# 1. Introduction

Our study examines the effect of uncertainties due to infectious disease outbreaks (*INF*) on the energy futures market. Our study is relevant as disease outbreaks, such as the current COVID-19 pandemic, can distort economic activities, causing economic and financial uncertainties, triggering increased unemployment. These resultant consequences may lead to international financial chaos that disturbs asset allocations and risk management and, most notably, financial stability (Bouri et al., 2020). According to Qin et al. (2020), the COVID-19 pandemic had a substantial negative net effect on oil prices by as much as 80%. Similarly, the severe acute respiratory syndrome (SARS) outbreak of 2003 also had a severe effect on oil prices. However, this often-held negative assertion may change as a pandemic outbreak may trigger a negative supply shock, driving up prices (Qin et al., 2020). Therefore, a study of this nature may help investors re-stabilize their portfolios, switching from risky assets to safe-haven assets in order to mitigate portfolio risks (Bouri et al., 2020).

A few studies have attempted to explain the interaction between pandemics and financial markets. Salisu & Adediran (2020) find that the infectious disease equity market volatility is a good predictor of energy market volatility in both in-sample and out-of-sample tests. Liu et al. (2020) examine the interaction among the COVID-19 pandemic, crude oil market, and the U.S. stock market, and find a negative connection between crude oil returns and stock returns. Interestingly, they also discover that the COVID-19 pandemic cannot exert a negative effect but has a statistically significant positive effect on crude oil returns and stock returns. Mazur et al. (2020) investigate the U.S. stock market performance during the crash of March 2020 triggered by COVID-19 and find that natural gas, food, healthcare, and software stocks earn high positive returns, while equity values in petroleum, real estate, entertainment, and hospitality sectors fall significantly. The study also finds that loser stocks exhibit extreme asymmetric volatility, which correlates negatively with stock returns. However, it is still unknown to what extent uncertainties due to infectious disease outbreaks affect the energy futures markets. In theory, outbreaks of infectious diseases will restrain energy demand, leading to a decline in energy prices and returns. There is also the possibility that investors’ expectations may be affected adversely, causing a fall in stock returns (Liu et al., 2020). Since investors take different actions to deal with possible risks and uncertainties, they may choose to delay investment decisions and investments, which eventually reduce stock returns.

This paper extends the literature on COVID-19 by focusing on the causal impact of uncertainties due to infectious disease outbreaks on the energy futures market. To this end, we utilize the novel non-parametric causality-in-quantiles approach recently developed by Balcilar et al. (2016). This approach can test the non-linear causality of the *k*th order across all quantiles of the entire distribution of commodity returns and is robust to the presence of misspecification errors, structural breaks, and frequent outliers, which are frequently found in financial time series (Balcilar et al., 2016). Furthermore, as a justification for using the non-parametric quantile-in-causality approach, we conduct a test for non-linearity by applying the Brock et al. (1996) test which validates the adoption of the non-linear causality-in-quantiles approach.

The rest of the paper is structured as follows. Section 2 provides a description of the methodology. Section 3 presents the discussion of data and empirical results, and Section 4 concludes.

# 2. Methodology

This paper follows the Balcilar et al. (2016) methodology, which is an extension of the Nishiyama et al. (2011) and the Jeong et al. (2012) non-linear causality frameworks. As noted by Jeong et al. (2012), the variable xt (*INF*) does not cause yt (energy future returns) in the σ-quantile with respect to the lag-vector of {yt−1,…, yt−q, xt−1,xt−q} if

Qσ(yt|yt−1, …,yt−q, xt−1,…,xt−q)=Qσ(yt|yt−1, …,yt−q)(1)

While xt causes yt in the σth quantile with respect to {yt−1,…, yt−q, xt−1,xt−q} if

Qσ(yt|yt−1, …,yt−q, xt−1,xt−q)≠Qσ(yt|yt−1, …,yt−q)(2)

