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J. Risk Financial Manag. 2021, 14, 440 13 of 15 the triangularity property of distances. If you take three points, A, B and C, the distance between any of those points is smaller than the total of the other two distances. Furthermore, note that Sρ is a ”squared integral". The second line in Equation (8) also tells us that it is a simple expectation of 1 − ( f2/ f1)1/2. This is equal to metric developed by Bhathacharya as a measure of relations between two variables. By algebra, we can derive the Diff-in-diff estimator as the entropy metrics between stocks and crytos during the COVID era subtract the entropy metric between them during pre-COVID period: βˆDID = Sρ(fsi,t2, fcj,t2)− Sρ(fsi,t1, fcj,t1). Table 6. DID decomposition. Distribution pre-COVID COVID era Change 5. Conclusions Stock Sρ(fsi,t1, f0) Sρ(fsi,t2, f0) Sρ(fsi,t2, fsi,t1) Crypto Sρ(fcj,t1, f0) Sρ(fcj,t2, f0) Sρ(fcj,t2, fcj,t1) Difference Sρ(fsi,t1, fcj,t1) Sρ(fsi,t2, fcj,t2) Sρ(fsi,t2, fcj,t2)−Sρ(fsi,t1, fcj,t1) This paper investigates the similarity and co-dependence between cryptocurrencies, daily returns and stock daily returns, before and after the COVID-19 outbreak in early 2020. Data exhibited different features before and after the COVID-19 outbreak. There is a similarity between the Bitcoin and NASDAQ stock market index with or without COVID- 19. The similarity and dependence between cryptos and stock market indexes has become stronger after the COVID-19 outbreak. Our findings are robust to model misspecification, and avoid linear measures of dependence and correlation. The entropy profiles method and time series models play different roles in forecasting the cryptocurrency return volatility, and these approaches are complimentary. The time series models elaborate the dynamic movement of returns, on average (conditional mean models). The entropy profiles method is a nonparametric approach which reveals the evolution of the entire distributions and their quantiles. In this paper, we have several findings: Firstly, we found that during the pre-COVID period, NASDAQ and Bitcoin return’s distributions are the most similar. Secondly, we can see during the COVID era, the distances between all asset returns have declined by 75% or more, and most of these changes are caused by changes of stock return distributions. We also found that the asset group with the closest similarity with Bitcoin are Coal, Steel and Mining industries during pre-COVID period, and Business Supplies, Utilities, Tobacco Products and Restaurants, Hotels, Motels industries, compared to several others during COVID era. Finally, through non-linear co-dependence test, we found that during the COVID era, the densities of stocks and cryptos became more similar and less independent. These results are meaningful because we revealed the similarity and dependence structure between crypto and stock distributions. This can be useful in applying existing theories on stocks to cryptos. As for future directions of this study, we plan to examine newer data as we have observe the effective vaccines rollout, stock market volatility and the crypto prices peak to a new high in 2021. We believe that the examination of newer data will drive more promising and effective policy implications. Author Contributions: Conceptualization, E.M. and X.W.; methodology, E.M. and X.W.; software, X.W.; validation, E.M. and X.W.; formal analysis, E.M. and X.W.; investigation, E.M. and X.W.; resources, E.M. and X.W.; data curation, E.M. and X.W.; writing—original draft preparation, X.W.; writing—review and editing, E.M.; visualization, X.W.; supervision, E.M.; project administration, E.M.; funding acquisition, N/A. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding.PDF Image | Contrasting Cryptocurrencies with Other Assets
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