The relationship between geopolitical risk and crude oil prices: evidence from nonlinear and frequency domain causality tests

ABSTRACT

This paper aims to investigate the possible explanatory effect of geopolitical risks on the oil price changes from February 1986 to December 2019 by employing the nonlinear bivariate Granger causality test and frequency domain Granger causality test based on the geopolitical risk index. The results suggest that there exists a nonlinear causality from geopolitical risks to crude oil prices. Moreover, geopolitical risks have a short-term impact on oil prices, less than 12 months. Actual geopolitical events have a smaller and less lasting impact on oil prices than pure geopolitical risks.

KEYWORDS:

• Geopolitical risk
• oil prices
• nonlinear analysis
• Granger causality
• frequency domain

JEL Codes:

• C22
• G15
• G18

1. Introduction

Due to its financial and political characteristics, crude oil is viewed as a strategic resource with tight ties to daily life, the economy, and national security, and its price and output play a significant role in the global economy (Su et al., 2021; Sweidan, 2021). Therefore, it has been a popular subject of research for so many years (Cunado et al., 2020). As an internationally traded commodity, crude oil is comparable to other commodities whose prices are affected by supply-side and demand-side factors (Kilian & Park, 2009; Kilian, 2009). According to British Petroleum (B.P.) data, crude oil accounted for one-third of the world’s energy consumption in 2016. Understanding the causes of oil price fluctuations is crucial, as the price of oil has a significant impact on economic output (Hamilton, 1996), inflation (Chen, 2009; Darby, 1982), stock markets (Kilian, 2009; Narayan & Narayan, 2010; Park & Ratti, 2008; Wen et al., 2019), exchange rate markets (Basher et al., 2016; Mensi et al., 2017), money market (Su et al., 2021), industrial production fluctuation (Kisswani, 2021) and fundamental industries (Jiao et al., 2012; Li et al., 2017). In addition, non-fundamental factors such as economic policy uncertainty, market speculation, investor interest, and political risk tend to have a significant impact on the crude oil price (He et al., 2019; Narayan et al., 2017; Yao et al., 2017; Zhang, 2013). Consequently, ensuring the availability of oil in the arteries of industrialised nations has become a crucial concern (Sweidan, 2021). In recent years, however, due to its financial and political characteristics (Su et al., 2021), the Geopolitical Risk Index (GPR) is frequently cited as the deciding factor in investment decisions by commercial investors, financial journalists, and central bank governors (Caldara & Iacoviello, 2019). In addition to economic and policy uncertainty, GPR is also a form of uncertainty (Gkillas et al., 2022; Olanipekun & Alola, 2020). Consequently, the relationship between oil and GPR has attracted the attention of numerous researchers from the perspective of international space and national political patterns.

Huge fluctuations in crude oil prices have been correlated with a shifting and unpredictable worldwide landscape, which has been marked by significant geopolitical events. The oil market is closely related to the global political landscape, particularly when dealing with the evolving GPR; oil exporting countries will also be constrained by investment options and should consider taking some reflective measures (Olanipekun & Alola, 2020); and the relationship between the two affects national interests and global development (Su et al., 2021). Due to the scarcity of oil and spatial heterogeneity, a number of conflicts and wars broke out, including the Libyan War (Ji & Guo, 2015), the Gulf War, and the Gulf War (Lieber, 1992). These conflicts have had a substantial effect on oil prices. The escalation of geopolitical and economic tensions is currently the greatest worldwide threat (Selmi et al., 2020). As the proxy of geopolitical risk, Figure 1 shows the GPR index surges around major events like the Gulf War, 9/11, the 2003 Iraq invasion, the Arab Spring, the Russia-Ukraine crisis of 2014, and the Paris terrorist attacks of 2015. The emergence of GPR effectively characterises geopolitical risks (Alqahtani & Klein, 2021; Smales, 2021). During periods of high geopolitical risk, the corresponding international crude oil prices have risen or declined substantially; namely, geopolitical risks are connected with the price of crude oil (see Figure 1). Specifically, we estimate the Spearman rank correlation coefficient between the GPR index and Brent oil price is 0.215, demonstrating a statistically significant association between geopolitical risk and international crude oil. For instance, in April 1986, after a U.S. air assault in Libya, the price of oil dropped significantly. After the 2015 terrorist events in Paris, the oil price also dropped significantly. Particularly, isolationist and protectionist beliefs and behaviours in some nations have resulted in penalties and damage to numerous economies (such as U.S. sanctions against Venezuela and Iran) and have even impeded the trend of globalisation (Selmi et al., 2020). All of these events contributed to a period of unstable crude oil prices, which continue to undermine the economic stability of oil-producing and oil-consuming countries. How do geopolitical threats affect oil prices under these conditions, and how long do these effects last? Answering these questions is crucial for risk management, hedging, and investment decision-making.

Figure 1. The benchmark GPR index and the real WTI and Brent oil price changes (1986:01-2019:12).

In light of the dominance of GPR in significant global risk shocks and the significance of oil as an asset, recent research has begun to examine the relationship between the two (Gkillas et al., 2022; Salisu et al., 2021; Wang et al., 2021). However, the literature that emphasises the impact of GPR on the analysis of dynamic changes in oil prices is scarce at this time (Alqahtani & Klein, 2021; Bouoiyour et al., 2019), especially GPR may have a huge global impact on investor sentiment and energy and financial markets (Selmi et al., 2020); At the same time, the existing research on the relationship between GPR and oil price fluctuations yields a mixed result (Gkillas et al., 2022).

To achieve this, this study uses the GPR index developed by Caldara and Iacoviello (2017) as a proxy for geopolitical risk. On the basis of this information, both nonlinear causality tests and a Granger causality test in the frequency domain were conducted to determine whether geopolitical risks have a substantial predictive effect on the movement of real crude oil prices. We find that geopolitical risks have a solid nonlinear predictive effect on oil prices, but this effect is only relevant over the short run. In addition, pure geopolitical risks, such as nuclear tensions, war threats, and terrorist threats, have a greater and longer-lasting impact on oil prices than actual bad geopolitical events.

The following are the key contributions of this paper: First, this study employs nonlinear Granger causality tests to assess the existence of a nonlinear causal link between global geopolitical risks and oil prices, indicating that big political events have had a strong nonlinear effect on oil price fluctuations. This is the first study to explore nonlinearities in the relationship between global geopolitical risks and oil prices, as far as we know. Second, using the Granger causality test in the frequency domain, this study examines whether the explanatory role of geopolitical risks on oil prices differs under different time-frequency conditions (such as in the short or long term). The weighted sum of sinusoidal components can describe a stationary process with a specific frequency. Instead of calculating a single Granger causality measure for the entire relationship, the frequency domain Granger causality test model calculates the Granger causality of each frequency component separately. This makes it possible to determine whether the predictive effect of geopolitical risks is short- or long-term. Third, we distinguish the effect of real geopolitical events from pure geopolitical risks on changes in oil prices. By adopting the GPR estimated by Caldara and Iacoviello (2017), in contrast to existing literature such as Caldara and Iacoviello (2017), whose focus on the difference of impact magnitude between GPT (pure geopolitical risks) and GPA (actual geopolitical events) on the real activity, we discover that GPT has a higher and more permanent impact on oil prices than GPA.

