postgraduate program «applied risk management»

SCHOOL OF ECONOMICS AND POLITICAL SCIENCES

DEPARTMENT OF ECONOMICS

Postgraduate Program

«APPLIED RISK MANAGEMENT»

MASTER THESIS

AN EMPIRICAL ANALYSIS ON THE

BEHAVIOR OF CRYPTOCURRENCIES’

RETURN AND VOLATILITY

DIMITRA TSOGKA

DIMITRIOS KENOURGIOS

ATHENS, GREECE

OCTOBER, 2021

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Acknowledgment

Through the writing of this master thesis, I have received a great deal of support and assistance.

I would like to thank my Supervisor, Professor Dimitrios Kenourgios, whose assistance and

guidance on the formulation of the major aim, questions, and methodologies was valuable. The

feedback was really significant for the improvement of the thesis.

I would also like to thank my family, who are continuously supporting me in each aspect and

attempt in my life, including this postgraduate program and they are always there for me.

Finally, I would like to thank my boyfriend, for providing me with unfailing support and

continuous encouragement throughout the years of my studies and through the process of

writing this thesis.

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Abstract

Cryptocurrencies have become famous nowadays and more and more studies on all their

aspects are being published.

In this thesis, two main studies are conducted with one sub-study to follow.

The first study refers to the examination of the interconnection of four popular cryptocurrencies

namely Bitcoin, Ethereum, Litecoin and Cardano. The study employs the Ordinary Least

Squares (OLS) Model and basic tests are conducted.

The second study is related to the examination of the volatility dynamics of the four

cryptocurrencies namely Bitcoin, Ethereum, Litecoin and Cardano in relation to the nine

popular indices namely S&P 500, Dow Jones, Gold Price, Crude Oil Price WTI, Dow Jones

Conventional Electricity, Dow Jones Real Estate, Baltic Dry index (BDI), Barclays US

Aggregate Bond Index, S&P Goldman Sachs Commodity Index. For this study the employment

of a multivariate GARCH model is necessary. Thus, the Diagonal BEKK model has been

selected to be used.

Furthermore, a sub-study is presented, which deals with the volatility dynamics of Bitcoin and

the nine aforementioned indices for the pre-COVID-19 period and the COVID-19 period.

This thesis has shown not only that there is positive relationship between the Bitcoin and other

cryptocurrencies returns but also that cryptocurrencies present the highest weekly loss, highest

average return, and highest volatility among all the studied indices and that the most stable

index is the Barclays US Aggregate Bond. It can also be determined that the influence of the

past common information of the variables is less significant than the persistence of covariance

between all cryptocurrencies and indices. Also, the results show that S&P 500 index previous

information strongly affects the cryptocurrencies’ returns and vice versa, while for the Gold

and cryptocurrencies case, previous information has the least impact on their returns comparing

to the other indices. As for the comparison of the pre-COVID-19 and COVID-19 periods

volatility spillover between Bitcoin and indices, the covariance ARCH coefficients seem to be

different in each period, nevertheless, for both periods the greatest covariance GARCH

coefficient is spotted between Bitcoin and S&P Goldman Sachs Commodity Index, while the

lowest is observed for the Bitcoin and Baltic Dry Index for the pre-COVID-19 period, while

for the Bitcoin and Dow Jones for the COVID-19 period. The effect of COVID-19 crisis seems

to have changed the behaviour of financial and cryptocurrency markets.

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Contents

Acknowledgment ................................................................................................................. 2

Abstract ............................................................................................................................... 3

List of Tables ....................................................................................................................... 6

List of Figures...................................................................................................................... 7

1. Introduction ................................................................................................................. 8

2. Historical Background .............................................................................................. 11

3. Technological Background and Safety...................................................................... 14

3.1. Blockchain Technology ...................................................................................... 14

3.2. Smart Contracts ................................................................................................. 18

3.3. Ways Bitcoin is Obtained................................................................................... 19

3.4. Double Spending ................................................................................................ 21

4. Reasons why cryptocurrencies were developed and their potential to substitute

traditional currency .......................................................................................................... 22

5. Government position and legal point of view ........................................................... 23

6. Literature Review ...................................................................................................... 24

7. Empirical Methodology ............................................................................................. 26

7.1. Relationship between Bitcoin and other Cryptocurrencies ................................... 26

7.1.1. Classical Linear Regression Model.................................................................. 26

7.1.1. Unit Root Test ............................................................................................ 28

7.1.2. Cointegration .............................................................................................. 29

7.2. Relationship between Cryptocurrencies and Indices ........................................ 32

7.2.1. ARCH Models ............................................................................................ 32

7.2.2. Testing for ARCH Effects .......................................................................... 33

7.2.3. GARCH Models ......................................................................................... 34

7.2.4. Extensions of the Basic GARCH Models ................................................... 35

7.2.5. Multivariate GARCH Models .................................................................... 37

8. Data Description ........................................................................................................ 39

8.1. Cryptocurrencies Description............................................................................ 40

8.1.1. Bitcoin ......................................................................................................... 40

8.1.2. Ethereum .................................................................................................... 42

8.1.3. Cardano ...................................................................................................... 44

8.1.4. Litecoin ....................................................................................................... 46

8.2. Indices Description............................................................................................. 49

8.2.1. S&P 500 ...................................................................................................... 49

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8.2.2. Dow Jones Industrial Average ................................................................... 49

8.2.3. Gold Futures ............................................................................................... 50

8.2.4. Crude Oil WTI ........................................................................................... 50

8.2.5. Dow Jones Conventional Electricity .......................................................... 51

8.2.6. Dow Jones Real Estate ............................................................................... 51

8.2.7. Baltic Dry index (BDI) ............................................................................... 52

8.2.8. Barclays US Aggregate Bond Index ........................................................... 53

8.2.9. S&P Goldman Sachs Commodity .............................................................. 53

8.3. Sample Data Frequency ..................................................................................... 54

9. Analysis Results ......................................................................................................... 55

9.1. OLS Regression Between Bitcoin and Selected Cryptocurrencies ................... 55

9.1.1. Data Presentation and Descriptive Statistics ............................................. 56

9.1.2. Unit Root Test ............................................................................................ 60

9.1.3. OLS Regression .......................................................................................... 61

9.1.4. OLS Regression Tests................................................................................. 62

9.1.5. Engle-Granger Cointegration Test ............................................................ 68

9.2. Diagonal BEKK Model for Selected Cryptocurrencies and Indices ................. 69

9.2.1. Diagonal BEKK Model for Bitcoin and Indices ........................................ 70

9.2.2. Diagonal BEKK Model for Ethereum and Indices.................................... 77

9.2.3. Diagonal BEKK Model for Cardano and Indices ..................................... 84

9.2.4. Diagonal BEKK Model for Litecoin and Indices....................................... 91

9.3. Diagonal BEKK Model for Bitcoin and Indices: Pre and During COVID-19

period …………………………………………………………………………………………………………………………….98

10. Conclusions .......................................................................................................... 110

Bibliography .................................................................................................................... 113

Appendix – BEKK Model Conditional Variances and Covariances Graphs ................ 117

A. Bitcoin and Indices .............................................................................................. 117

B. Ethereum and Indices .......................................................................................... 121

C. Cardano and Indices ............................................................................................ 125

D. Litecoin and Indices ............................................................................................. 129

E. Bitcoin and Indices for the Pre-COVID-19 and COVID-19 periods .................. 133

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List of Tables

Table 1: Critical Values for DF tests .................................................................................... 29

Table 2: Critical Values for Engle-Granger Cointegration Test on Regression Residuals with

no Constant in Test Regression............................................................................................ 31

Table 3: Market Cap Percentage of the Studied Cryptocurrencies ........................................ 48

Table 4: Descriptive Statistics of Cryptocurrency Returns ................................................... 58

Table 5: Correlation Matrix among Studied Cryptocurrencies .............................................. 59

Table 6: ADF test for Variables ........................................................................................... 60

Table 7: OLS Regression results .......................................................................................... 61

Table 8: White Heteroskedasticity Test ............................................................................... 62

Table 9: OLS with Heteroskedasticity Corrected ................................................................. 63

Table 10: Breusch-Godfrey Autocorrelation Test ................................................................. 64

Table 11: Augmented Dickey-Fuller (ADF) test on studied Cryptocurrency Pairs ................ 68

Table 12: Bitcoin & Indices Descriptive Statistics ............................................................... 70

Table 13: GARCH BEKK Model Results for Bitcoin and Indices ........................................ 72

Table 14:GARCH BEKK Model Results for Bitcoin and Indices ......................................... 73

Table 15: GARCH BEKK Model Results for Bitcoin and Indices ........................................ 74

Table 16: Ethereum & Indices Descriptive Statistics ............................................................ 77

Table 17: GARCH BEKK Model Results for Ethereum and Indices .................................... 79



Table 20: Cardano & Indices Descriptive Statistics .............................................................. 84

Table 21: GARCH BEKK Model Results for Cardano and Indices ...................................... 86



Table 24: Litecoin & Indices Descriptive Statistics .............................................................. 91

Table 25: GARCH BEKK Model Results for Litecoin and Indices ...................................... 93



Table 28: Bitcoin & Indices Descriptive Statistics for the Pro-COVID 19 period ................. 98

Table 29: Bitcoin & Indices Descriptive Statistics for the COVID 19 period ........................ 99

Table 30: GARCH BEKK Model Results for Bitcoin and Indices for the pro-COVID 19

period................................................................................................................................ 101


period................................................................................................................................ 102


period................................................................................................................................ 103

Table 33: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period

......................................................................................................................................... 104


......................................................................................................................................... 105


......................................................................................................................................... 106

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List of Figures

Figure 1: Applying business logic with smart contracts ........................................................ 18

Figure 2: Difficulty and Computing Power (Hash rate) since 2009 ....................................... 20

Figure 3: Searching Popularity of "antminer" and "bitmain" ................................................ 21

Figure 4: Illustration of Heteroskedasticity .......................................................................... 27

Figure 5: BTC Cryptocurrency ............................................................................................ 40

Figure 6: Bitcoin Price Historical Chart ............................................................................... 41

Figure 7: Bitcoin Market Cap Historical Chart ..................................................................... 41

Figure 8: ETH Cryptocurrency ............................................................................................ 42

Figure 9: Ethereum Price Historical Chart ........................................................................... 43

Figure 10: Ethereum market Cap Historical Chart ............................................................... 43

Figure 11: ADA Cryptocurrency ......................................................................................... 44

Figure 12: Cardano Price Historical Chart ........................................................................... 44

Figure 13: Cardano Market Cap Historical Chart ................................................................. 45

Figure 14: LTC Cryptocurrency .......................................................................................... 46

Figure 15: Litecoin Price Historical Chart............................................................................ 46

Figure 16:Litecoin Market Cap Historical Chart .................................................................. 47

Figure 17:Total Market Cap of All Existing Cryptocurrencies ............................................. 47

Figure 18: S&P 500 Price Historical Chart .......................................................................... 49

Figure 19: Dow Jones Price Historical Chart ....................................................................... 50

Figure 20: Gold Futures Price Historical Chart .................................................................... 50

Figure 21: Crude Oil WTI Price Historical Chart ................................................................. 51

Figure 22: Dow Jones Conventional Electricity Price Historical Chart ................................. 51

Figure 23: Dow Jones Real Estate Price Historical Chart ..................................................... 52

Figure 24: Baltic Dry Index Price Historical Chart ............................................................... 52

Figure 25: Barclays US Aggregate Bond Index Price Historical Chart ................................. 53

Figure 26: S&P Goldman Sachs Commodity Price Historical Chart ..................................... 54

Figure 27: Bitcoin Price Time Series ................................................................................... 56

Figure 28: Ethereum Price Time Series ................................................................................ 56

Figure 29: Cardano Price Time Series ................................................................................. 57

Figure 30: Litecoin Price Time Series .................................................................................. 57

Figure 31: Studied Cryptocurrencies Returns Price Series .................................................... 58

Figure 32: Normality of Residuals Test ............................................................................... 66

Figure 33: CUSUM Test ..................................................................................................... 67

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1. Introduction

A Cryptocurrency is a digital currency which differs from the traditional currency and assets

and is being used as an exchange medium. The transactions security and verification method

which Is used is cryptography and this makes it difficult to double-spend (Neeraj Kumar,

Shubhsni Aggarwal, 2020).

Blockchain technology is used in order to provide security, transparency, traceability and

immutability to transactions (Mohil Maheshkumar Patel, Sudeep Tanwar, Rajesh Gupta, Neeraj

Kumar, 2020). Cryptocurrencies are secured via blockchain, using a system of both private and

digital keys (Houben Robby, Alexander Snyers, 2018).

Nowadays, cryptocurrencies have been established as an alternative investment to traditional

assets. The number of crypto coins has been significantly increased, so some cryptocurrency

development companies have started to develop cryptocurrency indices to control the market

growth. Also, investors have commenced thinking of creating and investing in portfolios

consisted of different cryptocurrencies. (Nektarios Aslanidis, Aurelio F. Bariviera, Alejandro

Perez-Laborda, 2021).

The relationship of cryptocurrency market with the US stock market and commodity market is

known, so it will be useful to manage investors’ portfolios and determine investment portions

on cryptocurrencies for a secure and profitable investment. Investing on cryptocurrencies has

been popular nowadays among investors, so the examination of the dynamic relationship

between cryptocurrency and commodity and stock market is of the interests of economic

entities (Jong-Min Kim, Seong-Tae Kim, Sangjin Kim, 2020).

The variability in Bitcoin volatility, the existence of period with high volatility and period with

low volatility, indicates that the application of Generalized Autoregressive Conditional

Heteroskedasticity (GARCH) type models. In addition, the use of multivariate GARCH models

allows the test of sensitivity of Bitcoin in relation to other variables of economic and financial

scene (Ángeles Cebrián-Hernández, Enrique Jiménez-Rodríguez, 2021).

The COVID-19 pandemic is a health crisis which has been translated into an economic shock,

the first one after Bitcoin’s inauguration in 2009. The study of the COVID-19 crisis in

cryptocurrency world is interesting and shall be examined (Christy Dwita Mariana, Irwan Adi

Ekaputra, Zaafri Ananto Husodo, 2021).

The purpose of this study is to examine the behaviour of cryptocurrencies’ return and volatility.

More specifically the following approaches are used:

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The first is the examination of the relationship between the returns of some main

cryptocurrencies namely Bitcoin, Ethereum, Cardano and Litecoin with the employment of the

Ordinary Least Squares (OLS) Model for the common sample period from 31 December 2017

to 16 August 2021, with the use of daily observations.

The second is the study of the volatility dynamics of the same four cryptocurrencies in relation

to the nine popular indices namely S&P 500, Dow Jones, Gold Price, Crude Oil Price WTI,

Dow Jones Conventional Electricity, Dow Jones Real Estate, Baltic Dry index (BDI), Barclays

US Aggregate Bond Index, S&P Goldman Sachs Commodity Index, with the employment of a

multivariate GARCH model, the Diagonal BEKK model. The sample size is different for each

cryptocurrency – indices study and is related to the data availability for each cryptocurrency.

