Interdependence between the BRICS Stock Markets and the Oil Price since the Onset of Financial and Economic Crises

Abstract

In this paper, we use a copula to examine the relationship and dynamic dependence structure between the crude oil market and the BRICS countries’ stock indices expressed through financial crises, from the 2008 global financial crisis to COVID-19, based on daily data. We characterize the long-term relationship as well as the short-term dynamics and represent the interdependence between them. We also study the short-run conditional links through the considered variables under the effects of long-run interactions and the asymmetric volatility spillover relationship. In addition, we establish that the volatility transmission is stubborn and that the impact of the crises and our empirical findings prove that there is fractional co-integration between crude oil and financial markets. We notice that there are lengthy correlations between the variables, as we detect significant bidirectional causal links. In particular, we see positive short-run links and use an optimal copula coefficient to measure the risk spillovers between oil markets and financial markets that represent the dependence structure. For robustness purposes, based on a sliding-window analysis, we complement our investigation with VaR analysis.

1. Introduction

The financial market issues and trades long-term securities that mobilize both domestic and international savings. Securities, which can be issued by either public or private legal entities, are transferable and susceptible to quotation in return for a financing resource with a maturity exceeding 1 year (Rajamani et al. 2022; Khilar et al. 2022; Huang 2016). Bonds and shares represent the two basic types that are traded. All other securities are dissociations, assemblies or derivatives of these two. In fact, this market consists of three sections: the primary market, in which issues are made and issuers and subscribers participate; the secondary market, in which old securities are traded between buyers and sellers; and the derivatives market, in which accompanying operations are carried out through intermediaries. On the primary market, subscribers, which may be companies or states, issue securities, shares or bonds to buyers. On the secondary market, securities that were previously issued on the primary market are traded. The prices can then deviate from the issue value set by the issuer.

The secondary market has two important functions (Jay Ritter 1998). First, it makes resales liquid. This increased liquidity renders securities more attractive to subscribers and thus makes it easier for companies to issue new securities (Qamar et al. 2016). Second, it determines a price for each security, which provides the investor with a valuation of the assets and tells the company at what price it can issue new securities on the primary market. The higher the stock price is, the higher the price at which the company can issue new shares on the primary market, and therefore the more funds it can raise in the primary market. Furthermore, we note that the secondary market constitutes what is called the stock exchange, in which exchanges and quotations are performed. A quotation is an operation that involves fixing daily the value of the securities previously introduced into the stock exchange. To conclude, the whole game of market participants consists of using the market to maximize the overall profitability of their securities portfolio by obtaining or offering liquidity in a judicious manner (DeMarzo et al. 2021). The size and form of the interventions depend, as for stock market speculators, on expectations of the subsequent price of liquidity. Fundamentally, financial markets are exposed to financial risks caused by various large-scale macroeconomic forces and the likelihood of sectoral failures.

The trade-off between risk and return is a fundamental issue in asset pricing theory, which has received much attention from researchers, market investors and analysts. In general, investors require a larger expected return from a security that is riskier (Liu et al. 2020). The financialization of commodity markets is currently a topic of considerable interest to both financial market investors and policy makers. Oil can be considered the most important commodity. In fact, oil derivative contracts always register the highest trading volume of all commodity derivative contracts. They are used heavily for hedging purposes and international portfolio diversification.

Kollias et al. (2013) argued that higher oil prices may be taken as a signal of better business performance and, therefore, lead to higher equity prices. Hamilton (2009) presented a similar argument asserting that increasing oil prices may signal higher economic growth, thereby improving sentiments. The other interactions of stock and oil can be attributed to the emergence of commodity financialization and the use of commodities, including oil as a substitute for traditional financial products in building portfolios or as a hedge against crises (Mensi et al. 2016; Rouatbi et al. 2021).

This research focuses on the BRICS stock markets. As a result, such markets diverged from developed markets (e.g., the US and European markets) during the early stages of the global financial crisis, but re-emerged in the post-crisis period (Aloui et al. 2011; Kenourgios et al. 2011; Alexakis et al. 2016; Burzala 2016; Mollah et al. 2016). As evidence, the five BRICS markets (Brazil, Russia, India, China, South Africa) have been ranked among the world’s potential economic powers given their strong performance during the 2008 global financial crisis.

As their stock market indices have outperformed those of developed economies, they have attracted considerable interest from both domestic and international traders. The BRICS have emerged as a systemic and influential bloc in the recent history of economic and political alliances. They continue to play an increasingly important role in the global economy. In 1990, their weight in the world GDP barely reached 10%, but it had risen to 25.5% by 2018. By the count of the World Trade Organization (WTO), the BRICS’ position in the export world in 2011 more than doubled. BRICS exports grew by more than 500% in just over a decade, while total world exports grew by 195% in the same period. Concomitantly, the size, significance and swift growth of the BRICS economies pose ecological challenges for the whole world. The BRICS are made up of oil-exporting countries (Russia and Brazil) and oil-importing countries (India, China and South Africa).

The question raised is whether investment in oil is both profitable and safe. In addition, what economic conditions would be favorable for such an investment?

The report offers three important contributions to the existing literature. Firstly, it addresses a question that has been largely overlooked by previous studies—how oil price drops and sudden changes affect crises differently. This is a significant gap in the literature, as earlier studies have mostly focused only on changes in oil prices. To build on previous research, this study introduces a new decomposition method suggested by Ready (2018). Secondly, the study uses multi-threshold models to differentiate between small and large oil price changes. These models provide a more precise representation of how crises respond compared to traditional nonlinear autoregressive distributed models (NARDL), which only consider oil price changes. To our knowledge, this study is the first to use these models to explore asymmetry in the BRICS context. Finally, the study’s findings are useful for decision-makers who need to manage state currency depreciation, as well as for investors who want to create a holding of crude oil and a variety of commodities for profits in the shadows and over the long term.

The remainder of the paper is structured as follows: Section 2 provides a brief overview of the relevant literature. In Section 3, we introduce the main concept that we employ in our analysis. Section 4 presents a detailed discussion of the methodology and data sources that we utilized for this study. In Section 5, we describe the data and present our empirical results. Finally, we conclude this paper by offering recommendations based on our findings.

2. Literature Review

The vulnerability of BRIC countries, like that of other emerging markets, is linked to macroeconomic and global market conditions. (Chittedi 2010; Grima and Caruana 2017; Bouri et al. 2019). The integration of BRIC countries with developed economies is growing, according to several studies, while there is proof of considerable financial linkages between BRIC countries and first-world economies (Lehkonen and Heimonen 2014; Kristjanpoller and Bouri 2019).

Multiple studies have addressed the relationships between stock indices, which have been analyzed in hasty examinations using multivariate settings and may be defended by economic aggregates and the spurring behavior in financial markets (e.g., Forbes and Rigobon 2002; Uebele and Ritschl 2009; Aloui et al. 2011, 2013; Gjika and Horváth 2013; Agnieszka Cyndecka 2017). It is therefore necessary to look for reliable measures of co-movements (Bu et al. 2019). Hence, alternative approaches have been used to measure correlated movements in international stock markets.