The remaining sections of the paper are structured as follows: Section 2 briefly reviews the relevant studies in the existing literature. Section 3 presents the data and the methodology. Section 4 contains the empirical results. In Section 5, the conclusion and policy recommendations are provided.

2. Relevant literature review

In recent decades, numerous scholars have conducted extensive research on the causes of oil price fluctuations (Chu et al., 2022; Ngoma, 2022). Due to the importance of oil prices to the financial system (Hamilton, 1996), some studies have conducted extensive research on the relationship between macroeconomic variables and oil prices (Lee et al., 2021), the most prominent of which is an investigation into the relationship between the stock market and oil price (Alamgir & Amin, 2021; Apergis & Miller, 2009; Bashir, 2022; Dutta, 2017; Fasanya et al., 2021; Hwang & Kim, 2021; Tian et al., 2021). For example, using smooth transition vector autoregression, Hwang and Kim (2021) examined the response of U.S. stock market returns to oil price shocks and the degree of asymmetry in their performance at different periods of the business cycle. The findings indicate that the response of U.S. stock returns to categorisation shocks is uneven over the whole business cycle, but the impact of demand-driven shocks on U.S. stock returns is stronger and more enduring. In recent years, several researchers have also begun to focus on the correlation between oil prices and bond yields, which represents a significant new development (Azhgaliyeva et al., 2021; Chen et al., 2022; Demirer et al., 2019; Kang et al., 2014; Lee et al., 2021, 2022; Morrison, 2019; Su et al., 2022; Tule et al., 2017). For instance, Morrison (2019) examined the impact of oil price innovations on emerging market sovereign total bond returns and confirmed that oil prices have a statistically significant impact on the total sovereign bond returns of investment portfolios of key oil exporting and importing nations around the globe. Through the structural vector autoregression method, Lee et al. (2022) examined the link between oil shocks, geopolitical uncertainty, and green bond returns. The results indicate that specific oil demand has a substantial effect on geopolitical risks and the dynamics of green bonds, and that positive changes in geopolitical risks are accompanied by a fall in oil prices and an increase in the return on green bonds. Geopolitical behaviour has a negative effect on oil prices, whereas geopolitical threats have a substantial positive impact on the return on green bonds.

Since the onset of the COVID-19 pandemic, oil prices have generally decreased and uncertainty has increased. Several studies have begun investigating the relationship between COVID-19 and oil prices to inform macroeconomic policymaking (Khalfaoui et al., 2022). In the past two years, research on the impact of COVID-19 on oil prices has become an active subject of study (e.g. Benlagha & El Omari, 2022; Bourghelle et al., 2021; Gil-Alana & Monge, 2020; Jia et al., 2021; Khalfaoui et al., 2022; Ren et al., 2021; Sharif et al., 2020). Using the coherence wavelet method and the wavelet-based Granger causality tests, Sharif et al. (2020) analysed the correlation between the recent spread of COVID-19 in the United States, the impact of oil price fluctuations, the stock market, geopolitical risks, and the uncertainty of U.S. economic policy within the time-frequency framework. The findings indicate that the effect of COVID-19 on geopolitical risk is significantly greater than its effect on economic uncertainty in the United States.

In light of the substantial impact of GPR on the global economy (Antonakakis et al., 2017) and the significance of the oil price market (Selmi et al., 2020), many recent studies are analysed in terms of the relationship between GPRs and oil prices (Dey et al., 2020; Ivanovski & Hailemariam, 2022; Li et al., 2020; Qian et al., 2022; Su et al., 2021). Using the autoregressive Markov region switching model, Qian et al. (2022) studied the predictability of GPR to the volatility of the oil market and confirmed that GPR could lead to high volatility of the oil market, GPR has useful information to predict the volatility of the oil market, and GPR is effective in the long-term prediction range. In recent years, it has been prevalent to utilise dummy variables in econometric models to examine the impact of GPRs on oil prices (Ewing & Malik, 2017; Lee et al., 2010). For example, Ewing and Malik (2017) employed the GARCH model with dummy variables to estimate breakpoints and found that bad and good news affect the volatility of crude oil prices. Others have suggested time-frequency decomposition methods (Martina et al., 2011; Zhang et al., 2008, 2009), such as wavelet decomposition and Empirical Mode Decomposition (EMD). For instance, Using the EMD model, Zhang et al. (2009) assessed the effect of political events on crude oil price volatility and found that severe events had a considerable impact on oil price fluctuations. Early research often focused on the fluctuations in crude oil prices preceding and following important political events (Kesicki, 2010; Monge et al., 2017). However, the time-frequency decomposition method depends primarily on oil price data and cannot properly quantify the effect of political risk variables on crude oil prices (Chen et al., 2016). Therefore, some scholars propose the use of multi-dimensional variables to construct a comprehensive index to reflect the effect of political risk on oil price changes. Hammoudeh et al. (2013) built an index system that incorporated risk ratings, stock market data, and crude oil prices to explore the nexus in BRICS countries, arguing that the oil price is more sensitive to economic than financial risk. Chen et al. (2016) and Lee et al. (2017) estimated the impact of political risk on oil price fluctuations using the International Country Risk Guide (ICRG) index as a proxy for political risk. Noting that the ICRG index is restricted to regional or individual countries, it cannot be utilised to distinguish between the direct effect of actual global geopolitical events and global pure geopolitical risks on oil prices.

Although a wealth of evidence addresses the relationship between political issues and oil prices, major study routes have not yet been investigated under the empirical framework. First, the existing literature concentrates on political risk in specific regions rather than global geopolitical risks, but the risk created by political events, particularly threats that have not yet materialised, may have a different effect on crude oil prices. Second, early empirical research is founded on the premise of linearity and frequently employs linear models. However, economic events and regime change could result in structural modifications to the pattern of oil prices and the relationship between geopolitical risks and oil prices (Lee et al., 2010; Noguera, 2013). Moreover, the political risk may have a nonlinear effect on oil prices, and investment returns based on a nonlinear relationship are likely to be higher. Third, if causation is restricted to a single data frequency, it is impossible to evaluate long-term components (low frequencies) and short-term components (high frequencies) independently (Lemmens et al., 2008). Therefore, it is impossible to determine if the explanatory effect of geopolitical risks on oil prices has a short-term or long-term time horizon. It is usually assumed that high-frequency components are more susceptible to structural break and nonlinear forces, indicating that the conventional linear model is inadequate (Geng et al., 2017). Therefore, investors must alter their investment plan by elucidating the time horizon of geopolitical risks’ impact on the price of crude oil.