Thus, for Bitcoin and indices the studied period is 18 July 2010 – 15 August 2021, for Ethereum

and indices is 02 August 2015 – 15 August 2021, for Litecoin and indices is 14 September 2014

– 15 August 2021 and for Cardano and indices is 24 September 2017 – 15 August 2021, with

the use of weekly observations.

The last one is the examination of the volatility dynamics of Bitcoin and the nine indices for

the pre-COVID-19 period and the COVID-19 period. The sample period is 18 July 2010 – 15

August 2021 and weekly observations are used.

The first part of this thesis which is related to the interconnection of the returns of main

cryptocurrencies is an extension of the study of Silva et al. (2018) who examined the

interconnection of Bitcoin with other seven main cryptocurrencies for the period of August

2015 to September 2017. In this study the studied period is extended, with increased number of

observations and more recent data.

The second part of the thesis extends the studies of Corbet et al. (2018b), Katsiampa (2018) and

Katsiampa (2019) by employing an asymmetric Multivariate GARCH model, namely Diagonal

BEKK model, which tests the dynamic conditional volatility and the correlation between main

cryptocurrencies with other market indices and assets. Previous studies have examined the

dynamic conditional volatility and correlation among cryptocurrencies with the Diagonal

BEKK model or between cryptocurrencies and market indices and assets with the use of other

techniques. In subject thesis, a combination of these approaches has been made and additional

indices with a wider sample size range and more recent data have been used.

The additional analysis that has been conducted and refers to the examination of the volatility

dynamics between Bitcoin and selected indices for the pre-COVID-19 and COVID-19 period.

This study consists an extension on the existed studies and examines the current situation during

COVID-19 period. The results of this study are useful for the literature as they show how this

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health global crisis affect the economic world and more specifically the volatility spillovers of

Bitcoin and famous market indices and assets.

This thesis includes, except for the introductory chapter, the historical and technological

background of the cryptocurrencies’ creation and development for a better understanding of the

subject of study, the reasons why cryptocurrencies were developed and also the governmental

position and the legal point of view of the cryptocurrencies and their use. In addition, a literature

review is conducted and the empirical methodology that is followed is presented. This

theoretical background is followed by the analysis results. At the end, the conclusions are

presented as well as suggestions on further research.

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2. Historical Background

Cryptocurrency history starts one decade ago and up to present its evolution is almost

unexpected.

On 18 August 2008 the bitcoin.org domain was developed and later the same year the bitcoin

designer named “Satoshi Nakamoto” published a paper that intrigued the public. This first

reference to the bitcoin cryptocurrency was the following: “Bitcoin: A peer-to-peer Electronic

Cash System”. Subject paper describes how the digital coins transactions could be secured from

double-spending and other threats, and they so can be trustworthy (Nakamoto, 2008).

“Satoshi Nakamoto” is presumed not to be the real name of the designer of bitcoin protocol and

many efforts have been made to find out who actually is this person. Even the multiple

approaches, the identity of bitcoin designer remains still unknown.

On 12 January 2009, Nakamoto made the first bitcoin transaction, by transferring 10 bitcoins

to a computer programmer Hal Finney.

On 22 May 2010 the first bitcoin real life transaction was made, and a bitcoin user paid 10,000

bitcoins for two pizzas. Later this year, on 15 August 2010, bitcoin was hacked and a transaction

of 184 billion bitcoins was made. Bitcoin developer Jeff Garzik, when noticed this not really

common transaction realized that there were in a big problem.

In 2011, new cryptocurrencies (Litecoin, Namecoin and Swiftcoin) appeared, while bitcoin

was said to be used in the dark web, a fact that drove bitcoin price to a low level. From January

this year until June, the Electronic Frontier Foundation was accepting bitcoins, however this

stopped due to legal concerns on cryptocurrencies. In 2003, this concern was surpassed, and

bitcoin were again accepted. Also, on 16 April 2011 an articled called “Online Cash Bitcoin

Could Challenge Governments, Banks” was published at Time Magazine, being the fist time

that cryptocurrency entered the traditional media (Brito, 2011).

In 2012, cryptocurrencies started to become popular, and a fictionalised trial was made in the

third season of “The Good Wife” US drama. On 20 June same year, Coinbase was created.

Coinbase is a digital currency exchange, where users can buy, sell, store, use and earn

cryptocurrency.

On 29 October 2013, the first bitcoin ATM available for public, was settled in Canada. At the

end of 2013, 67 different cryptocurrencies were in existence in total.

On 19 March 2014, the MT. Gox, a cryptocurrency exchange platform based in Japan went

bankrupt, and as a result lots of investors lost their money. The same year, Microsoft accepted

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cryptocurrency as a payment method for games (Bigmore, 2018). At the end of 2014, 517

different cryptocurrencies were in existence in total.

On 23 February 2015, a new cryptocurrency, Ethereum was developed. At the same time,

Bitstamp, a bitcoin exchange based in Europe, was hacked. Bitstamp, came back to normality

a few days later and ensured customers that they their investments had not been affected

(Bigmore, 2018). At the end of 2015, 577 different cryptocurrencies were in existence in total.

By the end of the 2016, the bitcoin ATMs around the world were almost doubled (from 600 at

the beginning of the year to 900 at the end). Swiss National Railway, Steam computer software

website and Uber (in Argentina) were accepting cryptocurrency payments (Bigmore, 2018). At

the end of 2016, 663 different cryptocurrencies were in existence in total.

On 07th March 2017, the price of bitcoin surpasses the one-ounce gold price. The bitcoin was

split into bitcoin and bitcoin cash, when the “Bitcoin Cash hard fork” took place. On 20 April

2017, a law came in effect in Japan according to which bitcoin was recognized as a legal

payment method and at the same time Skandiabanken in Norway accepted bitcoin as an

investment asset and payment system. In May-June over 1,000 cryptocurrencies were listed,

and relevant market worth was more than $100bilion. In November market word raised over

$250bilion, while at the end of the year it reached $500bilion. At the end of 2017, 1353 different

cryptocurrencies were in existence in total.

In February 2018, cryptocurrency market worth was more than $824bilion and in only a month

it reached the lowest level ever seen since 2017, $248bilion. In September that year, market cap

fell to $186bilion. At the end of 2018, 2073 different cryptocurrencies were in existence in total.

In September 2019, the bitcoin ATMs worldwide were 5,457. At the end of 2019, 2388 different

cryptocurrencies were in existence in total.

At the end of 2020, 4118 different cryptocurrencies were in existence in total.

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Chart 1: Number of Cryptocurrencies per Year (Author’s Graph with data collected from (CoinMarketCap, 2021))

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3. Technological Background and Safety

There are several technologies on which cryptocurrencies are based and according to which

they operate.

3.1. Blockchain Technology

One of the first applications of blockchain technology are cryptocurrencies, and specifically

Bitcoin. Also, the same technology is used by Ethereum, one of the most popular

cryptocurrencies after Bitcoin. Blockchain technology could be described as follows:

Blockchain is an encrypted archiving mechanism that is essential for the network reliability.

Actually, reliability is achieved by recording in chronological order and publishing all

transactions made on the system. In particular, blockchain is expected to largely replace the

existing trading system, emphasizing the necessity of digitizing the way transactions are

recorded. The key to moving into this new era is to develop a commonly accepted trading

system, which will improve the existing security of the assets traded by users, but also ensure

the protection of the system itself.

Distributed Ledger Technology is a broader category of technology that includes the blockchain

and is a digitized transaction ledger which stems from a computer science research conducted

by Haber and Stornetta who introduced the cryptographic benefits of hash-linked,

chronologically ordered and timestamped records (Stuart Haber, W. Scott Stornetta, 1991)

(Stuart Haber, W. Scott Stornetta, 1997).

The blockchain technology refers to the unchangeable public ledgers, which are constructed

using decentralized techniques and generally is not based on any central authority that controls

the network. Blockchain records the transactions take place in the system and shares the

transactions’ content with all the users in the network. This process of recording and publication

of the transactions made in a system is called “peer-to-peer network”. The main characteristic

of subject network is the consensus, which means that is being updated with the agreement of

all users in the network, and the fact that any information or update that is included in the system

cannot be deleted or modified.

More specifically, blockchain is consisted of blocks, that record every single transaction in the

system in chronological order. The chain starts from the first block that is created and in order

to approve any subsequent transaction, the verification of all the blocks included in the chain is

required. Once this new transaction is verified, it is added as a new block at the end of the

blockchain (Harish Natarajan, Solvej Krause, Helen Gradstein, 2017).

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The approval of the transactions could be completed via the following consensus mechanisms:

o Proof of work

This mechanism is used by Bitcoin. Mining process is being used for each block verification.

Algorithms that attach a unique hash to each block depending the information that block

includes, are used to approve the data included in each block. This mechanism requires users

to approve the hashes of transactions each time in order for the blockchain assets to be updated.

As a result, process is being expensive and time-consuming and requires large computational

power.

o Proof of Stake

This mechanism is used by Ethereum. Proof of Stake makes the mining process simpler, as

actually mining does not take place. Instead, users can approve their transactions and modify

blockchain on the basis of their own share or “stake” in the currency. By using this mechanism,

decentralized verification process is more straight forward, and large amount of computational

power can be saved. Also, operating costs are limited comparing to other mechanisms.

o Proof of Authority” (PoA), “Proof of Importance” (PoI), “Proof of History” (PoH)

These mechanisms are recently developed, and they focus to be less costly and power intensive

and to be more time-efficient (PwC, 2018).

Public and private key

As mentioned above, the blockchain is an encrypted mechanism of recorded transactions in

chronological order. So, this technology is based on public key encryption to protect user

accounts from attacks by unauthorized hackers. The network then, as an encryption technique,

provides a private key to each user which is its unique digital signature as a means of completing

any transaction on the network, as well as a public key known to all users of the system and

allows the control of the involved users and the validity of every new transaction.

For instance, if a transaction between two people (A, B) is to be conducted, and person A

intends to send a sum of money to person B, the following process will be followed. According

to the above encryption technique, in order to add a new block to the chain, person A should

create a new transaction, including public keys of A, B people in it. After that, person A signs

the transaction via his private key and finally, to verify the transaction, the network uses person

A's public key and adds the new block to the blockchain.

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Blockchain Network

Blockchain network can be divided into the below categories:

Public network

In public network, the transactions made are neither managed nor supervised by any existing

central authority, a fact that upgrades the independence of the network. In addition, any user of

the network can have access to the information and execution of the transactions and thus

participate in the process of their verification and completion. Also, this kind of network does

not require authorized user to verify and validate transactions, everyone in the network is

encouraged to perform as per the contract to reach the best result of the network. Thus, public

blockchain’s important trait is the transparency, as transactions could be verified by any user

(Rebecca Yanga, Ron Wakefielda, Sainan Lyua, Sajani Jayasuriyaa, Fengling Hanb, Xun Yib,

Xuechao Yangb, Gayashan Amarasinghea, Shiping Chenc, 2020).

This blockchain network category is mainly supported by digital currencies such as Bitcoin and

Ethereum, at it provides a combination of financial incentives and also facilitates the

verification of transactions of each network through specialized cryptographic mechanisms

such as proof of work (PoW) or proof of stake (PoS), where through computer algorithms they

perform the transaction verification process and so the addition of new blocks to the chain.

However, the above encryption mechanisms can process a limited volume of transactions

simultaneously and their operating costs are high, a fact that consists of the main disadvantage

of this category of networks. As a result, financial institutions such as banks, which maintain a

large number of users in their databases are directed to the second category of networks, namely

private or permissioned networks.

Finally, these who participate in the proof of work or proof of stake for the validation of new

blocks in the blockchain are called miners of the network and this process is called mining.

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Private network

Private blockchain networks include two sub-categories, the fully private blockchain networks

and the consortium blockchain networks. In the first type of network, all written authorizations

for the completion of transactions are retained in one structure, while in the second, the

consortium process is defined by a predetermined number of nodes.

Private networks have a limited number of authorized users (trusted parties) for the transactions’

verification process. In order for a new user to entry the verification process, the approval of

existing members or the central authority of the system is required. Thus, in private blockchain

networks, the central authority has partial or full control of the system and as a result, has the

ability to impose regulations related to transactions or network rules. Moreover, the

differentiation in the mechanisms used for transactions verification in a private blockchain

network, is cost and time-efficient related to those of public blockchain networks. In this way,

the legal transparency of the parties involved in the verification process of a private network is

established. Last but not least, private blockchains are characterized by robust data privacy and

modifications could be made once all nodes come into agreement that the data can be amended

by consensus (Rebecca Yanga, Ron Wakefielda, Sainan Lyua, Sajani Jayasuriyaa, Fengling

Hanb, Xun Yib, Xuechao Yangb, Gayashan Amarasinghea, Shiping Chenc, 2020).

Federated Network

This blockchain network type, being a partially decentralized one, is a combination of public

and private network. Its main difference is traced on the fact that a specific user has the

verification right and there is no need for every user to verify the transaction processes (Rebecca

Yanga, Ron Wakefielda, Sainan Lyua, Sajani Jayasuriyaa, Fengling Hanb, Xun Yib, Xuechao

Yangb, Gayashan Amarasinghea, Shiping Chenc, 2020).

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3.2. Smart Contracts

The smart contracts logic had been firstly appeared 20 years ago by Szabo, however their

application on blockchain is very recent (Szabo, 1997).

The smart contacts are the software programs, basically they are computer codes, that are

uploaded to a ledger and their function is to check whether specific conditions are met and

subsequently to generate instructions for downstream processes i.e. payment instructions. Once

Smart Contracts are accepted onto the ledger, they cannot change (Cermeno, 2016).

The following diagram describes their function:

Figure 1: Applying business logic with smart contracts (Source: BBVA Research, based on Jo Lang / R3 CEV)

Smart contracts do enforce the blockchain functionality, as they allow the transition from the

consensus on data stream achievement to the consensus on computation achievement. (Ahmed

Kosba, Andrew Miller, Elaine Shi, Zikai Wen, Charalampos Papamanthou, 2016)

Nevertheless, there are some potential problems which have to be resolved, like the

expansibility, as it is not really possible for each node to process every transaction as the users

and so contracts are increasing, the precision and accuracy of the code as users have to trust

contracts’ correct functions and that they do not charge excessive fees due to not necessary

computations, and finally the connection of the smart contracts with their legal counterpart.

(Gareth W. Petersz, Efstathios Panayi, 2018)

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3.3. Ways Bitcoin is Obtained

Bitcoins could be obtained by three different ways:

1. Legally registered exchanges

They can be used for exchange of traditional currency with cryptocurrency.

2. Mining

A significant aspect of mining is that depending the computing power that is currently available

in the network, the bitcoin system adjusts with an automated way the difficulty of Proof-of-

Work algorithm to confirm that the block are generated every 10 minutes on average. This

adjustment is not done immediately, as it takes 2016 blocks and after these blocks and based

on the previous mining time all nodes will amend the difficulty of Proof-of-Work algorithm.

The mining speed or service rate could be almost stable when considering a time frame of a

week or a month. Nevertheless, for a smaller time frame, the mining speed changes based on

how often the difficulty adjust in a given period.

