Table $4.5$ shows the descriptive statistics for the NBP futures and spot return series. For these two return series, the means are larger than the medians, implying that these distributions have a larger number of outliers in the right tail than in the left tail. As Fig. $4.2$ illustrates, because the spot price series has a large spike, its return series has extreme minimum and maximum values. The range of the spot return series is wider than that of the futures return series and the standard deviation of the spot return series is larger than that of the futures return series. This result implies that the spot return series tends to be more volatile than the futures return series. Both distributions had positive skewness, implying that each distribution has a long right tail. These distributions had kurtosis values of over 3, indicating that each distribution has a sharp peak and fat-tail. The Jarque-Bera test rejects the normal distribution hypothesis for both series.

Figure $4.8$ plots the NBP spot and futures return series. In Fig. 4.2, we can observe that when the spot price series spikes, the spot return series also spikes.

Table $4.6$ shows the estimation results for each multivariate GARCH model for the NBP return series. All the parameters of these variance equations are statistically significant at the $1 \%$ level, whereas all the parameters of the mean equations are not statistically significant, even at the $10 \%$ level.

Figure $4.9$ plots the covariance series calculated using each multivariate GARCH model. Each model can represent the conditional covariance between these return series, and these covariance series are synchronized.

Figure $4.10$ plots the OHR series calculated using the estimated diagonal VECH model. The OHR series is about $75 \%$ throughout this period. Referring to Fig. 4.2, we can see that the OHR series plummets to about $-80 \%$ in March 2018 when only the spot price series spikes. The OHR series is negative in August 2016 and October 2019 when the spot price series separates and moves downward relative to the futures price series. The OHR series spikes to nearly $200 \%$ in June 2017 when only the spot price series swoops down.

When firms procure energy with a high price fluctuation risk, they often trade futures to hedge the risk. Such firms must work to curb the volatility of the portfolio return, which consists of a spot and its futures. By dividing the covariance of the spot and futures return series by the variance of the future return series, we can obtain the ratio of futures positions to spot positions that minimizes volatility, which is defined as the $\mathrm{OHR}$. In other words, by estimating the multivariate GARCH model that formulates the conditional covariance and variance, we can obtain the conditional OHR series, which helps us construct a timely optimal portfolio.

This chapter estimates three types of bivariate GARCH models consisting of the spot and futures return series in the US and UK natural gas markets to calculate OHR and HE. We adopt the diagonal VECH, diagonal BEKK, and CCC models as multivariate GARCH models. The OHR series fluctuates drastically depending on the spot and futures market conditions. All the multivariate GARCH models here can capture the time dependence of the covariance and variance in the same way. However, comparing the average HE values reveals that constructing the portfolio using the diagonal BEKK model is the best hedging strategy for both the $\mathrm{HH}$ and NBP markets.

While this chapter adopts primitive multivariate models (i.e., the diagonal VECH model, diagonal BEKK model, and CCC model), various multivariate GARCH models exist. Section $4.4$ explains the VECH and BEKK models. Advancing the CCC model, Engle [7] proposes a dynamic conditional correlation (DCC) model that assumes that the correlation coefficient is conditional. Moreover, Cappiello et al. [4] propose the ADCC model, which incorporates the asymmetry in which the correlation tends to be stronger after a negative return than after a positive return. We recommend that readers consider the optimal hedging strategy by applying these various methods to compare the HE values for the securities and commodities of interest.

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