**1. Introduction**

Our paper is aimed at analyzing the return and volatility spillover among natural gas, crude oil, and the electricity utility sector indices across North America and Europe using the methods of Diebold and Yilmaz [1] and Barunik and Krehlik [2] in time and frequency domains. In current commodity markets, energy futures play a major role in economic activities. In particular, as natural gas is cleaner and produces fewer greenhouse emissions than fossil fuels, such as oil and coal, the importance of natural gas has been increasing in the global energy market. Natural gas can be used in many areas, including residential, commercial, industrial, power generation, and vehicle fuels. According to the IEA, natural gas grew 4.6% in 2018, accounting for almost half of the increase in global energy demands (https://www.iea.org/fuels-and-technologies/gas). In both national policy scenarios (gas demand increases by more than a third) and sustainable development scenarios (gas demand will increase slowly by 2030 and return to current levels by 2040), natural gas continues to outperform coal and oil. Meanwhile, crude oil is used to generate electricity, which is also an important raw material for the chemical industry. Thus, we chose the United States (US) and Canada in North America, and Germany, France, the United Kingdom (UK), and Italy in Europe, which are the Group of Seven (G7) member countries, to investigate the spillover among the two energies and electricity utility stocks. According to the BP Statistical Review of World Energy 2019 [3], the consumption shares of natural gas

and crude oil in 2018 were as follows: US (16.6%), Canada (2.5%), Germany (2.3%), France (1.7%), the UK (1.4%), and Italy (1.1%). These six countries are major consumers of natural gas and crude oil.

With the development of financial globalization, the financial community is paying increasingly more attention to the transmission of dynamic return links and volatility throughout the capital market. In the situation of a market crash or crisis, portfolio managers and policymakers need to take some actions to prevent the risk of transmission. Therefore, empirical research on the intensity of spillovers provides insights for accurate predictions of returns and volatility. In particular, investors should know, especially in recent years, how fluctuations in natural gas, crude oil, and electricity utilities stock indices a ffect the risk and value of their investment portfolios. In addition, from 2009 to 2019, several major events influenced these two energy markets and the stock market. These extreme events led to fluctuations in the return and volatility spillover among the three markets in North America and Europe. Hence, understanding the return and volatility spillover caused by financial shocks is not only essential for investors in terms of risk managemen<sup>t</sup> and portfolio diversification but also for policymakers in developing appropriate policies to avoid impacts from future extreme events. Tian and Hamori [4] have also indicated that policymakers need to understand the transmission mechanism of volatility shock spillover that leads to financial instability.

The main contributions of our study can be summarized as follows. First, as far as we know, this is the first study to investigate the return and volatility spillover among natural gas, crude oil, and the electricity utility sector indices in North America and Europe, respectively, using the Diebold and Yilmaz [1] method for time domain and the Barunik and Krehlik [2] method for frequency domain. Second, we separately analyzed the return and volatility spillover in North America and Europe between the two energy futures and electricity utility stocks to determine the similarities and di fferences of the spillover e ffects in the two regions. Third, we employ a rolling analysis to examine the dynamics of the connectedness of return and volatility in time and frequency domains.

The remainder of our paper is described as follows. Section 2 provides a literature review. Section 3 describes the empirical techniques. In Section 4, we explain the data and the descriptive statistics through a preliminary analysis. In Section 5, we report the empirical results of the spillover e ffects and the moving window analysis. Finally, we conclude our analysis in Section 6.

#### **2. Literature Review**

There are numerous studies in the literature investigating spillover e ffects on the relationship between crude oil and stock markets. Arouri et al. [5] used the generalized vector autoregression (VAR)–generalized autoregressive conditional heteroskedasticity (GARCH) model to analyze the volatility spillover between oil and the stock markets in Europe and the US. Using the sector data, they found that oil and sector stock returns have significant volatility spillover. Soytas and Oran [6] used the Cheung–Ng approach (Cheung and Ng [7]) to analyze the volatility spillover between the world oil market and electricity stock returns in Turkey. They found new information that was not found through conventional causality tests using aggregated market indices. Arouri et al. [8] used a recent generalized VAR–GARCH model to investigate the return and volatility spillover between the oil and stock markets among Gulf Cooperation Council (GCC) countries from 2005 to 2010. They found that the return and volatility spillover between them was significant enough for investors to diversify their portfolios. Nazlioglus et al. [9] investigated the volatility transmission between crude oil and some agricultural commodity markets (wheat, corn, soybean, and sugar). They found that the good price crisis and risk transmission has significantly a ffected the dynamics of volatility spillover. Nakajima and Hamori [10] analyzed the relationship among electricity prices, crude oil prices, and exchange rates. They found that exchange rates and crude oil Granger cause electricity prices neither in mean nor in variance.