In order to analyze the dependence structure of the international stock market, we can see that there is some kind of link between fluctuation, volatility, correlation dynamics and tail dependence. Financial modelers are responsible for measuring the dependencies between market returns estimated with a multivariate model. Numerous papers have shown that the correlation between two variables can vary considerably from independence to nonlinear dependence, with complex forms that take all of the characteristics into account (e.g., Hoese and Huschens 2013; Diaz et al. 2016; Huang et al. 2021; Al-Awadhi et al. 2020). Analysts have relied on the simplest dependency measure, the “Pearson correlation”, which is capable of capturing the inherent dependency structure presented between the series and leading to the optimal portfolio.

Among recent academic research, a growing body of literature has studied the asymmetric effect of oil prices on BRICS stock markets. Caporale et al. (2015) investigates the time-varying impact of oil price uncertainty on stock prices in China. Li and Guo (2022) used NARDL to estimate the asymmetric relationship between oil and BRICS stock markets. In the context of Nigeria, Okere et al. (2021) found that while the linear ARDL model showed a positive association between the Nigerian stock market and crude oil prices in both the long and short run, the short-run correlation was notably stronger. Specifically, there was a significant positive correlation in the short run, while the long-run correlation was positive but not statistically significant. However, the nonlinear ARDL results demonstrate that the Nigerian stock market is significantly impacted by positive crude oil price shocks, resulting in an increase in performance. Conversely, negative crude oil price shocks have a considerable positive effect on the Nigerian stock market’s performance.

Given these studies, mixed results exist in the literature regarding the impact of oil prices on BRICS during crises periods, due to different methodologies and scopes and the inability to clarify the role of skewness (Raheem et al. 2020). Gaye Gencer and Demiralay (2016) proceeded to model the daily and one-week market vagaries with the conditional volatilities obtained from the FIAPARCH models and revealed that the results of the Student-t skewed distribution are the best for predicting the one-day VaR for all stock markets. According to Gaye Gencer and Demiralay (2016), this paper used long memory models with the aim of analyzing the long-range dependence and volatility clustering in oil and stock markets in the BRICS countries during the global financial crisis.

Numerous studies have applied long memory models (Narayan et al. 2016; Kumar et al. 2019) and have used ARFIMA-FIAPARCH models to analyze the behavior of BRICS stock markets and crude oil prices during periods of crisis (Dimitriou et al. 2013). They found evidence of long memory in both markets and that the volatility of these markets increased significantly during the crisis period.

The consequences of the COVID-19 crisis and other crises on the complexity of financial markets and/or macroeconomic relationships have been discussed in several articles.

There is a huge literature on BRIC, both in quiet times and in turbulent times (Dash and Singh 2019; Karanasos et al. 2020). The vulnerability of BRIC markets to changes in global economic and financial conditions has also been the subject of studies (Lehkonen and Heimonen 2014; Grima and Caruana 2017; Bouri et al. 2019; Ayadi et al. 2021).

Moreover, oil markets have a most important effect on the economy and finance. Indeed, crude oil is used as an industrial raw material for transportation and industry. On the one hand, it is considered an investment that acts as a portfolio diversifier and asset hedge during calm economic times. On the other hand, it is considered a secure asset through periods of recession and unstable political situations. Furthermore, an increase in the price of oil leads to a higher production cost, which results in an increase in the inflation rate. This phenomenon therefore raises doubts about the appropriateness of crude oil markets to invest in safe assets and negatively affects portfolio risk management, asset allocations and financial instrument prices (Husain et al. 2019; Malik and Umar 2019).

The coronavirus (COVID-19) pandemic, which spread worldwide, was undoubtedly a source of systematic risk, causing dramatic fluctuations in the financial markets (Zhang et al. 2020; Athari and Hung 2022). This crisis has created unprecedented socioeconomic burdens for markets around the world, including Brazil, Russia, India, China and South Africa, the BRICS countries. At the heart of the financial contagion caused by COVID-19 were the Chinese financial markets (Corbet et al. 2021; Liu et al. 2017). Therefore, the effect of COVID-19 on the BRICS stock markets, one of the largest financial markets, should be analyzed from the perspective of pairwise dependencies. In 2020, Goodell (2020) identified the potential economic and social impacts of COVID-19, including challenges to health, employment productivity, tourism, foreign direct investment, banking and insurance, government spending, and financial market impacts. Aloui et al. (2011) examined the measure of the current global crisis and the contagion effects induced by conducting an empirical investigation of the extreme financial interdependences.

Hence, empirical studies have been conducted to identify whether random variables distributed according to unit roots are involved in an equilibrium situation. Only by using co-integration tests (Bernard and Roy 2003) can the relationships between unit root variables be studied. The co-integration phenomenon allows us to detect the presence of a regular correlation over a long time horizon between two non-stationary variables, by integrating latency variables and exogenous variables. The presence of such a correlation indicates that a relationship between the reference market and the target market will allow it to outperform the reference market, thanks to the use of extra information from other markets.

As a result, we discarded the assumption of a performing market, as current prices reflect past prices and are sufficiently informative. In addition, we analyzed the risk management mechanisms and safe-haven properties of oil under different conditions. The authors tested for the possibility of an equilibrium relationship between the variables using the classical co-integration tests developed by Engle and Granger (1987) and Louis Dupont (2009), as crude oil is not assumed to be stationary (Baillie and Bollerslev 1994; Crowder 1994; Diebold et al. 1994). Yet theories based on long-term oil markets have emphasized real factors as the main determinants of their equilibrium level. (Ratti and Vespignani 2016; Ready 2018; Beckmann et al. 2020; Charfeddine and Barkat 2020; Eleyan et al. 2021).

The process of modeling the relationship between oil prices and the transmission of shocks is essential to the financial decision-making of international investors, and helps to minimize the risk associated with stock indices and to build an optimal portfolio. In the past, the growth of oil prices after World War II was the main driver of recession (Hamilton 1983).

It is crucial to observe the impacts of these major events on the connectedness between stock and commodity markets, particularly when the trajectories of stock and commodity prices are characterized by high uncertainty. These interactions between the equity and commodity markets are not the same according to financial, energy or health crises. Indeed, the volatility of equity and commodity prices varies according to the evolution of the price of oil. Boubaker and Raza (2017) observed overall higher co-movements in China, India and South Africa.

The study period (from 4 January 2010 through 20 October 2021) covers several turbulent periods and crises, including all sharp fluctuations in the commodity futures markets and major global events such as the GFC in 2008–2009, the European crisis in 2010–2012, the oil price crash in 2014–2015 and the global health crisis (COVID-19) in 2020. Ji et al. (2020) applied the wavelet-based grapevine copula to the multivariate dependence structure at different frequencies between Chinese stock markets, crude oil, and safe havens including gold, the Swiss franc (CHF) exchange rate, and the Japanese yen (JPY) exchange rate. This shows that the multivariate distribution fluctuates over time and with frequency.

Using a VaR model, Hammoudeh and Aleisa (2004) proved that there is strong bidirectional causality between the Saudi stock market and oil prices. The findings also indicated that the other CCG markets are not directly affected by oil price changes. Zarour (2006) also used a VaR model to examine the link between oil price changes and CCG stock markets and revealed that only the Saudi and Omani markets are predictive of upward oil price changes.

Portfolio risk management must therefore be analyzed from the perspective of its impact during the recent COVID-19 crisis. There is a large body of work that assesses portfolio risk management capacity in the context of five distinct economic and financial events, namely the GFC (12 September 2008 to 31 December 2012), the EDC (4 January 2010 until 31 December 2012), the oil crisis/Brexit (21 August 2015 to 29 September 2019) and the COVID-19 pandemic (2 January 2020 until 9 November 2021).

