**1. Introduction**

In light of the increasing attention being paid to environmental sustainability, energy systems are gradually transitioning from a dependence on non-renewable resources to the use of environment-friendly resources. This will have a grea<sup>t</sup> impact on day-to-day life, economies, businesses, manufacturers, and governments. Compared to coal or petroleum, natural gas has many qualities that makes it burn more efficiently. It also generates fewer emissions of most types of air pollutants, including carbon dioxide. With the expansion of gas pipelines, the increasing number of gas liquefaction plants, and the exploitation of natural gas fields, it is reasonable to consider that the natural gas trade will become more globalized. Natural gas has become a major part of the world's energy consumption, demand, and supply in recent years. In 2018, for example, natural gas consumption rose by 5.3%, one of the fastest rates of growth since 1984. With the continuing rapid expansion in liquefied natural gas (LNG), the inter-regional natural gas trade grew by 4.3%, which was more than double the 10-year average [1]. As reported in the Global Energy Perspective 2019: Reference Case [2], natural gas will be the only fossil fuel whose share of total energy demand continues to increase until 2035, and China will represent nearly half of the global demand growth. Other developing countries are also expected to increase their demand for natural gas.

Brazil, Russia, India, China, and South Africa (BRICS), a group of five fast-growing developing countries, play an important and expanding role in the world economy. In recent years, BRICS have represented an increasing share of global economic growth. According to the International Monetary Fund (IMF), as of 2018, the combined gross domestic product (GDP) of these five nations accounted for 23.2% of the gross world product (GWP). Given the growth of BRICS and the fact that energy is

a crucial ingredient for economic development, these countries' relationship with natural gas will only become closer. According to the BP Statistical Review of World Energy [1], in 2018, the total consumption of natural gas in BRICS was 835.8 billion cubic meters (BCM), which accounted for 21.7% of the total global consumption. In terms of imports, China became the second largest importer of LNG, with imports increasing from 4.6 BCM in 2008 to 73.5 BCM in 2018. India was the fourth largest importer, with imports increasing from 11.3 BCM to 30.6 BCM over the same period. In terms of exports, Russia was the largest exporter of pipeline gas. It also accounted for nearly 6% of total LNG exports. As the trade of natural gas is usually settled in US dollars, it is meaningful to study the relationship between the natural gas price and the BRICS's exchange rates.

Against this backdrop, this paper investigates the interdependence between the natural gas price and the BRICS's exchange rate. In doing so, this study is expected to o ffer valuable insights for market operators, investors, and economists. We use the Henry Hub natural gas futures as the data for the natural gas price. There are two reasons behind this choice of dataset: First, the shale gas revolution in America has dramatically increased US production of shale gas since 2007. World Energy Outlook 2018 [3], produced by the International Energy Agency (IEA), has predicted that natural gas production in America will increase from 976 BCM in 2017 to 1328 BCM in 2040 and that this increase will be mainly due to the growth in shale gas production. Therefore, the Henry Hub natural gas price, which usually represents pricing for the North American natural gas market, has a grea<sup>t</sup> influence on the global energy market. We assume that this influence will become stronger over time. The second reason is that there are multiple natural gas price indexes in the world, such as the Japan Korea Marker and the UK National Balancing Point (NBP); however, we cannot predict which price index has strong connectedness with the BRICS's exchange rates. Therefore, we select the Henry Hub price given its characteristics of high liquidity and large trading volume.

Our contribution to the literature is twofold. First, we apply the connectedness methodology from Diebold and Yilmaz [4–6], which allows us to know how pervasive the risk is throughout the entire market by quantifying the contribution of each variable to the system. We also apply the time–frequency version of connectedness proposed by Baruník and Kˇrehlík [7] to find the connectedness between di fferent variables in the short, medium, and long term. Second, to the best of our knowledge, there is not much research on the relationship between the natural gas price and exchange rates. Nevertheless, there are many studies that analyzed the relationship between crude oil prices and foreign exchange rates, and almost all of them show that exchange rates are highly connected to the oil price. For example, in our previous research on the relationship between the West Texas Intermediate (WTI) crude oil price and BRICS's exchange rates using the copula method, we found a significant negative dependence between the two variables. Considering the globalization of natural gas trade, high demand growth (1.6% per year), and the expansive market share in the global energy market (World Energy Outlook 2018 [3] has predicted that, by 2030, natural gas will overtake coal and become the second largest source of energy after oil.), it is reasonable to compare the relationship between the crude oil price and exchange rates with that between the natural gas price and exchange rates. Therefore, in this study, we also aim to determine whether BRICS's exchange rates are closely linked to the natural gas price, as they are to the oil price.