Despite the many well-documented studies on the spillover between crude oil and the stock market, there are relatively few studies on the natural gas and financial markets. Ewing et al. [11] analyzed the volatility spillover between oil and natural gas markets using the GARCH model. Acaravci et al. [12] investigated the long-term relationship between natural gas prices and stock prices using the vector error correction model developed by Johansen and Juselius [13].

Diebold and Yilmaz [1,14,15] developed the methodology of analyzing the connectedness in time domain based on the variance decomposition of the forecast error to assess the share of forecast error variation in its magnitude and direction; Barunik and Krehlik [2] then extended this connectedness to frequency domain to show the spillover effect from different frequency ranges. Many researchers have applied these empirical techniques to investigate the connectedness between markets in time domain or both in time and frequency domains.

Maghyereh et al. [16] analyzed the connectedness between oil and equities in 11 major stock exchanges in time domain. They found a robust transmission from the crude oil market to the equity market, which grew stronger from mid-2009 to mid-2012. Duncan and Kabundi [17] investigated the domestic and foreign sources of volatility spillover in South Africa in time domain. In addition, Liow [18] characterized the conditional volatility spillover among G7 countries in regard to public real estate, stocks, bonds, money, and currency, both domestically and internationally in time domain. Sugimoto et al. [19] examined the spillover effects on African stock markets during the global financial crisis and the European sovereign debt crisis in time domain.

Toyoshima and Hamori [20] researched the connectedness of return and volatility in the global crude oil markets in time and frequency domains. They found that the Asian currency crisis (1997–1998) and the global financial crisis (2007–2008) generated an increase in return and volatility spillover effects. Lovcha et al. [21] characterized the dynamic connectedness between oil and natural gas volatility in frequency domain. Ferrer et al. [22] analyzed the return and volatility connectedness of the stock prices of US clean energy companies, crude oil prices, and important financial variables in time and frequency domains.

We use a rolling analysis to examine the spillover of return and volatility in North America and Europe separately in time and frequency domains. Zhang and Wang [23] also analyzed the return and volatility spillover between the Chinese and global oil markets and employed a moving-window analysis to better understand and capture the dynamics of return and volatility spillover in time domain.

Finally, we investigate some prior studies similar to ours. Similar to our study, Oberndorfer et al. [24] focused on investigating the volatility spillover across energy markets and the pricing of European energy stocks by using the GARCH model, and they found that oil price is the main index for energy price developments in the European stock market. Kenourgios et al. [25] investigated the contagion effects of the global financial crisis (2007–2009) across assets in different regions. Kenourgios et al. [26] also investigated the contagion effects of the global financial crisis (2007–2009) in six developed and emerging regions by applying the FIAPARCH model. Baur [27] also studied different channels of financial contagions across 25 major countries and found that the crisis significantly increased the co-movement of returns. Singh et al. [28] examined price and volatility spillovers in the stock markets of North America, Asia, and Europe and found that a greater regional influence exists among the Asian and European stock markets. Balli et al. [29] analyzed the return and volatility spillovers and their determinants in emerging Asian and Middle Eastern countries. They found that developed financial markets have significant spillover effects on emerging financial markets and shocks originated in the US play a dominant role.

#### **3. Empirical Techniques**

#### *3.1. Diebold–Yilmaz Method*

Our study employs the method proposed by Diebold and Yilmaz [1] for measuring spillover in a generalized VAR framework. This approach was designed to measure the connectedness concept built on the basis of the generalized forecast error variance decomposition (GFEVD) of a VAR approximating model. First, we conceived an *N*-variable VAR (*p*) model, as (1) below.

$$y\_t = \sum\_{i=1}^{p} \Phi\_i y\_{t-i} + \varepsilon\_t \tag{1}$$

where *yt* is the *N* × 1 vector of the observed variables at time *t*, and Φ is the *N* × *N* coefficient matrix. The error vector ε*t* is independent and identically distributed, and white noise (0, ) with covariance matrix is possibly non-diagonal.