According to Cong et al. (2008), Value-at-Risk (VaR) is a statistical tool used to measure the potential loss in value of an asset or portfolio of assets over a given time period with a certain level of confidence. It is often used by investors and financial institutions to evaluate and manage risk. Numerous studies show that during times of crisis, VaR can be a useful tool to check the robustness of investment strategies or risk management plans (Alexander and Sheedy 2008; Jorion 2007; Ren et al. 2020).

3. Key Components That Form the Basis of Copulas

The following section presents an overview of the copula functions utilized to measure the association between fluctuations in oil prices and stock market performance. It also outlines the approach taken and the goodness-of-fit test employed to determine the optimal copula for the study.

3.1. Copula Properties

A copula function is used to merge various univariate distributions into a coherent multivariate distribution, preserving the original multivariate distribution’s information.1 Among its advantages, a copula has great flexibility in the implementation of multivariate analysis; it establishes a relationship between the distribution functions of random variables. A copula allows a wider selection of joint distributions of financial series. Indeed, it does not only take linear relationships between variables into account; in other words, it models statistical relationships that are not necessarily linear. In addition, it allows for less restrictive joint probability distributions, taking better account of stylized facts, and enables researchers to study risk management and determine risk measures.

Sklar’s theorem (Sklar 1959) states that the joint distribution function $F$ of two continuous stochastic variables, $x_1$ and $x_2$, can be decomposed into two distributions of individual variables, $F_1$ and $F_2$, and a copula $C$, characterizing the correlation structure between the constituents.

Formally, let $x=(x_1,x_2)$ be a two-dimensional random vector with joint distribution function $F(x_1,x_2)$ and marginal distributions $F_i$, $i=1,2$. A copula is present $C(u_1,u_2)$ in such a manner that:

$$F(x_1,x_2)=C(F_1(x_1),F_2(x_2)).$$

(1)

If both marginal distributions $F_i$ are continuous, then the copula $C(u_1,u_2)$ is uniquely defined according to the theorem.

A key feature of copulas is their ability to detect tail dependence. The upper (right) tail dependence $\left({\lambda }_U\right)$ refers to the positive likelihood of positive outliers to cohere, while the lower (left) tail dependence $\left({\lambda }_L\right)$ refers to the negative likelihood of negative outliers to cohere. Thus, we can formally define ${\lambda }_U$ and ${\lambda }_L$ respectively as

$$\begin{array}{ll}{\lambda }_U& =\underset{u\to 1}{{\displaystyle \lim }} P\left(x_1\ge F_1^{-1}\left(u\right)|x_2\ge F_2^{-1}\left(u\right)\right),\hfill \\ & =\underset{u\to 1}{{\displaystyle \lim }}\frac{1-2u+C(u,u)}{1-u}.\end{array}$$

(2)

$$\begin{array}{ll}{\lambda }_L=& \underset{u\to 0}{{\displaystyle \lim }} P\left(x_1\le F_1^{-1}\left(u\right)|x_2\le F_2^{-1}\left(u\right)\right),\hfill \\ & =\underset{u\to 0}{{\displaystyle \lim }}\frac{C(u,u)}{u}.\end{array}$$

(3)

where $F_1^{-1}\left(u\right)$ and $F_2^{-1}\left(u\right)$ are the marginal quantile functions.

In this study, multiple copula functions, including normal, Student-t, Gumbel, Clayton, and Frank, were examined to analyze the relationship between oil price fluctuations and stock market performance. Additionally, the analysis included the application of survival copulas.

3.1.1. The Normal Copula

The normal copula is one of the copula functions that have been examined in this study, along with the Student-t, Gumbel, Clayton and Frank copulas. The normal copula is specifically the copula for the multinormal distribution, and it is mathematically defined as follows:

$$C_N\left(u_1,u_2\right)={\displaystyle \underset{-\infty }{\overset{{\varphi }^{-1}\left(u_1\right)}{\int }}{\displaystyle \underset{-\infty }{\overset{{\varphi }^{-1}\left(u_2\right)}{\int }}\frac{1}{2\pi \sqrt{\left(1-2{\theta }^2\right)}}\exp \left(-\frac{s^2-2\theta st+t^2}{2\left(1-{\theta }^2\right)}\right)} dsdt}.$$

(4)

where $-1\le \theta \le 1$ is the coefficient measuring linear association. ${\phi }^{-1}$ is the opposite of the standardized normal distribution of the unitary variable. The normal copula exhibits no tail dependence: ${\lambda }_{L_N}={\lambda }_{U_N}=0$.

3.1.2. The Student-t Copula

The Student-t copula is expressed by:

$$C_t\left(u_1,u_2\right)={\displaystyle \underset{-\infty }{\overset{t_v^{-1}\left(u_1\right)}{\int }}{\displaystyle \underset{-\infty }{\overset{t_v^{-1}\left(u_2\right)}{\int }}\frac{1}{2\pi \sqrt{\left(1-2{\theta }^2\right)}}\exp {\left(1+\frac{s^2-2\theta st+t^2}{v\left(1-{\theta }^2\right)}\right)}^{-\frac{v+2}{2}}} dsdt}.$$

(5)

where $-1\le \theta \le 1$ is the coefficient measuring linear association. $t_v^{-1}$ is the opposite of the standard distribution featuring Student-t, with $v>2$. The copula of the latter exhibits the same strength of dependence between variables in both the upper and lower tails of the distribution; thus, it possesses an identical tail dependence: ${\lambda }_{L_t}={\lambda }_{U_t}=-2t_{v+1}\left(\sqrt{v+1}\frac{\sqrt{1-\theta }}{\sqrt{1+\theta }}\right)$.

3.1.3. The Gumbel Copula

The Gumbel copula (Gumbel 1960) is a high-amplitude copula. It is a copula that exhibits different levels of dependence between variables in the upper and lower tails of the distribution, and displays a more pronounced dependence in the upper tail compared to the lower tail. This copula is defined as:

$$C_G\left(u_1,u_2\right)=\exp \left(-{\left[{\left(-\ln \left(u_1\right)\right)}^{\theta }+{\left(-\ln \left(u_2\right)\right)}^{\theta }\right]}^{{\theta }^{-1}}\right).$$

(6)

where $\theta \in \left[1,+\infty \right[$, ${\lambda }_{L_G}=0$ and ${\lambda }_{U_G}=2-2^{-\theta }$.

3.1.4. The Clayton Copula

The Clayton copula, introduced by Clayton in 1978, is another example of a copula that exhibits different levels of dependence between variables in the upper and lower tails of the distribution, with a stronger dependence in the lower tail compared to the upper tail. This copula is given by:

$$C_C\left(u_1,u_2\right)=\max \left\{{\left(u_1^{-\theta }+u_2^{-\theta }-1\right)}^{-{\theta }^{-1}},0\right\}.$$

(7)

where $\theta \in \left[-1,+\infty \right[\backslash \left\{0\right\}$, ${\lambda }_{L_C}{=2}^{-{\theta }^{-1}}$ and ${\lambda }_{U_C}=0$.

