**3. Methodology**

#### *3.1. Spillover Index*

A correlation coefficient only represents a simultaneous phenomenon between variables. Therefore, our analysis on the relationship between variables needs to go further. We begin by calculating the index proposed by Diebold and Yilmaz [3] which can capture the spillover effect between variables.

We assume the following covariance stationary four-variable VAR (*p*):

$$\mathbf{x}\_{t} = \sum\_{l=1}^{p} \Phi\_{l} \mathbf{x}\_{t-l} + \varepsilon\_{t\prime} \tag{2}$$

where *xt* is the four-dimensional vector of return or volatility, which must be a stationary series; Φ*l* are the 4 × 4 coefficient matrices; *p* is the lag length based on the Schwarz information criterion; and ε*t* is an independently and identically distributed sequence of four-dimensional random vectors with zero mean and covariance matrix <sup>E</sup><sup>ε</sup>*t*ε*t* = Σ.

We can represent the above VAR model in the following VMA:

$$\mathbf{x}\_{l} = \sum\_{l=0}^{\infty} A\_{l} \varepsilon\_{l-l} \tag{3}$$

where *Al* = *pl*=<sup>1</sup>Φ*lAt*−*l*, *A*0 being a 4 × 4 identity matrix with *Al* = 0 for *l* < 0.

We define the spillover effect from the *m*th to the *l*th market up to *H*-step-ahead as the following equation by using the *H*-step-ahead forecast error variance decompositions:

$$\Theta\_{lm} = \frac{\sigma\_{mm}^{-1} \sum\_{h=0}^{H-1} \left( \mathbf{e}\_{l}^{\prime} A\_{h} \Sigma \mathbf{e}\_{m} \right)^{2}}{\sum\_{h=0}^{H-1} \mathbf{e}\_{l}^{\prime} A\_{h} \Sigma A\_{h}^{\prime} \mathbf{e}\_{l}} \tag{4}$$

where σ*mm* is the standard deviation of the error term for the *m*th equation and *el* is the selection vector, with one as the *l*th element and zeros otherwise.

*Energies* **2019**, *12*, 3927

We normalize each entry of the variance decomposition matrix by the row sum, that is, 4, as the pairwise connectedness:

$$
\overleftarrow{\partial\_{lm}} = \frac{\partial\_{lm}}{\sum\_{m=1}^{4} \partial\_{lm}} = \frac{\partial\_{lm}}{4} \tag{5}
$$

We define the sum of pairwise connectedness as total connectedness:

$$S = \frac{\sum\_{l=1}^{4} \sum\_{m=1, l \neq m}^{4} \overleftarrow{\partial\_{lm}}}{4} \tag{6}$$

The numerator is the sum of the spillover effects, excluding the spillover effects on itself. In other words, the total connectedness means the sum of the relative proportion of the portfolio's response to a shock.

Moreover, we measure the directional spillover effects received by the *l*th market from all other markets as:

$$S\_{l\cdot} = \frac{\sum\_{l=1, l\neq m}^{4} \overleftarrow{\partial\_{lm}}}{4} \tag{7}$$

Similarly, we measure the directional spillover effects transmitted by the *l*th market to all other markets as:

$$S\_{\cdot l} = \frac{\sum\_{l=1, l \neq m}^{4} \overleftarrow{\partial\_{ml}}}{4} \tag{8}$$

#### *3.2. Spectral Analysis*

Based on the Fourier transform utilized by Baruník and Kˇrehlík [25], we spectrally decompose Diebold and Yilmaz's [3] indexes into short-term (1 to 5 business days) factors, medium-term (6 to 20 business days) factors, and long-term (from 21 business days onward) factors.