The rest of this paper is organized as follows: A brief review of relevant literature is provided in Section 2. Section 3 introduces the empirical methodology used in this study. Section 4 reports empirical results. Section 5 gives the conclusion. Finally, a robustness analysis is presented in the Appendix A.

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

As we have mentioned above, there is not much literature that has analyzed the relationship between the natural gas price and exchange rates, as far as we know. However, there are many studies on the relationship between the exchange rate and other variables, such as the oil price and the stock market. Chen and Chen [8] investigated the long-term relationship between di fferent crude oil price

indexes and G7 countries' exchange rates using the monthly panel data between January 1972 and December 2005. They found that oil prices may account for the movements of the real exchange rate and there is a link between oil prices and real exchange rates. Additionally, from the results of panel predictive regression, they found that the crude oil price has the ability to forecast the future exchange rate. Andries, et al. [9] identified the patterns of co-movement of the interest rate, stock price, and exchange rate in India using wavelet analysis. They used the data span from July 1997 to December 2010. The empirical results showed that exchange rates, interest rates, and stock prices are linked to each other and that the stock price fluctuations lag behind both the exchange rates and interest rates. Brahm et al. [10] used monthly data to investigate the relationship between the crude oil price and exchange rates in the long term and short term, respectively. The data span was from January 1997 to December 2009. Empirical results indicated exchange rates Granger-caused crude oil prices in the short term, whereas crude oil Granger-caused exchange rates in the long term. Furthermore, based on impulse response analysis, exchange rate shock had a significant negative e ffect on crude oil prices. Jain and Pratap [11] explored the relationship between global prices of crude oil and gold, the stock market in India, and the USD–INR exchange rate using the DCC-GARCH (dynamic conditional correlation-generalized autoregressive conditional heteroscedasticity) model. They also examined the lead lag linkages among these variables using symmetric and asymmetric non-linear causality tests. They used daily data from the period of 2006 to 2016, finding that a fall in the value of the Indian Rupee and the benchmark stock index was caused by a fall in gold and crude oil prices.

On the empirical side, the methodology used in this paper has already been applied in many fields. Maghyereh et al. [12] used implied volatility indices (VIX) of the daily close price of crude oil in 11 countries. They found that the connectedness between oil and equity was dominated by the transmissions from the oil market to equity markets and most of the linkages between these two markets were established from mid-2009 to mid-2012, a period that witnessed the start of the global recovery. Lundgren et al. [13] studied the renewable energy stock returns and their relation to the uncertainty of currency, oil price, stocks, and US treasury bonds. They used data covering the period from 2004 to 2016, and found that the European stock market depends on renewable energy stock prices. Singh et al. [14] employed a dynamic and directional network connectedness between the implied volatility index (VIX) of the exchange rates of nine major currency pairs and the crude oil using the data between May 2017 and December 2017. They found that crude oil a ffected currencies more than currencies a ffected crude oil, but the reverse was true during the crude oil crisis period. Furthermore, their results revealed that EUR–USD is more sensitive to crude oil price fluctuation than others. Ji et al. [15] combined empirical mode decomposition with a connectedness methodology, and examined the dynamic connectedness among crude oil, natural gas, and refinery products using daily data between 3 January 2000 and 15 September 2017. Employing a constant analysis, they found that crude oil and its refinery product tend to be a net transmitter, while the natural gas tends to be a net receiver. In time-varying analysis, they found that the total connectedness generally increased until the 2014 crude oil crash, and then decreased sharply. Lovch and Perez-Laborda [16] used the connectedness method and frequency decomposition method to investigate the relationship between the natural gas and crude oil price during the period from 1994 to 2018. They found that the volatility connectedness varied over time; the connectedness became weak after the financial crisis; and the volatility had long-run e ffects, except during some specific periods, when volatility shocks transmitted faster but dissipated in the short-run.