To circumvent the limitations above, we employ the GPR as a proxy for geopolitical risk in our models, which has proven to be an accurate representation of global geopolitical risk with strong explanatory power in terms of economic output and stock market outcomes (Apergis et al., 2017). By generating two indices, the index may distinguish between the direct effect of adverse geopolitical events and the effect of pure geopolitical risks. In particular, Geopolitical Acts (GPA) and Geopolitical Threats (GPT) are GPR components that can be used to isolate the influence of pure geopolitical risks (Caldara & Iacoviello, 2018; Dissanayakea et al., 2018; Qian et al., 2022). GPA refers to the occurrence of adverse geopolitical events, such as the Paris bombing attack in November 2015 and the start of the Iraq war in 2003, which is constructed by the groups directly mentioning adverse events, capturing the beginning and actual unfolding of the war and other events; GPT is constructed by the groups directly mentioning risks, focusing on capturing various GPT, such as nuclear threats, war threats, and terrorist threats, ref. Moreover, the two indices are highly interdependent and correlated (Caldara, Iacoviello, 2022; Caldara, Conlisk, 2022; Gong & Xu, 2022). Therefore, GPT captures the potential for geopolitical action, whereas GPA represents the existing state of geopolitical events, and the influence of GPA has a smaller effect on actual activities than the impact of GPT (Baur & Smales, 2018; Gu et al., 2021). In terms of the energy price study, Gu et al. (2021) found that GPA and GPT shocks have differing effects on the oil market, although both have a smaller influence on the oil market than the EPU shock. In addition, Qian et al. (2022) discovered that GPT is more useful than GPA for predicting oil. Neither GPT nor GPA can fully represent swings in oil prices during expansion, and GPT improves more than GPA during the recession.

In contrast, Mei et al. (2020) revealed that GPA contributes more to long-term oil output forecasting than GPT. In addition, Liu et al. (2021) discovered that, in contrast to the negligible effect of GPA on energy fluctuations, GPT actively encourages energy fluctuations; that is, the effect of GPR on energy fluctuations is more likely to be conveyed via GPT. In terms of commodity market study, Gong and Xu (2022) demonstrated that GPA reveals more than GPT the influence of GPR on the connectivity effects between commodity markets. In addition, Yang et al. (2022) discovered that GPT and GPA had positive and negative effects on the global commodity market, time variability and substantial short-term effects. In terms of digital currency price research, Colon et al. (2021) discovered that the cryptocurrency market could be used as a safe haven against GPT during bull markets but against GPA during extreme bear markets.

In conclusion, with the expansion of data on geopolitical risks in recent years, more literature has begun empirically examining the influence of geopolitical risk on the international crude oil market, which provides a good reference for our research. However, most literature focused on the linear relationship between geopolitical risks and global crude oil prices. Few studies explored the nonlinear impact of geopolitical risks on global crude oil prices, particularly the heterogeneity of causality at various time frequencies. This research employs the nonlinear bivariate Granger causality (G.C.) test and the frequency domain Granger causality test to evaluate the potential explanatory effect of geopolitical risks on global crude oil price fluctuations.

3. Data and method

3.1. Data

The GPR index consists of monthly data from February 1986 through December 2019, according to Caldara and Iacoviello (2017).¹ It is determined by tallying the number of geopolitical tension-related articles published in 11 top international newspapers each month (as a share of the total number of news articles). The search discovers papers whose citations contain six-word groups (Table 1). As previously stated, the GPR index is separated into the GPA and GPT index based on groups of search terms.

Table 1. Specifications of the GPR.

GPT	Group 1 includes words that clearly mention geopolitical risk, and military-related tensions involving large regions of the world and a U.S. involvement.
	Group 2 includes words related to nuclear tensions.
	Group 3 includes words related to war threats
	Group 4 includes words related to terrorist threats.
GPA	Groups 5 and 6 aim at capturing press coverage of actual adverse geopolitical events (as opposed to just risks), which can be expected to lead to increases in geopolitical uncertainty, such as terrorist acts or the beginning of a war.
GPR	Search groups 1 to 6 above

The specific index construction process and the inclusion indicators can see Caldara and Iacoviello (2017).

The Energy Information Administration (EIA) of the USA² provides the spot price of Brent crude oil from May 1987 to December 2019 and the spot price of WTI crude oil from February 1986 to December 2019. Real crude oil price is Brent and WTI crude oil prices deflated by the U.S. consumer price index (CPI). We calculate real oil price changes by $100\ast ln(p_t/p_{\mathit{t}-1})$, where $p_t$ is the real oil price at time t, deflated by the CPI available from the Bureau of Labour Statistics.³

3.2. Econometric modelling framework

To determine if geopolitical risks and crude oil prices have a nonlinear causal relationship, we first investigate the effect of geopolitical risks on oil prices using the nonlinear Granger causality test (Diks & Panchenko, 2006; Hiemstra & Jones, 1994). On this premise, the Granger causality test in the frequency domain established by Lemmens et al. (2008) is utilised to determine whether the predictive effect of geopolitical risks on the oil price is short-term or long-term.

3.2.1. Nonlinear Granger causality test

Diks and Panchenko (2006) provide a new nonparametric test (hereafter, D&P test) for Granger causality based on the study of Hiemstra and Jones (1994) (hereafter, H&J test). The null hypothesis is that $X_t$ does not contain additional information about $Y_t$:

(1) $H_0:Y_{\mathit{t}+1}\left|(X_t^{l_X};Y_t^{l_Y})\right.\sim Y_{\mathit{t}+1}|Y_t^{l_Y}$(1)

where $l_x$ and $l_y$respectively denote the lag length of $X_t$ and $Y_t$. The null hypothesis becomes a statement about the invariant distribution of the $(l_x+l_{\mathit{y}}+1)$ dimensional vector, $W_t=(X_t^{l_x},Y_t^{l_y},Z_t)$, $Z_t=Y_{\mathit{t}+1}$, the conditional distribution of $\mathit{Z}$given $(\mathit{X},\mathit{Y})=(\mathit{x},\mathit{y})$ is the same as that of $\mathit{Z}$ given $\mathit{Y}=\mathit{y}$ under the null hypothesis. Further, Eq. (1) can be restated in terms of ratios of joint distributions. Specifically, the joint probability density function $f_{\mathit{x},\mathit{y},\mathit{z}}(\mathit{x},\mathit{y},\mathit{z})$ and its marginal density function should satisfy:

(2) $\frac{{\mathit{f}}_{\mathit{x},\mathit{y},\mathit{z}}(\mathit{x},\mathit{y},\mathit{z})}{f_y(\mathit{y})}=\frac{{\mathit{f}}_{\mathit{x},\mathit{y}}(\mathit{x},\mathit{y})}{f_y(\mathit{y})}\cdot \frac{{\mathit{f}}_{\mathit{y},\mathit{z}}(\mathit{y},\mathit{z})}{f_y(\mathit{y})}$(2)

This states that $\mathit{X}$ and $\mathit{Z}$ are independent conditionally on $\mathit{Y}=\mathit{y}$ for each fixed value of $\mathit{y}$. Re-specify the null hypothesis of no nonlinear Granger causality as:

(3) $\mathit{q}=\mathit{E}[f_{\mathit{x},\mathit{y},\mathit{z}}(\mathit{X},\mathit{Y},\mathit{Z})f_y(\mathit{Y})-f_{\mathit{x},\mathit{y}}(\mathit{X},\mathit{Y})f_{\mathit{y},\mathit{z}}(\mathit{Y},\mathit{Z})]=0$(3)

where $\mathit{n}$ is the sample size, and ${\hat{f}}_{\mathit{w}}$ is a local density estimator of a $d_w$-variate random vector $\mathit{W}$at $W_i$ denoted by ${\hat{f}}_{\mathit{w}}(W_i)=\frac{{(2\mathit{\epsilon })}^{-\mathit{dw}}}{\mathit{n}-1}\sum _{\mathit{j},\mathit{j}\ne \mathit{i}}{I}_u^W$, which is based on indicator functions $I_u^w=\mathit{I}(∥W_i-W_j∥<{\epsilon }_n)$, with $\mathit{I}(\cdot )$the indicator function and bandwidth ${\epsilon }_n$. A natural estimator $\mathit{q}$ based on indicator functions is