Definitively, Qσ(yt|∙)=σth quantile of yt depending on *t and* 0<σ<1. We denote Vt−1≡(yt−1, …,yt−q), Ut−1≡ (xt−1,…, xt−q), and Wt=(Ut,Vt); and Fyt|Wt−1(yt|Wt−1) and Fyt|Vt−1(yt|Vt−1) represents the conditional distribution of yt given Wt−1 andVt−1, respectively. Also, Fyt|Vt−1(yt|Vt−1) is assumed to be absolutely continuous in yt for almost all Wt−1. If we proceed by denoting Qσ(Wt−1)≡Qσ(yt|Wt−1) and Qσ(Vt−1)≡Qσ(yt|Vt−1), then we have Fyt|Wt−1{Qσ(yt|Wt−1)}=σ with a probability of one. The hypothesis to be tested based on the specified definitions in Equations (1) and (2) are;

H0=P{Fyt|Wt−1{Qσ(yt|Wt−1)}=σ}=1,(3)

H1=P{Fyt|Wt−1{Qσ(yt|Wt−1)}=σ}<1,(4)

Following Jeong et al. (2012), the distance measure J={τtE(τt|Wt−1)fW(Wt−1)}, where τt and fz(Wt−1) are the regression error and marginal density function of Zt−1, respectively. The regression error in Equation (3) can only be true if and only if E[1{yt≤Qσ(Vt−1)|Wt−1)}=σ or, equivalently, 1{yt≤Qσ(Vt−1)}=σ+τt, where 1{∙} is the indicator function. Thus, Jeong et al. (2012) specify the distance measure, G≥0, as:

G=E[{Fyt|Wt−1{Qσ(yt|Wt−1)}−σ}2fW(Wt−1)](5)

We will have a situation where G=0 if and only if the null in Equation (3) is true, while we will have G>0 otherwise in Equation (4). To test for the *J*-statistic, feasible kernel-function of Equation (6) is used:

ˆGT=1T(T−1)s2qT∑t=q+1 T∑r=q+1,r≠tK(Wt−1−Zs−1s)^τt^τs,(6)

Where K(∙) denotes the kernel function with bandwidth *s*. *T*, *q*, ^τt is the sample size, lag-order and estimate of the regression error, respectively. The estimate of the regression error is computed as thus:

^τt=1{yt≤^Qσ(Yt−1)}−σ(7)

Also, we further use the non-parametric kernel method to estimate the σth conditional quantile of yt given Vt−1 as ˆQσ(Vt−1)=ˆF−1yt|Vt−1(σ|Vt−1), where the Nadarya-Watson Kernel estimator is specified as follows :

ˆF yt|Vt−1(yt|Vt−1)=∑Tr=q+1,r≠tN(Vt−1−Vr−1s)1(yr≤yt)∑Tr=q+1,r≠tN(Vt−1−Vr−1s)(8)

Where N(∙) is the kernel function and *s* is the bandwidth.

Balcilar et al. (2016) extend the above frameworks to account for causality in higher order moments, such that

yt=h(Vt−1)+ϑ(Ut−1)τt,(9)

Where τt is the white noise process and h(∙) and ϑ(∙) equals the unknown functions that satisfy pertinent conditions for stationarity. Although, this specification allows no granger-type causality testing from Ut−1 to yt, it could detect the “predictive power” from Ut−1 to y2t when ϑ(∙) is a general non-linear function. Thus, the study re-formulates Equation (9) to account for the null and alternative hypothesis for causality in variance in Equations (10) and (11):

H0=P{Fy2t|Wt−1{Qσ(yt|Wt−1)}=σ}=1,(10)

H1=P{Fy2t|Wt−1{Qσ(yt|Wt−1)}=σ}<1,(11)

The feasible test statistic for testing of the null hypothesis in Equation (10) is obtained, and then replace yt in Equations (6) – (8) with y2t (that is, volatility). With the inclusion of the Jeong et al. (2012) approach, the study overcomes the issue that causality in mean implies causality in variance. Specifically, the study interprets the causality in higher-order moments through the use of the following model:

yt=h(Ut−1,Vt−1)+τt,(12)

Thus, we specify the higher order quantile causality as

H0=P{Fykt|Wt−1{Qσ(yt|Wt−1)}=σ}=1,for k=1,2,…,k,(13)

H1=P{Fykt|Wt−1{Qσ(yt|Wt−1)}=σ}<1,for k=1,2,…,k.(14)

Overall, we test that