#### **3. Empirical Methodology**

In this paper, we employ two methods to establish the nature of the relationship between exchange rates and natural gas price. The first method is provided by Diebold and Yilmaz (DY) [4–6], whose approach calculates the connectedness between di fferent objects by introducing variance decomposition into vector autoregression (VAR) models. The second method is based on Baruník and Kˇrehlík (BK) [7], who proposed a new framework to estimate connectedness by using a spectral representation of variance decomposition. In conclusion, the DY framework describes the connectedness as "when shocks are arising in one variable, how would other variables be changing?", whereas the BK framework estimates the connectedness in short-, medium-, and long-term financial cycles.

#### *3.1. Connectedness Table*

Based on Diebold and Yilmaz [6], a simplified connectedness table is presented in Table 1, which gives a clear picture of aggregated and disaggregated connectedness.


**Table 1.** Connectedness table.

Source: Diebold and Yilmaz (2015).

In the table, *xi* is the interested variable, whereas *dij* is the pairwise directional connectedness from *xj* to *xi*, which shows what percentage of the h-step-ahead forecast error variance in *xi* is due to the shocks in *xj* (Equation (1)). We can simply understand *dij* as how much future uncertainty of *xi* is due to the shocks in *xj*:

$$\mathbb{C}\_{i \leftarrow j} = d\_{\hat{l}}.\tag{1}$$

The column "From" is the total directional connectedness from *xj* to others (Equation (2)), and the row "To" means the total directional connectedness from others to *xi* (Equation (3)):

$$C\_{\leftarrow \leftarrow j} = \sum\_{\substack{i=1 \\ i \neq j}}^{N} d\_{ij\prime} \tag{2}$$

$$\mathbb{C}\_{i \gets \cdots} = \sum\_{\substack{j=1 \\ j \neq i}}^{N} d\_{ij}. \tag{3}$$

We were also interested in net pairwise directional connectedness (Equation (4)) and net total directional connectedness (Equation (5)), which are expressed as a negative value to indicate a net recipient and a positive value to indicate a net transmitter:

$$\mathbf{C}\_{i\dot{j}} = \mathbf{C}\_{\dot{j}\leftarrow\dot{i}} - \mathbf{C}\_{i\leftarrow\dot{j}\prime} \tag{4}$$

$$C\_i = C\_{\cdots \leftarrow i} - C\_{i \leftarrow \cdots} \tag{5}$$

Finally, the total connectedness (Equation (6)), calculated by the grand total of the off-diagonal entries of *dij*, is given in the lower-right cell of the connectedness table:

$$C = \frac{1}{N} \sum\_{\substack{i,j=1 \\ i \neq j}}^{N} d\_{ij}. \tag{6}$$

#### *3.2. Generalized Forecast Error Variance Decomposition (GFEVD)*

Diebold and Yilmaz [4] measured connectedness based on forecast error variance decompositions from VAR models, which were introduced by Sims [17] and Koop et al. [18]. However, the calculation of variance decomposition requires orthogonalized shocks and depends on ordering the variables, so Diebold and Yilmaz [5] exploited the generalized forecast error variance decomposition (GFEVD) of Pesaran and Shin [19] to solve those problems. In this paper, we employ the method of GFEVD to calculate the connectedness.

We will give a brief introduction to GFEVD, followed by an explanation of Lütkepohl [20] and Diebold and Yilmaz [6].

For easy understanding, we first consider a VAR (1) process with N-variable:

$$\begin{aligned} y\_t &= v + A\_1 y\_{t-1} + u\_t, \quad t = 0, \pm 1, \pm 2 \dots \\ E(u\_t) &= 0 \\ E(u\_t u\_t') &= \Sigma\_u \\ E(u\_t u\_s') &= 0, \quad t \neq s. \end{aligned} \tag{7}$$

If the generation mechanism starts at time *t* = 1, we get:

$$\begin{aligned} y\_1 &= v + A\_1 y\_0 + u\_1 \\ y\_2 &= v + A\_1 y\_1 + u\_2 = v + A\_1 (v + A\_1 y\_0 + u\_1) + u\_2 \\ &= (I\_N + A\_1) v + A\_1^2 y\_0 + A\_1 u\_1 + u\_2 \\ &\dots \\ y\_t &= (I\_N + A\_1 + \dots + A\_1^{t-1}) v + A\_1^t y\_0 + \sum\_{m=0}^{t-1} A\_1^m u\_{t-m} \dots \end{aligned} \tag{8}$$