(4) ${\mathrm{T}}_{\mathrm{n}}(\mathit{\epsilon })=\frac{(\mathit{n}-1)}{\mathit{n}(\mathit{n}-2)}\sum _i({\hat{f}}_{\mathit{x},\mathit{y},\mathit{z}}(\mathit{X},\mathit{Y},\mathit{Z}){\hat{f}}_{\mathit{y}}(\hat{Y})-{\hat{f}}_{\mathit{x},\mathit{y}}(X_{\mathit{i},}{Y}_i){\hat{f}}_{\mathit{y},\mathit{z}}(Y_{\mathit{i},}{Z}_i))$(4)

For $l_x=l_{\mathit{y}}=1$, if ${\epsilon }_n=\mathit{C}{n}^{-\mathit{\beta }}$$\mathit{C}>0$ and $\mathit{\beta }\in (\frac{1}{4},\frac{1}{3})$. The test statistic in Eq.(4) satisfies:

(5) $\sqrt{\mathit{n}}\frac{(T_n({\epsilon }_n)-\mathit{q})}{S_n}\stackrel{d}{⟶}\mathit{N}(0,1)$(5)

where $\stackrel{d}{⟶}$denotes convergence in distribution and$S_n$is an estimator of the asymptotic variance of $T_n(\cdot )$. The critical value of the test statistic is 1.28, which rejects the null hypothesis at the 10% level of significance and supports evidence in favour of a nonlinear Granger causality among variables.

3.2.2. Frequency domain Granger causality test

The frequency domain Granger causality test over the spectrum is proposed by Lemmens et al. (2008). Given $X_t$ and $Y_t$ are two stationary time series with length T. The goal is to test whether $X_t$ Granger-causes $Y_t$ at a given frequency $\mathit{\lambda }$. Measuring for G.C. in the frequency domain is performed on the univariate innovations series, ${\mu }_t$ and $v_t$, derived from filtering the $X_t$ and $Y_t$ as univariate ARMA processes, which are white-noise processes with zero means, possibly correlated with each other at different leads and lags. Let $x_2^2$ and $S_v(\mathit{\lambda })$ be the spectral density function of ${\mu }_t$ and $v_t$ at a frequency $\mathit{\lambda }\in [0,\mathit{\pi }]$, defined by:

(6) $S_{\mu }(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\sum _{\mathit{k}=-\mathrm{\infty }}^{\mathrm{\infty }}{\gamma }_{\mu }(\mathit{k})e^{-\mathit{i\lambda k}}$(6)

(7) $S_v(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\sum _{\mathit{k}=-\mathrm{\infty }}^{\mathrm{\infty }}{\gamma }_V(\mathit{k})e^{-\mathit{i\lambda k}}$(7)

where ${\gamma }_{\mu }(\mathit{k})=\mathit{Cov}({\mu }_{\mathit{t},}{\mu }_{\mathit{t}-\mathit{k}})$and ${\gamma }_v(\mathit{k})=\mathit{Cov}(v_{\mathit{t},}{v}_{\mathit{t}-\mathit{k}})$ represent the autocovariances of ${\mu }_t$ and $v_t$ at a lag $\mathit{k}$. The idea of spectral representation is that each time series may be decomposed into a sum of uncorrelated components, each related to a particular frequency $\mathit{\lambda }$. Let $S_{\mathit{\mu v}}(\mathit{\lambda })$ be the cross-spectrum between ${\mu }_t$ and $v_t$ series, which defined as:

(8) $S_{\mathit{\mu v}}(\mathit{\lambda })=C_{\mathit{\mu v}}(\mathit{\lambda })+\mathit{i}{Q}_{\mathit{uv}}(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\sum _{\mathit{k}=-\mathrm{\infty }}^{\mathrm{\infty }}{\gamma }_{{\mathit{ }}_{\mathit{\mu v}}}(\mathit{k})e^{\mathit{i\lambda k}}$(8)

where the called spectrum$C_{\mathit{\mu v}}(\mathit{\lambda })$ and quadrature spectrum $Q_{\mathit{uv}}(\mathit{\lambda })$ are respectively the real and imaginary parts of the cross-spectrum and $\mathit{i}=\sqrt{-1}$. Here ${\gamma }_{\mathit{\mu v}}(\mathit{k})=\mathit{Cov}({\mu }_{\mathit{t},}{v}_{\mathit{t}-\mathit{k}})$ represents the cross-covariance of ${\mu }_t$ and $v_t$ at lag $\mathit{k}$. The spectrum $Q_{\mathit{uv}}(\mathit{\lambda })$ between the two series ${\mu }_t$ and ${\nu }_t$ at a frequency $\mathit{\lambda }$ can be interpreted as the covariance between the two series ${\mu }_t$ and ${\nu }_t$ that is attributable to cycles with frequency $\mathit{\lambda }$. The cross-spectrum can be estimated non-parametrically by:

(9) ${\hat{S}}_{\mathit{\mu v}}(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\left\{\sum _{-\mathit{M}}^M{w}_k{\hat{\gamma }}_{{ }_{\mathit{\mu v}}}(\mathit{k})e^{-\mathit{i\lambda k}}\right\}$(9)

with the empirical cross-covariances ${\hat{\gamma }}_{\mathit{\mu v}}(\mathit{k})=\hat{C}\mathit{ov}({\mu }_{\mathit{t},}{v}_{\mathit{t}-\mathit{k}})$and window weights $w_k$, for $\mathit{k}=-\mathit{M},\dots ,\mathit{M}$. equation (9)(9) ${\hat{S}}_{\mathit{\mu v}}(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\left\{\sum _{-\mathit{M}}^M{w}_k{\hat{\gamma }}_{{ }_{\mathit{\mu v}}}(\mathit{k})e^{-\mathit{i\lambda k}}\right\}$(9) is called the weighted covariance estimator, and the weights $w_k$ are selected as the Bartlett weighting scheme i.e. $1-\left|\mathit{k}\right|/\mathit{M}$. The constant $\mathit{M}$ determines the maximum lag order considered. The spectra of equations (6)(6) $S_{\mu }(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\sum _{\mathit{k}=-\mathrm{\infty }}^{\mathrm{\infty }}{\gamma }_{\mu }(\mathit{k})e^{-\mathit{i\lambda k}}$(6) and (7(7) $S_v(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\sum _{\mathit{k}=-\mathrm{\infty }}^{\mathrm{\infty }}{\gamma }_V(\mathit{k})e^{-\mathit{i\lambda k}}$(7) ) are estimated in a similar way. This cross-spectrum allows us to compute the coefficient of coherence $h_{\mathit{\mu v}}(\mathit{\lambda })$ defined as $h_{\mathit{\mu v}}(\mathit{\lambda })=\frac{\left|{\mathit{S}}_{{\mathit{ }}_{\mathit{\mu v}}}\left.(\mathit{\lambda })\right|\right.}{\sqrt{S_{\mu }(\mathit{\lambda })S_v(\mathit{\lambda })}}$.

