**5. Conclusions**

We adopt the approach of Diebold and Yilmaz [3] to examine the spillover effects between global natural gas price indexes. Moreover, we employ the Fourier transform utilized by Baruník and Kˇrehlík [25] for spectrally decomposing Diebold and Yilmaz's [3] index.

We use daily data from 2 February 2009 to 28 February 2019. We employ HH, NBP, TTF, and JKM as the North American, British, Continental European, and Asia-Pacific market price indexes, respectively.

The results of the analyses for the entire period show that the total connectedness of return and volatility is 22.9% and 32.8%, respectively. Volatility is more highly integrated than returns in the global natural gas market. However, total connectedness is mostly dependent on the pairwise connectedness between NBP and TTF. Therefore, we cannot conclude that the global natural gas market is integrated.

The results of the spectral analyses indicate that the spillover effects of the return series almost depend on short-term factors, that is, events within 5 business days. If we further expand the natural gas pipeline network to produce more LNG, the total connectedness of the return series and the ratio of short-term factors should be higher with the increase in arbitrage trading. By contrast, the spillover effect of the volatility series is almost dependent on the long-term factors, that is, accumulated shocks for more than a month. This result represents the long-term memory of volatility.

The results of the dynamic analyses with moving window samples are consistent with the results above. The spillover effects of the return and volatility series tend to be dependent on events within a week and more than a month ago, respectively. However, regional climate, demand and supply, and incidents might make spillover effects unstable. Unfortunately, the spillover index for around 10 years does not show that the liquidity of the global natural gas market tends to increase.

For practitioners, this study implies that constantly monitoring market spillovers is significant. The information obtained by this study's approach is also of high value for energy portfolio rebalancing, including derivatives. Moreover, if we monitor the index as a predictive indicator, the information helps with early risk aversion.

Finally, clarification of the spillover effects between the crude oil and natural gas markets will be required for future studies, because many natural gas trading contracts still have their prices linked to crude oil prices.

**Author Contributions:** Data curation, T.N.; formal analysis, Y.T.; retry analysis, T.N.; writing, T.N.; editing, T.N.; supervision, T.N.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank the anonymous reviewers, whose valuable comments helped improve an earlier version of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