If all eigenvalues of *A*1 have modulus less than 1 (VAR process is stable), we have:

$$\begin{array}{rcl}(I\_N + A\_1 + \cdots + A\_1^{t-1})v \to & (I\_N - A\_1)^{-1}v \quad \text{as } t \to \infty\\A\_1^t y\_0 \to 0 \quad \text{as } t \to \infty \dots \end{array} \tag{9}$$

Then, we can rewrite Equation (7) as:

$$y\_t = \mu + \sum\_{m=0}^{\infty} A\_1^m u\_{t-m\nu} \quad t = 0, \pm 1, \pm 2 \dots \tag{10}$$
 
$$\text{where } \mu \equiv \left(I\_N - A\_1\right)^{-1} v.$$

Secondly, let us consider a VAR (*p*) process:

$$y\_t = v + A\_1 y\_{t-1} + \dots + A\_p y\_{t-p} + u\_t, \ t = 0, \pm 1, \pm 2, \dots \tag{11}$$

By using matrices, we can rewrite the VAR (*p*) process as a VAR (1) process:

$$\begin{aligned} Y\_t &= \boldsymbol{\upsilon} + A\boldsymbol{Y}\_{t-1} + \boldsymbol{U}\_t\\ Y\_t &= \begin{bmatrix} y\_1\\ y\_2\\ \dots\\ \boldsymbol{Y}\_{t-p+1} \end{bmatrix} \quad \boldsymbol{\upsilon} &= \begin{bmatrix} \boldsymbol{\upsilon} \\ \boldsymbol{0} \\ \dots \\ \boldsymbol{N}(\boldsymbol{p} \times \mathbf{1}) \end{bmatrix} \quad \text{(Np \times 1)} \quad \boldsymbol{=} \begin{bmatrix} \boldsymbol{\upsilon} \\ \boldsymbol{0} \\ \dots \\ \boldsymbol{0} \end{bmatrix} \\\ \boldsymbol{A} &= \begin{bmatrix} A\_1 & A\_2 & \dots & A\_{p-1} & A\_p \\ \boldsymbol{I}\_N & \boldsymbol{0} & \dots & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{I}\_N & \dots & \boldsymbol{0} & \boldsymbol{0} \\ \dots & \dots & \dots & \dots & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \dots & \boldsymbol{I}\_N & \boldsymbol{0} \end{bmatrix}, \quad \boldsymbol{U}\_t \quad \text{(12)} \quad \boldsymbol{=} \begin{bmatrix} \boldsymbol{u}\_t \\ \boldsymbol{0} \\ \dots \\ \boldsymbol{0} \end{bmatrix}. \end{aligned} \tag{12}$$

Similar to Equation (10), Equation (12) can be rewritten as:

$$\mathcal{Y}\_t = \mu + \sum\_{m=0}^{\infty} A^m U\_{t-m} \quad t = 0, \pm 1, \pm 2 \dots \tag{13}$$

By pre-multiplying a *N* × *Np* matrix *J* ≡ [*IN* :0: ... : 0], we get:

$$\begin{array}{rcl} \mathcal{Y}\_{t} = \mathsf{J}\mathcal{Y}\_{t} = \mathsf{J}\mu + \sum\_{m=0}^{\infty} \mathsf{J}A^{m} \mathsf{U}\_{t-m} = \mathsf{J}\mu + \sum\_{m=0}^{\infty} \mathsf{J}A^{m} \mathsf{J}^{\prime} \mathsf{J}\mathsf{U}\_{t-m} \\ & = \mu + \sum\_{m=0}^{\infty} \Phi\_{m} \mathsf{u}\_{t-m} \\ \end{array} \tag{14}$$
 
$$\begin{array}{rcl} \mu & = \mathsf{J}\mu, \quad \Phi\_{m} \\ (\mathsf{N}\times\mathsf{1}) & (\mathsf{N}\times\mathsf{N}) \\ \end{array} \equiv \mathsf{J}A^{m}\mathsf{J}\prime, \quad \begin{array}{rcl} \mathsf{u}\_{t} \\ (\mathsf{N}\times\mathsf{1}) & = \mathsf{J}\mathsf{U}\_{t}. \end{array} \tag{14}$$