Under the null hypothesis $h_{\mathit{\mu v}}(\mathit{\lambda })=0$, the estimated squared coefficient of coherence at the frequency $\mathit{\lambda }$, with $0<\mathit{\lambda }<\mathit{\pi }$ when appropriately rescaled, converges to a chi-squared distribution with 2 degrees of freedom, denoted by $x_2^2$.

(10) $2(\mathit{n}-1){\hat{h}}_{\mathit{\mu \nu }}^2(\mathit{\lambda })\stackrel{d}{\to }{\chi }_2^2$(10)

where $\stackrel{d}{\to }$ stands for convergence in distribution, $\mathit{n}=\mathit{T}/(\sum _{\mathit{k}=-\mathit{M}}^M{w}_k^2)$. The null hypothesis $h_{\mathit{\mu \nu }}(\mathit{\lambda })=0$ versus $h_{\mathit{\mu \nu }}(\mathit{\lambda })>0$ is then rejected if ${\hat{h}}_{\mathit{\mu \nu }}(\mathit{\lambda })>\sqrt{\frac{{\chi }_{2,1-\alpha }^2}{2(\mathit{n}-1)}}$,

with ${\chi }_{2,1-\alpha }^2$ being the $1-\mathit{\alpha }$ quantile of the chi-squared distribution with 2 degrees of freedom. Then decompose the cross-spectrum (Eq. (6)) into three parts: (i) $S_{\mathit{\mu }\leftrightarrow \mathit{\nu }}$, the instantaneous relationship between ${\mu }_t$ and ${\nu }_t$; (ii) $S_{\mathit{\mu }\to \mathit{\nu }}$, the directional relationship between ${\nu }_t$ and lagged values of ${\mu }_t$; and (iii) $S_{\mathit{\nu }\to \mathit{\mu }}$, the directional relationship between ${\mu }_t$ and lagged values of ${\nu }_t$, i.e.

(11) $S_{\mathit{\mu \nu }}(\mathit{\lambda })=\left[S_{\mathit{\mu }\leftrightarrow \mathit{\nu }}+S_{\mathit{\mu }\to \mathit{\nu }}+S_{\mathit{\nu }\to \mathit{\mu }}\right]=\frac{1}{2\mathit{\pi }}\left[{\gamma }_{\mathit{\mu \nu }}(0)+\sum _{\mathit{k}=-\mathrm{\infty }}^{-1}{\gamma }_{\mathit{\mu \nu }}(\mathit{k})e^{-\mathit{i\lambda k}}+\sum _{\mathit{k}=1}^{\mathrm{\infty }}{\gamma }_{\mathit{\mu \nu }}(\mathit{k})e^{-\mathit{i\lambda k}}\right]$(11)

The proposed spectral measure of G.C. is based on the key property that ${\mu }_t$ does not Granger-cause ${\nu }_t$ if and only if ${\gamma }_{\mathit{\mu \nu }}(\mathit{k})=0$ for all $\mathit{k}<0$. The goal is to test the predictive content of ${\mu }_t$ relative ${\nu }_t$ to which is given by the second part of Eq. (11), that is: $S_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\left[\sum _{\mathit{k}=-\mathrm{\infty }}^{-1}{\gamma }_{\mathit{\mu \nu }}(\mathit{k})e^{-\mathit{i\lambda k}}\right]$. The Granger coefficient of coherence is then given by $h_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })=\frac{\left|{\mathit{S}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })\right|}{\sqrt{S_{\mu }(\mathit{\lambda })S_{\nu }(\mathit{\lambda })}}$. Therefore, in the absence of G.C., $h_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })=0$ for every $\mathit{\lambda }$ in$[0,\mathit{\pi }]$. The Granger coefficient of coherence takes values between zero and one. Granger coefficient of coherence at a frequency $\mathit{\lambda }$ is estimated by

(12) ${\hat{h}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })=\frac{\left|{\hat{S}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })\right|}{\sqrt{{\hat{S}}_{\mathit{\mu }}(\mathit{\lambda }){\hat{S}}_{\mathit{\nu }}(\mathit{\lambda })}}$(12)

with $\left|{\hat{S}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })\right|$ as in equation (9)(9) ${\hat{S}}_{\mathit{\mu v}}(\mathit{\lambda })=\frac{1}{2\mathit{\pi }}\left\{\sum _{-\mathit{M}}^M{w}_k{\hat{\gamma }}_{{ }_{\mathit{\mu v}}}(\mathit{k})e^{-\mathit{i\lambda k}}\right\}$(9) , but with all weights $w_k=0$, for $\mathit{k}\ge 0$. The distribution of the estimator of the Granger coefficient of coherence is derived from the distribution of the coefficient of coherence (equation (10)(10) $2(\mathit{n}-1){\hat{h}}_{\mathit{\mu \nu }}^2(\mathit{\lambda })\stackrel{d}{\to }{\chi }_2^2$(10) ). Under the null hypothesis of${\hat{h}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })=0$, the distribution of the squared estimated Granger coefficient of coherence at a frequency $\mathit{\lambda }$, with $0<\mathit{\lambda }<\mathit{\pi }$ is given by $2(\mathit{n}{ }^{\mathrm{\prime }}-1){{\hat{h}}^2}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })\stackrel{d}{\to }{\chi }_2^2$, where $\mathit{n}$ is now replaced by $\mathit{n}{ }^{\mathrm{\prime }}=\mathit{T}/(\sum _{\mathit{k}=-\mathit{M}}^{-1}{w}_k^2)$. Since the weights $w_k$, with a positive index k, are set equal to zero when computing ${\hat{S}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })$, in effect only the $w_k$ with negative indices are considered. The null hypothesis ${\hat{h}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })=0$ versus ${\hat{h}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })>0$is rejected if ${\hat{h}}_{\mathit{\mu }\to \mathit{\nu }}(\mathit{\lambda })>\sqrt{\frac{{\chi }_{2,1-\alpha }^2}{2(\mathit{n}{ }^{\mathrm{\prime }}-1)}}$.

4. Empirical results

4.1. The preparative work of empirical estimation

Before employing the nonlinear bivariate Granger causality test and frequency domain Granger causality test to investigate the potential explanatory effect of geopolitical risks on oil price fluctuations, more preparatory work was required.

4.1.1. Presentation of statistical results

To demonstrate why WTI and Brent are together used as proxy variables for worldwide crude oil prices, we present descriptive data on oil prices and geopolitical risks (Table 2). The mean for the change in Brent crude oil price is bigger than that of WTI price, and the variance of Brent price is greater than that of WTI price, confirming that there is a substantial difference between the two crude oil markets in North America and Europe.

Table 2. Descriptive statistics.

	Brent	WTI	lnGPR	lnGPA	lnGPT
Mean	0.1104	0.0548	4.5249	4.4429	4.5428
Median	0.8597	0.9124	4.4955	4.4049	4.5272
Maximum	46.9032	54.0164	6.2394	6.7500	5.8928
Minimum	−55.1035	−56.8145	3.6648	3.3481	3.6025
Std. Dev.	9.8358	9.7260	0.3346	0.4628	0.3255
Skewness	−0.6190	−0.6918	1.2531	1.2023	0.7922
Kurtosis	9.3757	10.6297	7.6720	6.7308	5.2427
Jarque-Bera	722.3602***	1067.2340***	500.1060***	350.5223***	134.1527***
Observations	411	426	427	427	427

*, **and ***indicate significance at the 10%, 5% and 1% level, respectively.

