*4.1. Data*

For this study, we collected daily data from Bloomberg, including the Henry Hub natural gas futures price (GASF), and the nominal dollar-denominated exchange rates for the Brazilian Real (BRL), Russian Ruble (RUB), Indian Rupee (INR), offshore Chinese Yuan (CNH), and South African Rand (ZAR). We used the offshore Chines Yuan instead of the onshore Chinese Yuan (CNY) for the reason that China has reformed its exchange rate regime twice, once in 2005 and the other in 2010. Before and after each reform, CNY kept its exchange rate steady for a long time, with almost no fluctuation or only change in a narrow range. Therefore, we chose CNH, which has more fluctuations, to conduct our analysis. In order to match the data availability for CNH, we used the data sample period from 23 August 2010 to 20 June 2019.

The stationary return series were obtained from Equation (18), and are in percentage points:

$$r\_{i,t} = 100 \times \ln\left(\frac{p\_{i,t}}{p\_{i,t-1}}\right). \tag{24}$$

The return series for the natural gas price and exchange rate over time are plotted in Figure 1. Figure 1 shows that the GASF return had the highest volatility compared to the others. We consider that the natural gas price was largely affected by temperature, so most fluctuations occurred concentratedly during the winter season. A small number of fluctuations were recorded in the middle of the year, such as in 2012, when hot weather forecasts and elevated cooling demands created a grea<sup>t</sup> demand for natural gas. The RUB return fluctuated drastically at the end of 2014, when the crude oil crash happened, and caused the financial crisis in Russia.

Table 2 provides summary statistics for all return series. CNH has the lowest mean of all return series, as well as standard deviation. Therefore, in some way, CNH remained stable under governmen<sup>t</sup> regulations. The GASF had the highest standard deviation, as shown in Figure 1. The distribution of all return series significantly deviated from normal, as demonstrated by the Jarque–Bera test.

**Figure 1.** Daily return. Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. (**<sup>a</sup>**–**f**) refer to BRL, RUB, INR, CNH, ZAR, and GASF return series, respectively.


**Table 2.** Summary statistics for daily returns.

Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. The sample period is from 23 August 2010 to 20 June 2019. The JB-Test refers to the Jarque–Bera test for normality. \*\*\* indicates rejection of the null hypothesis that the data are normally distributed at the 1% level of significance.

We were interested in not only the connectedness of the return series, but also the volatility connectedness, because volatility can provide a measure of risk and is particularly crisis-sensitive [25]. As volatility is unobserved and must be estimated, we used generalized autoregressive conditional heteroscedasticity (GARCH) models to obtain the volatilities of BRL, INR, CNH, and GASF return series, and Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) models to obtain the volatility of RUB and ZAR return series (for the sake of brevity, the results of the GARCH model and GJR-GARCH model are omitted).

The plots of volatility are presented in Figure 2. For simplicity's sake, we used a di fferent scale for the y-axis in RUB and GASF. The GASF fluctuated violently and most fluctuations accumulated during the winter season, which is consistent with the return series. The volatilities of the five exchange rates reached a high level at the end of 2011, compared to the period before and after, when the eurozone debt crisis reached its peak. The BRL's volatility was turbulent after 2010, especially between 2015 and 2017, when Brazil experienced a severe economic crisis and faced a dramatic economic recession. The volatility of INR reached its peak at the end of 2013, as the Indian rupee had depreciated greatly. The description statistics for volatility are reported in Table 3. Similar to the return series, GASF has the highest standard deviation, whereas CNH has the lowest. All volatilities were skewed and had high kurtosis, indicating that the distributions showed obvious non-normality characteristics. The Jarque–Bera test also verifies our opinion.

**Figure 2.** Volatility. Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. (**<sup>a</sup>**–**f**) refer to the volatility of BRL, RUB, INR, CNH, ZAR, and GASF return series, respectively.


**Table 3.** Summary statistics for volatilities of daily returns.

Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. The volatilities of BRL, INR, CNY, and GASF return series were calculated by the generalized autoregressive conditional heteroscedasticity (GARCH) (1,1) model, and the volatilities of RUB and ZAR return series were calculated by the Glosten–Jagannathan–Runkle (GJR)-GARCH (1,1) model. The sample period is from 23 August 2010 to 20 June 2019. \*\*\* indicates rejection of the null hypothesis that the data are normally distributed at the 1% level of significance.

#### *4.2. Connectedness and Frequency Decomposition*

As the calculation of the connectedness index is based on the VAR model, we conducted an augmented Dickey–Fuller (ADF) test for the unit root before applying the data to the VAR model.