Finally, we ge<sup>t</sup> a moving average (MA) representation of the VAR(*p*) process:

$$\begin{aligned} y\_t &= \mu + \sum\_{m=0}^{\infty} \Phi\_m u\_{t-m} \\ E(u\_t) &= 0 \\ E(u\_t u\_t') &= \Sigma\_u \\ E(u\_t u\_s') &= 0, \quad t \neq s. \end{aligned} \tag{15}$$

The h-step GFEVD can be expressed as:

$$\alpha\_{ij,h}^{\mathcal{S}} = \frac{\sigma\_{jj}^{-1} \sum\_{m=0}^{h-1} \left( e\_i^{\prime} \Phi\_m \Sigma\_n e\_j \right)^2}{\sum\_{m=0}^{h-1} \left( e\_i^{\prime} \Phi\_m \Sigma\_n \Phi\_m^{\prime} e\_j \right)^{\prime}},\tag{16}$$

where *ei* is the *i*-th column of *IN* and <sup>σ</sup>*jj* is the *j*-th diagonal element of Σ*<sup>u</sup>*.

Because the sums of the forecast error variance contribution are not necessarily in agreement, we contribute our generalized connectedness indexes as:

$$d\_{ij} = \widehat{\alpha\_{ij}^{\mathcal{S}}} = \frac{\alpha\_{ij,h}^{\mathcal{S}}}{\sum\_{j=1}^{N} \alpha\_{ij,h}^{\mathcal{S}}}.\tag{17}$$

#### *3.3. Spectral Representation of GFEVD*

Based on the DY framework, the BK framework defines the general spectral representation of GFEVD and uses it to define the frequency-dependent connectedness measure, which is inspired by the previous research of Geweke [21–23] and Stiassny [24].

*Energies* **2019**, *12*, 3970

We still consider the MA representation of the VAR(*p*) process (Equation (15)). The BK framework provides a frequency response function (Equation (18)), which can be obtained as a Fourier transform of the coefficient Φ*m*: 

$$\Psi(e^{-i\lambda}) = \sum\_{m} e^{-i\lambda m} \Phi\_{m\_{\prime}} \quad i = \sqrt{-1}. \tag{18}$$

The generalized causation spectrum over frequencies λ ∈ (−π, π) is defined as:

$$(f(\lambda))\_{j,k} = \frac{\sigma\_{kk}^{-1} \left| (\Psi(e^{-i\lambda})\Sigma\_u)\_{j,k} \right|^2}{(\Psi(e^{-i\lambda})\Sigma\_u \Psi'(e^{+i\lambda}))\_{j,j}},\tag{19}$$

where (*f*(λ))*j,k* represents the portion of the spectrum of *xj* at a given frequency λ due to shocks in *xk*. In order to obtain a natural decomposition of variance decomposition to frequencies, a weighting function is defined as:

$$\Gamma\_{j}(\lambda) = \frac{(\Psi(e^{-i\lambda})\Sigma\_{\mathfrak{u}}\Psi'(e^{-i\lambda}))\_{j,j}}{\frac{1}{2\pi}\int\_{-\pi}^{\pi}(\Psi(e^{-i\lambda})\Sigma\_{\mathfrak{u}}\Psi'(e^{-i\lambda}))\_{j,j}d\lambda},\tag{20}$$

where Γ*j* (λ) represents the power of the *j*-th variable at a given frequency.

The entire range of frequencies' influence of GFEVD from *xj* to *xk* is expressed as:

$$
\omega\_{jk}^{\rm ov} = \frac{1}{2\pi} \int\_{-\pi}^{\pi} \Gamma\_j(\lambda) (f(\lambda))\_{j,k} d\lambda. \tag{21}
$$

Additionally, the GFEVD on specified frequency band *d* = (*a*, *b*), *a*, *b* ∈ (−π, π), *a* < *b*, is defined as:

$$
\omega^{d}\_{j\bar{k}} = \frac{1}{2\pi} \int\_{d} \Gamma\_{\bar{j}}(\lambda) (f(\lambda))\_{j\bar{k}} d\lambda. \tag{22}
$$

As in Section 3.2, we contribute our scaled GFEVD on frequency band d as below, to make sure that the sums of variance contribution are in agreement:

$$d\_{ij} = \widetilde{\alpha^d\_{jk}} = \frac{\alpha^d\_{jk}}{\sum\_k \alpha^{\infty}\_{jk}}.\tag{23}$$

#### **4. Empirical Results**