According to Hiemstra and Jones (1994), Diks and Panchenko (2006), and Lemmens et al. (2008), all variables must be stationary for the models of nonlinear G.C and frequency domain G.C. employed in this paper. The unit root and stationary test results are shown in Table 3. Note that both the Augmented Dicky-Fuller (ADF) test (1979) and Philip’s Peron (P.P.) test (1988) reject the null hypothesis of a unit root in all time-series variables at a significance level of 1%, suggesting that all variables that satisfy the condition for constructing the nonlinear G.C. are stationary.

Table 3. Unit root and stationary test.

	ADF		PP
	No Trent	Trend	No Trent	Trend
lnGPR	−5.306(2) ***	−5.301(2) ***	−7.604(8) ***	−7.603(8) ***
lnGPT	−5.567(2) ***	−5.559(2) ***	−9.414(10) ***	−9.411(10)***
lnGPA	−6.864(0)***	−6.859(0)***	−6.854(9)***	−6.852(9)***
WTI	−16.155(0)***	−16.133(0)***	−15.986(19)***	−15.958(19)***
Brent	−13.964(1)***	−13.947(1)***	−14.783(20)***	−14.756(20) ***

*, ** and ***indicate the rejection of the null hypothesis (unit root) at the 10%, 5% and 1% levels, respectively. The numbers in parentheses are the optimal lag order in the ADF and Newey-west bandwidth in the P.P. test based on the Schwarz Info criterion. WTI and Brent denote the oil price changes.

Using a simple unit root test without structural breaks is likely to diminish the test’s size and explanatory power, leading to erroneous empirical results. For this reason, Carrion-I-Silvestre et al. (2009)‘s unit root test is utilised to avoid such problems. The empirical unit root results of Carrion-I-Silvestre et al. (2009) are presented in Table 4, which includes various structural breaks for each series. The results of the tests reveal that all variables are also stationary.

Table 4. Unit root and stationary test with multiple breaks (GLS-based unit root test).

	$\mathit{M}{Z}_{\alpha }$		$\mathit{M}{Z}_t$
lnGPR	−106.336**	[−23.988]	−7.292**	[−3.444]
lnGPT	−47.576**	[−22.826]	−4.877**	[−3.373]
lnGPA	−97.047**	[−23.982]	−6.966**	[−3.444]
WTI	−60.768**	[−23.284]	−5.508**	[−3.388]
Brent	−125.799**	[−22.649]	−7.931**	[−3.331]

** indicate the rejection of the null hypothesis (unit root) at 5%. The values in brackets indicate critical values obtained through bootstrapping, as in Carrion-I-Silvestre et al. (2009).

4.1.2. BDS test

If two-time series and their relationships have considerable nonlinear properties, it is inappropriate to employ linear models to fit the nexus (Balcilar et al., 2017). To demonstrate the applicability of our models, the second stage entails determining if the relationship between GPR and crude oil price is nonlinear. Similar to Wang and Wu (2012), we use the BDS test (Broock et al., 1996) to examine the probability of nonlinearity for oil price changes, geopolitical risks, and the oil price changes equation in the VAR model incorporating geopolitical risks. As one of the most popular tests for nonlinearity, the BDS test examines whether data series increments are independent and identically distributed (i.i.d.). The BDS test results are presented in Table 5. As demonstrated in panel 1, the null of i.i.d. at various embedding dimensions (m) for each geopolitical risk index and oil price change series is rejected firmly at the 1% significance level. These findings prove the nonlinear features for real oil price fluctuations and geopolitical risks. In addition, the residuals of the oil price fluctuations equation in the VAR model containing (relative) geopolitical risks pass the BDS test at the 1% significance level, as shown in panels 2, 3, and 4. It suggests that the relationship between the GPR and oil price is nonlinear and that the conventional Granger causality tests based on a linear framework are likely to be misspecified.

Table 5. BDS tests.

m	2	3	4	5	6
Panel 1 BDS test for each variable
Brent	0.033*	0.050*	0.059*	0.060*	0.059*
WTI	0.033*	0.052*	0.064*	0.065*	0.063*
lnGPR	0.101*	0.169*	0.215*	0.241*	0.252*
lnGPT	0.079*	0.125*	0.154*	0.167*	0.170*
lnGPA	0.107*	0.178*	0.223*	0.250*	0.262*
Panel 2 BDS test for the residuals of the oil equation of the VAR model with lnGPR
Brent-VAR(4)	0.028*	0.044*	0.048*	0.050*	0.052*
WTI-VAR(3)	0.021*	0.038*	0.045*	0.047*	0.047*
Panel 3 BDS test for the residuals of the oil equation of the VAR model with lnGPT
Brent-VAR(4)	0.028*	0.044*	0.049*	0.051*	0.053*
WTI-VAR(3)	0.021*	0.038*	0.045*	0.048*	0.049*
Panel 4 BDS test for the residuals of the oil equation of the VAR model with lnGPA
Brent-VAR(4)	0.028*	0.043*	0.048*	0.051*	0.052*
WTI-VAR(4)	0.024*	0.038*	0.043*	0.043*	0.042*

The table shows the BDS statistic. The * indicates significance at the 1% level. The parameter m is the embedding dimension. The lag parameters for the VAR model are selected based on the Akaike information criterion (AIC). m stands for the embedded dimension.

4.1.3. Structural break tests

Following Balcilar et al. (2017), since the linear Granger causality test cannot be considered robust and reliable, we turn to the Bai and Perron test (Bai & Perron, 1998, 2003) for providing strong evidence of nonlinearity and structural break in the relationship between geopolitical risks (including GPR, GPT, and GPA) and the movement of real oil prices (including WTI oil prices and Brent oil prices). According to Table 6, there are structural breaks in the relationship between geopolitical risks and real oil prices, which suggests that Granger causality tests based on a linear framework are likely to be misspecified. In particular, the Bai and Perron tests reveal five structural breaks with the dates for the GPR – WTI real oil price equation namely 1990M06, 1996M11, 2002M03, 2008M09, 2015M04, and the GPA – WTI real oil price equation and GPT – WTI equation have five breaks as well. These dates overlap with several significant oil market events. In 1991, for instance, the Gulf war persisted, and the price of crude oil surged dramatically during this time. However, according to the geopolitical risks – Brent real oil price equation, there are five different structural breaks between geopolitical risks and the Brent oil price, with 1994M11, 2000M04, 2005M09, 2010M10 and 2015M12 being the dates for geopolitical risks (GPR). The international price of crude oil changed significantly during the period covered by the preceding events. As with the BDS test, Bai and Perron test has identified nonlinearity, structural breaks in the link between geopolitical risks and the movement of real oil prices suggest that Granger causality tests based on a linear framework are likely to be misspecified. We now focus on the nonlinear Granger causality tests in light of the substantial evidence of either nonlinearity.

