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J. Risk Financial Manag. 2021, 14, 440 11 of 15 Daily Log-Return p-Value 0.16 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.09 <2.22 × 10−16 *** 0.19 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.20 <2.22 × 10−16 *** 0.12 <2.22 × 10−16 *** 0.21 <2.22 × 10−16 *** 0.26 <2.22 × 10−16 *** 0.23 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.19 <2.22 × 10−16 *** 0.22 <2.22 × 10−16 *** 0.21 <2.22 × 10−16 *** 0.27 <2.22 × 10−16 *** 0.09 <2.22 × 10−16 *** 0.09 <2.22 × 10−16 *** 0.22 <2.22 × 10−16 *** 0.22 <2.22 × 10−16 *** 0.19 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.13 <2.22 × 10−16 *** 0.17 <2.22 × 10−16 *** 0.18 <2.22 × 10−16 *** 0.24 <2.22 × 10−16 *** 0.10 <2.22 × 10−16 *** 0.24 <2.22 × 10−16 *** 0.25 <2.22 × 10−16 *** 0.20 <2.22 × 10−16 *** S_rho 0.04 0.07 0.03 0.05 0.04 0.04 0.04 0.04 0.04 0.07 0.04 0.05 0.04 0.04 0.04 0.06 0.05 0.09 0.05 0.03 0.04 0.05 0.04 0.03 0.04 0.04 0.08 0.03 0.05 0.04 p-Value 0.0808 0.1010 0.2121 0.0202 * 0.0909 0.4040 0.1212 0.1717 0.1414 0.0101 * 0.2020 0.0202 * 0.0808 0.1111 0.1212 0.0202 * 0.0505 <2.22 × 10−16 *** 0.0101 * 0.3939 0.1313 0.1111 0.1717 0.3535 0.1515 0.2020 <2.22 × 10−16 *** 0.2626 0.1010 0.1010 Food Beer Smoke Games Books Hshld Clths Hlth Chems Txtls Cnstr Steel Fabpr Elceq Autos Carry Mines Coal Oil Util Telcm Servs Buseq Paper Trans Whlsl Rtail Meals Fin Other S_rho We also calculated the Sρ with select asset classes with themselves before and after the COVID-19 outbreak (see column 2 in Table 4). It is clear that for all industry groups during COVID era, the asset distributions diverge from their own pre-COVID distributions, and the distribution divergence of industry groups are more significant comparing with cryptos’ (shown in Table 3). Table 4. Entropy measure between Bitcoin and different Industries. Pre-COVID and COVID Era with Itself Pre-COVID with Bitcoin S_rho p-Value 0.22 <2.22 × 10−16 *** 0.21 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.10 <2.22 × 10−16 *** 0.15 <2.22 × 10−16 *** 0.21 <2.22 × 10−16 *** 0.12 <2.22 × 10−16 *** 0.17 <2.22 × 10−16 *** 0.15 <2.22 × 10−16 *** 0.11 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.07 <2.22 × 10−16 *** 0.13 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.12 <2.22 × 10−16 *** 0.15 <2.22 × 10−16 *** 0.09 <2.22 × 10−16 *** 0.02 <2.22 × 10−16 *** 0.11 <2.22 × 10−16 *** 0.22 <2.22 × 10−16 *** 0.20 <2.22 × 10−16 *** 0.16 <2.22 × 10−16 *** 0.14 <2.22 × 10−16 *** 0.18 <2.22 × 10−16 *** 0.15 <2.22 × 10−16 *** 0.19 <2.22 × 10−16 *** 0.18 <2.22 × 10−16 *** 0.20 <2.22 × 10−16 *** 0.16 <2.22 × 10−16 *** 0.20 <2.22 × 10−16 *** COVID Era with Bitcoin Difference −0.18 −0.14 −0.11 −0.05 −0.11 −0.17 −0.08 −0.13 −0.11 −0.04 −0.10 −0.02 −0.09 −0.10 −0.08 −0.08 −0.05 0.07 −0.05 −0.18 −0.16 −0.11 −0.10 −0.15 −0.11 −0.15 −0.10 −0.17 −0.11 −0.16 Note: Entries marked with *** have empirical p-values < 0.01 and * 0.05 ≤ p < 0.10 under the null of independence of returns. 3.4. Testing General Nonlinear Co-Dependence The above test of (Renyi 1961) may be employed for testing stochastic indepen- dence of any two random variables X and Y. Let f1 = f(xi,yi) be the joint density and f2 = g(xi) ∗ h(yi) be the product of the marginal densities. The unknown density functions are replaced with nonparametric kernel estimates. The methodology is as before, with the null of independence imposed in the bootstrap resampling implementation of the test. Bandwidths are obtained via likelihood cross-validation by default for the marginal and joint densities. The results are in Table 5. There is significant dependence only between Bitcoin and NASDAQ before the COVID-19 outbreak. During COVID era, independence is comfortably rejected for all pairings. The two situations represent very radical changes in the status of cryptos for portfolio diversification.PDF Image | Contrasting Cryptocurrencies with Other Assets
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