3.1.5. The Frank Copula

The Frank copula, proposed by Frank in 1979, is a copula that exhibits the same strength of dependence between variables in both the upper and lower tails of the distribution, and can be expressed as follows:

$$C_F\left(u_1,u_2\right)=-{\theta }^{-1}\ln \left(1+\frac{\left(\exp \left(-\theta u_1\right)-1\right)\left(\exp \left(-\theta u_2\right)-1\right)}{\exp \left(-\theta \right)-1}\right).$$

(8)

where $\theta \in \left]-\infty ,+\infty \right[\backslash \left\{0\right\},$ ${\lambda }_{L_F}=0$ and ${\lambda }_{U_F}=0$.

3.1.6. The Survival Copulas

The survival or rotated copulas are the copula of $\left(1-u_1\right)$ and $\left(1-u_2\right)$ instead of $u_1$ and $u_2$, respectively. A survival copula is a function that quantifies the unequal dependence between the upper and lower tails of the distribution referring to the original function. It is the left-tail dependence of the Gumbel copula that is taken into account by the Gumbel survival copula instead of the right-tail dependence, and the right-tail dependence that is taken into account by the Clayton survival copula instead of the left-tail dependence of the Clayton copula.

• Clayton copula of the survival copula

Survival Clayton copula is obtained by deriving from the Clayton copula. The formula for this copula is:

$$C_{SC}\left(u_1,u_2\right)=u_1+u_2-1+{\left({\left(1-u_1\right)}^{-\theta }+{\left(1-u_2\right)}^{-\theta }-1\right)}^{-{\theta }^{-1}}.$$

(9)

where $\theta \in \left[0,+\infty \right[\backslash \left\{0\right\}$.

• Gumbel copula of the survival copula

The survival Gumbel copula is a type of copula that captures the left-tail dependence of the Gumbel copula instead of the right-tail dependence. This copula is given by:

$$C_{SG}\left(u_1,u_2\right)=u_1+u_2-1+\exp \left(-{\left[{\left(-\ln \left(1-u_1\right)\right)}^{\theta }+{\left(-\ln \left(1-u_2\right)\right)}^{\theta }\right]}^{{\theta }^{-1}}\right).$$

(10)

where $\theta \in \left[0,+\infty \right[\backslash \left\{0\right\}$.

3.2. Estimating the Parameters That Govern the Copula’s Behavior

For this paper, we employed the canonical maximum likelihood (CML) method developed by Genest et al. (1995). This method is considered quasi-parametric, as it does not rely on any assumption regarding the parametric form of the marginal distributions. However, it is important to note that inaccurate estimation of the marginal distribution may impact the parameters of the copula. For approximating more precisely, the CML method does not require specifying the marginal densities, and instead utilizes integral transformations of empirical probabilities to determine the uniform marginals $\left[0,1\right]$ required to determine the copula parameters. A two-step procedure is used for the estimation:

The first step of the estimation process involves transforming the sample $\left(x_{1t},x_{2t}\right),$ $t=1,\dots ,T$ into uniformly distributed variables $\left({\widehat{u}}_{1t},{\widehat{u}}_{2t}\right)$ employing the empirical cumulative distribution function (CDF) distribution ${\widehat{F}}_i\left(•\right)$, denoted as stated:

$${\widehat{F}}_i\left(•\right)=\frac{1}{T+1}{\displaystyle \sum _{t=1}^T{1}_{\left[X_{it}\le •\right]}},\forall i=1,2.$$

(11)

The function is denoted by $1_{\left[X_{it}\le •\right]}$, which is an indicator measure.

In the second step, the values of copula parameters are determined using the following method:

$$\widehat{\theta }=\underset{\theta }{argmax}{\displaystyle \sum _{t=1}^T\ln \left(c\left({\widehat{u}}_{1t},{\widehat{u}}_{2t}\right);\theta \right)}.$$

(12)

where ${\widehat{u}}_{1t}={\widehat{F}}_1\left(x_{1t}\right)$ and ${\widehat{u}}_{2t}={\widehat{F}}_2\left(x_{2t}\right)$ are samples simulated from copula data.

In copula parameter estimation, it is common to encounter the challenge of selecting the optimal copula that fits the dataset most effectively. To accomplish this, we used several information criteria, namely the log-likelihood (LL), the Akaike information criterion (AIC) and the Bayes information criterion (BIC). Furthermore, we used the goodness-of-fit test.

3.3. Goodness-of-Fit Test

A particular copula $C_0$ was utilized to appropriately model the dependence structure of a multivariate distribution. This statement indicates that the goodness-of-fit test proposed by Genest et al. (2009) in 2009 was used to determine if the selected copula provided an appropriate fit for the given dataset. This test is based on comparing the estimated copula with the empirical copula using a measure of difference. The null hypothesis of this test is $H_0:C\in C_0$ for the same class $C_0$ of copulas, and the statistic of this test $S_T$ is based on the distance of Cramér–von Mises, given by:

$$S_T={\displaystyle \underset{0}{\overset{1}{\int }}{\left|\tilde{K}\left(t\right)\right|}^2{k}_{\widehat{\theta }}\left(t\right) }dt.$$

(13)

where $k_{\widehat{\theta }}\left(t\right)$ is the density function associated with $K_{\theta }\left(t\right)$, $\widehat{\theta }$ is the estimator of $\theta$ and $\tilde{K}\left(t\right)$ is Kendall’s process, given by:

$$\tilde{K}\left(t\right)=\sqrt{T}\left(\widehat{K}\left(t\right)-K\left(t\right)\right),\forall 0\le t\le 1.$$

(14)

where $K\left(t\right)$ denotes the univariate distribution function and $\widehat{K}\left(t\right)$ is the empirical distribution function given by:

$$\widehat{K}\left(t\right)=\frac{1}{T}{\displaystyle \sum _{i=1}^T{1}_{\left[U_i\le t\right]}}.$$

(15)

where $U_i=1/\left(T-1\right){\displaystyle {\sum }_{j=1}^T{1}_{\left[X_{1j}\le X_{1i}, X_{2j}\le X_{2i}\right]}}$ for each $i=1,\dots ,T.$

When the observed value $S_T$ surpasses the percentile $\left(1-\alpha \right)\mathrm{th}$ of its distribution, the null hypothesis is defeated. The multiplier approach presented in Kojadinovic and Yan (2011) was used to calculate the p-values related to the test statistic.

3.4. Exchange Point Testing Method

The validation of the selected copula requires a condition of stability of the estimated copula parameter. We must therefore verify that the parameter of the selected copula is constant in the long run. To do this, we used the change point test method developed by Dias and Embrechts (2004).

Note that the size of the sample is $T$ and the observed date $\left(x_{11},x_{21}\right),\dots ,\left(x_{1T},x_{2T}\right)$. Note the interest in testing the null hypothesis of no change point: $H_0:{\theta }_1=\cdots ={\theta }_T$ versus the alternative hypothesis of a change point $H_1:{\theta }_1=\cdots ={\theta }_{k^{\ast }}\ne {\theta }_{k^{\ast }+1}=\cdots ={\theta }_T$, where $k^{\ast }$ is the unique unknown change point position. Whether under the null or alternative hypothesis, all parameters are treated as unknown. If $k^{\ast }=k$ is known, the method for structuring the likelihood ratio statistic to test $H_0$ is explained below:

$$\begin{array}{ll}-2\log {\Lambda }_k& =2\left[{\displaystyle \sum _{i=1}^k\log C\left({\widehat{\theta }}_k;F_1\left(x_{1i}\right),F_2\left(x_{2i}\right)\right)}\right.\hfill \\ & +{\displaystyle \sum _{i=k+1}^T\log C\left({\widehat{\theta }}_{k^{\ast }};F_1\left(x_{1i}\right),F_2\left(x_{2i}\right)\right)}\\ & \left.-{\displaystyle \sum _{i=1}^T\log C\left({\widehat{\theta }}_T;F_1\left(x_{1i}\right),F_2\left(x_{2i}\right)\right)}\right].\end{array}$$

(16)

where ${\widehat{\theta }}_k$, ${\widehat{\theta }}_{k^{\ast }}$ and ${\widehat{\theta }}_T$ are estimated from data using Maximum Likelihood Estimation (MLE).

