Essays on Diversification: Cryptocurrencies and Other Assets Open Access

Wu, Xi (Spring 2022)

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There are three essays in this doctoral dissertation that study assets similarity and investment diversification. The purpose of this dissertation is to make clear a central theme: Diversification makes it desirable to identify new asset classes. The decision of what is a new asset class is based on its comparison with other assets. This comparison can be based on traditional methods which compare predictive models, such as GARCH and other time series models, summary data., etc. But these methods may not be able to pick up other deeper aspects of laws that generate the return series. These deeper characteristics include nonlinearities, asymmetries, tail behaviors and other distribution characteristics that may be obscured and even twisted by model artifacts. For example, a conditional mean regression or time series model, the workhorse of financial analysis, is by design not able to look beyond the conditional mean of the returns data and its pattern and evolution. For highly volatile assets such as cryptocurrencies, but more generally for all non-Gaussian and nonlinear financial assets, these model-based comparisons and assessments are unnecessarily limiting. Assessment of "entire" distributions, made possible by entropy metrics, addresses these limitations. In addition, if the underlying laws are linear / Gaussian, nothing is lost. Entropy metrics become equivalent to these traditional conditional mean and variance models assessments.

In the first essay "Cryptocurrency Return Forecast Using Time Series Models and Entropy Approach", I investigate the notion of "similarity” between assets, in this case, cryptocurrencies with a set of other assets, including stock market indices. I use a novel approach based on entropy. This is in contrast to the traditional approach of comparing model fits, such as those based on time series models, including the GARCH model and ARIMA model. The approaches that based on time series models, which I also examine, reveal predictive structures and moments of underlying probability laws that generate returns. Entropies compare entire distributions of asset returns and capture all statistical aspects of returns and their differences. Both approaches are meant to identify new asset classes for diversification purposes. Sometimes assets are diverse in more ways, nonlinear and in higher moments and tails, than typical conditional mean and quantile models can reveal. Finally, I find max entropy closest industry portfolios to cryptocurrencies, which ensures shrinkage towards maximum diversification of portfolio weights. My findings will be useful in exploring the prediction of cryptocurrencies returns based on stock market performance.

In the second essay "Contrasting Cryptocurrencies with Other Assets: Full Distributions and the COVID Impact", I investigate any similarity and dependence based on the full distributions of cryptocurrency assets, stock indices and industry groups. I characterize full distributions with entropies to account for higher moments and non-Gaussianity of returns. Divergence and distance between distributions are measured by metric entropies, and rigorously tested for statistical significance. I assess stationarity and normality of assets, as well as the basic statistics of cryptocurrencies and traditional asset indices, before and after COVID-19 pandemic outbreak. These assessments are not subjected to possible misspecifications of conditional time series models which are also examined for their own interests. I find that NASDAQ daily return has the most similar density and co-dependence with Bitcoin daily return, generally, but after COVID-19 outbreak in early 2020, even S\&P500 daily return distribution is statistically closely dependent on, and indifferent from Bitcoin daily return. All asset distances have declined by 75\% or more after COVID-19 outbreak. I also find that the highest similarity before COVID-19 outbreak is between Bitcoin and Coal, Steel and Mining industries, and after COVID-19 outbreak it is between Bitcoin and Business Supplies, Utilities, Tobacco Products and Restaurants, Hotels, Motels industries, compared to several others. This study shed light on examining distribution similarity and co-dependence between cryptocurrencies and other asset classes, especially demystify effects of the important timely topic, COVID-19.

In the third essay "Do Cryptocurrencies and Other Assets Converge? A Clustering Analysis of Asset Returns", I examine the prospects for clustering, or convergence of asset classes. In the first instance, I examine if a set of cryptocurrencies form identifiable clusters within this class. Using entropy metric to assess "similarity” of entire distributions, I implement Agglomerative Hierarchical Clustering technique to examine whether or not cryptocurrencies are converging to "clubs" with similar distributions of returns. To arrive at a more convincing conclusion, I also apply the K-means Clustering to justify our results. I discover that cryptocurrencies share similar geographic locations and similar functions tend to converge to same clusters. I also observe another potential explanation to our results called the "Coinbase effect". In the second stage, I examine if these clusters include other asset classes, such as commodities. I find cryptocurrencies and commodities are separated into different clusters using entropy metric as cluster proximity, which is consistent with intuitive assumptions. I also find that the cluster that contains the distributions of Coal (COAL) and Petroleum and Natural Gas (OIL) have smaller distance to cryptocurrency distributions. To conclude, my work will help to enhance the profiling of the clusters with additional insights. As a result, this work offers a description of the market and a methodology that can be reproduced by investors that want to understand the main trends on the market and that look for cryptocurrencies with different financial performance.All these three essays help to reveal the relationship between cryptocurrency returns and other asset returns. And I believe my findings will be useful in exploring the prediction of cryptocurrency returns based on stock market performance, and I verify that cryptocurrency is indeed an "orthogonal” assets that provide new opportunities to diversify risk.

Table of Contents

1 Preface 1

1.1 From Money and Fiat Currencies to Cryptocurrencies 2

1.2 Brief Introduction to Cryptocurrency Market 3

1.3 Motivation and Contribution of the Dissertation 5

2 Cryptocurrency Return Forecast Using Time Series Models and Entropy Approach 12

2.1 Introduction 14

2.2 Data and Basic Characteristics 17

2.3 Time Series Method 20

2.3.1 ARIMA Model 20

2.3.2 GARCH Model 23

2.4 Entropy Profiles Method 28

2.4.1 Introduction to Information Theory and Entropy 28

2.4.2 Using Entropy to Test Equality of Univariate Densities 32

2.4.3 Using Entropy to Test Equality of Densities between Industries 38

2.4.4 Testing Density Equality Based on Conditional Distribution 43

2.4.5 Maximum Entropy Approach to Portfolio Selection 47

2.5 Conclusion 60

3 Contrasting Cryptocurrencies with Other Assets: Full Distributions and the COVID Impact 68

3.1 Introduction 70

3.2 Data and Basic Characteristics 74

3.3 Entropy Profiles Method 78

3.3.1 Brief Introduction to Information Theory and Entropy 78

3.3.2 Using Entropy to Test Equality of Univariate Densities 81

3.3.3 Similarity with Select Asset Classes 86

3.3.4 Testing General Nonlinear Co-dependence 89

3.4 Difference-in-Differences Analysis 89

3.5 Three-Period Analysis and the Vaccine Effect 92

3.6 Conclusion 100

4 Do Cryptocurrencies and Other Assets Converge? A Clustering Analysis of Asset Returns 108

4.1 Introduction 110

4.2 Empirical Methodology 114

4.2.1 Entropy Measures of Distributional Distance 114

4.2.2 Cluster Analysis 115

4.3 Data 118

4.4 Results 128

4.4.1 Clustering Analysis of Cryptocurrencies 128

4.4.2 Clustering Analysis of Cryptocurrencies and Commodities 135

4.4.3 Comparing with K-means Clustering Results 141

4.5 Conclusions 147

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