However, it is well-known that the unit root hypothesis can be rejected if the data series contain structural break(s) [26–28]. As our sample period is long, from 2010 to 2019, which is almost 9 years, and several big events happened during the sample period, such as the 2014 crude oil crush, which may have had an impact on the economies and caused structural breaks, it is well-founded to consider that structural breaks may exist. Therefore, a Bai–Perron test for structural breaks was conducted. The p-values of the Bai–Perron test for the return series are presented in Table 4. All numbers indicate the acceptance of null hypothesis that no break exists. These results confirm the reliability of the ADF test.

**Table 4.** Bai–Perron breakpoint test on return series.


Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. Each number indicates the p-value of the Bai–Perron breakpoint test.

The results of the ADF test are presented in Table 5. All results show that no unit root exists. The p lags of the VAR model were chosen by the Akaike information criterion (AIC). Return series used the VAR (1) model, whereas volatilities used the VAR (2) model (for the sake of brevity, the results of the VAR model are omitted).


**Table 5.** Augmented Dickey–Fuller (ADF) test on return series and volatilities.

Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. Each number indicates that Dickey–Fuller is the ADF test statistic. \*\*\* indicates rejection of the null hypothesis that a unit root is present in the time series at the 1% level of significance.

The connectedness index based on DY and its spectral representation based on BK of short-, medium-, long-term are reported in Tables 6–9, respectively. The frequency band of short term, medium term, and long term in Table 6 roughly corresponds to 1 day to 5 days, 5 days to 21 days, and more than 21 days, respectively. The diagonal elements in both tables represent the own-market connectedness and are not important in our paper. We have more interest in the off-diagonal elements, which indicate pairwise connectedness between two variables: the values of the To row, which show when one variable receives a shock; how much influence would be exerted on other variables; the values of the "From" column, which measure the composition of one variable's change; and the values of the "Net" row, which reveal whether a variable is a net recipient or a net transmitter. The "GAS-FX" column, which exhibits the net pairwise directional connectedness between GAS and the five exchange rates, is the most critical for our study.


**Table 6.** Connectedness among the natural gas future price and BRICS's exchange rates.

Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. From column reports the total directional connectedness from others to *xi*. To row reports the total directional connectedness from *xi* to others. Net row reports the net total directional connectedness. GAS-FX column reports the net pairwise connectedness between the GASF and exchange rates, which is calculated by the GASF to others minus the others to GASF. The number in red represents the largest value in this system. The number in bold means the total connectedness. All results are expressed as a percentage.

**Table 7.** Connectedness among the natural gas future price and exchange rates in the frequency domain (short term).



**Table 7.** *Cont.*

Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price. From column reports the total directional connectedness from others to *xi*. To row reports the total directional connectedness from *xi* to others. Net row reports the net total directional connectedness. GAS-FX column reports the net pairwise connectedness between the GASF and exchange rates, which is calculated by the GASF to others minus the others to GASF. The number in red represents the largest value in this system. The number in bold means the total connectedness. The frequency band of short term roughly corresponds to 1 day to 5 days. All results are expressed as a percentage.



Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas future price. From column reports the total directional connectedness from others to *xi*. To row reports the total directional connectedness from *xi* to others. Net row reports the net total directional connectedness. GAS-FX column reports the net pairwise connectedness between the GASF and exchange rates, which is calculated by the GASF to others minus the others to GASF. The number in red represents the largest value in this system. The number in bold means the total connectedness. The frequency band of medium term roughly corresponds to 5 days to 21 days. All results are expressed as a percentage.


**Table 9.** Connectedness among the natural gas future price and exchange rates in the frequency domain (long term).

Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas future price. From column reports the total directional connectedness from others to *xi*. To row reports the total directional connectedness from *xi* to others. Net row reports the net total directional connectedness. GAS-FX column reports the net pairwise connectedness between the GASF and exchange rates, which is calculated by the GASF to others minus the others to GASF. The number in red represents the largest value in this system. The number in bold means the total connectedness. The frequency band of long term roughly corresponds to more than 21 days. All results are expressed as percentages.

As shown in Table 6, the total connectedness of the return series is 22.484%, which is almost twice as much as the connectedness between volatilities (11.697%), but both of them are modest. In this system, no matter what the connectedness from the return series or volatilities was, the shocks transmitted from ZAR to BRL contributed the largest value. The BRL was a net transmitter in return connectedness, but a net recipient in volatility connectedness. The RUB was opposite to BRL in that it was a net recipient in the return case, but a net transmitter in the volatility case. Furthermore, INR, CNH, and GASF were net receivers in both cases, whereas ZAR was a net transmitter. By obtaining the absolute value of the "Net" row, we found that INR had the strongest influence (11.517%) in all return series and ZAR was the most powerful variable (16.677%) in volatilities.