Yet, if $k^{\ast }$ is unknown, the high value of $Z_T$ will rule out the null hypothesis of no change point.

$$Z_T=\underset{1\le k\le T}{\max }\left(-2\log {\Lambda }_k\right).$$

(17)

Dias and Embrechts (2004) proposed detailed critical thresholds for different levels of significance.

4. Empirical Results

4.1. The Description of the Data

This database included daily crude oil prices and stock indices of BRICS countries for the period from 14 November 2010 to 29 November 2022, giving a total of T = 3483 observations.2 The Energy Information Administration (EIA) determines the price of oil using the Brent crude oil price as a proxy. The data expressed in US dollars used the most significant stock market index for each country as an indicator, extracted from the Bloomberg database.

The data was converted into logarithmic form and treated as first differences to produce series that exhibited stock market returns and oil price changes. Unit root tests, including those with structural breaks, confirmed the stationarity of all returns. Therefore, using the logarithmic difference could make the series stationary. To study the statistical and econometric features of each time series, we examined the logarithm of the returns.

The stock market series and return showed in Figure 1 appear to oscillate randomly around zero, while the variance fluctuates over time with alternation of volatile and tranquil periods; the descriptive statistics of the yield series are presented in

Table 1. Descriptive analysis of returns series.

Countries	DLBRA	DLRUS	DLIND	DLCHI	DLAFR	DLOIL
Mean	0.0008	0.0004	0.0006	0.0005	0.0007	0.0005
Maximum	0.2197	0.2410	0.2065	0.1909	0.0812	0.1758
Minimum	−0.1865	−0.2433	−0.1565	−0.1616	−0.0360	−0.1723
Std. Dev.	0.0222	0.0244	0.0168	0.01839	0.0199	0.0227
Skewness	−0.1968	−0.8368	−0.301	−0.0698	−0.2904	−0.1222
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Kurtosis	13.7022	21.0023	15.1034	12.6939	9.4539	9.0567
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Jarque–Bera	3157.7456	4879.5456	3023.5663	2851.9997	1118.0023	1307.0091
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
$Q\left(10\right)$	17.9923	15.0232	14.0435	11.1728	14.0654	11.0962
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
$Q^2\left(10\right)$	2568.5726	3578.5342	2867.3429	4582.8094	3578.5947	45,643.876
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
ARCH$\left(10\right)$	117.3457	157.9453	139.1563	217.0763	179.1523	217.9937
p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

.

The presented table reveals that the daily average returns of the Brazilian stock market were the highest (0.0008%), while the average price returns of Russia were comparatively lower, following crude oil and China. The Indian and South African stock markets exhibited higher returns than Russia but lower than Brazil. It is noteworthy that the minimum and maximum returns range was the widest for Russia, indicating a higher level of risk in the Russian stock market. This is also evident in the standard deviations. In other words, the analysis reveals that the Russian stock market and crude oil exhibit greater risk compared to the other stock markets. In contrast, India’s stock market experienced the least volatility among the BRICS nations. All the kurtosis was leptokurtic. Mensi et al. (2018) found that China has the highest skewness among the stock markets, indicating frequent small gains and sudden extreme losses. This finding is in line with similar studies (Yoon et al. 2019), which also report excess kurtosis for all energy and stock markets exceeding three. In particular, the Russian stock market has the highest kurtosis.

The Jarque–Bera test yielded compelling evidence to reject the null hypothesis of normality for all series. This finding clarifies the considerable disparity observed between the minimum and maximum values, as well as the calculated mean. The deviation from a normal distribution in the returns suggests a higher occurrence of small negative shocks and large positive shocks compared to what would be expected in a normal distribution.

All series show significant serial correlation as per the Ljung–Box test, and the existence of heteroscedasticity was discovered by the ARCH-LM test, resulting in a rejection of the null hypothesis at the 1% level. Given the heteroscedasticity, we aimed to identify the appropriate type of ARCH model for each stock market return.

4.2. Marginal Distributions

The selected estimation method followed a two-stage approach: modeling of the marginal distributions and the dependence structure.

For the first stage, it focused on analyzing the marginal distributions of the variables.

To achieve this, we conducted a preliminary examination of the individual distributions of each variable, considering their specific characteristics and properties. This analysis helps to understand the behavior and properties of each marginal distribution separately, then how it contributes to the overall dependence structure.

For the second stage, we focused on modeling the dependence structure between variables.

This involves exploring the relationships, associations, and patterns of dependence among the variables. We examined how the variables interacted with each other and determined the strength and direction of their dependencies.

Various statistical techniques and models can be employed to capture the dependence structure, such as copulas. The goal is to understand and quantify the interrelationships between variables, which can provide valuable insights for further analysis and decision-making.

Moreover, there was no obvious restriction in the spectral densities associated with the data, thus signaling the distinct likelihood of long memory of both mean and variance to occur.3 To address this issue, we employed the autoregressive fractionally integrated moving average-fractionally integrated asymmetric power autoregressive conditionally heteroskedastic (ARFIMA-FIAPARCH) model developed by Tse (1998) for estimation purposes. This model is specifically designed to capture long memory effects and asymmetries in the data, providing a suitable framework for analyzing and understanding the dynamics of the series. Apart from capturing long-run dependence, the ARFIMA-FIAPARCH model offers an additional advantage over the ARFIMA-FIGARCH model. It is capable of capturing certain stylized properties commonly observed in oil price changes and stock market returns, including fat tails and leverage effects. These characteristics play a crucial role in accurately representing the dynamics of the data and providing a more comprehensive understanding of the underlying processes. In fact, in the conditional mean equation, the ARFIMA $\left(p,d_m,q\right)$ process is provided by:

$$\begin{array}{r}\varphi (L)(1-L{)}^{d_m}(u_t-\mu )=\theta (L){\epsilon }_t,\\ {\epsilon }_t\left|I_{t-1}\right.\sim N(0,h_t).\end{array}$$

(18)

where $\mu >0$ is a constant, $0\le d_m<1/2$ is the fractional integration parameter, $L$ is the lag operator, $\varphi (L)=1+{\varphi }_{1}L+\dots +{\varphi }_p{L}^p$ and $\theta (L)=1+{\theta }_{1}L+\dots +{\theta }_q{L}^q$ are polynomials of order $p$ and $q$ respectively. The parameters p and q represent the orders of an autoregressive (AR) model and a moving average (MA) model, respectively. For the model to be stationary and invertible, the roots of the characteristic equation corresponding to the AR component should be distinct and located outside the unit circle. Similarly, the roots of the characteristic equation corresponding to the MA component should also be distinct and lie outside the unit circle. $I_{t-1}$ is the information set available at time $t-1$. The innovations of the ARFIMA process are assumed to be normally distributed.