Table 6. Bai and Perron test for the geopolitical risks – WTI real oil price equation: $\mathit{WT}{I}_t=\mathit{c}+\mathit{W}{T}_{\mathit{t}-1}+\mathit{g}{r}_{\mathit{t}-1}+{\epsilon }_t$ and the geopolitical risks-Brent real oil price equation: $\mathit{Bren}{t}_t=\mathit{c}+\mathit{Bren}{t}_{\mathit{t}-1}+\mathit{g}{r}_{\mathit{t}-1}+{\epsilon }_t$, where $\mathit{gr}$denotes the variables of geopolitical risks (lnGPR, lnGPT, lnGPA). We build an AR (4) model based on the S.C. criterion.

The geopolitical risks - WTI real oil price				The geopolitical risks-Brent real oil price
Breaks	F-statistic	Scaled F-statistic	Critical Value^a	Breaks	F-statistic	Scaled F-statistic	Critical Value^a
Panel 1: GPR
0 vs. 1	1.892	7.567	16.190	0 vs. 1	1.955	15.644	23.7
1 vs. 2*	3.898	15.591*	13.770	1 vs. 2*	2.612	20.893*	20.62
2 vs.3*	4.009	16.038*	12.170	2 vs. 3*	2.676	21.407*	18.69
3 vs. 4*	3.594	14.376*	10.790	3 vs. 4*	3.649	29.193*	16.96
4 vs. 5*	3.744	14.975*	9.090	4 vs. 5*	2.703	21.621*	14.77
Break dates: 1991M06, 1996M11, 2002M03, 2008M09,2015M04				Break dates: 1994M11, 2000M04, 2005M09, 2010M10, 2015M12
Panel 2: GPT
0 vs. 1	1.000	5.999	20.08	0 vs. 1	1.847	14.773	23.7
1 vs. 2	2.785	16.711	17.37	1 vs. 2	2.385	19.081	20.62
2 vs.3*	2.857	17.143*	15.58	2 vs. 3	3.355	26.842*	18.69
3 vs. 4	2.282	13.694	13.9	3 vs. 4*	3.266	26.131*	16.96
4 vs. 5*	2.079	12.476*	11.94	4 vs. 5*	3.019	24.149*	14.77
Break dates: 1991M06, 1996M11, 2002M04, 2008M01, 2015M06				Break dates: 1994M11, 2000M04, 2006M02, 2011M03, 2016M04
Panel 3: GPA
0 vs. 1	0.748	5.982	23.7	0 vs. 1	2.055	16.441	23.7
1 vs.2*	3.039	24.309*	20.62	1 vs. 2*	2.606	20.845*	20.62
2 vs. 3*	2.555	20.440*	18.69	2 vs. 3*	2.755	22.041*	18.69
3 vs. 4*	2.495	19.962*	16.96	3 vs. 4*	3.206	25.649*	16.96
4 vs. 5*	2.313	18.502*	14.77	4 vs. 5*	2.402	19.217*	14.77
Break dates:1992M02,1999M01,2005M07, 2010M10, 2016M01				Break dates: 1992M08, 1997M11, 2002M12, 2008M01, 2015M12

* denotes to significance at the 5% level; a denotes to Bai and Perron (1998, 2003) critical values.

4.2. Nonlinear Granger causality tests

Nonlinear causality between geopolitical risks and oil price fluctuations are investigated using the nonlinear Granger causality test developed by Hiemstra and Jones (1994) and the nonlinear Granger causality test developed by Diks and Panchenko (2006), respectively, to ensure the reliability of the results. The empirical results of the nonlinear Granger causality test are presented in Table 7. Panel 1 of Table 7 displays the findings for the GPR index, while Panels 2 and 3 provide the results for the GPT index and GPA index, respectively. The null hypothesis of no nonlinear causality from geopolitical risks, including the GPR, GPT, and GPA, to oil price fluctuations (WTI and Brent) is rejected at a significance level of 5% for the lag length from the first lag to the fifth lag. Therefore, sufficient statistical evidence supports the notion that geopolitical risks have a strong nonlinear predictive effect on the variations in the worldwide oil price. It demonstrates that the recent turbulence in the Middle East and Russia–Ukraine war have played a significant nonlinear role in oil price fluctuations. In addition, the test results for the nonlinear Granger causality indicates no substantial nonlinear Granger causality from oil prices to geopolitical risks.

Table 7. Nonlinear Granger causality tests.

$\mathit{WT}{I}_t=\mathit{c}+\mathit{W}{T}_{\mathit{t}-1}+\mathit{g}{r}_{\mathit{t}-1}+{\epsilon }_t$	H&J (P-value)	D&P (P-value)	H&J (P-value)	D&P (P-value)
Panel 1: GPR	$ln\mathit{GPR}\to \mathit{WTI}$		$ln\mathit{GPR}\to \mathit{Brent}$
1	2.144**(0.016)	2.617***(0.004)	2.802***(0.003)	2.971*** (0.001)
2	2.612***(0.004)	2.534*** (0.006)	3.193*** (0.001)	3.245*** (0.001)
3	2.686***(0.004)	2.697*** (0.004)	3.516*** (0.000)	3.169*** (0.001)
4	2.878***(0.002)	2.373*** (0.009)	3.536*** (0.000)	2.659*** (0.004)
5	2.641***(0.004)	2.595*** (0.005)	3.097*** (0.001)	3.023*** (0.001)
Panel 2: GPT	$ln\mathit{GPT}\to \mathit{WTI}$		$ln\mathit{GPT}\to \mathit{Brent}$
1	1.697**(0.045)	1.680**(0.046)	2.225** (0.013)	2.142**(0.016)
2	1.657**(0.049)	1.539**(0.062)	2.198***(0.013)	2.487*** (0.006)
3	1.637**(0.051)	2.095**(0.018)	2.521*** (0.005)	2.355*** (0.009)
4	2.186**(0.014)	2.024**(0.021)	2.51***8(0.005)	1.813**(0.035)
5	2.204**(0.014)	2.219**(0.013)	2.132**(0.016)	1.741**(0.041)
Panel 3: GPA	$ln\mathit{GPA}\to \mathit{WTI}$		$ln\mathit{GPA}\to \mathit{Brent}$
1	2.226**(0.013)	2.578*** (0.005)	2.731*** (0.003)	2.484***(0.007)
2	2.807***(0.003)	2.479*** (0.007)	2.727*** (0.003)	2.664*** (0.004)
3	2.713***(0.003)	2.586*** (0.005)	2.930*** (0.002)	2.986*** (0.001)
4	2.889***(0.002)	2.511*** (0.006)	3.378*** (0.000)	2.462*** (0.007)
5	2.894***(0.002)	2.606*** (0.005)	2.874*** (0.002)	2.506*** (0.006)

*, **and ***indicate rejection of the null hypothesis at the 10%, 5% and 1% level, respectively; $l_X=l_Y$ denotes the lag length.

4.3. Frequency domain Granger causality

It extends the nonlinear nexus to examine the causality between two variables at various time frequencies. By doing so, we may determine the duration of geopolitical risks’ impact on the price of crude oil and the severity of geopolitical risks’ effects at different time frequencies. We further investigate the nonlinear association between geopolitical risks and oil prices at low and high frequencies.