Also, one more significant aspect of the mining is the miners that are in the system. When

bitcoin mining started and until 2017, the mining process was like a hobby for many people in

computer science, using CPU or GPU on their personal computers. After 2017, when the price

of Bitcoin raised significantly, mining process became more competitive and intense, as it can

be seen from the following hash rate diagram.

20

Figure 2: Difficulty and Computing Power (Hash rate) since 2009 (blockchain.com, 2021)

The use of CPU or GPU on a personal computer was no more profitable and special computer

equipment was necessary. The bitcoin mining was then started using dedicated mining ASIC

equipment which can only be used for this purpose.

As more and more professional miners are joining the mining network, this process will be

unprofitable for amateurs. This trend of the technology change on the mining process can also

be seen from the Google trend data on the two biggest producers of ASIC mining equipment

“antminer” and “bitmain”, as per following figure.

0.00E+00

2.00E+07

4.00E+07

6.00E+07

8.00E+07

1.00E+08

1.20E+08

1.40E+08

1.60E+08

1.80E+08

2.00E+08

0

5E+12

1E+13

1.5E+13

2E+13

2.5E+13

3E+13

Has

h R

ate

Dif

ficu

lty

difficulty hash-rate

21

Figure 3: Searching Popularity of "antminer" and "bitmain" (GoogleTrend, 2021)

3. Exchange of good and services

Goods and services could be exchanged with cryptocurrencies instead of other goods/ services

and/or traditional currency.

3.4. Double Spending

Double-spending is a potential problem of digital currencies, which describes the risk that

digital currency is spent twice. This stems from the fact that it is relatively easy for someone

who understands the technology behind cryptocurrencies, to find out the way they are being

produced. Nevertheless, as subject algorithms need great computational power, it is almost sure

that it will fail. It is important to mention that such attacks have already taken place (Carlos

Pinzόn, Camilo Rocha , 2016).

0

20

40

60

80

100

120

2009

-01

2009

-06

2009

-11

2010

-04

2010

-09

2011

-02

2011

-07

2011

-12

2012

-05

2012

-10

2013

-03

2013

-08

2014

-01

2014

-06

2014

-11

2015

-04

2015

-09

2016

-02

2016

-07

2016

-12

2017

-05

2017

-10

2018

-03

2018

-08

2019

-01

2019

-06

2019

-11

2020

-04

2020

-09

2021

-02

2021

-07

Inte

rest

ove

r Ti

me

(%)

22

4. Reasons why cryptocurrencies were developed and their potential to substitute

traditional currency

The reasons of the creation of cryptocurrencies have been studied since their nascence. The

intention behind the development of cryptocurrencies was the creation of a new way of payment

that could be used internationally, decentralized (peer-to-peer) and without having any financial

institution behind it. The global economic crisis of 2008 played of course its role. The massive

collapse of the banking sector during the period of the financial crisis and the insecurities in

financial institutions led to the rapid development of cryptocurrencies (Ahmad Chokor, Elise

Alfieri, 2021). At the same time, people had the need to use alternative types of money, because,

after the economic crisis, a number of them lost the faith and the trust to the conventional money.

Moreover, these virtual currencies were designed in order to have a global character, across

national borders, without limits and also to be stored for later use like the conventional money.

However, cryptocurrencies are considered inadequate to satisfy all of these goals, at least to a

high degree. Many times, they are used in cases of fraud and manipulation, they encourage and

facilitate the crime (Joshua R. Hendrickson, William J. Luther, 2021), due to the anonymity,

thus they should be surrounded by a legal scheme, which will provide more safety for the users.

So, although the fact that a number of people lost their faith, after the global economic crisis,

to the political and economic system, cryptocurrencies have failed to be established, 13 years

later. Furthermore, cryptocurrencies present fluctuations and volatility and for that reason the

store of money is not an easy task, since there is not stability and in other words, a constant

value. Of course, the investments to the new, according to many supporters of cryptocurrencies,

gold, to the «digital gold» (Zigah, 2020) (gold and cryptocurrencies have some commons, like

the limited supply) are not without risk (Marek Dabrowski, Lukasz Janikowski, 2018), due to

the uncertainties.

Cryptocurrencies, for all the above reasons, may not be ready yet to substitute traditional

currency, but the perspective is still powerful.

23

5. Government position and legal point of view

Although the fact that there is a global recognition of the need to be regulated cryptocurrencies,

no consensus has been achieved concerning how to classify them. Governments attempt -

unilaterally- to define, regulate and manage cryptocurrencies, but in order to be formed a

common legislative framework there is a requirement for agreement, since the market is

universal.

There are many reasons behind the regulation of cryptocurrencies. However, they constitute, as

it has already mentioned, an international market with different aspects and this makes it very

difficult for a single regulatory power to be implemented across borders.

The approaches of cryptocurrency’s regulation could be divided based on three main objectives:

a) the cryptocurrency market is characterized by volatility, thus the price stability, via the

regulation, is essential, b) protecting consumers against unlawful activities is also crucial,

through the regulation, c) several factors can make the cryptocurrency market illiquid, thus

limiting the ability of its participants to buy or sell crypto assets. So, implementing new

regulations for this specific market could lead to significant revenues for the governments

(Ahmad Chokor, Elise Alfieri, 2021).

Some governments consider cryptocurrencies as assets, while others consider them as currency

and in general as a way of payment. For instance, the Australian Taxation Office (ATO) has

decided that cryptocurrency is a commodity, not a currency and for that reason it corresponds

to the tax instructions provided by the relevant authorities in other countries, such as Canada

and Singapore (O. Bolotaeva, A. Stepanova, S. Alekseeva, 2019).

Some governments emphasize in regulation of the market for fiscal reasons, others for

restrictive reasons and others for creation banning policies, fact that entails different angles of

view. In any case, at the moment, has not been tested yet a regulation for the turnover of

cryptocurrencies, which is one of the main problems of the lack of adequate regulation of the

market.

El Salvador became into September of 2021 the first country that adopted Bitcoin as legal tender

fact that it is connected with a figuration of a legal framework. Nevertheless, it is still premature

to be supported that even with the function of El Salvador’s experiment it has established a

commonly accepted practice for the regulation of the market.

24

6. Literature Review

Cryptocurrencies have been popular the last years and so except for their investing and public

popularity they have also obtained academic attention, with more and more articles and

literature in general to be developed.

From the early years of the Bitcoin development, many academics have been focused on the

study of the determinants of cryptocurrency price as well as on the price discovery process.

Kristoufek (2013) and Panagiotidis et al. (2018) concluded that Bitcoin prices are connected

with Bitcoin search queries on Google Trends and Wikipedia. The latter has also shown that

gold returns is also an important variable for Bitcoin returns. Georgoula et al. (2015) showed

that the Wikipedia search queries, and the hash rate have a positive effect on Bitcoin Price

considering a short-run analysis. Georgoula et al. (2015) also showed that Twitter feeds are

positively correlated with Bitcoin price.

Cryptocurrency volatility has also been research object. Univariate GARCH type models have

been employed by Katsiampa (2017), who found that the Component GARCH model is the

best fit to Bitcoin price returns comparing to other GARCH models studied, by Chu et al. (2017)

who studied twelve different GARCH models on the seven most popular cryptocurrencies and

concluded to the one that fits better, by Liu et al. (2017) who studied several distributions under

GARCH model to see which fits better and Takaishi (2018) who investigated the daily volatility

asymmetry with the employment of GARCH type models. However, multivariate models are

necessary to be employed for the examination of the co-movements of cryptocurrencies.

Silva et. Al (2018) studied the interconnection of Bitcoin with other seven main

cryptocurrencies for the period of August 2015 to September 2017 and concluded that there is

a correlation between their returns, and their behaviour could be explained together.

Also, there have been several studies examining the interconnection among cryptocurrencies

and other financial assets. Yermack (2015) concluded that there is no correlation between

Bitcoin and widely used currencies and gold, while Baur et al. (2018) concluded that there is

no correlation between Bitcoin and traditional assets such as stocks, bonds, and commodities

during normality periods. Lee et al. (2018) also showed that return correlations between

cryptocurrencies and traditional assets are low which is in line with Corbet et al. (2018b) study

where evidence of relative isolation of cryptocurrencies from other financial and economic

assets was found. Nevertheless, the connection of cryptocurrency market with stock and other

assets market is not fully explored yet.

25

In COVID-19 era, a lot of studies have been focused on the impact of the pandemic to the

cryptocurrency market. Naeem et al. (2021) have studied the spillover effect among seven

major cryptocurrencies using VAR model, for the pre and post-COVID-19 crisis and they

showed that there is low network integration among cryptocurrencies for the pre-COVID period,

while there are tangled clusters among them for the post-COVID period. Jabotinsky et al. (2021)

showed that the COVID cases and deaths worldwide in the early days of pandemic are

positively correlated with cryptocurrencies market capitalization. Also, Kumar (2021)

confirmed with the study that there exists unidirectional causal relation from COVID-19

confirmed and death cases to cryptocurrency price returns. The study of Yousaf et al. (2021)

which examined the return and volatility spillovers between S&P 500 and cryptocurrencies

during pre-COVID and COVID periods, concluded that there is no important return and

volatility spillovers between US stock and cryptocurrency markets during the pre-COVID

period, however there is unidirectional return transmission and volatility spillover form S&P

500 to Litecoin only. Lastly, Mnif at al. (2021) has stated that COVID-19 has positive impact

on the cryptocurrency market efficiency.

26

7. Empirical Methodology

In subject thesis two different studies will be carried out. For the first one, the relationship

between the Bitcoin returns and the returns of the other selected cryptocurrencies will be studied,

with the employment of OLS (Ordinary Least Squares) model. For the second one, the

relationship between the selected cryptocurrencies returns and the selected indices returns will

be studied, with the employment of multivariate GARCH (Generalized Autoregressive

Conditional Heteroskedasticity) models.

7.1. Relationship between Bitcoin and other Cryptocurrencies

As mentioned above, for the study of the relationship between Bitcoin and other selected

cryptocurrencies will be carried out. The OLS model will be used for subject analysis, based

on the background and methodology described below.

7.1.1. Classical Linear Regression Model

The assumptions of the Classical Linear Regression Model are the below:

𝐸(𝑢𝑡) = 0

𝑣𝑎𝑟(𝑢𝑡) = 𝜎2 < ∞

𝑐𝑜𝑣(𝑢𝑖 , 𝑢𝑗 ) = 0

𝑐𝑜𝑣(𝑢𝑡 , 𝑥𝑡 ) = 0

𝑢𝑡 ~ 𝑁(0, 𝜎2)

The above are required to approve that the estimation technique (i.e. Ordinary Least Squares)

has a number of desirable properties and that it is proper to conduct the hypothesis tests for the

estimation of coefficients.

𝐸(𝑢𝑡) = 0

This assumption is that the average value of the errors is zero. This assumption will never

be violated if there is a constant term in the regression.

𝑣𝑎𝑟(𝑢𝑡) = 𝜎2 < ∞

27

This is the assumption of homoskedasticity, meaning that the variance of errors is constant.

If the variance of errors is not constant, then there is heteroskedasticity. The residuals

�̂�𝑡 should be calculated and plotted against the explanatory variables 𝑥2𝑡 .

Figure 4: Illustration of Heteroskedasticity

𝑐𝑜𝑣(𝑢𝑖 , 𝑢𝑗 ) = 0 𝑓𝑜𝑟 𝑖 ≠ 𝑗

This assumption is that the covariance between the error terms over the time is zero, so the

errors are not correlated with one another, and they are “autocorrelated” or “serially correlated”.

The autocorrelation test is conducted on residuals �̂�𝑡.

𝑐𝑜𝑣(𝑢𝑡 , 𝑥𝑡 ) = 0

Subject assumption is that the 𝑥𝑡 is non-stochastic, meaning that the OLS estimator is consistent

and unbiased in the presence of stochastic regressors. It is necessary for the regressors not to

be correlated with the error term of the estimated equation.

𝑢𝑡 ~ 𝑁(0, 𝜎2)

The last assumption is that the disturbances are normally distributed. One test which is usually

used to check the normality of disturbances is the Bera-Jarquue, which uses the property of a

normal distributed random variable that the mean and variance (fist two moments) characterise

the whole distribution. Skewness and kurtosis, which are the third and fourth moments of

distribution, measure the extent to which the distribution is not symmetric about its mean and

the size of the tails 9how fat they are) respectively. The normal distribution has a coefficient of

28

kurtosis of 3 and is not skewed. A leptokurtic distribution is more peaked than the normal

distribution with same mean and variance and has fatter tails.

7.1.1. Unit Root Test

Each cryptocurrency should undergo a unit root test in order to check if the time series is

stationary or non-stationary. Dickey-Fuller test and the augmented Dickey-Fuller test could be

performed to check the stationarity or non-stationarity of the variables and if they follow the

unit root process. Each cryptocurrency time series will be tested individually for a unit root.

The equation of Dickey-Fuller test is the following:

𝛥𝑦𝑡 = 𝑎𝑦𝑡−1 + 𝑢𝑡 (7.1.1)

Where:

𝑢𝑡: noise variable

The null hypothesis of a unit root existence is the 𝑎 = 0, while the 𝑎 < 0 is the alternative

hypothesis of no unit root existence and thus time series stationarity (Brooks, 2018).

As for the augmented Dickey-Fuller method, the only difference with the Dickey-Fuller is that

there are lags added into the model. Lags are included in order to eliminate autocorrelation of

the noise variable 𝑢𝑡 and the dependent variable (David A. Dickey, Wayne A. Fuller, 1979).

The equation augmented of Dickey-Fuller test is the following:

𝛥𝑦𝑡 = 𝑎𝑦𝑡−1 + ∑ 𝛾𝑖𝛥𝑦𝑡−𝑖 +

𝜌

𝑖=1

𝑢𝑡 (7.1.2)

Where:

𝜌: number of lags of the dependent variable

The hypothesis that is tested in the augmented Dickey-Fuller test is exactly the same with the

Dickey-Fuller test.

The number of lags could be decided based on the frequency of the used data (i.e. for monthly

data 12 lags should be used). Also, an information criterion shall be used for the decision of the

29

lags number, so as the number of lags that will be chosen to minimise the value f this

information criterion (Brooks, 2018).

Table 1: Critical Values for DF tests

Significance Level 10% 5% 1%

CV for constant but no trend -2.57 2.86 -3.43

CV for constant and trend -3.12 -3.41 -3.96

Note: CV is the Conditional Variance

7.1.2. Cointegration

A Cointegration test is being conducted in order to find out if there is a correlation between

several time series in the long term (CFI, 2021).

Ganger (1981) was the first to signalize that a vector of variables, all of which are stationary

after differencing, can have linear combinations that are stationary in levels. Cointegration was

later studied by Engle and Granger (1987), who introduced a method based on regression that

could be used to analyse time series data with common trends. Also, they pointed out that in

case correlation between the two non-stationary time series is important, this does not

necessarily show that they are significantly connected.