The FIAPARCH process is employed to model the conditional variance, expressed as follows:

$$h_t^{\delta /2}=w+\left(1-{\left(1-\beta \left(L\right)\right)}^{-1}\varpi \left(L\right){\left(1-L\right)}^{d_v}\right){\left(\left|{\epsilon }_t\right|-\gamma {\epsilon }_t\right)}^{\delta }.$$

(19)

where $w>0$ is a constant, $0\le d_v<1$ is the fractional integration parameter, $\beta (L)=1+{\beta }_{1}L+\dots +{\beta }_P{L}^P$ and $\varpi (L)=1+{\varpi }_{1}L+\dots +{\varpi }_Q{L}^Q$ are polynomials of order $P$ and $Q$, respectively, whose roots are distinguished and located outside the unit radius. $\delta >0$ is the power term used in the Box–Cox transformation of the conditional standard deviation. $h_t^{\delta /2}$. $-1<\gamma <1$ is the leverage coefficient that is based on the asymmetric effect. When $\gamma >0$, negative shocks induce greater volatility compared to positive shocks. When $\gamma <0$, the magnitude of the shocks is stated by the term $\left(\left|{\epsilon }_t\right|-\gamma {\epsilon }_t\right)$. When $\gamma =0$ and $\delta =2$, the process of Equation (19) shrinks to a FIGARCH $(P,d_v,Q)$ process.

Figure 2 displays the estimated residuals and standardized residuals for the stock market series throughout the analyzed time period.

Table 2. The estimates generated by the ARFIMA-FIAPARCH model.

	DLBRAZIL	DLRUSSIA	DLINDIA	DLCHINA	DLAFRICA	DLOIL
$\left(p,d_m,q\right)$	$\left(0,d_m,0\right)$	$\left(0,d_m,0\right)$	$\left(1,d_m,1\right)$	$\left(1,d_m,1\right)$	$\left(0,0,0\right)$	$\left(1,0,1\right)$
$\left(P,d_v,Q\right)$	$\left(1,d_v,1\right)$	$\left(1,d_v,1\right)$	$\left(1,d_v,1\right)$	$\left(1,d_v,1\right)$	$\left(1,d_v,1\right)$	$\left(1,d_v,1\right)$
$\mu$	0.000 ***	0.000 ***	0.001 ***	0.001 ***	0.000 ***	0.002 ***
	(3.462)	(3.983)	(4.455)	(3.982)	(4.323)	(3.764)
$d_m$	0.068 ${}^{\ast \ast \ast }$	0.078 ${}^{\ast \ast \ast }$	0.132 ${}^{\ast \ast \ast }$	0.094 ${}^{\ast \ast \ast }$	0.104 ${}^{\ast \ast \ast }$	0.152 ${}^{\ast \ast \ast }$
	$\left(2.967\right)$	$\left(2.689\right)$	$\left(3.672\right)$	$\left(3.498\right)$	$\left(3.577\right)$	$\left(2.987\right)$
${\varphi }_1$			−0.758 ${}^{\ast \ast \ast }$	−0.549 ${}^{\ast \ast \ast }$		−0.645 ${}^{\ast \ast \ast }$
			$\left(-6.588\right)$	$\left(-3.276\right)$		$\left(-9.365\right)$
${\theta }_1$			0.720 ${}^{\ast \ast \ast }$	0.612 ${}^{\ast \ast \ast }$		0.771 ${}^{\ast \ast \ast }$
			$\left(5.837\right)$	$\left(2.782\right)$		$\left(9.511\right)$
$\omega$	0.000 ${}^{\ast \ast \ast }$	0.000 ${}^{\ast \ast \ast }$	0.000 ${}^{\ast \ast \ast }$	0.000 ${}^{\ast \ast \ast }$	0.000 ${}^{\ast \ast \ast }$	0.000 ${}^{\ast \ast \ast }$
		$\left(2.761\right)$
$d_v$	0.467 ${}^{\ast \ast \ast }$	0.486 ${}^{\ast \ast \ast }$	0.4830 ${}^{\ast \ast \ast }$	0.532 ${}^{\ast \ast \ast }$	0.463 ${}^{\ast \ast }$	0.482 ${}^{\ast \ast \ast }$
	$\left(3.616\right)$	$\left(3.623\right)$	$\left(7.938\right)$	$\left(8.724\right)$	$\left(2.310\right)$	$\left(6.736\right)$
$\gamma$	0.278 ${}^{\ast \ast }$	0.395 ${}^{\ast \ast \ast }$	0.242 ${}^{\ast \ast \ast }$			0.637 ${}^{\ast \ast \ast }$
	$\left(2.079\right)$	$\left(2.295\right)$	$\left(2.618\right)$			$\left(3.040\right)$
$\delta$	1.595 ${}^{\ast \ast \ast }$	1.938 ${}^{\ast \ast \ast }$	2.112 ${}^{\ast \ast \ast }$			1.821 ${}^{\ast \ast \ast }$
	$\left(8.117\right)$	$\left(10.600\right)$	$\left(15.180\right)$			$\left(8.942\right)$
${\beta }_1$	0.462 ${}^{\ast \ast \ast }$	0.698 ${}^{\ast \ast \ast }$	0.748 ${}^{\ast \ast \ast }$	0.768 ${}^{\ast \ast \ast }$	0.739 ${}^{\ast \ast \ast }$	0.517 ${}^{\ast \ast \ast }$
	$\left(3.044\right)$	$\left(10.960\right)$	$\left(10.720\right)$	$\left(11.320\right)$	$\left(6.763\right)$	$\left(3.548\right)$
${\varpi }_1$	0.192 ${}^{\ast \ast \ast }$	0.338 ${}^{\ast \ast \ast }$	0.095 ${}^{\ast \ast \ast }$	0.169 ${}^{\ast \ast \ast }$	0.259 ${}^{\ast \ast }$	0.238 ${}^{\ast \ast \ast }$
	$\left(3.040\right)$	$\left(4.646\right)$	$\left(3.819\right)$	$\left(4.027\right)$	$\left(2.462\right)$	$\left(2.778\right)$
Skw	0.375	−0.478	−0.902	0.984	1.853	−0.576
Ex. Kurt	2.325	2.478	4.351	3.568	1.467	0.789
$Q\left(20\right)$	11.458	17.356	17.341	11.342	14.458	16.734
$Q^2\left(20\right)$	15.456	18.457	22.277	16.350	13.283	12.4564

Note: The values in parenthesis are the t-Student. Skw is Skewness. Ex. Kurt is Excess of Kurtosis. $Q\left(20\right)$ is the Ljung-Box statistic for serial correlation in returns for order $20$. $Q^2\left(20\right)$ is the Ljung-Box statistic for serial correlation in squared returns for order $20$. ${}^{\ast \ast }$ and ${}^{\ast \ast \ast }$ denote significance at the 5% and 1% levels respectively.

showcases the estimates of the ARFIMA-FIAPARCH model obtained using the quasi-minimum likelihood method. The significance of the two fractional integration parameters, $d_m$ and $d_v$, is observed in all series. This indicates the existence of long-run dependence in the average returns of the BRICS stock markets and the volatility of both the BRICS stock market returns and oil price changes.

