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J. Risk Financial Manag. 2021, 14, 440 7 of 15 computes the nonparametric metric entropy (normalized Hellinger, or (Granger et al. 2004) for testing the null of equality of two univariate density (or probability) functions. For continuous variables, where f1 = f (x) and f2 = f (y) are the marginal densities of the random variables X and Y. The second expression is in a moment from which is often replaced with a sample average, especially for theoritical developments. If the density of X and the density of Y are equal, this metric will yield the value zero, and is otherwise positive and less than one. We use Sρ to test the distance between crypto density and stock market index density. Some properties this entropy measure Sρ are given in (Granger et al. 2000) and (Giannerini et al. 2015). In particular, the modulus of Sρ is between 0 and unity; Sρ is equal to or has a simple relationship with the (linear) correlation coefficient in the case of a bivariate normal distribution; Sρ is metric, that is, it is a true measure of distance and not just of ”divergence". This is especially important in our applications where triangularity property is required in meaningful comparative assessments of several distances and asset classes. Software for nonparametric kernel smoothing implementation of this metric is made available in R (NP package) among others. For the kernel function, we employ the widely used nonparametric second-order Gaussian kernel, while bandwidths are selected via likelihood cross-validation (Nakayama and Silverman 1986). Bootstrap is conducted via resampling with replacement from the pooled empirical distributions of X and Y under the null hypothesis of equality. We estimate the metric Sρ for the daily returns data for x = Returncrypto and y = Returnstock. Table 2 shows the Sρ values and the corresponding p-values. As was noted in (Granger et al. 2000) and (Skaug and Tjostheim 1996), the asymptotic distribution of Sρ is unreliable for practical inference. We therefore compute p-values by resampling the statistic under the null of equality. Examining Table 2, we see that Sρ is smallest between x = Bitcoin and y = NASDAQ, both during the pre-COVID and COVID era periods, which indicates that the distance between the densities of Bitcoin daily returns and NASDAQ daily returns is smaller than other combinations. The p-value shows that the result is significant. By visualizing the result in Figures 2–5, we can also see the Bitcoin daily returns density and the NASDAQ stock market index daily returns density have similar shapes. While during COVID era, also S&P500 returns distribution is statistically closely dependent on, and indifferent from Bitcoin’s. Comparing Sρ before and after the COVID-19 outbreak, we conclude that the values of Sρ decrease generally in all cases, sometimes dramatically. This suggests that the densities of crypto and stock index returns became more similar with the advent of COVID-19. This mostly due to a large change in the distribution of major stock indices, but also party due to a smaller movement in crypto distributions. Table 3 reveals the entropy metric Sρ of the assets themselves pre-COVID and COVID era. By doing so, we can see if the difference between the cryptos and stocks is partly due to specific asset change caused by the effect of COVID-19. The results show that the distributions of S&P500 and NASDAQ changed dramatically and significantly before and after the COVID-19 outbreak, which indicates that the changes of Sρ between cryptos and stocks may mainly caused by the changes of stocks’ distributions. We will dive deeper on this part in Section 4. Sρ = 1 (f1/2 − f1/2)2dx 212 1 f1/2 (8) = (1− 2 )2dF1(x), 2 f1/2 1PDF Image | Contrasting Cryptocurrencies with Other Assets
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