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J. Risk Financial Manag. 2021, 14, 440 12 of 15 Table 5. Independence test. Pre-COVID (Aug. 2015–Jan. 2020) COVID Era (Feb. 2020–Sep. 2020) S_rho p-Value 0.0148 2.22 × 10−16 *** 0.0172 2.22 × 10−16 *** 0.0163 2.22 × 10−16 *** 0.0178 2.22 × 10−16 *** Difference 0.0063 0.0096 0.0091 0.0117 Daily Log-Return S&P500 and Bitcoin S&P500 and Ethereum NASDAQ and Bitcoin NASDAQ and Ethereum S_rho 0.0085 0.0076 0.0072 0.0061 p-Value 0.0303 * 0.5758 0.0101 * 0.6061 Note: Entries marked with *** have empirical p-values < 0.01 and * 0.05 ≤ p < 0.10 under the null of independence of returns. 4. Difference-in-Differences Analysis Difference in differences (Diff-in-diff) is a statistical technique used in econometrics and quantitative research that attempts to mimic an experimental research design using observational study data, by studying the differential effect of a treatment on a “treatment group” versus a “control group” in a natural experiment. It calculates the effect of a treat- ment on an outcome by comparing the average change over time in the outcome variable for the treatment group, compared to the average change over time for the control group (Card and Krueger 1993). Before we construct our Diff-in-diff model, we would like to emphasize that the entropy metrics exhibit linear decomposition property. The reason why we can decompose Sρ is that it is a metric, which means it satisfies the triangularity property of distances. Therefore, we can write the entropy metric between stock and crypto during COVID era as the summation of the entropy metric between them during pre-COVID period plus a time trend λt and plus the COVID effect. Sρ(fsi,t2, fcj,t2) = Sρ(fsi,t1, fcj,t1)+λt +COVID+εi,j, (9) where Sρ(fsi,t2, fcj,t2) stands for the entropy metric between stock i and crypto j dur- ing COVID era, and Sρ(fsi,t1, fcj,t1) stands for the entropy metric between stock i and cryptojduringpre-COVIDperiod.λt isthetimetrenddefinedbyλt =Sρ(fsi,t2,fsi,t1)+ Sρ(fcj,t2, fcj,t1), which measures the entropy metric of both stock i and crypto j from pre- COVID period to COVID era with itself. COVID is the effect of exogenous shock provided by COVID-19 to the entropy metrics. εi,j is the residual term. Since we have already calculated the distribution distances between assets in the pre- vious sections, from Equation (9), we can easily estimate the COVID effect on the entropy metrics,sayCOVID.UsingentropymetricsS betweenBitcoinandotherassets(including ρ S&P500, NASDAQ, the the 30 industry portfolios), we can estimate the COVID effect COVID = −0.30. This indicates that after the broke out of COVID-19 pandemic, the distri- butions of stocks and cryptos became more similar and less independent, quantitatively, the entropy metrics decrease by −0.30 in average. Next, we follow Card and Krueger (1994) to construct our Diff-in-diff model: Sρ(fAi,tj, f0) = β0 +β1 ∗Covid+β2 ∗Crypto+βDID ∗(Covid∗Crypto)+ε, (10) where the dependent variable Sρ(fAi,tj, f0) is our variable of interest, it stands for the entropy metric between asset i’s distribution at time j, fAi,tj , and a benchmark distribution f0. Crypto and Covid are dummy variables. Crypto equals to 1 if the asset is a crypto, while it equals 0 if the asset is stock. Covid equals to 1 if during the COVID era and it equals to 0 if during the pre-COVID period. The coefficient for the interaction term, Covid ∗ Crypto, is the Diff-in-diff estimator. In this way, we construct our Diff-in-diff model for entropy metric. We come up with a new method to use our nonparametric entropy metric to estimate the Diff-in-diff estimator. In Table 6, we show the decomposition of the Diff-in-diff analysis. The reason why we can decompose Sρ is that it is a metric, which means it satisfies PDF Image | Contrasting Cryptocurrencies with Other Assets
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