The Ljung–Box statistical test, applied to both yields and squared yields with a lag of 20, reveals no evidence of serial correlation or heteroscedasticity. This indicates that the ARFIMA-FIAPARCH method is suitable for modeling the dynamics of the marginal distribution. The presence of skewness and excess kurtosis in the standardized residuals indicates that the tail of the conditional distribution is thicker than that of a normal distribution. This observation supports the earlier findings discussed in the previous section. Consequently, we move forward to analyze the values of the innovations obtained from the ARFIMA-FIAPARCH model.

4.3. Dependence Structure over the Entire Period

Throughout the entire duration $\left[1,T\right]$, we examined the dependency structure between adjusted oil price movements and adjusted stock market returns for each country using the aforementioned copulas.4 We selected the strongest copula, having the smallest coefficient, $\mathrm{LL}$,5 AIC and BIC. The selection of the copula was further validated by the goodness-of-fit test presented in

Table 3. The copula parameters and measures of tail dependence are examined for the complete time span of the dataset.

	Selected Copula	$\widehat{\mathit{\theta }}$	${\widehat{\mathit{\lambda }}}_{\mathit{L}}$	${\widehat{\mathit{\lambda }}}_{\mathit{U}}$	LL	AIC	BIC	${\widehat{\mathit{S}}}_{\mathit{T}}$ [p-Values]
BRAZIL	Clayton	0.0199	0.000	0	−10.301	−20.609	−20.590	0.015 [0.251]
RUSSIA	Gumbel	1.145	0	0.161	−12.440	−24.889	−24.880	0.030 [0.390]
INDIA	Survival Gumbel	1.131	0.150	0	−16.790	−33.590	−33.581	0.024 [0.322]
CHINA	Survival Gumbel	1.146	0.165	0	−14.669	−29.356	−29.340	0.024 [0.360]
SOUTH AFRICA	Survival Gumbel	1.170	0.194	0	−16.253	−32.471	−32.475	0.034 [0.237]

Notes: ${\widehat{S}}_T$ represents the Cramér–von Mises statistic calculated using Equation (13), and the corresponding p-values are enclosed within parentheses. The p-values were obtained using the multiplier approach outlined in Kojadinovic and Yan (2011).

.

Based on the observed returns, the selected copulas exhibit a significant fit to the data at the 1% level of significance. Table 3 provides details on the chosen copula, its corresponding parameter, and the tail correlation coefficients.

The findings indicate that the copula values are positive across all countries, implying a positive correlation between oil price increases and stock price appreciation. The positive relationship between oil prices and stock prices can be attributed to shocks that stimulate global consumption of industrial raw materials, resulting in higher prices for both oil and stocks (Kilian and Park 2009). This positive relationship between oil prices and stock prices can also be justified by the positive linkage of both variables to the global economic scenario (Aloui et al. 2013). The global economic conditions can impact both oil prices and stock prices, leading to a positive correlation between the two.

4.4. Testing the Change Point

In this analysis, we considered the alternative hypothesis of a single change point as an alternative proposition to the null hypothesis that the selected copulas have no change points. Empirical investigations demonstrate that the observed results of $-2\log {\Lambda }_k$, computed using Equation (16), exceeded the predefined threshold values.6 Given these circumstances, the null hypothesis can be rejected in favor of the alternative hypothesis of a single change.

Table 4. Points Change Dates.

Countries	${\widehat{\mathit{T}}}_1$
BRAZIL	1 March 2020
RUSSIA	1 June 2020
INDIA	29 June 2020
CHINA	9 May 2020
SOUTH AFRICA	2 July 2020

presents an overview of the dates corresponding to the identified change points.

The observed changes in 2020, which exhibited a similar pattern across countries, may coincide with the onset of the COVID-19 pandemic.

4.5. An Analysis of the Dependency Structure over Different Sub-Periods

Subsequently, we revisited the analysis of the dependency structure between filtered oil price fluctuations and filtered stock market returns for each country during two distinct sub-periods: a pre-financial crisis period characterized as a quiet period $\left[1,{\widehat{T}}_1\right]$, and a post-financial crisis period referred to as a crisis period $\left[{\widehat{T}}_1,T\right]$. For each respective time interval or segment, we found that the copula utilized for the entire period remained consistent.7 

Table 5. The copula parameter estimates and tail dependence coefficients for each specific time interval.

Countries	Quiet Period (Pre-Financial Crisis Period)			Crisis Period (Post-Financial Crisis Period)
	${\widehat{\theta }}_1$	${\widehat{\lambda }}_{L,1}$	${\widehat{\lambda }}_{U,1}$	${\widehat{\theta }}_2$	${\widehat{\lambda }}_{L,2}$	${\widehat{\lambda }}_{U,2}$
RUSSIA	0.000	0.000	0	0.204	0.034	0
INDIA	0.071	0.000	0	0.107	0.003	0
CHINA	1.123	0	0.147	1.109	0	0.133
SOUTH AFRICA	1.112	0.135	0	1.131	0.154	0
BRAZIL	1.121	0.144	0	1.160	0.183	0

displays the obtained results.

During the crisis period, it is observed that the copula parameters and tail dependence coefficients are higher compared to the calm period. This indicates an increased correlation between oil prices and stock market returns during the crisis, suggesting the presence of a spillover effect. These findings have significant implications for investors, particularly in the context of the COVID-19 crisis. The observed contagion effect indicates that the potential benefits of diversification through crude oil investments are reduced during this period. Investors should be mindful of the heightened correlation between crude oil prices and stock market returns, which can limit the effectiveness of diversification strategies. It is important for investors to carefully assess the risks and adjust their investment approaches accordingly, taking into account the changing dynamics of the market during such crisis situations. According to Boubaker and Raza (2017), there is evidence of generally higher co-movements in China, India and South Africa. However, these findings differ from the results reported by Büyüksahin et al. (2009) and Büyükşahin and Robe (2014). The disparities in the findings could be attributed to various factors, such as differences in the time periods examined, methodologies employed, or specific characteristics of the data analyzed. These findings support the conclusions of Martín-Barragan et al. (2015), demonstrating that correlations between oil and stock markets tend to generate contagion during times of crisis, while remaining stable and close to zero during calm periods. This suggests that an increase in the correlation between these markets can serve as an early indicator of a potential emerging crisis and positive spillover effects in the distribution of aggregate demand. Consequently, utilizing oil prices as a forecasting tool for economic activity measures holds promising potential, as highlighted by Bachmeier et al. (2008).

4.6. Value-at-Risk Study

This section examined how the dynamic dependence of the proposed copula model can improve the accuracy of market risk predictions for a portfolio consisting of equally weighted energy and BRICS equity markets. As a measure of market risk for the portfolio, by using value-at-risk (VaR), there is a focus on the portfolio, as highlighted by Chen et al. (2019).

To estimate the VaR, Monte Carlo simulations were used as an alternative to the analytical method, which is only applicable for Gaussian copula models. By utilizing Monte Carlo simulations, copula functions can be considered to capture the dependence structure between the two variables. This approach permits the construction and simulation of random scenarios from the joint distribution of the variables, incorporating the chosen marginals and the desired type of dependence structure.

The VaR represents a forecast of a specific percentile, typically in the lower tail, of the distribution of portfolio returns over a given time period. By using the copula model and Monte Carlo simulations, we can generate a range of scenarios and estimate the potential losses the portfolio may experience at different confidence levels.