Later, Murray (1994) presented the cointegration and the error connection with an example of

a drunk person and her dog, which came out of a bar in order to wander aimlessly, following a

random walk. The drunk walk illustrates a random walk, same to the dog walk. Nevertheless,

sometimes the drunk person is looking for the dog, calling his name and requesting to come

closer to her. Now, neither dog nor drunk person follows a random walk; each has added the

so-called “error-correction mechanism” to her or his steps. Their paths are non-stationary,

nevertheless their long-run relationship will be stationary (Murray, 1994).

A time series is non-stationary and has a unit root if it is integrated of order 1 (the I(1) process).

Two or more non-stationary time series are considered cointegrated if they move together over

the time and so they have a common stochastic trend. These variables have a long run

relationship, which could be violated in the short-run, however it will be re-established and

return to the long-run equilibrium.

In this thesis, the cointegration between Bitcoin, Ethereum, Cardano, XRP, Litecoin and Stellar

will be tested. Fourteen pairs of all above cryptocurrencies will be made and tested. The first

cryptocurrency of each check will be symbolized with 𝑋𝑡 and the other with 𝑌𝑡 . In case both 𝑋𝑡

and 𝑌𝑡 are I(1) processes, then the are considered cointegrated if there exists a I(0) stationary

linear combination between them.

30

Engle-Granger Cointegration Test

Engle-Granger is the most popular cointegration test, which utilizes a single equation and

consists of a two-step method.

The first step is the check that all variables are I(1) following by regression with the use of the

OLS (Ordinary Least Squares) technique. The OLS regression is as below:

𝑌𝑡 = 𝛽1 + 𝛽2𝛸𝑡 + 𝑢𝑡 (7.1.3)

The residuals of the cointegrated regression shall be saved and checked that they are I(0), in

order to proceed to the next step. If residuals are I(1), a model should be estimated including

only the first differences.

The residuals test for stationarity is being conducted via the augmented Dickey-Fuller test on

the residuals. Relevant regression is the below:

𝛥�̂�𝑡 = 𝜓�̂�𝑡−1 + 𝑣𝑡 (7.1.4)

Where:

𝑣𝑡: is the independent and identically distributed error

In this case the critical values are different than the Dickey-Fuller or the augmented Dickey-

Fuller test, as they test a series of raw data, while here a test on residuals is conducted.

The new set of critical values has been introduced by Engle and Granger (1987) and they are

larger in absolute value than the Dickey-Fuller critical values.

31

Table 2: Critical Values for Engle-Granger Cointegration Test on Regression Residuals with no Constant in Test Regression

Number of Variables in System Sample Size T 0.01 0.05 0.10

2

50 -4.32 -3.67 -3.28

100 -4.07 -3.37 -3.03

200 -4.00 -3.37 -3.02

3

50 -4.84 -4.11 -3.73

100 -4.45 -3.93 -3.59

200 -4.35 -3.78 -3.47

4

50 -4.94 -4.35 -4.02

100 -4.75 4.22 -3.89

200 -4.70 -4.18 -3.89

5

50 -5.41 -4.76 -4.42

100 -5.18 -4.58 -4.26

200 -5.02 -4.48 -4.18

The null and the alternative hypothesis for the unit root on residuals is the below:

𝐻0: �̂�𝑡 ~ 𝐼(1)

𝐻1: �̂�𝑡 ~ 𝐼(0)

So, the null hypothesis is that there is a unit root – and therefore there is not a stationary linear

combination of non-stationary variables and there is no cointegration, while the alternative

hypothesis is that there is not a unit root in the potentially cointegrated regression residuals –

and therefore there is a stationary linear combination of non-stationary variables and there is

cointegration.

The second step is the use of the residuals obtained from the previous step as one variable in

the error correction model (Brooks, 2018).

32

7.2. Relationship between Cryptocurrencies and Indices

The second analysis includes the study of the relationship between cryptocurrencies returns and

common indices returns. For this study, the employment of multivariate GARCH models is

necessary. The description of ARCH and GARCH models is presented below.

7.2.1. ARCH Models

The Autoregressive Conditionally Heteroskedastic (ARCH) model is a particular non-linear

model which is being widely used. The assumption in this model is that the variance of errors

is not constant, so there is heteroskedasticity.

Another characteristic of the financial asset returns that provides a motivation for the ARCH

class models, is the “volatility clustering” or “volatility pooling”. This characteristic is the

tendency of large changes in asset prices returns) to follow large changes and small changes to

follow small changes. This phenomenon can be parameterised with ARCH models.

Main definition for the understanding of the model is the one of the conditional variance of a

random variable 𝑢𝑡 , which may be denoted 𝜎𝑡2 and is presented as follows:

𝜎𝑡2 = 𝑣𝑎𝑟(𝑢𝑡 ,𝑢𝑡−1 ׀ 𝑢𝑡−2 , … ) = 𝐸[(𝑢𝑡 − 𝐸(𝑢𝑡))2׀ 𝑢𝑡−1, 𝑢𝑡−2, … ] (7.2.1)

It is usually considered that 𝐸(𝑢𝑡) = 0, so the above equation is changed to the following:

𝜎𝑡2 = 𝑣𝑎𝑟(𝑢𝑡 ,𝑢𝑡−1 ׀ 𝑢𝑡−2 , … ) = 𝐸[(𝑢𝑡)2׀ 𝑢𝑡−1 , 𝑢𝑡−2, … ] (7.2.2)

It is easily understood from the last equation that the conditional variance of a normally

distributed random variable 𝑢𝑡 with zero mean is equal to the conditional expected value of 𝑢𝑡 .

In ARCH model, the conditional variance is calculated as:

𝜎𝑡2 = 𝑎0 + 𝑎1𝑢𝑡−1

2 (7.2.3)

This is the ARCH(1) model as the conditional variance depends on one lagged squared error.

The ARCH(q) model, which is the general case of an ARCH model, is described by the equation

below:

𝜎𝑡2 = 𝑎0 + 𝑎1𝑢𝑡−1

2 + 𝑎2𝑢𝑡−22 + 𝑎3𝑢𝑡−3

2 + ⋯ + 𝑎𝑞𝑢𝑡−𝑞2 (7.2.4)

33

The mean equation can take almost any form, depending on the user is. Nevertheless, an

example could be the following:

𝑦𝑡 = 𝛽1 + 𝛽2𝑥2𝑡 + 𝛽3𝑥3𝑡 + 𝛽4𝑥4𝑡 + 𝑢𝑡 (7.2.5)

The conditional variance 𝜎𝑡2, is not logical to be zero by definition, as all the variables of the

equation are squares of lagged errors. The non-negativity condition for ARCH(q) models is that

all coefficients should not be negative, thus: 𝑎𝑖 ≥ 0 ∀ 𝑖 = 0,1,2, . . , 𝑞.

7.2.2. Testing for ARCH Effects

This test is being conducted in order to check whether ARCH effects are present in the residuals

of the estimated model.

The null hypothesis for the test is that the coefficients of all q lags of the squared residuals are

not significantly different than zero.

So, the null and alternative hypothesis are as follows:

𝐻0: 𝑎1 = 0 𝑎𝑛𝑑 𝑎2 = 0 𝑎𝑛𝑑 𝑎3 = 0 𝑎𝑛𝑑 … 𝑎𝑛𝑑 𝑎𝑞 = 0

𝐻1: 𝑎1 ≠ 0 𝑜𝑟 𝑎2 ≠ 0 𝑜𝑟 𝑎3 ≠ 0 𝑜𝑟 … 𝑜𝑟 𝑎𝑞 ≠ 0

34

7.2.3. GARCH Models

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model allows the

conditional variance to be dependent upon previous own lags, so the variance equation has the

following form:

𝜎𝑡2 = 𝑎0 + 𝑎1𝑢𝑡−1

2 + 𝛽1𝜎𝑡−12 (7.2.6)

This is the GARCH(1,1) model. The 𝜎𝑡 is the conditional variance and is the estimate variance

for one period ahead and is calculated based on any past relevant information. Thus, the current

variance 𝜎𝑡 is calculated as the weighted average of the 𝑎0 which is the long-term average value,

the 𝑎1𝑢𝑡−12 , which is the volatility information form previous period and the 𝛽1𝜎𝑡−1

2 , which is

the fitted variance of model based on previous period.

The GARCH(p,q) model, which is the general case of an GARCH model, is described by the

equation below:

𝜎𝑡2 = 𝑎0 + 𝑎1𝑢𝑡−1

2 + 𝑎2𝑢𝑡−22 + 𝑎3𝑢𝑡−3

2 + ⋯ + 𝑎𝑞𝑢𝑡−𝑞2 + 𝛽1𝜎𝑡−1

2

+ 𝛽2𝜎𝑡−22 + 𝛽3𝜎𝑡−3

2 + ⋯ + 𝛽𝑝𝜎𝑡−𝑝2

𝜎𝑡2 = 𝑎0 + ∑ 𝑎𝑖𝑢𝑡−𝑖

2

𝑞

𝑖−1

+ ∑ 𝛽𝑗𝜎𝑡−𝑗2

𝑝

𝑗=1

(7.2.7)

The unconditional variance 𝑢𝑡 is constant and given by the below form for the case that 𝑎1 +

𝛽 < 1 :

𝑣𝑎𝑟(𝑢𝑡) = 𝑎0

1 − (𝑎1 + 𝛽) (7.2.8)

If 𝑎1 + 𝛽 ≥ 1 the unconditional variance is not defined and there would be non-stationarity in

variance.

35

7.2.4. Extensions of the Basic GARCH Models

Many of the GARCH model extensions have been developed to limit the problems of a

standard GARCH(p,q) model.

These problems are the below:

The condition of non-negativity may be violated. In this case the coefficients should

be forced to be non-negative by inserting in the model artificial constraints.

Even though GARCH models can account for volatility clustering and leptokurtosis,

they do not have the ability to account for leverage effects.

GARCH model does not allow for direct feedback between the conditional variance

and conditional mean.

Asymmetric GARCH Models

GARCH models enforce a symmetric behaviour of volatility to positive and negative shocks,

which comes from the fact that the conditional variance does not include any sign of the

lagged residuals (as it is squared). Nevertheless, it is supported that the negative shocks in

financial markets are possible to make volatility increase by more than a positive shock of the

same magnitude. Such asymmetries, in case of equity returns, are refer to the leverage effects.

The “volatility feedback” hypothesis provides a different view, meaning that given dividends

are constant, if the expected returns increase when stock price volatility increases, then stock

prices shall reduce when volatility increases.

The GJR model and the exponential GARCH (EGARCH) models are two well-known

asymmetric GARCH models.

GJR Model

This model was formulated in 1993 by Glosten, Jagannathan and Runkle and this is how its

name occurred. It is also known as the TGRCH (Threshold GARCH) model.

GJR model has an additional term for the possible asymmetries. The conditional variance is:

𝜎𝑡2 = 𝑎0 + 𝑎1𝑢𝑡−1

2 + 𝛽1𝜎𝑡−12 + 𝛾1𝑢𝑡−1

2 𝐼𝑡−1 (7.2.9)

36

Where:

𝐼𝑡−1 = 1 𝑖𝑓 𝑢𝑡−1 < 1 or 𝐼𝑡−1 = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

For a leverage effect: 𝛾 > 0

The non-negativity condition is now: 𝑎0 > 0, 𝑎1 > 0, 𝛽 ≥ 0 𝑎𝑛𝑑 𝑎1 + 𝛾 ≥ 0

EGARH Model

This model was formulated in 1991 by Nelson. The following is one possible form of the

equation:

ln (𝜎𝑡2) = 𝜔 + 𝛽 ln(𝜎𝑡−1

2 ) + 𝛾 𝑢𝑡−1

√𝜎𝑡−12

+ 𝑎 [|𝑢𝑡−1|

√𝜎𝑡−12

− √2

𝜋 ] (7.2.10)

The advantage of this model is that there is no need to make the non-negativity artificially, as

even in case that the parameters are negative, the 𝜎𝑡2 will be positive, as the ln is modelled.

Also, asymmetries can occur under this model, since 𝛾 will be negative if the relationship

between volatility and return is negative.

GARCH In-Mean Model

In finance there is the assumption that investors should be awarded for the risk they take, by

additional return. This could be operationalized if the return of a security is let to be partly

determined by its risk. An example of GARCH-M model is the following:

𝑦𝑡 = 𝜇 + 𝛿𝜎𝑡−1 + 𝑢𝑡 , 𝑢𝑡~𝑁(0, 𝜎𝑡2) (7.2.11)

𝜎𝑡2 = 𝑎0 + 𝑎1𝑢𝑡−1

2 + 𝛽1𝜎𝑡−12 (7.2.12)

If 𝛿 is positive and statistically significant, then additional risk (increase of the conditional

variance), causes the mean return increase and the 𝛿 is considered as risk premium.

37

7.2.5. Multivariate GARCH Models

Multivariate GARCH models are generally the same with the univariate models, with the

addition of the specific equation for the covariances movement over the time. Some of the

multivariate GARCH models which are being used are the VECH model, the diagonal VECH,

the DCC model and the BEEK and diagonal BEKK models. Below the BEKK model which

was used for the thesis is being analyzed.

Diagonal BEKK Model

Following Corbet et al. (2019), Katsiampa (2018, 2019), it is used a simple specification of

the conditional mean equation, as subject study is conducted to mainly examine the

cryptocurrency-index volatility co-movement and therefore their conditional covariance

matrix.

So, the conditional mean equation of the variables price returns is the following:

𝑟𝑡 = 𝜇 + 휀𝑡 (7.2.13)

Where:

𝑟𝑡: the vector of price returns

𝜇: the vector of parameters that estimate the mean of the return series

휀𝑡: the vector of residuals with a conditional covariance matrix 𝐻𝑡 given the available

information set 𝐼𝑡−1.

All three components of the conditional mean equation are (2x1) vectors, as two variables

(one cryptocurrency and one index) are used in each model.

The Diagonal BEKK model is used in this thesis for the conditional covariance matrix 𝐻𝑡 ,

being a special case of the unrestrictive BEKK (Baba, Engle, Kraft, Kroner) model. One of its

advantages over the BEKK model is that the number of estimated parameters is significantly

reduced, while it still maintains the positive definiteness of 𝐻𝑡 .

The covariance matrix is given 𝐻𝑡 = [ℎ11 ℎ12

ℎ21 ℎ22]

38

The conditional covariance matrix of the Diagonal BEKK model is given as:

𝐻𝑡 = 𝐶′𝐶 + 𝐴′휀′휀𝛢 + 𝐵′𝐻𝑡−1𝐵

𝐻𝑡 = [𝑐11 𝑐12

0 𝑐22] ′ [

𝑐11 𝑐12

0 𝑐22] + [

𝑎11 𝑎12

𝑎21 𝑎22] ′ [

휀1,𝑡−1

휀2,𝑡−1] ′ [

휀1,𝑡−1

휀2,𝑡−1] [

𝑎11 𝑎12

𝑎21 𝑎22]

+ [𝛽11 𝛽12

𝛽21 𝛽22] ′ [

ℎ11,𝑡−1 ℎ12,𝑡−1

ℎ21,𝑡−1 ℎ22,𝑡−1] [

𝛽11 𝛽12

𝛽21 𝛽22]

(7.2.14)

Where:

C: the parameter matrix

A: the coefficient matrix of ARCH effect

B: the coefficient matrix of GARCH effect

The matrix A examines the ARCH effects from past return to current conditional variances,

while the matrix B examines the GARCH effect from past conditional variance to current

conditional variances. Both matrixes A and B are assumed to be diagonal, thus all the off-

diagonal elements are zero.