The Fourier transform of Equation (2) is as follows:

$$f(\omega)\_{lm} = \frac{\sigma\_{mm}^{-1} \left( \left( \left( \sum\_{h=0}^{H-1} e^{-i\omega h} A\_h \right) \Sigma \right)\_{lm} \right)^2}{\left( \left( \sum\_{h=0}^{H-1} e^{-i\omega h} A\_h \right) \Sigma \left( \sum\_{h=0}^{H-1} e^{i\omega h} A\_h \right)' \right)\_{ll}} \tag{9}$$

This is the spillover effect from the *m*th market to the *l*th market up to *H*-step-ahead, which is expressed by angular frequency ω.

We define the weighting function, the ratio of ω component to all-frequency components concerning the spillover to the *l*th variable, as follows:

$$\Gamma\_{I}(\omega) = \frac{\left(\left(\sum\_{h=0}^{H-1} e^{-i\omega h} A\_{h}\right) \Sigma \left(\sum\_{h=0}^{H-1} e^{i\omega h} A\_{h}\right)\right)\_{\mathcal{U}}}{\frac{1}{2\pi} \int\_{-\pi}^{\pi} \left(\left(\sum\_{h=0}^{H-1} e^{-i\lambda h} A\_{h}\right) \Sigma \left(\sum\_{h=0}^{H-1} e^{i\lambda h} A\_{h}\right)\right)\_{\mathcal{U}} d\lambda} \tag{10}$$

The spillover index from the *m*th market to the *l*th market in all bands is expressed as:

$$(\theta\_{\infty})\_{lm} = \frac{1}{2\pi} \int\_{-\pi}^{\pi} \Gamma\_l(\omega) f(\omega)\_{lm} d\omega \tag{11}$$

The spillover index from the *m*th market to the *l*th market in band *d* is expressed as:

$$(\theta\_d)\_{lm} = \frac{1}{2\pi} \int\_d \Gamma\_l(\omega) f(\omega)\_{lm} d\omega \tag{12}$$

We convert (<sup>θ</sup>*d*)*lm* to the relative contribution <sup>θ</sup>*<sup>d</sup>lm* as:

$$\left(\widetilde{\theta\_d}\right)\_{lm} = \frac{(\theta\_d)\_{lm}}{\sum\_{l=1}^4 (\theta\_\infty)\_{lm}}\tag{13}$$

We calculate the total spillover index in the *d* band as:

$$\mathbf{C}\_{d} = \frac{\sum\_{l=1}^{4} \sum\_{m=1, l \neq m}^{4} \left(\overline{\theta}\_{d}\right)\_{lm}}{\sum\_{l=1}^{4} \sum\_{m=1}^{4} \left(\overline{\theta}\_{d}\right)\_{lm}} \tag{14}$$

Naturally, this is consistent with Diebold and Yilmaz's [3] spillover index.

#### *3.3. Rolling Analysis*

It is insufficient to focus on static spillover indicators, which are calculated by the Diebold and Yilmaz [3] and Baruník and Kˇrehlík [25] for the entire period. To capture the dynamics of the spillover effects, we employ the rolling window approach as a sampling method. As shown in Figure 4, this study fixes the moving window sample size to 300 trading days and offsets the window by one business days every time we perform an analysis.

**Figure 4.** Illustration of moving window procedure.

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

#### *4.1. Spillover Index*

The spillover analyses results of the return series are reported in Table 5. The spillover index from HH to the other variables and from the other variables to HH are 0.42% and 0.57%, respectively. HH fluctuates independently of the other variables.

The spillover indexes from NBP to TTF and from TTF to NBP are both above 40%. While the total connectedness is 22.9%, the spillover indexes from the other variables to TTF and from TTF to the other variables are 10.7% and 10.8%, respectively. Moreover, the spillover indexes from the other variables to NBP and from NBP to the other variables are 10.5% and 11.1%, respectively. TTF and NBP have a high presence as both the sources and destinations of spillover effects, because these two natural gas

markets in Europe are almost integrated, although HH and JKM fluctuate relatively independently. However, only the spillover index from NBP to JKM is not at a negligible level, at 2.65%.

