

A glance at petroleum-importing countries illustrates that optimal hedge ratios between petroleum and the Financials sectors vary from 0.454 for Chile to 0.061 for Pakistan, suggesting that a long position of $1 in a petroleum asset should be hedged with short positions of 45.4 cents and 6.1 cents in the Financials sector index of the first and second countries, respectively. The results indicate that it is not desirable to hedge a $1 long position in petroleum with short positions in the Energy sector indices of the Eurozone and the United Kingdom as the associated costs are high. On the contrary, the smallest hedge ratio was obtained between petroleum and the Utilities sector of Japan. More specifically, a $1 investment in petroleum should be shorted by 2.4 cents in the stock sector index to minimise risks. Among petroleum importers, the best hedging opportunities are provided by sector indices of China and Pakistan, whereas the greater costs associated with hedging petroleum risk exposure are offered by sector indices of the Eurozone and the United Kingdom. In addition, it is worth mentioning that, on average, hedge ratios are also small for India, Indonesia, Japan, South Korea, and Turkey. These findings are moderately consistent with those of Yousaf and Hassan (2019) and Belhassine and Karamti (2021), although the authors use different methodologies and aggregate market indices. Comparing sectors, it can be noted that the lowest optimal hedge ratio is between the Utilities sector index of Japan and petroleum, where a $l long position in the sector index should be hedged with a short position of only 1.3 cents in petroleum.

To conclude, the results support the argument stating that the risk-adjusted performance of a portfolio composed of stock sector indices is ameliorated by the addition of a petroleum asset (Arouri et al., 2012). The obtained optimal portfolio weights and hedge ratios differ substantially, thereby confirming the heterogeneity of sectors within petroleum-exporting and -importing countries. It appears that sector indices of Saudi Arabia and China provide lower hedging costs, reflecting their efficiency in hedging exposure to petroleum price risks.

6. Conclusions

Generally, the effects of the petroleum price volatility on sectors of net petroleum-exporting and -importing economies are not expected to be symmetric. From the standpoint of portfolio management, it is essential to be enlightened about the heterogeneity of stock sectors’ sensitivities. Thus, to conduct a sectoral-level examination focusing on the extensive range of markets with different levels of development, following Mateus et al. (2017) and Bagirov and Mateus (2022), a unique approach to manually built sector indices utilising firm-level data is employed. This study scrutinises the volatility transmissions between spot prices of the Brent crude petroleum and self-constructed stock sector indices of seven petroleum-exporting and nineteen petroleum-importing nations using the sample of 5768 unique equities listed in the Basic Materials, Consumer Cyclicals, Consumer Non-Cyclicals, Energy, Financials, Healthcare, Industrials, Technology, Telecommunications Services, and Utilities sectors from 3 January 2005 to 28 September 2018. Empirically, the VAR-GARCH process, where the variance equation follows the specification of Ling and McAleer (2003), is applied to capture bilateral effects. The work is further extended by analysing optimal portfolio weights and hedge ratios computed based on the estimates of models for each petroleum and stock sector index pair.

The empirical results provide evidence of the significant presence of shock and volatility interactions between petroleum and the stock sector indices of petroleum exporters and importers. However, it should be emphasised that the direction and magnitude of transmissions vary across the studied markets and sectors, thereby indicating the importance of the sector-level analysis. Interestingly, spillovers mostly occur from the stock sector indices of petroleum-exporting countries, including importers with large petroleum production levels, to petroleum, pointing to the advantages of monitoring the performance of these countries’ stock markets for potential transformations in the petroleum market. On the other hand, transmissions usually take place from petroleum to stock sector indices in the case of petroleum-importing countries. The volatility cross effects are more evident for sector indices of Canada and Mexico (petroleum exporters), and Brazil, the United States, Australia, Israel, Japan, Pakistan, and South Korea (petroleum importers). The greater number of detected shocks or innovations indicate that past short-term shocks originating from stock sector indices (or petroleum) generally induce more prominent effects on the conditional volatility of petroleum (or stock sector indices) than past volatilities and therefore should be considered by investors and policymakers. For the stock sector indices of countries which are predominantly developing, where some or no interactions were observed, it appears that past own shocks or innovations and volatilities are more important in forecasting the future levels of volatility. Hence, one can conclude that sectors of emerging markets are less sensitive to petroleum price shocks and volatility.

The optimal portfolio weights and hedge ratios vary substantially from one sector to another in petroleum-exporting and -importing countries. The results suggest that formed portfolios should comprise more sector indices, except some that should have greater holdings of petroleum assets. The analysis of hedge ratios indicates that stock sector indices of Saudi Arabia (net exporter) and China (net importer), where no or little evidence of volatility spillovers was documented, provide better opportunities for hedging exposure to petroleum risks. In sum, adding a petroleum asset to a diversified portfolio of stock sector indices enhances its risk-adjusted performance, which is in line with Arouri et al. (2011a, 2012).

Overall, these findings have important implications for international investors and policymakers. Petroleum is among the factors influencing the volatility of stock sector indices in the studied net petroleum-exporting and -importing countries. Therefore, detecting and understanding the vulnerability of sector indices to fluctuations in petroleum prices are vital for the efficient diversification of investment portfolio holdings and management of risks and thus can be utilised to establish investment strategies. Furthermore, policymakers in countries where significant volatility transmissions from petroleum to stock sector indices are reported should account for these effects when implementing appropriate practices.

Building upon the findings of the present study, there are several directions for future research that are worth exploring. First, given the unprecedented turbulence experienced by financial markets due to the global health crisis, ongoing geopolitical tensions, and regional instabilities, it would be insightful to extend the estimation period of the volatility spillover analysis beyond 2018, although the core outcomes of the current study are expected to remain robust. Second, applying dynamic conditional correlation models could provide more granular insights into how correlations, portfolio weights, and hedge ratios change over time in response to market shocks. Finally, dividing the data into sub-samples and testing out-of-sample hedging effectiveness can help evaluate the robustness of hedging strategies across different market conditions, thereby identifying periods where traditional approaches may be less effective.