The values of the conditional variances depend on their past values and their past squared

residuals, while the values of the conditional covariances are influenced by their past values

and the past cross-product residuals.

The error distribution is selected to be multivariate normal.

The log-likelihood function of the model is:

𝑙(𝜃) = −𝑇𝑁

2log 2𝜋 −

1

2 ∑(𝑙𝑛|𝐻𝑡| + 휀𝑡

′𝛨𝑡−1휀𝑡)

𝛵

𝑡=1

(7.2.15)

Where:

𝜃: all the unknown parameters

𝑁: the number of variables

𝑇: the number of observations

39

8. Data Description

Cryptocurrencies are being used more and more frequent with their popularity to be increased

dramatically. Due to this increasing interest of cryptocurrencies, the quantification of their

variation is of high importance and need.

It is known that cryptocurrencies are highly volatile compared to traditional currencies.

As for the traditional currencies, the most known models which have been used for the

exchange rates are maybe based on Generalized Autoregressive Conditional Heteroskedasticity

(GARCH) models. Nevertheless, there is not much work on the GARCH-type models fitting to

the exchange rates of cryptocurrencies.

GARCH modelling of Bitcoin, the first and the most popular cryptocurrency, has previously

taken place. However, in the subject thesis, the modelling of more cryptocurrencies will be

attempted (Jeffrey Chu, Stephen Chan, Saralees Nadarajah, Joerg Osterrieder, 2017).

For the empirical approach, all the available cryptocurrencies are explored and narrowed down

to the following four cryptocurrencies, which are included in the twenty largest

cryptocurrencies in terms of Market Capitalization (as of 16 August 2021):

Bitcoin

Ethereum

Cardano

Litecoin

Above cryptocurrencies will be studied in relation to the following indices:

S&P 500

Dow Jones

Gold Price

Crude Oil Price WTI

Dow Jones Conventional Electricity

Dow Jones Real Estate

Baltic Dry index (BDI)

Barclays US Aggregate Bond Index

S&P Goldman Sachs Commodity

40

8.1. Cryptocurrencies Description

Below is the basic information related to Cryptocurrencies Structure as well as Market

Capitalization.

8.1.1. Bitcoin

As already stated in the theoretical part of this thesis, Bitcoin was introduced by Satoshi

Nakamoto in 2008 and he created the Bitcoin network in 2009. It is a decentralized electronic

currency and is based on the “Proof to Work” mechanism, that solves the double-spend issue.

The anonymity is the basic characteristic of Bitcoin, as users do not need to reveal their identity

while they are using it for transactions. Under this system, each miner has an individual ledger

for all the transactions, and they obtain a consensus on the state of transactions every 10 minutes

and then they update their ledgers (Kwok Ping Tsang, Zichao Yang, 2021).

Bitcoin is the first generation blockchain protocol.

The following figure represents the Bitcoin (BTC) cryptocurrency:

Figure 5: BTC Cryptocurrency (Neeraj Kumar; Shubhani Aggarwal, 2021)

Bitcoin is currently the most popular and the first cryptocurrency in terms of Market

Capitalization.

The price as well as the Market Cap historical charts of Bitcoin are the following:

41

Figure 6: Bitcoin Price Historical Chart (CoinMarketCap, 2021)

Figure 7: Bitcoin Market Cap Historical Chart (CoinMarketCap, 2021)

As of 16 August 2021, Bitcoin Market Cap is the 42.51% of the Total Cryptocurrency Market

Cap.

42

8.1.2. Ethereum

Ethereum is also a very popular and leading cryptocurrency on the market, which went live in

July 2015.

Ethereum main differences in relation to the Bitcoin are the following:

While Bitcoin included only transaction information in a block, Ethereum additionally includes

user identity and product details. Also, in the Ethereum technology, many new blocks can be

simultaneously produced in the Blockchain system. Moreover, the “gas” concept is introduced

by Ethereum, which is the running cost which is needed to insert various types of information

in a block and could be purchased with Ethereum. Subsequently, information can be inserted

in the Ethereum block with the use of this purchased gas, which is limited for each block. Thus,

the block size cannot be increased with the same was with a Bitcoin block. Last but not least,

Ethereum block size is small and cannot handle more that 30kB of information, so the average

block generation time is also smaller that the one of Bitcoin. Ethereum average block-

generation time is approximately 15 minutes per block, while relevant Bitcoin time is 10

minutes per block (Han-Min Kim, Gee-Woo Bock, Gunwoong Lee, 2021).

Shubhani A. and Neeraj K. (2021, p.227-266) state that “Ethereum is a decentralized software

platform that enables smart contracts and decentralized applications to be built and run without

any downtime, fraud, control, or central authority” (Neeraj Kumar; Shubhani Aggarwal, 2021).

Ethereum is the second generation blockchain protocol.

The following figure represents the Ethereum (ETH) cryptocurrency:

Figure 8: ETH Cryptocurrency (Neeraj Kumar; Shubhani Aggarwal, 2021)

Ethereum is currently the second cryptocurrency in terms of Market Capitalization.

43

The price as well as the Market Cap historical charts of Ethereum are the following:

Figure 9: Ethereum Price Historical Chart (CoinMarketCap, 2021)

Figure 10: Ethereum market Cap Historical Chart (CoinMarketCap, 2021)

As of 16 August 2021, Ethereum Market Cap is the 18.18% of the Total Cryptocurrency Market

Cap.

44

8.1.3. Cardano

Cardano is a public blockchain-based decentralized platform, based on a research-first driven

approach, that is used to develop smart contracts to deliver more advanced elements and also

to send and receive digital funds. Through cryptography, the digital currency can be used for

secure and quick payments (Neeraj Kumar; Shubhani Aggarwal, 2021).

The following figure represents the Cardano (ADA) cryptocurrency:

Figure 11: ADA Cryptocurrency (Neeraj Kumar; Shubhani Aggarwal, 2021)

Cardano is currently the third cryptocurrency in terms of Market Capitalization.

The price as well as the Market Cap historical charts of Cardano are the following:

Figure 12: Cardano Price Historical Chart (CoinMarketCap, 2021)

45

Figure 13: Cardano Market Cap Historical Chart (CoinMarketCap, 2021)

As of 16 August 2021, Cardano Market Cap is the 3.29% of the Total Cryptocurrency Market

Cap.

46

8.1.4. Litecoin

Litecoin was created in 2011 by former Google employee Charlie Lee and is considered as an

alternative Bitcoin. Lee wanted to decrease the required time for a new transaction confirmation

and modify the currency mining procedure to make sure that anyone could participate. Lee said

"My view is that people would use litecoin every day to buy things. It would be only the chosen

payment method" (William Aparecido Maciel da Silva, Nicolle Caroline Brasil Martins, Ingrid

de Andrade Miranda, Ingrid de Andrade Miranda, Donizete Reina, 2020).

The following figure represents the Litecoin (LTC) cryptocurrency:

Figure 14: LTC Cryptocurrency (Neeraj Kumar; Shubhani Aggarwal, 2021)

The price as well as the Market Cap historical charts of Litecoin are the following:

Figure 15: Litecoin Price Historical Chart (CoinMarketCap, 2021)

47

Figure 16:Litecoin Market Cap Historical Chart (CoinMarketCap, 2021)

As of 16 August 2021, Litecoin Market Cap is the 0.59% of the Total Cryptocurrency Market

Cap.

The following graph depicts the percentage of each cryptocurrency Market Capitalization of

the total Cryptocurrency Market Capitalization.

Figure 17:Total Market Cap of All Existing Cryptocurrencies (CoinMarketCap, 2021)

The total Market Capitalization chart of all existing Cryptocurrencies can be found below:

48

Chart 2: Market Cap Percentage of the Studied Cryptocurrencies (Author’s Graph with data collected from (CoinMarketCap, 2021))

The table below includes the relevant detailed data of the previous chart:

Table 3: Market Cap Percentage of the Studied Cryptocurrencies (CoinMarketCap, 2021)

Cryptocurrency Market Cap

(as of 16/08/2021) Percentage of Total Market Cap

(as of 16/08/2021)

Bitcoin 864,345,726,183 USD 42.51%

Ethereum 369,734,989,544 USD 18.18%

Cardano 66,807,041,752 USD 3.29%

Litecoin 11,939,005,327 USD 0.59%

Other 720,442,110,424 35.43%

Total 2,033,268,873,230 USD 100.00%

Bitcoin

Ethereum

Cardano Litecoin

Other

MARKET CAP (AS OF 16/08/2021)

49

8.2. Indices Description

Moreover, a few details for the selected indices are presented below:

8.2.1. S&P 500

The S&P 500 (or Standard & Poor's 500) Index is a market-capitalization-weighted index of

the 500 largest U.S. publicly traded companies. It is a float-weighted index, that means that the

number of shares which are available for public trading are used for the adjustment of company

market capitalizations.

This index is commonly accepted as the best measurement of large-cap U.S. equities, and this

is the reason why a lot of funds have been created to track the performance of S&P 500.

Figure 18: S&P 500 Price Historical Chart (Investing.com, 2021)

8.2.2. Dow Jones Industrial Average

Dow Jones Industrial Average is a market average created by Dow Jones (or more specifically

Dow Jones & Company), which is one of the world’s largest business and financial news

company. This company was founded in the 19th century by Charles Dow, Edward Jones, and

Charles Bergstresser.

The Dow Jones Industrial Average assists investors understand the overall direction of stock

prices. This index groups together the prices of 30 of the most traded stocks on the New York

Stock Exchange (NYSE) and the Nasdaq.

50

Figure 19: Dow Jones Price Historical Chart (Investing.com, 2021)

8.2.3. Gold Futures

Gold Futures are contracts that are traded on exchanges and a buyer agrees to purchase a

specific quantity of the commodity at a specific price at a date in the future.

Investors positions could be short or long on these future contracts.

Figure 20: Gold Futures Price Historical Chart (Investing.com, 2021)

8.2.4. Crude Oil WTI

West Texas Intermediate (WTI) crude oil id ne of the three main benchmarks in oil pricing,

along with Brent and Dubai Crude. WTI is a specific grade of crude oil and is known as light

sweet oil as it contains approximately 0.34% sulphur that makes it “sweet” and its low density

makes it “light”. It comes mainly from Texas and it is elivered in the Midwest and Gulf of

Mexico via pipelines.

WTI is the underlying commodity of the New York Mercantile Exchange's (NYMEX) oil

futures contract and is considered a high-quality oil that is easily refined.

51

It is important, as it constitutes a price reference for the sellers and buyers of crude oil.

Figure 21: Crude Oil WTI Price Historical Chart (Investing.com, 2021)

8.2.5. Dow Jones Conventional Electricity

Dow Jones Conventional Electricity Index price history is presented on the following graph:

Figure 22: Dow Jones Conventional Electricity Price Historical Chart (Investing.com, 2021)

8.2.6. Dow Jones Real Estate

Dow Jones Real Estate index has been created in order to track the performance of Real Estate

Investment Trusts (REIT) and other companies that make direct real estate investments or

indirect real estate investments through development, management or ownership, including

property agency.

It is a representative of the Real Estate Supersector and it is a float market cap weighted.

52

Figure 23: Dow Jones Real Estate Price Historical Chart (Investing.com, 2021)

8.2.7. Baltic Dry index (BDI)

Baltic Dry Index provides a benchmark for the price of shipping of major raw materials by sea.

This index consists of three sub-indices that are used to measure different sizes of bulk carriers:

Capesize (typically iron or coal cargo transportation of approximately 150,000 tonnes),

Panamax (typically grain or coal cargo transportation of approximately 60,000 - 70,000 tonnes)

and Supramax (cargo transportation of approximately 48,000 - 60,000 tonnes). Twenty-three

shipping routes are taken into consideration for this index, carrying coals, iron ore, grains, and

other commodities. The Baltic Dry Index is daily reported by Baltic Exchange in London.

Figure 24: Baltic Dry Index Price Historical Chart (Trading Economics, 2021)

53

8.2.8. Barclays US Aggregate Bond Index

The Barclays US Aggregate Bond Index is a broad-based fixed-income index which is used as

a benchmark to measure the relative performance.

This index tracks the performance of U.S. investment-grade bond market and includes

government Treasury securities, corporate bonds, mortgage-backed securities (MBS) and asset-

backed securities (ABS), to simulate the existing bonds in the market.

In this thesis, the iShares Barclays Aggregate Bond ETF price is used, as this is the ETF that

mirrors the Barclays US Aggregate Bond Index, and this is the one that investors who want to

invest in securities look.

Figure 25: Barclays US Aggregate Bond Index Price Historical Chart (Investing.com, 2021)

8.2.9. S&P Goldman Sachs Commodity

The S&P Goldman Sachs Commodity index is a composite index of commodities and tracks

the performance of the commodities market. Subject index is often used as a benchmark for

commodities investments. It is weighted by world production and includes the physical

commodities that have active, liquid future markets.

Prior to the purchase if the index from Standard & Poor’s, it was called just Goldman Sachs

Commodity Index.

Any commodity that satisfies some eligibility criteria and other conditions, can be included in

this index, with no limitation to the number of commodities.

54

Figure 26: S&P Goldman Sachs Commodity Price Historical Chart

8.3. Sample Data Frequency

For the OLS study among cryptocurrencies, daily historical data will be used. Subject price

history has been exported from the Investing.com website (Investing.com, 2021). The daily

closing prices of each cryptocurrency are gathered in order to be used in the analysis. The

historical data range to be used has been selected to be from 31 December 2017 to 16 August

2021. The starting date has been moved to 31 December 2017 as this is the Cardano limited

trading history.

For the GARCH modelling, it is important to mention that couple of cryptocurrencies and

indices will be studied. For the examination of each cryptocurrency with all the indices, the

dataset is determined by the cryptocurrency price data availability. In this regard for the dataset

of Bitcoin and indices, the studied period is 18 July 2010 – 15 August 2021, for the Ethereum

and indices is 02 August 2015 – 15 August 2021, for the Litecoin and indices is 14 September

2014 – 15 August 2021 and finally for the Cardano and indices is 24 September 2017 – 15

August 2021.

Since the cryptocurrency price data is available for 7 days per week and the indices data just

for 5 days per week, in this part of the study weekly prices are selected to be used.

The total number of observations is for 578 Bitcoin, 315 for Ethereum, 361 for Litecoin and

203 for Cardano.

All the price data for both cryptocurrencies and indices are listed in US dollars.

55

9. Analysis Results

9.1. OLS Regression Between Bitcoin and Selected Cryptocurrencies

At the first part of this thesis, the relationship between the Bitcoin returns and the returns of the

rest selected cryptocurrencies (Bitcoin, Ethereum, Cardano, Litecoin) will be studied.