The spillover indexes of the volatility series are reported in Table 6. The total connectedness of the volatility series is 32.8%, while the total connectedness of the return series is 22.9%. Risks tend to be transmitted between markets rather than returns. This mutual relationship is similar to the tendency of the return series. However, the characteristic of volatility series is that the influence of HH on the other variables is larger. Additionally, we hardly observe the spillover effects from JKM to the other variables.


**Table 5.** Spillover index between return series.

**Table 6.** Spillover index between volatility series.


#### *4.2. Spectral Analysis*

The spectral analyses results of the return series spillover are reported in Table 7. The total connectedness from 1 to 5 business days, from 6 to 20 business day, and over 21 business days is 17.3%, 4.12%, and 1.49%, respectively. The short-term factors contribute most to the return spillover. The spillover effects of the return series are mostly explained by events within about one week. Arbitrage might contribute to these short-term spillover effects.

**Table 7.** Spectral analyses of spillover index between return series.


Table 8 presents the spectral analyses results of the volatility series spillovers. The total connectedness from 1 to 5 business days, from 6 to 20 business days, and over 21 business days is 0.10%, 2.37%, and 30.3%, respectively. Contrary to the return series, the long-term factors contribute most to the volatility spillover. Most of the spillover effect of the volatility series is caused by events that occurred more than one month ago. The reason might be the long-term memory of volatility.


**Table 8.** Spectral analyses of spillover index between volatility series.

#### *4.3. Rolling Analysis*

#### 4.3.1. Total Connectedness

Figure 5 shows the total connectedness of the return series and the results of its spectral analyses. The total connectedness has not fluctuated significantly for 10 years and there is no significant change in the composition ratio of the total connectedness. The short-term factors have almost caused return spillovers.

**Figure 5.** Total connectedness of return series.

Figure 6 represents the total connectedness of the volatility series and the results of its spectral analyses. As with the static analysis results, the long-term factors almost caused volatility spillovers.

Total connectedness spikes in February 2014, February 2018, and November 2018. We assume these spikes are caused by sudden fluctuations in each series. The price of HH spikes in February 2014 because of a grea<sup>t</sup> cold wave. The same phenomenon occurs every winter, although the scale is

small. Additionally, the JKM, NBP, and TTF prices began to decline at around same time, because the spot market, which has been tight since the Great East Japan Earthquake, shifts and becomes loose. Therefore, the spillover index spikes.

The oversupply strengthens from January to February 2018, because of the expectation to expand USA's LNG export capacity along with the production of crude oil and natural gas. As a result, each gas price index falls sharply. Conversely, in November, the HH increases sharply because the storage stock levels in the USA fall significantly below recent levels. These facts cause the two spikes in 2018.

**Figure 6.** Total connectedness of volatility series.

#### 4.3.2. Pairwise Connectedness

Figure 7 traces the pairwise connectedness of the return series and the results of its spectral analyses. In all combinations, the spillover effects of the return series continue to depend on short-term factors. Compared to the total connectedness, which is stable at around 30%, the pairwise connectedness between any variables is low and stable, although the pairwise connectedness from NBP to JKM and from TTF to JKM temporarily reaches around 3%. The mutual spillover effects offset each other because they are almost at the same level. However, when the Asia-Pacific spot market becomes tight or loose, the European market might be relatively stronger than the Asia-Pacific one.

**Figure 7.** Pairwise connectedness of return series.

Figure 8 shows the pairwise connectedness of the volatility series and the results of its spectral analyses. Under all combinations, the spillover effects of the volatility series continue to depend on long-term factors. The spillover effects from HH to NBP and from TTF to JKM are periodically strong. The HH volatility spikes caused by the cold wave in North America affect the global market. In the European market, the spillover effect from TTF to NBP is stably high. TTF might be more prone to risks than NBP, because the former is more closely linked to intra-European and international trading using pipelines and LNG.

**Figure 8.** Pairwise connectedness of volatility series.