In Tables 7–9, we note that the sum of total connectedness in the short term, medium term, and long term is equal to the total connectedness shown in Table 6, which is in agreemen<sup>t</sup> with the definition of frequency decomposition for connectedness. It is interesting to find that the total connectedness from the return series is highest in the short term (17.756%), followed by the medium term (3.480%) and long term (1.248%). By contrast, from volatilities, the value is highest in the long term (11.232%), followed by the medium term (0.370%) and short term (0.095%), which means that the uncertainty transmitted by the shock has a long-term impact on the market, rather than the shock itself.

From the values of the GAS-FX column, we found that the net pairwise connectedness between GAS and the exchange rate was higher in volatilities than return series, but both were very weak. All values were almost zero. The possible reason for this is that the GASF data we have chosen are for Henry Hub natural gas, which could be seen as representative of the North American natural gas market. However, as the natural gas pipeline in North America can hardly reach any BRICS countries, and the distance between North America and BRICS countries makes the transportation cost of LNG expensive, whether as an import or export, North American natural gas is not the primary selection for BRICS countries. Our opinion is also supported by statistics from the BP Statistical Review of World Energy [1]. Whether natural gas is traded by a pipeline or LNG, the quantity being directed from the US, Mexico, and Canada to BRICS countries is very low. Consequently, we could say that the natural gas price is unrelated to the exchange rate in BRICS countries.

#### *4.3. Rolling-Window Analysis*

We also conducted a rolling-window analysis to investigate the time-varying connectedness between GAS and exchange rates. The window size was 300 (we also obtained the dynamic connectedness from a window size of 400 and obtained similar results to the result produced from the 300 window size). Figure 3 plots the dynamic total connectedness. From Figure 3, we can see that the total connectedness from the return series begins with a high level (around 40%) in the first few windows, and then falls after 2011 (around 25%), when South Africa joined the BRICS group and the period in which the European debt crisis was at its peak. After a temporary rise in late 2012, the connectedness falls again at the beginning of 2013 (around 20%). From 2013 to mid-2015, the connectedness fluctuates between 20% and 25%, and then rises again to over 30% after mid-2015. The connectedness drops dramatically between 2017 and 2018, from almost 35% to around 20%, and then recovers slowly. The trend of dynamic connectedness from volatilities is similar to that of return series. There are several unusual peaks and troughs in the plot, which we think are related to big events, like the Russian financial crisis (2014), the Brazilian economic recession (2015), and the US–China trade war (2018). Figure 4 presents the frequency decomposition of dynamic connectedness. We find that, whether in the short, medium, or long term, the trend of return connectedness is similar to the dynamic total connectedness. However, for volatilities, in the short and medium term, the connectedness exhibits almost no change (except for some abrupt rises and falls), and the long term has a similar trend to total connectedness. We think that long-term connectedness exerts the most influence in the case of volatility.

**Figure 3.** Dynamic connectedness: (**a**) total connectedness of return series and (**b**) total connectedness of volatilities. BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively.

**Figure 4.** Frequency decomposition of dynamic connectedness: (**a**) frequency decomposition of total connectedness for return series and (**b**) frequency decomposition of total connectedness for volatilities. Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price.

The time-varying net pairwise connectedness between GAS and exchange rates return series is plotted in Figure 5. Like the result in Section 4.2, all values were low and almost all of them were below 2.5% in terms of the absolute value, so were negligible. The results from volatilities are presented in Figure 6. There are also several sudden rises and falls, which is consistent with the plot of total connectedness. In the net pairwise connectedness of GAS-BRL, GAS-INR, and GAS-ZAR pairs, except for the abnormal value at some points, the values were almost insignificant; thus, we can hardly say that the GAS has an influence on the exchange rate or vice versa. However, in GAS-RUB and GAS-CNH pairs, there are some significant positive or negative periods during our data span. Before 2014, GAS was a net transmitter to RUB, and then turned into a net recipient after 2014, when Russia was undergoing an economic crisis caused by the oil price crash. After 2016, GAS was a net transmitter to CNH, when Australia became the largest supplier of LNG to China instead of Qatar, and the trade kept increasing after that.

**Figure 5.** Net pairwise connectedness of return series. (**<sup>a</sup>**–**<sup>e</sup>**) refer to the net pairwise connectedness between GASF and BRL, RUB, INR, CNH, and ZAR return series, respectively. Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price.

**Figure 6.** Net pairwise connectedness of volatility. (**<sup>a</sup>**–**<sup>e</sup>**) refer to net pairwise connectedness between the volatility of GASF and BRL, RUB, INR, CNH, and ZAR, respectively. Note: BRL, RUB, INR, CNH, and ZAR are Brazilian Real, Russian Ruble, Indian Rupee, offshore Chinese Yuan, and South African Rand, respectively. GASF is the Henry Hub natural gas futures price.