The final sample is the aforementioned four cryptocurrencies which belong among to the 20

cryptocurrencies with the highest market capitalization and is shown in Equation below.

𝐵𝑇𝐶 = 𝑎 + 𝛽1𝐸𝑇𝐻 + 𝛽2𝐴𝐷𝐴 + 𝛽3𝐿𝑇𝐶 + 휀 (9.1.1)

Where:

𝑎 ∶ is the intercept parameter

𝛽: is the slope of the control variables

𝐵𝑇𝐶: Bitcoin return

𝐸𝑇𝐻: Ethereum return

𝐴𝐷𝐴: Cardano return

𝐿𝑇𝐶: Litecoin return

휀: disturbance (error or residue)

The returns of the selected cryptocurrencies are calculated as the natural logarithm of the

arithmetic return.

Based on the above, the calculation for the return of cryptocurrencies is as follows:

𝑅𝑖 = ln (

𝑃𝑡

𝑃𝑡−1) = 𝑙𝑛(𝑃𝑡) − 𝑙𝑛(𝑃𝑡−1) (9.1.2)

Where:

𝑅𝑖: the cryptocurrencies return

𝑃𝑡: number of daily closing cryptocurrencies in period 𝑡

𝑃𝑡−1: number of daily closing cryptocurrencies in period 𝑡 − 1

56

9.1.1. Data Presentation and Descriptive Statistics

First, the price and natural logarithmic returns figures of all cryptocurrencies are presented

below:

Figure 27: Bitcoin Price Time Series

Figure 28: Ethereum Price Time Series

57

Figure 29: Cardano Price Time Series

Figure 30: Litecoin Price Time Series

58

Figure 31: Studied Cryptocurrencies Returns Price Series

Moreover, the descriptive statistical analysis of test variables selected was performed and the

results are reported on the following Table.

Table 4: Descriptive Statistics of Cryptocurrency Returns

Cryptocurrency Returns Mean Median Standard Deviation Min Max

BTC 0.0009065 0.001304 0.04159 -0.4973 0.1774

ADA 0.0008082 0.00 0.07981 -0.6931 0.4055

ETH 0.001097 0.001742 0.05433 -0.5896 0.2308

LTC -0.000184 -0.001142 0.05702 -0.4867 0.2864

Note: This table presents the descriptive statistics for the daily return series of BTC: Bitcoin,

ETH: Ethereum, ADA: Cardano, LTC: Litecoin for the sample 31 December 2017 to 16 August

2021.

As it is reported, all the average returns of the variables are close to zero, which means that the

basic assumption of finance is met. It is observed that the standard deviation is lower for the

BTC and higher for the ADA. Standard deviation is a total risk measure, so the ADA seems to

be more “risky” according to our analysis. BTC presents a low risk, based on the sample size

of subject study.

59

The highest average return is observed for the cryptocurrency ETH, while its standard deviation

is not the greatest one.

The lowest average return is observed for the cryptocurrency LTC, where the average return is

negative for the studied sample size.

The cryptocurrency with the most significant maximum return value is the ADA, while the

cryptocurrency with the lowest minimum value is the same. It seems that ADA cryptocurrency

has the highest volatility among the other studied cryptocurrencies.

In continuation to the above descriptive analysis of selected cryptocurrencies, analysis is moved

on the correlation test performance between the variables, as shown in the Table below:

Table 5: Correlation Matrix among Studied Cryptocurrencies

BTC ADA ETH LTC

BTC 1

ADA 0.5702 1

ETH 0.8229 0.6357 1

LTC 0.7978 0.5866 0.825 1

Note: This table presents the correlation Matrix for the daily return series of BTC: Bitcoin,

ETH: Ethereum, ADA: Cardano, LTC: Litecoin for the sample 31 December 2017 to 16 August

2021.

It can be noticed that the return of the logarithm of cryptocurrencies exhibit relatively high

correlation (all are above 0.50), which may be a signal of multicollinearity between the

variables.

The highest correlation is met among LTC and ETH, while the lowest correlation among the

BTC and ADA.

Among the explanatory variables, the lowest correlation was between LTC and ADA variables.

This may demonstrate that low correlation is possible to stem from high speculation that exists

in the market cryptocurrencies, as outlined by Ciaian et al. (2016).

60

9.1.2. Unit Root Test

The first step prior to the start of the OLS regression, is to check that all variables that are going

to be used in the regression are stationary.

In this regard, the Augmented Dickey- Fuller test was conducted for all the dependent and

independent variables.

Table 6: ADF test for Variables

Coefficient Standard error t-ratio p-value

BTC -1.08831 0.0274035 -39.7100 1.22E-11 ***

LTC -1.09427 0.0273952 -39.9400 5.17E-11 ***

ADA -1.1701 0.0271162 -43.1400 1.00E-04 ***

ETH -1.09245 0.0274036 -39.8700 3.16E-11 ***

Note: This table presents the results of the t-statistics of Augmented Dickey-Fuller (ADF) test

of the daily return series of BTC: Bitcoin, ETH: Ethereum, ADA: Cardano, LTC: Litecoin for

the sample 31 December 2017 to 16 August 2021.

∗∗∗ Indicates statistical significance at the 1% level.

∗∗ Indicates statistical significance at the 5% level.

∗ Indicates statistical significance at the 10% level.

The *** in the table indicates the rejection of null hypothesis of the 1% significance level.

As it is concluded from the table above, the Augmented Dickey-Fuller tests on variables, has

shown that all variables are stationary.

The stationarity of the returns is what was expected, as returns should be stationary, while prices

shall be non-stationary.

61

9.1.3. OLS Regression

In the OLS regression, the Bitcoin returns was the dependent variable, while the Ethereum,

Litecoin and Cardano returns were the independent variables.

The OLS regression results are included in the Table below:

Table 7: OLS Regression results

Coefficient Standard Error t-ratio p-value

Constant 0.000521685 0.000603309 0.8647 0.3874

LTC 0.265859 0.0189044 14.0600 5.80E-42 ***

ADA 0.0213442 0.0098897 2.1580 0.0311 **

ETH 0.379764 0.0208188 18.2400 1.78E-66 ***

Mean Dependent Variable 0.000907 S.D. Dependent Variable 0.041587

Sum Suared Residuals 0.000907 S.E.of regression 0.021934

R-squared 0.635077 Adjusted R-squared 0.721810

F(3, 1320) 1145.250 P-value (F) 0.000000

Log-Likelihood 3180.609 Akaike criterion -6353.218

Schwarz criterion -6332.464 Hannan-Quinn -6345.438

Rho 0.014424 Durbin-Watson 1.969057

Note: This table presents the results of the Ordinary Least Squares (OLS) Regression of the

equation: 𝐵𝑇𝐶 = 𝑎 + 𝛽1𝐸𝑇𝐻 + 𝛽2𝐴𝐷𝐴 + 𝛽3𝐿𝑇𝐶 + 휀 , which includes the daily return

series of BTC: Bitcoin as the dependent variable and the daily returns of ETH: Ethereum, ADA:

Cardano and LTC: Litecoin as the independent variables for the sample period from 31

December 2017 to 16 August 2021.




From this table is visible that all the independent variable coefficients are statistically

significant.

Also, all coefficients are positive, which shows the positive relationship between the Bitcoin

and other cryptocurrencies returns. In other words, when Litecoin, Ethereum and Cardano

returns are increasing, the Bitcoin return is also increasing. The opposite happens when Litecoin,

Ethereum and Cardano returns are reducing, the Bitcoin return is also reducing.

So, there is positive relationship between the studied cryptocurrencies returns.

62

9.1.4. OLS Regression Tests

The following test were conducted to check that the OLS regression hypothesis are violated or

not.

o Heteroskedasticity Test – White Test

o Autocorrelation Test

o Normality of Residuals

o CUSUM Test

o ARCH Test

The results are shown below:

o Heteroskedasticity Test – White Test

Table 8: White Heteroskedasticity Test

coefficient std error t-ratio p-value

Constant 0.000248088 2.97E+00 8,357 1.61E-16 ***

LTC 0.000668163 0.000836282 0.799 0.4245

ADA -0.0016762 0.000447812 -3.743 0.0002 ***

ETH 0.00400327 0.000940047 4,259 2.20E-05 ***

sq_LTC 0.0509907 0.00719747 7,085 2.27E-12 ***

X2_X3 -0.00615439 0.011329 -0.5432 0.5871

X2_X4 -0.0886929 0.019062 -4.653 3.60E-06 ***

sq_ADA 0.00222322 0.00303148 0.7334 0.4635

X3_X4 -0.0260159 0.0120396 -2.161 0.0309 **

sq_ETH 0.12317 0.0142121 8,667 1.29E-17 ***

R-squared 0.32636

Alternative Statistics: TR^2= 432.100494

With p-value = P(chi-square>432.100494) = 0

Note: This table presents the results of the White Test, which is an Heteroskedasticity test. This

test is conducted on the OLS regression with the daily return series of BTC: Bitcoin as the

dependent variable and the daily returns of ETH: Ethereum, ADA: Cardano and LTC: Litecoin

as the independent variables for the sample period from 31 December 2017 to 16 August 2021.




63

The null hypothesis for the Heteroskedasticity Test is that there Homoskedasticity.

Here the P value is zero, which means that the null hypothesis is rejected, so the alternative

hypothesis is valid. Thus, the model has heteroscedasticity.

In order for the heteroskedasticity to be correct, the robust standard errors choice is selected.

The OLS regression with heteroskedasticity corrected is as follows:

Table 9: OLS with Heteroskedasticity Corrected

Coefficient Standard Error t-ratio p-value

Constant 0.000521685 0.000611378 0.8533 0.3937

LTC 0.265859 0.0388847 6,837 1.23E-11 ***

ADA 0.0213442 0.0117254 1,820 0.0689 *

ETH 0.379764 0.0468576 8,105 1.20E-15 ***

Mean Dependent Variable 0.000907 S.D. Dependent Variable 0.041587 Sum Suared Residuals 0.635077 S.E.of regression 0.021934 R-squared 0.722441 Adjusted R-squared 0.721810 F(3, 1320) 210.374700 P-value (F) 0.000000 Log-Likelihood 3180.609000 Akaike criterion -6353.218000 Schwarz criterion -6332.464000 Hannan-Quinn -6345.438000 Rho 0.014424 Durbin-Watson 1.969057

Note: This table presents the results of the Ordinary Least Squares (OLS) Regression of the

equation: 𝐵𝑇𝐶 = 𝑎 + 𝛽1𝐸𝑇𝐻 + 𝛽2𝐴𝐷𝐴 + 𝛽3𝐿𝑇𝐶 + 휀 , which includes the daily return

series of BTC: Bitcoin as the dependent variable and the daily returns of ETH: Ethereum, ADA:

Cardano and LTC: Litecoin as the independent variables for the sample period from 31

December 2017 to 16 August 2021. Robust standard errors have been selected.




o Autocorrelation Test

The Breusch-Godfrey Test was conducted in order to check the existence of autocorrelation.

The number of lags that were used were 60, meaning two months, as the cryptocurrencies trade

seven days per week.

64

Table 10: Breusch-Godfrey Autocorrelation Test

coefficient std error t-ratio p-value

Const 3.82E-01 0.00059968 0.006363 0.9949

d_l_LITECOIN 0.00330329 0.0192937 0.1712 0.8641

d_l_CARDANO 0.00282949 0.0100979 0.2802 0.7794

d_l_ETHEREUM -0.00588603 0.0214741 -0.2741 0.7841

uhat_1 0.0140084 0.0282977 0.495 0.6207

uhat_2 0.0349534 0.0282384 1,238 0.216

uhat_3 -0.0110548 0.0282659 -0.3911 0.6958

uhat_4 -0.075246 0.0282248 -2.666 0.0078 ***

uhat_5 0.0167469 0.028367 0.5904 0.5551

uhat_6 0.0758069 0.0282757 2,681 0.0074 ***

uhat_7 0.0348426 0.028369 1,228 0.2196

uhat_8 -0.0167435 0.0283757 -0.5901 0.5553

uhat_9 -0.0140109 0.0283961 -0.4934 0.6218

uhat_10 0.0746025 0.0283744 2,629 0.0087 ***

uhat_11 -0.0122663 0.0284952 -0.4305 0.6669

uhat_12 0.0237071 0.028465 0.8329 0.4051

uhat_13 0.0154353 0.028417 0.5432 0.5871

uhat_14 0.0171486 0.0284073 0.6037 0.5462

uhat_15 -0.0151064 0.0284643 -0.5307 0.5957

uhat_16 -0.0342073 0.0285254 -1.199 0.2307

uhat_17 0.0145911 0.0284687 0.5125 0.6084

uhat_18 -0.0250876 0.0284503 -0.8818 0.3781

uhat_19 0.0285957 0.028519 1,003 0.3162

uhat_20 0.0430241 0.0284627 1,512 0.1309

uhat_21 0.0174657 0.0285143 0.6125 0.5403

uhat_22 0.0272057 0.0285537 0.9528 0.3409

uhat_23 -0.0121657 0.0285693 -0.4258 0.6703

uhat_24 -0.00491768 0.0285935 -0.172 0.8635

uhat_25 0.0521482 0.0286186 1,822 0.0687 *

uhat_26 -0.0357975 0.0285634 -1.253 0.2103

uhat_27 0.0143195 0.0286128 0.5005 0.6168

uhat_28 0.0450534 0.0285839 1,576 0.1152

uhat_29 0.0279802 0.0285858 0.9788 0.3279

uhat_30 -0.0559828 0.028642 -1.955 0.0509 *

uhat_31 -0.0162949 0.0286296 -0.5692 0.5693

uhat_32 -0.0417843 0.0286482 -1.459 0.1449

uhat_33 0.0346219 0.0285949 1,211 0.2262

uhat_34 -0.00737883 0.0286298 -0.2577 0.7967

uhat_35 -0.0241158 0.0286193 -0.8426 0.3996

uhat_36 -0.0381268 0.0285954 -1.333 0.1827

uhat_37 0.0288217 0.0285794 1,008 0.3134

uhat_38 0.0100429 0.0286523 0.3505 0.726

uhat_39 -0.047875 0.028592 -1.674 0.0943 *

uhat_40 0.0258588 0.0286265 0.9033 0.3665

65

uhat_41 -0.00506599 0.0286434 -0.1769 0.8596

uhat_42 0.0162191 0.0286335 0.5664 0.5712

uhat_43 0.0305576 0.0285896 1,069 0.2853

uhat_44 0.0761691 0.0286311 2,660 0.0079 ***

uhat_45 -0.00225513 0.028667 -0.07867 0.9373

uhat_46 -0.00102807 0.0286994 -0.03582 0.9714

uhat_47 -0.0180032 0.0286669 -0.628 0.5301

uhat_48 0.0171356 0.0287524 0.596 0.5513

uhat_49 -0.0724893 0.0287592 -2.521 0.0118 **

uhat_50 -0.011937 0.0288251 -0.4141 0.6789

uhat_51 0.0216541 0.0287557 0.753 0.4516

uhat_52 0.00519358 0.0288474 0.18 0.8572

uhat_53 -0.00123529 0.0288251 -0.04285 0.9658

uhat_54 -0.0155548 0.0287585 -0.5409 0.5887

uhat_55 0.0243718 0.0286954 0.8493 0.3959

uhat_56 0.00688813 0.0287866 0.2393 0.8109

uhat_57 0.0265415 0.0286233 0.9273 0.354

uhat_58 0.0143989 0.02865 0.5026 0.6153

uhat_59 -0.0110509 0.0287225 -0.3847 0.7005

uhat_60 -0.00663468 0.0286821 -0.2313 0.8171

Unadjusted R-squared 0.057187

Test Statistics: LMF= 1,273,767

With p-value = P(F(60,1260)> 1.27377) = 0.0809

Alternative Statistics: TR^2= 75,715,440

With p-value = P(chi-square>75.7154) = 0.083

Ljung-Box Q' = 764,773

With p-value = P(chi-square> 76.4773) = 0.0743

Note: This table presents the results of the Breusch-Godfrey Test, which is an Autocorrelation

test. This test is conducted on the OLS regression with the daily return series of BTC: Bitcoin

as the dependent variable and the daily returns of ETH: Ethereum, ADA: Cardano and LTC:

Litecoin as the independent variables for the sample period from 31 December 2017 to 16

August 2021.