In this model, the VAR process can also transform into the vector moving average (VMA) (∞), as represented in (2). It is effective to use with the (*N* × *N*) matrix lag-polynomial |*In***-**Φ**1***z*−· · · <sup>−</sup>Φ*pz<sup>p</sup>*| = 0 with the *In* identity matrix. Assuming that the roots of |Φ(*z*)| lie outside the unit circle,

$$y\_t = \psi(L)\,\varepsilon\_t \tag{2}$$

where ψ(*L*) is the (*N* × *N*) matrix of infinite lag polynomials that can be calculated from ψ(*L*) = [ψ(*L*)]−**1**. However, as the order of the variables in the VAR system may influence the impulse response or variance decomposition results, to eliminate the influence from the ordering of the variables in the variance decomposition, Diebold and Yilmaz [1] applied the generalized VAR framework developed by Koop et al. [30] and Pesaran and Shin [31]. On the basis of this framework, the *H*-step-ahead GFEVD can be written in the form of (3):

$$(\theta\_H)\_{jk} = \frac{\sigma\_{kk}^{-1} \Sigma\_{h=0}^H (\left(\Psi \mu \Sigma\right)\_{jk})^2}{\Sigma\_{h=0}^H \left(\Psi \mu \Sigma \Psi\_h'\right)\_{jj}} \tag{3}$$

where ψ*h* is an *N* × *N* coefficient matrix of polynomials at lag *h*, and σ<sup>−</sup><sup>1</sup> *kk* = (**Σ**)*kk*. (<sup>θ</sup>*H*)*jk* indicates the contribution of the *k*th variable of the model to the variance of the forecast error of element *j* at horizon *h*. To sum the elements in each row of the GFEVD to total 1, each entry is normalized by the row sum as

$$
\stackrel{\cdots}{\theta}\_{jk}^{H} = \frac{\Theta\_{jk}^{H}}{\Sigma\_{K=1}^{N} \Theta\_{jk}^{H}} \tag{4}
$$

**~** θ *H jk* measures the pairwise spillover from *k* to *j* at horizon *H* and also measures the spillover effect from market *k* to *j*. We can aggregate this to measure the total spillover of the system. The total spillover can be measured by the pairwise spillover. The connectedness can be seen as the share of variance in the forecasts contributed to by errors (Diebold and Yilmaz [1]).

$$S^H = 100 \times \frac{\Sigma\_{j\neq k}^N \tilde{\boldsymbol{\theta}}\_{jk}^H}{\Sigma \boldsymbol{\theta}} = 100 \times \left(1 - \frac{\mathrm{Tr}\left\{\tilde{\boldsymbol{\theta}}^H\right\}}{\Sigma \boldsymbol{\theta}}\right) = 100 \times \left(1 - \frac{\mathrm{Tr}\left\{\tilde{\boldsymbol{\theta}}^H\right\}}{N}\right) \tag{5}$$

where *Tr*{.} is the trace operator. The total spillover calculates the total spillover across all markets in the form of (5). There are two measures by Diebold and Yilmaz [1] that show the relative importance of each variable in the system:

$$\sum\_{j \neq i} \sum\_{k=1}^{n} \overline{\Theta}\_{k,j}^{H}$$

Directional Spillover (From): *<sup>S</sup>Hk*← = 100 × ே ; the directional spillover (from) is the spillover that market k receives from all other markets. ஊೕసభಿ ࣂ෩ೖೕ ಹ

Directional Spillover (To): *S<sup>H</sup>*←*k* = 100 × ೕಯ ே ; the directional spillover (to) is the spillover that market k transmits to all other markets.

**~**

#### *3.2. Barunik and Krehlik Method*

Following Barunik and Krehlik [2], we describe the frequency dynamics (for the short-term, the medium-term, and the long-term) of spillover and the spectral formulation of variance decomposition.

