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J. Risk Financial Manag. 2021, 14, 440 2 of 15 Naeem et al. (2021) quantified the spillover effects among seven cryptos to explore the spillover characteristics cryptos, and discovered that Bitcoin, Litecoin, and Ripple are the dominant transmitters to return spillover. These studies inspired us to investigate how the density similarities between cryptos, stocks and industry groups will be affected by the COVID-19 outbreak. Statistical similarity and co-dependence are central to the analysis of market efficiency and allocation. Most existing studies focus on Bitcoin returns and “correlation” analysis. For example, (Baur et al. 2018) show that Bitcoin returns are essentially uncorrelated with traditional asset classes such as stocks and bonds, which points to diversification possibilities. Other studies investigate the determinants of Bitcoin returns. Li and Wang (2017) suggest that measures of financial and macroeconomic activity are drivers of Bitcoin returns. Kristoufek (2015) considers financial uncertainty, Bitcoin trading volume in Chinese Yuan and Google trends as potential drivers of Bitcoin returns. Recently, many studies examine whether Bitcoin belongs to any existing asset classes, with many comparing it to gold, others to precious metals or to speculative assets (Baur et al. 2018). Some have classified Bitcoin as a new asset class within currency and commodity groups (Dyhrberg 2016). Another area of interest is forecasting Bitcoin volatility, since such forecasts represent an important ingredient in risk assessment and allocation, and derivatives pricing theory. Balcilar et al. (2017) analyze the causal relation between trading volume and Bitcoin returns and volatility. They find that volume cannot help to predict the volatility of Bitcoin returns. Bouri et al. (2017) find no evidence for asymmetry in the conditional volatility of Bitcoins when considering the post December 2013 period and investigate the relation between the VIX index and Bitcoin volatility.Al-Khazali et al. (2018) consider a model for daily Bitcoin returns and show that Bitcoin volatility tends to decrease in response to positive news about the US economy. Scant attention has been paid to the full distributions of these assets. An exception is (Osterrieder and Lorenz 2017) and (Begusic et al. 2018) who have studied the unconditional distribution of Bitcoin returns and found that it has more probability mass in the tails than that of foreign exchange and stock market returns. Findings that are based on models of return and volatility, possibly with conditional covariates, are in effect assessing if similar mechanisms apply to different asset class returns. While this is an aspect of similarity, it does not respond, and indeed may impinge on the assessment of similarity of return outcomes/distributions. Similar distributions may arise from different evolutions and mechanisms over time. Our objective in this paper is to revisit some stylized facts of cryptocurrency markets and employ econometrics models for accurate volatility forecasts. In contrast to previous studies that use time series models to forecast crypto returns, in this paper we use entropy profiles of different asset classes and indices, as well as the cryptos. We test for similarity between cryptocurrency and stock returns in a manner that captures nonlinearities and higher moments, nonparametrically. We consider both Bitcoin and Ethereum as leading crypto which have large volume and relatively long histories. We use nonparametric en- tropy metrics to test equality between crypto density and stock market index returns. Time series models (ARIMA and GARCH), in contrast, impose a (traditionally) restrictive linear structure on the return data. This may produce non robust inferences and conclusions. Efficient market analysis is based on (typically) linear relation between a given asset and market returns. In this paper, we examine the general definition of dependence between crypto return and stock market returns. Stochastic independence is tested and degree of dependence is measured with entropy metrics. The rest of the paper is organized as follows: Section 2 presents the data analysis and some stylized facts. In Section 3, we calculate nonparametric entropy metrics to test the density equality between two cryptos (Bitcoin and Ethereum), two stock market indexes (S&P500 and NASDAQ) and 30 commodity industry groups. We conduct equality tests on both marginal distributions and conditional distributions for two periods (pre- COVID and COVID era) and compare the results. In Section 4, we consider a Diff-in-diffPDF Image | Contrasting Cryptocurrencies with Other Assets
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