The null hypothesis for this test is that there is no autocorrelation, meaning that the current error

is not related any of its previous values. The null hypothesis is valid, as the P value is greater

than 5%, so there is no autocorrelation in our model.

66

o Normality of Residuals

The normality of residuals test gave the following figure:

Figure 32: Normality of Residuals Test

The p-value of the normality of residuals test is equal to zero, so the null hypothesis of the

normality of residuals is being rejected.

0

5

10

15

20

25

30

-0.1 -0.05 0 0.05 0.1

Density

residual

relative frequency

N(7.0843e-18,0.021934)Test statistic for normality:

Chi-square(2) = 355.996 [0.0000]

67

o CUSUM Test

The CUSUM test, checks the parameter stability and constancy.

Figure 33: CUSUM Test

o ARCH Test

The ARCH test is a white noise test for squared time series, so it is the investigation of a higher

order (non-linear) of autocorrelation.

A time series exhibiting conditional heteroskedasticity (or correlation in the squared series) is

said to have autoregressive conditional heteroskedastic (ARCH) effects.

The ARCH test with 60 lags used, resulted to P-value of zero (0).

The null hypothesis is that there is no ARCH effect, while the alternative that there is ARCH

effect.

Here it is concluded that the null hypothesis is rejected, so there exist ARCH effects.

-150

-100

-50

0

50

100

150

0 200 400 600 800 1000 1200 1400

Observation

CUSUM plot with 95% confidence band

68

9.1.5. Engle-Granger Cointegration Test

At this stage, the Engle-Granger cointegration test among the cryptocurrencies has been

conducted. As the Engle-Granger test is being performed between two variables, pair of all

studied cryptocurrencies are created, for the test. Also, the cryptocurrency prices are used for

the performance of this test, instead of the return, as the non-stationarity of variables are

necessary.

The Engle-Granger cointegration test has the following process. Firstly, the test of the variables

for stationarity is being conducted, secondly, the regression is run where the residuals are stored

and finally the test of the residuals for stationarity is done, which is determined by ADF tests

on the residuals.

In order for evidence of cointegration relationship to be found, the unit root hypothesis should

not be rejected for the individual variable tests, however the unit root hypothesis for the

residuals should be rejected – both should apply.

In other words, the P-values of the individual variables tests as well as of the residuals should

be checked. The null hypothesis is that the series include unit-root, so they are non-stationarity

and not cointegrated.

In subject analysis, the P-values of the individual variables as well as for the residuals is shown

on the following Table.

Table 11: Augmented Dickey-Fuller (ADF) test on studied Cryptocurrency Pairs

ADF-(1) ADF-(2) ADF residuals

Bitcoin (1) -Litecoin (2) 0.9656 0.1561 0.4568

Bitcoin (1) -Ethereum (2) 0.9656 0.9941 0.6306

Bitcoin (1) -Cardano (2) 0.9656 0.9952 0.3131

Ethereum (1)-Litecoin (2) 0.9941 0.1561 0.8839

Ethereum (1) -Cardano (2) 0.9941 0.9952 7.21E-03***

Cardano (1) -Litecoin (2) 0.9952 0.1561 0.9087

Note: This table presents the results of the p values of t-statistics of Augmented Dickey-Fuller

(ADF) test of the daily return series of cryptocurrencies and the pairs for the sample 31

December 2017 to 16 August 2021.




The *** in the table indicates the rejection of null hypothesis of the 1% significance level.

69

It is observed that all the P-values of all individual variables are greater than 0.05, so the null

hypothesis is confirmed – it is not rejected, so individual variables are not stationary.

The Augmented Dickey Fuller test for the residuals of almost all studied pairs, estimates P-

values greater than 0.05, so the null hypothesis is confirmed – it is not rejected, so residuals are

not stationary.

For the Ethereum-Cardano pair, the Augmented Dickey Fuller test for the residuals estimates

P-values lower than 0.05, so the null hypothesis is not confirmed – it is rejected, so residuals

are stationary.

Taking into consideration that one of the two aforementioned criteria are not met except for the

case of Ethereum – Cardano pair, all the series/ pairs except for the Ethereum – Cardano, are

not cointegrated.

The Ethereum-Carano pair meets both criteria for cointegrated series, so it is cointegrated.

9.2. Diagonal BEKK Model for Selected Cryptocurrencies and Indices

In the second part of the analysis, the Diagonal BEKK model is used, in order to examine the

volatility dynamics of the four cryptocurrencies namely Bitcoin, Ethereum, Litecoin and

Cardano in relation to the nine indices namely S&P 500, Dow Jones, Gold Price, Crude Oil

Price WTI, Dow Jones Conventional Electricity, Dow Jones Real Estate, Baltic Dry index

(BDI), Barclays US Aggregate Bond Index, S&P Goldman Sachs Commodity Index.

The weekly price returns of the used variables are defined as follows:

𝑅𝑖,𝑡 = ln(𝑃𝑖,𝑡) − 𝑙𝑛(𝑃𝑖,𝑡−1) (8.2.1)

Where:

𝑅𝑖,𝑡: the weekly return of variable i on week t

ln(𝑃𝑖,𝑡): the logarithm of the variable i on week t

𝑙𝑛(𝑃𝑖,𝑡−1): the logarithm of the variable i on week t-1

70

9.2.1. Diagonal BEKK Model for Bitcoin and Indices

The empirical analysis starts with generating the descriptive statistics of the price returns of

Bitcoin and indices.

Table 12: Bitcoin & Indices Descriptive Statistics

AGG BDI BITCOIN CRUDE OIL WTI

DOW JONES

CONV.

ELECTRICITY

DOW

JONES

REAL ESTATE

DOW JONES GOLD S&P 500

S&P

GOLDM

AN SACHS

Mean 0.000135 0.001269 0.022664 -0.000283 0.001313 0.001299 0.002094 0.000709 0.002394 -7.28E-05

Median 0.000636 0.001340 0.010681 0.002322 0.002268 0.003678 0.003234 0.001723 0.003561 0.001537

Maximum 0.049123 0.470774 0.822906 0.275756 0.158057 0.204030 0.120840 0.112555 0.114237 0.080997

Minimum -0.052249 -0.335448 -0.715620 -0.346863 -0.182883 -0.283845 -0.189978 -0.101316 -0.162279 -0.145503

Std. Dev. 0.005562 0.096237 0.159956 0.051382 0.023902 0.028497 0.022944 0.022187 0.022197 0.027458

Skewness -0.662837 0.197143 0.766845 -0.687618 -0.694197 -1.481942 -1.290995 -0.143449 -1.056199 -0.882994

Kurtosis 26.50053 4.444344 9.417388 10.36552 18.91051 26.78635 16.21414 6.281865 11.99043 6.133003

J-B 13342.95 53.98495 1048.470 1352.089 6142.980 13837.69 4365.828 261.3752 2054.069 311.5039

Probability 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Sum 0.078129 0.733322 13.09962 -0.163607 0.758719 0.750823 1.210139 0.409554 1.383940 -0.042065

Sum Sq.Dev. 0.017850 5.343974 14.76305 1.523315 0.329645 0.468556 0.303751 0.284031 0.284292 0.435009

ADF p-

value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

LM

statistics 32.4812 3.7801 34.9621 120.9355 153.9909 26.8227 66.9986 3.2183 49.8520 20.4068

P-value 0.0000 0.0518 0.0000 0.0000 0.0000 0.0000 0.0000 0.0728 0.0000 0.0000

Observ.

Number 578

Note: This table presents the descriptive statistics for the weekly return series of AGG: Barclays

US Aggregate Bond, BDI: Baltic Dry Index, Bitcoin, Crude Oil WTI, Dow Jones Conventional

Electricity Index, Dow Jones Rea Estate Index, Dow Jones Index, Gold, S&P 500 index, S&P

Goldman Sachs Index for the sample period from 18 July 2010 to 15 August 2021. Std. dev

refers to the values of standard deviation. J-B is the Jarque-Bera test statistic of normality. LM

statistics and corresponding p-values, refer to the ARCH test. ADF is the Augmented Dickey-

Fuller test for unit root. The number of observations is 578.

The average return is positive for the Bitcoin and most of the indices for this examined

period, however it is negative for the S&P Goldman Sachs index and the Crude Oil WTI

71

index. The Bitcoin suffered the highest weekly loss of -71.56%, while the Bitcoin has the

highest average return of 2.27%.

The highest volatility is found for the Bitcoin, with volatility of 16.00%, while the lowest is

observed for the Barclays US Aggregate Bond index. The latter has the lowest average return,

which means that is the most stable index.

The Jarque-Bera normality test values are all greater than the critical value, so the null

hypothesis for the normal distribution is rejected. This is due to the leptokurtic kurtosis, with

kurtosis more than 3, in the return distributions (a normal distribution is not skewed and has a

coefficient of kurtosis of 3).

The skewness of almost all assets (except for Bitcoin and Barclays US Aggregate Bond) is

negative, indicating that they have a longer left tail. In contrast, the positive skewness of

Bitcoin and Barclays US Aggregate Bond indicated that than large positive price returns are

more common than the large negative returns.

Also, an Augmented Dockey-Fuler test for unit roots is being conducted for each variable, to

examine the stationarity of the price returns. The p-values, as shown in the table above, are all

zero, which means that the null hypothesis of unit root existence is rejected. Thus, all the

studied variables are stationary.

Moreover, Engle’s test for ARCH effects is used to examine whether volatility modelling is

required for these return series.

From the results presented in the table above, it is observed that all variables are stationary

and also that there is volatility clustering.

Based on the above, a multivariate GARCH model could be employed to study for the

conditional variances and covariances and so to test their volatility co-movements.

In the following table, there is the summary of the BEKK model results for Bitcoin-Index

pair.

72

Table 13: GARCH BEKK Model Results for Bitcoin and Indices

GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1

AGG BDI WTI

coefficient stand error prob coefficient stand error prob Coefficient stand error Prob

BITCOIN_RETURN 0.01406 0.00596 0.01830 0.01816 0.00623 0.00360 0.01770 0.00640 0.00570

INDEX_RETURN -0.00009 0.00021 0.67500 0.00446 0.00401 0.26630 0.00129 0.00178 0.47020

Log likelihood 2589.42000 897.21820 1314.97500

C(1,1) 0.00183 0.00019 0.00000 0.00182 0.00018 0.00000 0.00186 0.00019 0.00000

C(2,2) 0.00002 0.00000 0.00000 0.00314 0.00079 0.00010 0.00017 0.00006 0.00350

A1(1,1) 0.38326 0.03162 0.00000 0.41319 0.03172 0.00000 0.38851 0.02780 0.00000

A1(2,2) 0.56920 0.03150 0.00000 0.39836 0.05487 0.00000 0.34870 0.02660 0.00000

B1(1,1) 0.88195 0.01259 0.00000 0.86981 0.01242 0.00000 0.87891 0.01152 0.00000

B1(2,2) 0.35871 0.07467 0.00000 0.70413 0.07501 0.00000 0.89903 0.02238 0.00000

BTC ARCH COEF 0.14689 0.17073 0.15094

INDEX ARCH COEF. 0.32399 0.15869 0.12159

BTC GARH COEF. 0.77784 0.75656 0.77248

INDEX GARCH COEF. 0.12867 0.49579 0.80825

COV ARCH COEF. 0.21815 0.16460 0.13547

COV GARCH COEF. 0.31637 0.61245 0.79016

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Bitcoin with AGG: Barclays US Aggregate Bond

Index, BDI: Baltic Dry Index, WTI: Crude Oil WTI for the sample period from 18 July 2010 to 15 August 2021. C(1,1) and C(2,2) is the constant term

of the equation, A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of Bitcoin

and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH coefficient as B(1,1)2

and B(2,2)2 for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the

Covariance GARCH coefficient as B(1,1)*B(2,2).

73

Table 14:GARCH BEKK Model Results for Bitcoin and Indices


DJ CONV ELEC DJ REAL EST DJ

Coefficient stand error prob coefficient stand error prob coefficient stand error Prob

BITCOIN_RETURN 0.01705 0.00601 0.00460 0.01675 0.00587 0.00440 0.01675 0.00581 0.00390

INDEX_RETURN 0.00099 0.00082 0.22710 0.00153 0.00083 0.06380 0.00297 0.00069 0.00000

Log likelihood 1782.90200 1697.14500 1797.32300

C(1,1) 0.00177 0.00017 0.00000 0.00203 0.00021 0.00000 0.00186 0.00019 0.00000

C(2,2) 0.00006 0.00003 0.01690 0.00006 0.00002 0.00030 0.00005 0.00001 0.00000

A1(1,1) 0.39142 0.02981 0.00000 0.40404 0.03280 0.00000 0.38273 0.02977 0.00000

A1(2,2) 0.35567 0.04122 0.00000 0.46092 0.03789 0.00000 0.52208 0.04561 0.00000

B1(1,1) 0.88053 0.01193 0.00000 0.87194 0.01387 0.00000 0.88153 0.01243 0.00000

B1(2,2) 0.85783 0.04809 0.00000 0.84570 0.03042 0.00000 0.81422 0.03326 0.00000

BTC ARCH COEF 0.15321 0.16325 0.14648

INDEX ARCH COEF. 0.12650 0.21244 0.27257

BTC GARH COEF. 0.77533 0.76027 0.77710

INDEX GARCH COEF. 0.73587 0.71520 0.66296

COV ARCH COEF. 0.13922 0.18623 0.19982

COV GARCH COEF. 0.75534 0.73739 0.71776

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Bitcoin with Dow Jones Conventional Electricity

Index, Dow Jones Real Estate Index and DJ: Dow Jones Index, for the sample period from 18 July 2010 to 15 August 2021. C(1,1) and C(2,2) is the

constant term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term

of Bitcoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH coefficient

as B(1,1)2 and B(2,2)2 for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while

the Covariance GARCH coefficient as B(1,1)*B(2,2).