Notably, we measure spillovers in frequency domain using Fourier transform. Moreover, the frequency response function is obtained as a Fourier transform of the coefficients <sup>ψ</sup>*h***:**<sup>ψ</sup>*<sup>e</sup>*<sup>−</sup>*i*<sup>ω</sup> = <sup>Σ</sup>*he*<sup>−</sup>*i*ω*<sup>h</sup>*ψ*<sup>h</sup>*, where *i* = √−1. The generalized causation spectrum over frequencies ω ∈ (−π, π) is defined in the form of (6)

$$\left| \left( f(\omega) \right)\_{jk} = \frac{\sigma\_{kk}^{-1} \left| \left( \mathfrak{p} \left( e^{-i\omega} \right) \Sigma \right)\_{jk} \right|^2}{\left( \mathfrak{p} \left( e^{-i\omega} \right) \Sigma \mathfrak{p}^\prime \left( e^{+i\omega} \right) \right)\_{jj}} \tag{6}$$

where <sup>ψ</sup>*e*<sup>−</sup>*i*<sup>ω</sup> = <sup>Σ</sup>*he*<sup>−</sup>*i*ω*<sup>h</sup>*ψ*h* is the Fourier transform of the impulse response ψ*<sup>h</sup>*. It is vital to pay attention to (*f*(ω))*jk*, namely, the portion of the spectrum of the *j*th variable at a given frequency ω due to shocks in the *k*th variable. As domination holds the spectrum of the *j*th variable at a given frequency ω, we establish (6) for the quantity within the frequency causation. To obtain the generalized decomposition of the variance's decompositions to frequencies, we weight (*f*(ω))*jk* by the frequency share of the variance of the *j*th variable. This weighting function can be defined as (7)

$$\Gamma\_{\dot{\jmath}}(\omega) = \frac{\left(\psi(e^{-i\omega})\Sigma\psi'(e^{+i\omega})\right)\_{\dot{\jmath}\dot{\jmath}}}{\frac{1}{2\pi}\int\_{-\pi}^{\pi} (\psi(e^{-i\lambda})\Sigma\psi'(e^{+i\lambda}))\_{\dot{\jmath}\dot{\jmath}} d\lambda} \tag{7}$$

where the power of the *j*th variable at a given frequency sums through the frequencies to a constant value of 2<sup>π</sup>. When the Fourier transform of the impulse is a complex number value, the generalized factor spectrum is the squared coefficient of the weighted complex numbers and, therefore, a real number. In sum, we set the frequency band *d* = (*a*, *b*): *a*, *b* ∈ (−π, π), *a* < *b*.

The GFEVD under the frequency band d is

$$\theta\_{jk}(d) = \frac{1}{2\pi} \int\_{a}^{b} \Gamma\_{j}(\omega) (f(\omega))\_{jk} d\omega \tag{8}$$

However, GFEVD will still be normalized into (9). The scaled GFEVD on the frequency band *d* = (*a*, *b*): *a*, *b* ∈ (−π, π), *a* < *b* is shown below:

$$
\tilde{\theta}\_{jk}(d) = \frac{\theta\_{jk}(d)}{\Sigma\_k \theta\_{jk}(\infty)} \tag{9}
$$

where <sup>θ</sup>*jk*(*d*) is defined as the pairwise spillover at a given frequency band d. At the same time, it is possible to define the total spillover at frequency band *d*.

The frequency total spillover (frequency connectedness) on frequency band d can be defined as

$$S^F(d) = 100 \times \left( \frac{\tilde{\Sigma}\tilde{\theta}(d)}{\tilde{\Sigma}\tilde{\theta}(\infty)} - \frac{Tr\left\{\tilde{\theta}(d)\right\}}{\tilde{\Sigma}\tilde{\theta}(\infty)} \right) \tag{10}$$

**~**

where *Tr*{.} is the trace operator, and **Σ** θ(*d*) is the sum of all elements of the θ(*d*) matrix. The frequency total spillover decomposes the total spillover into the long-term, the medium-term, and the short-term, and these can sum into the total spillover S, as defined by Diebold and Yilmaz [1].

**~**

Equally, we can also define the two directional spillovers for frequency according to Diebold and Yilmaz [1].

Frequency Directional Spillover (From): *S<sup>F</sup> <sup>k</sup>*←(*d*) = 100 × ஊೕసభ ೕಯ ಿ ఏ෩ೖೕሺௗሻ ே ; the frequency directionalspillover (from) is the spillover that market k receives from all other markets at frequency band d. ಿ

Frequency Directional Spillover (To): *S<sup>H</sup>* ←*k* (*d*) = 100× ೕಯ ே ; the frequency directional spillover (to) is the spillover that market k transmits to all other markets at frequency band d.

ஊೕసభ

 ఏ෩ೖೕሺௗሻ
