We examine the spillover effect among five markets, namely the crude oil, natural gas, coal, electricity, and carbon credit markets. Then, we measure the risk of a portfolio consisting of these five indices.

First, we confirm the descriptive statistics and time plots for these price series as an overview. Then, we generate the return (see Sect. 3.4.1) and volatility series using the model (see Sect. 3.3.3) and confirm the descriptive statistics and time plots for these series. We test the stationarity of these variables using the augmented Dickey Fuller (ADF) unit root test (see Sect. 2.2.2) to confirm whether we can represent these variables by the vector moving average (VMA) model.

Second, by measuring the connectedness proposed by Diebold and Yilmaz [3] (see Sect. 3.3.1) and spectrally decomposing the connectedness as in Barunik and Krrehlik [1] (see Sect. 3.3.2), we grasp the spillover effects of returns and volatility among these markets.

Finally, estimating four types of copulas, (the Gaussian, $t$, Clayton, and Gumbel copulas, see Sect. 2.5.2), we measure the risk of the portfolio. This portfolio consists of long positions in crude oil, natural gas, coal, and carbon credits, and short positions in electricity. The risk of a long position is in the left tail of the return distribution, and the risk of a short position is in the right tail of the return distribution. We generate the distribution of daily returns for the portfolio by simulating each estimated copula 500,000 times. We then calculate the VaR and expected shortfall.

We confirm the representative statistics for all the price series. Table $5.1$ summarizes the descriptive statistics. For four indexes (TTF, Rotterdam, FrenchBI , and FIIA) other than Brent, the mean is larger than the median. These distributions contain

many outliers in the right tail. Although the range of these four variables was wide, the standard deviation was not large. The outliers in the right tail may be accidental. These four distributions have a positive skewness and a long right tail, consistent with the fact that these four variables have maximum values much larger than their respective means. These four distributions had kurtosis values greater than 3 , meaning that their distributions have sharp peaks and long fat-tails. Brent has the opposite characteristics. Its median is larger than its mean and its skewness is negative. Its distribution contained many outliers in the long left tail. This is consistent with the difference between its maximum and mean being smaller than the difference between its mean and minimum. The distribution had a kurtosis of less than 3, indicating that the distribution has a rounded peak and short, thin tails. For all variables, the Jarque-Bera test rejected the normal distribution hypothesis.

Figure $5.1$ shows the time plots of the five prices. In the short period after the spring of 2021, the four markets besides Brent rose sharply. These time plots are consistent with the descriptive statistics.

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