Considering the dynamic dependence captured by the copula model and employing Monte Carlo simulations will improve the accuracy of market risk forecasts for the BRICS portfolio. This provides valuable insights for portfolio managers and investors in terms of risk management and decision-making.

At time t, the VaR of a portfolio, with confidence level $1-\alpha$, is defined as $Va{R}_t\left(\alpha \right)=\inf \left\{x:F_{{\tilde{r}}_t}\ge \alpha \right\}$, where $F_{{\tilde{r}}_t}$ is the distribution function of the portfolio return ${\tilde{r}}_t$ and $\alpha \in \left(0,1\right)$; as a result, $Pr\left({\tilde{r}}_t\le VaR{\left(\alpha \right)}_t\right)=\alpha$.

To compute VaR using this technique, the following steps are involved. Firstly, simulate dependent uniform variates from the fitted copula model and converted them into standardized residuals by applying the inverse of the semi-parametric marginal cumulative distribution function (CDF) for each index. Then, use these simulated standardized residuals to calculate returns by incorporating the FIAPARCH volatility and the ARFIMA parameters observed in the original return series.

Once having the simulated return series $r_t$, proceed to compute the value of the global portfolio for each pair. To assess the accuracy of the VaR estimates, perform a backtest at the 99% and 99.5% confidence levels using the following procedure:

• Begin by estimating the model using the first 1800 observations.
• Simulate 2000 values of the standardized residuals.
• Estimate the VaR and count the number of losses that exceed the estimated VaR values.
• Repeat this procedure until the last observation, comparing the estimated VaR with the actual next-day value change in the portfolio.

Given the computational cost of this procedure, it is repeated only once every 75 observations.

Table 6. Off-sample measurement performance.

Countries		$\mathit{\alpha }=1\mathbf{\%}$		$\mathit{\alpha }=0.05\mathbf{\%}$
	Backtest	Proportion	Number	Proportion	Number
BRAZIL	Normal	0.097	85	0.092	79
	Clayton	0.089	78	0.085	70
RUSSIA	Normal	0.105	98	0.098	92
	Gumbel	0.096	89	0.087	82
INDIA	Normal	0.114	103	0.103	97
	Survival Gumbel	0.105	94	0.097	88
CHINA	Normal	0.092	94	0.086	86
	Survival Gumbel	0.087	86	0.080	78
AFRICA	Normal	0.086	83	0.077	76
	Survival Gumbel	0.078	75	0.071	69

allows the evaluation of the accuracy of the VaR estimates and assessment of their performance over time.

Turning to the tail dependence, there is evidence of significant tail dependence between oil price changes and stock market returns. The tail dependence indicates extreme co-movements and means that oil price changes and stock market returns crash or boom together in BRICS countries. As an additional description, it is relative to the oil position of the country, its oil consumption and the importance of oil to its national economy.

These findings hold significant implications for portfolio managers, as they can leverage the information on time–frequency co-movements to construct optimal portfolios. By considering the lagging or leading behavior of oil in different time horizons, managers can devise effective hedging strategies. The observed relationship also suggests that oil prices can serve as a valuable tool for predicting stock market movements in the BRICS countries, particularly in China and India, given their significant oil consumption. Policy makers can take advantage of this relationship to foster the development of the BRICS stock markets by targeting specific economic sectors that are closely linked to oil prices.

The BRICS economies, comprising Brazil, Russia, India, China, and South Africa, play significant roles as both consumers and producers in the global oil and natural gas markets. In 2019, China ranked second, India ranked third, Russia ranked sixth and Brazil ranked seventh among the largest oil consumers worldwide. Similarly, Russia and China ranked second and third, respectively, in terms of natural gas consumption in the same year. Russia, in particular, stands as the third-largest oil producer globally and possesses the largest natural gas reserves, estimated at 38 trillion cubic meters. These statistics highlight the substantial influence and importance of the BRICS economies in the global energy landscape.

5. Conclusions

In this paper, the dynamic dependence structure between the BRICS stock markets and the daily oil price was examined using the copula approach. First, The ARFIMA-FIAPARCH model was estimated for each stock return and for oil. Therefore, we focus on the dependence structure using copula analyses to investigate co-movement in the BRICS stock market. This analysis is crucial for understanding the power of co-movement in the BRICS stock market and its relationship to the price of crude oil, providing useful information for both short-term traders and long-term investors. In this research, a Gumbel copula was applied because it is the most frequently applied in market risk analysis; its usefulness derives from the fact that it can capture asymmetric tail dependence.

The long memory models show that, for each pair, there is strong evidence of asymmetric dynamic dependence and even dynamic dependence between series. Indeed, we found evidence of lower tail dependence in all the countries, which means that the stock market returns and oil price changes crash at the same time. Therefore, when investing in BRICS stock markets, investors should pay attention to signs of oil price variations when selecting their portfolios. Thus, the stability of the estimated copula parameter was analyzed by employing the change point test method suggested by Dias and Embrechts (2004). While long memory models have been shown to be useful for analyzing the behavior of BRICS stock markets and crude oil prices during periods of crisis, there are also limitations to these models. Multiple studies discussed the limits of long memory models (Canova and Ciccarelli 2013; Bekiros et al. 2015) and found that while the ARFIMA model can capture long memory and volatility clustering, it may not be able to account for structural breaks or trends in the data.

The empirical results reveal the existence of a change point for all countries. In particular, we observe that the copula parameters and tail dependence coefficients were significantly in excess of normal during the COVID-19 period, hence indicating the presence of a contagion effect. These results can help investors to diversify their portfolios and better manage risk. For robustness purposes, based on a rolling window analysis, a VaR analysis completes this paper.

Oil prices drive stock market developments in the BRICS countries and are therefore an important indicator of a stock market crash. Since the relationship between crude oil and stock markets is demand driven, policy makers need to closely monitor overall economic activity. This is because an increase or decrease in economic activity leads to an increase or decrease in energy consumption, which in turn affects the stock markets. Thus, policy makers can follow the evolution of the oil market through the financial market.

In other words, the BRICS markets are efficient, and the stakeholders anticipate shocks before they occur and cause a financial crisis. For instance, India and China are the indices in which the hedging opportunities with oil are the most advantageous under any scenario. The results are critical for policy makers, risk managers and international portfolio investors who operate with different investment scenarios. Portfolio managers and risk managers can use them to understand better the dynamics of the relationship between oil and equity markets, make more accurate forecasts of volatility across all markets and build hedging strategies according to their investment horizon. Policy makers will be able to identify the right energy policies, accurately define oil policies and forecast their impact on equity markets.

As limits, the estimators used in this paper are linear. The fact is, in spite of the fact that the link between the oil price and the BRICS stock market has been broadly studied, the findings are far from uniform. One of the reasons is that many researchers make the symmetric assumption of two variables, when in reality, deviation is more common. Nevertheless, in economics, nothing is fully linear. Most socioeconomic phenomena are path-dependent. It will be duly considered in our forthcoming research pursuits. As Hamilton (2011) explains, a nonlinear modeling approach would be preferable.

A possible extension for future research is to use NARDL, an asymmetric estimator, and the most advanced models, such as regime-switching processes or a dynamic D-vine copula model, to analyze multivariate dependence structures, as well as to use the structural VAR (SVAR) approach of Kilian and Park (2009) to assess the influence of different crude oil shocks (i.e., demand and supply shocks) on equity markets.