In Figure 2, the estimated Granger coefficients of coherence are plotted versus the frequency $\mathit{\lambda }\in (0,\mathit{\pi })$. This coefficient assesses whether and to what extent the geopolitical risks Granger cause the oil price at that frequency. The higher the estimated Granger coefficient of coherence is, the stronger the Granger causality at that particular frequency. If the coefficient is higher than the 5% critical value, the geopolitical risks significantly ‘Granger causes’ the oil price at the frequency $\mathit{\lambda }$. Note that the lag length $\mathit{M}=\sqrt{\mathit{T}}$ and the frequency can be translated into a cycle or periodicity of T months by $\mathit{T}=2\mathit{\pi }/\mathit{\lambda }$, where T is the period. In this paper, we distinguish between the long-term components (low frequencies) and the short-term components (high frequencies) of a time series. Numerous scholars, such as Wang et al. (2014), define the short term as the period being less than 12 months; otherwise, it is long term. Therefore, we define the long-term components to have a cycle larger or equal to 12 months, which corresponds to the frequency $\mathit{\lambda }\le 0.52$. The short-term components have a cycle smaller than 12 months, which corresponds to a frequency $\mathit{\lambda }>0.52$.

Figure 2. Granger coefficients of coherence for G.C. test from geopolitical risk to WTI oil price and Brent oil prices, respectively. The horizontal dashed line represents the critical value, at the 5% level, for the test for the null hypothesis of no G.C. The vertical dashed line represents the cut-off point ($\mathit{\lambda }=0.52$) between the short-term and long-term.

The top panel of Figure 2 shows Granger-causality results in the frequency domain from geopolitical risks to WTI oil prices. As can be seen, for GPR, GPT, and GPA, the null hypothesis of no causality cannot be rejected at the 5% significance level when the frequency $\mathit{\lambda }\le 0.52$ (in the long term). It reveals that geopolitical risks cannot lead to WTI oil price changes in the long term. However, when the frequency $\mathit{\lambda }>0.52$ (in the short term), there always exists larger Granger coefficients of coherence in the range $\mathit{\lambda }\in (1,3)$ to reject the null hypothesis of no causality. The result suggests a short-term predictive effect for geopolitical risks to WTI oil prices change. This finding is in line with Martina et al. (2011), who proved that some major events had affected the short-term but not the long-term oil market complexity by using an entropy method. One explanation for this finding is that geopolitical risks lead to great volatility in the global economy and further greatly impact the balance of supply and demand in the crude oil market (Glick & Taylor, 2010). In recent years, wars in the Middle East happened frequently, and ISIS terrorist activities in oil-producing countries seriously affected the supply of global crude oil and caused turbulence in crude oil prices. Meanwhile, some terrorist attacks have also occurred in the U.S. and European countries, such as the 911 event in 2001 and the Paris bombing attack in November 2015, all of which have frustrated the confidence of the market participants and also have had a negative impact on the economy. Nevertheless, the predictive effect of geopolitical risks cannot play a role in the long term due to automatic adjustments in the market. This suggests that corresponding short-term measures should be acted on quickly to deal with the influence of geopolitical risks. Our findings are similar to what Chen et al. (2016) found, notably that the political risk of OPEC countries has a significant and positive influence on Brent oil prices. It also supports the findings of Lee et al. (2017).

A greater Granger coefficient of coherence exists in a high-frequency domain for geopolitical threats derived from GPT than for GPA. Pure geopolitical risks have a bigger and longer-lasting effect on WTI oil price fluctuations than actual adverse geopolitical events. This is because certain large events, such as threat events, cause investors’ future expectations to become more uncertain, affecting their trading behaviour and ultimately having a stronger impact on crude oil prices (Noguera-Santaella, 2016). Obviously, geopolitical events affect the price of crude oil, but they should not induce investors to anticipate excessive future instability. Consequently, compared to GPT, GPA cannot have a greater and more lasting impact on oil prices.

Since Brent crude oil is also the major benchmark for world crude oil pricing, we select Brent oil price as the international oil price and re-estimate the frequency domain Granger causality test model. This provides a robustness test. As shown in the bottom panel of Figure 2, we obtain similar results with WTI oil prices. Overall, it confirms that geopolitical risks have only a short-term predictive effect on global oil price changes. Pure geopolitical risks have a larger and more persistent effect on oil prices than actual geopolitical events.

5. Concluding remarks

Using nonlinear causality tests and a Granger causality test in the frequency domain, this study investigated whether geopolitical risks have a significant predictive impact on the movement of real crude oil prices. According to the findings, geopolitical risks have a large nonlinear predictive effect on oil prices. The predictive effect of geopolitical risks, therefore, is only merely valid in the short term. Moreover, pure geopolitical risks have a bigger and longer-lasting impact on oil prices than actual adverse geopolitical events.

Our findings have important threefold implications for oil market participants. First, geopolitical risk information can predict oil prices in a nonlinear model but not in a linear model, indicating the necessity to optimise the estimated model from a nonlinear perspective that can better identify the role of geopolitical risk in the crude oil price formation mechanism. Second, market participants do not need to be unduly concerned about geopolitical risks, as they will only have a short-term (less than one year) impact on oil prices. This implies that when global geopolitical risks increase, the relevant ministries should undertake short-term stability policies in a timely manner to combat the volatility of oil prices. Finally, when devising various countermeasures, it is crucial to correctly differentiate between the impact of pure risk (such as war and terrorist attack threats) and actual political events on crude oil prices.

Our analysis reveals multiple avenues for future study. First, more research is needed to understand whether geopolitical risks emanating from certain nations affect oil prices differently. Second, whether there is a difference between the short- and long-term projections of geopolitical risks on oil prices can be investigated when the worldwide crude oil market is in various conditions (such as a bull market or a bear market). Third, the nonlinear G.C. from geopolitical risks to oil prices calls for a more fine-grained analysis of whether a threshold effect of geopolitical risks exists on oil prices.

Credit author statement

Yong Jiang: Conceptualisation, Methodology, Validation, Formal analysis, Investigation, Data curation, Supervision, Writing – original draft, Writing – review & editing, Software, Project administration, Funding acquisition. Yi-Shuai Ren: Conceptualisation, Methodology, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review & editing, Visualisation, Resources, Project administration, Funding acquisition. Xiao-Guang Yang: Resources, Review & editing, Visualisation, Funding acquisition. Chao-Qun Ma: Conceptualisation, Funding acquisition, Project administration, Funding acquisition. Olaf Weber: Review & editing. All authors read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grant [No. 72101120, 72104075, 71850012, 72192800, 72274056]; the National Social Science Fund of China under Grant [No. 19AZD014]; the Department of Science and Technology of Hunan province under Grant [No. 2018GK1020]; the Natural Science Foundation of Hunan Province [No. 2022JJ40106]; Youth project of Jiangsu Social Science Foundation [No. 21EYC001]; the Hunan social science achievement review committee under Grant [No. XSP21YBC087]; the third phase of Applied Economics of Nanjing Audit University for advantageous disciplines in Colleges and universities in Jiangsu Province project under Grant [No. [2018]87]; and the Hunan University Youth Talent Program.

Notes

1. http://www.policyuncertainty.com/gpr.html.

2. https://www.eia.gov/dnav/pet/pet_pri_spt_s1_d.htm.

3. https://www.bls.gov/.