74

Table 15: GARCH BEKK Model Results for Bitcoin and Indices


GOLD SP500 SP GOLDMAN

Coefficient stand error prob coefficient stand error prob coefficient stand error prob

BITCOIN_RETURN 0.01914 0.00642 0.00290 0.01647 0.00574 0.00410 0.01322 0.00504 0.00870

INDEX_RETURN 0.00044 0.00078 0.57520 0.00336 0.00068 0.00000 0.00006 0.00110 0.95940

Log likelihood 1739.90400 1807.71100 1614.99500

C(1,1) 0.00184 0.00021 0.00000 0.00186 0.00020 0.00000 0.00193 0.00020 0.00000

C(2,2) 0.00003 0.00001 0.00380 0.00004 0.00001 0.00010 0.00006 0.00003 0.02280

A1(1,1) 0.38142 0.03043 0.00000 0.38222 0.03140 0.00000 0.37236 0.03089 0.00000

A1(2,2) -0.30574 0.02900 0.00000 0.53934 0.04419 0.00000 -0.26620 0.03524 0.00000

B1(1,1) 0.88162 0.01243 0.00000 0.88232 0.01310 0.00000 0.88138 0.01289 0.00000

B1(2,2) 0.92133 0.01676 0.00000 0.81139 0.03232 0.00000 0.91979 0.02696 0.00000

BTC ARCH COEF 0.14548 0.14609 0.13865

INDEX ARCH COEF. 0.09348 0.29089 0.07086

BTC GARH COEF. 0.77726 0.77849 0.77682

INDEX GARCH COEF. 0.84885 0.65835 0.84601

COV ARCH COEF. -0.11662 0.20615 -0.09912

COV GARCH COEF. 0.81227 0.71590 0.81068

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Bitcoin with Gold price, S&P 500 Index and S&P

Goldman Sachs Commodity Index for the sample period from 18 July 2010 to 15 August 2021. C(1,1) and C(2,2) is the constant term of the equation,

A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of Bitcoin and Indices

respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH coefficient as B(1,1)2 and B(2,2)2

for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the Covariance GARCH

coefficient as B(1,1)*B(2,2).

75

The BEKK models above, include both index (or cryptocurrency) specific volatility and

index-bitcoin pair volatility spillover effects.

The log likelihood for all developed models is higher than 897 in any case, which makes the

null hypothesis to be rejected.

The constant of the conditional variance of the Baltic Dry Index (BDI) is the largest among

the rest conditional variances of the indexes, which suggests greater risk in this index. The

next higher constant of the conditional variance is this of the crude oil WTI, which also shows

a high risk in this index.

The constants of the rest indices are quite similar which suggests that information is quickly

shared between them.

The ARCH coefficients, meaning the coefficients A, measure the impact of previous

innovation. Among all indices, the Barclays US Aggregate Bond and the SP 500 indies have

the greatest ARCH effect. S&P Goldman Sachs has the lowest ARCH effect.

The GARCH coefficients, meaning the coefficients B, examine the persistence of the return

volatility. For example, periods with high volatility are followed by period with high volatility

and periods with low volatility are followed by periods with low volatility. Most GARCH

coefficients are above 0.6585, which proves the existence of volatility clustering. It shows

that there is higher possibility the extent of the present volatility movement to be related to

the previous volatility movement. The lowest GARCH coefficients are for Barclays US

Aggregate Bond and Baltic Dry Index, which shows that the relation between the current and

previous volatilities is not strong, there is not that high possibility the volatility movements to

be related.

From these results we conclude that there is strong evidence of GARCH effect and existence

of weaker ARCH effect. As a result, Bitcoin and indices shocks are influenced by past

information which is common to the respective assets.

With regards to the covariance coefficients, the ARCH coefficients reflect the effect of the

previous common information, while the GARCH coefficients give the persistence of their

return volatility regarding the covariance. The strongest ARCH effect is detected between the

Bitcoin and Barclays US Aggregate Bond, but also between Bitcoin and the S&P 500 index,

which shows that previous information of the one variable will affect the other. The lowest

ARCH effect is detected between Bitcoin and Gold.

76

The ARCH effect is lower than the GARCH effect, which shows that the influence of the past

common information of the variables is less significant than the persistence of covariance

between Bitcoin and indices.

The highest GARCH effect, and thus volatility clustering, is detected between Bitcoin and

Gold, while the lowest between Bitcoin and Barclays US Aggregate Bond.

The conditional variances and covariances graphs which were calculated by BEKK model are

presented in the Appendix of the thesis.

It can be concluded from the graphs that the covariance between the Bitcoin and Barclays US

Aggregate Bond, Dow Jones, S&P 500, Dow Jones Conventional Electricity, Crude Oil WTI

and Dow Jones Real Estate is mainly positive, while the covariance between the Bitcoin and

Baltic Dry Index, Gold price and S&P Goldman Sachs Commodity index is mainly negative.

The conditional covariance graphs are plotting the magnitude of volatility.

77

9.2.2. Diagonal BEKK Model for Ethereum and Indices

Below is the Table including the descriptive statistics of the price returns of Ethereum and

indices.

Table 16: Ethereum & Indices Descriptive Statistics

AGG BDI_RET

URN

CRUDEIL

WTI_RETURN

DJCONV

ELECTR_RETURN

DJREALEST

ATE_RETURN

DOWJONES_RETURN

ETH_RETURN

GOLD_RETURN

SP500_RETURN

SPGOLD

MAN_RETURN

Mean 0.000196 0.003625 0.001347 0.001338 0.001126 0.00222 0.026787 0.001305 0.002382 0.00096

Median 0.000631 0 0.004827 0.002296 0.003452 0.003223 0.01786 0.001445 0.004736 0.002077

Maximum 0.049123 0.470774 0.275756 0.158057 0.20403 0.12084 0.802512 0.101022 0.114237 0.080997

Minimum -0.052249 -0.295538 -0.346863 -0.182883 -0.283845 -0.189978 -0.530968 -0.09897 -0.162279 -0.145503

Std. Dev. 0.006120 0.103345 0.060909 0.028188 0.032455 0.026382 0.175095 0.020055 0.024481 0.029778

Skewness -0.557940 0.285997 -0.691649 -0.739104 -1.543548 -1.517048 0.537082 -0.098662 -1.354815 -0.985947

Kurtosis 31.92489 4.321392 9.118095 17.59951 27.85488 16.28069 5.547304 7.067787 13.36724 6.214891

Jarque-Bera 10997.36 27.21147 516.3979 2826.217 8233.247 2435.77 100.309 217.6889 1507.037 186.6885

Probability 0 0 0 0 0 0 0 0 0 0

Sum 0.061787 1.141968 0.424357 0.421317 0.354675 0.69937 8.43795 0.411155 0.750467 0.302361

Sum Sq.Dev. 0.011760 3.353575 1.164914 0.249492 0.330742 0.218542 9.626727 0.126294 0.188193 0.278424

ADF p-value 0 0 0 0 0 0 0 0 0 0

LM

statistics 14.8466 2.5816 74.3096 79.246 11.6627 36.3081 0.008603 5.3486 28.0441 22.375

P-value 0.0001 0.1081 0 0 0.0006 0 0.9261 0.0207 0 0

Observ.

Number 315


US Aggregate Bond, BDI: Baltic Dry Index, Crude Oil WTI, Dow Jones Conventional

Electricity Index, Dow Jones Rea Estate Index, Dow Jones Index, Ethereum, Gold, S&P 500

index, S&P Goldman Sachs Index for the sample period from 02 August 2015 to 15 August

2021. Std. dev refers to the values of standard deviation. J-B is the Jarque-Bera test statistic of

normality. LM statistics and corresponding p-values, refer to the ARCH test. ADF is the

Augmented Dickey-Fuller test for unit root. The number of observations is 315.

The average return is positive for the Ethereum and all the studied indexes for the examined

period. The Ethereum is the variable that suffered the highest weekly loss of -53.10%, while

the same is the one with the highest average return of 80.25%.

78

The highest volatility is found for the Ethereum, with volatility of 17.51%, while the lowest is

observed for the Barclays US Aggregate Bond. The latter has also the lowest average return,

which means that is the most stable index for the studied period.


hypothesis for the normal distribution is rejected.

The skewness of almost all assets (except for Ethereum and Baltic Dry Index) is negative,

indicating that they have a longer left tail. The positive skewness of Ethereum and Baltic Dry

Index indicated that than large positive price returns are more common than the large negative

returns.







It is observed that there is volatility clustering on almost all variables, except for the

Ethereum.

In the following table, there is the summary of the BEKK model results for Ethereum-Index

pair.

79

Table 17: GARCH BEKK Model Results for Ethereum and Indices


AGG BDI WTI

coefficient stand error prob coefficient stand error prob coefficient stand error Prob

ETHEREUM_RETURN 0.01633 0.00940 0.08220 0.01333 0.01006 0.18510 0.02182 0.01016 0.03170

INDEX_RETURN 0.00001 0.00026 0.95820 0.00655 0.00575 0.25450 0.00360 0.00284 0.20530

Log likelihood 1354.37000 405.66020 613.18500

C(1,1) 0.00365 0.00115 0.00150 0.00350 0.00110 0.00150 0.00190 0.00053 0.00030

C(2,2) 0.00001 0.00000 0.00000 0.00349 0.00111 0.00170 0.00044 0.00020 0.02620

A1(1,1) -0.33005 0.05249 0.00000 0.40821 0.05660 0.00000 0.18437 0.03587 0.00000

A1(2,2) 0.61589 0.04965 0.00000 0.44547 0.07666 0.00000 0.46681 0.05829 0.00000

B1(1,1) 0.87161 0.03350 0.00000 0.84905 0.03664 0.00000 0.94396 0.01247 0.00000

B1(2,2) 0.48644 0.11806 0.00000 0.67852 0.09698 0.00000 0.79041 0.06754 0.00000

ETH ARCH COEF 0.10893 0.16663 0.03399

INDEX ARCH COEF. 0.37932 0.19844 0.21792

ETH GARH COEF. 0.75970 0.72089 0.89105

INDEX GARCH COEF. 0.23662 0.46039 0.62475

COV ARCH COEF. -0.20328 0.18184 0.08607

COV GARCH COEF. 0.42398 0.57610 0.74612

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Ethereum with AGG: Barclays US Aggregate Bond

Index, BDI: Baltic Dry Index, WTI: Crude Oil WTI for the sample period from 02 August 2015 to 15 August 2021. C(1,1) and C(2,2) is the constant

term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Ethereum and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of

Ethereum and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Ethereum and Indices respectively and GARCH coefficient

as B(1,1)2 and B(2,2)2 for Ethereum and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while


80





ETHEREUM _RETURN 0.01625 0.00963 0.09160 0.02281 0.00990 0.02120 0.01974 0.00992 0.04650

INDEX_RETURN 0.00080 0.00124 0.51800 0.00161 0.00110 0.14160 0.00371 0.00096 0.00010

Log likelihood 879.48530 858.52970 894.57110

C(1,1) 0.00335 0.00115 0.00370 0.00189 0.00051 0.00020 0.00204 0.00061 0.00080

C(2,2) 0.00010 0.00004 0.01280 0.00009 0.00004 0.01630 0.00004 0.00002 0.00910

A1(1,1) 0.29575 0.05079 0.00000 0.15003 0.04080 0.00020 0.20335 0.03915 0.00000

A1(2,2) 0.43380 0.06469 0.00000 0.55520 0.06122 0.00000 0.63784 0.06185 0.00000

B1(1,1) 0.88906 0.03225 0.00000 0.95018 0.01221 0.00000 0.93753 0.01566 0.00000

B1(2,2) 0.78341 0.07271 0.00000 0.75647 0.07078 0.00000 0.78081 0.04315 0.00000

ETH ARCH COEF 0.08747 0.02251 0.04135

INDEX ARCH COEF. 0.18818 0.30825 0.40684

ETH GARH COEF. 0.79043 0.90285 0.87897

INDEX GARCH COEF. 0.61374 0.57225 0.60967

COV ARCH COEF. 0.12830 0.08330 0.12971

COV GARCH COEF. 0.69650 0.71879 0.73204

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Ethereum with Dow Jones Conventional Electricity

Index, Dow Jones Real Estate Index and DJ: Dow Jones Index for the sample period from 02 August 2015 to 15 August 2021. C(1,1) and C(2,2) is the

constant term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Ethereum and Indices respectively and the B1(1,1) and B(2,2) is the GARCH

term of Ethereum and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Ethereum and Indices respectively and GARCH

coefficient as B(1,1)2 and B(2,2)2 for Ethereum and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as

A(1,1)*A(2,2), while the Covariance GARCH coefficient as B(1,1)*B(2,2).

81




coefficient stand error prob coefficient stand error prob coefficient stand error prob

ETHEREUM _RETURN 0.01506 0.00990 0.12810 0.01846 0.00970 0.05690 0.01922 0.01006 0.05600

INDEX_RETURN 0.00057 0.00100 0.56740 0.00331 0.00095 0.00050 0.00238 0.00163 0.14400

Log likelihood 923.89110 919.00680 802.52060

C(1,1) 0.00276 0.00101 0.00620 0.00203 0.00061 0.00080 0.00302 0.00094 0.00130

C(2,2) 0.00001 0.00001 0.04670 0.00002 0.00001 0.01380 0.00010 0.00005 0.04630

A1(1,1) 0.25951 0.04795 0.00000 0.21909 0.03841 0.00000 0.27278 0.04297 0.00000

A1(2,2) 0.28514 0.02672 0.00000 0.64340 0.05661 0.00000 0.38173 0.05929 0.00000

B1(1,1) 0.91137 0.02736 0.00000 0.93437 0.01592 0.00000 0.90193 0.02465 0.00000

B1(2,2) 0.94131 0.01307 0.00000 0.79393 0.03885 0.00000 0.85989 0.05068 0.00000

ETH ARCH COEF 0.06734 0.04800 0.07441

INDEX ARCH COEF. 0.08131 0.41397 0.14572

ETH GARH COEF. 0.83060 0.87305 0.81347

INDEX GARCH COEF. 0.88606 0.63032 0.73942

COV ARCH COEF. 0.07400 0.14097 0.10413

COV GARCH COEF. 0.85788 0.74182 0.77556

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Ethereum with Gold price, S&P 500 Index and S&P

Goldman Sachs Commodity Index for the sample period from 02 August 2015 to 15 August 2021. C(1,1) and C(2,2) is the constant term of the equation,

A1(1,1) AND A(2,2) is the ARCH term of Ethereum and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of Ethereum and Indices

respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Ethereum and Indices respectively and GARCH coefficient as B(1,1)2 and B(2,2)2

for Ethereum and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the Covariance

GARCH coefficient as B(1,1)*B(2,2).

82


index-bitcoin pair volatility spill over effects.








innovation. Among all indices, the S&P 500, the Dow Jones and the Barclays US Aggregate

Bond indices have the greatest ARCH effect. In contrast, gold has the lowest ARCH effect.


volatility. Most GARCH coefficients are above 0.5722, which proves the existence of

volatility clustering. It shows that there is higher possibility the extent of the present volatility

movement to be related to the previous volatility movement. The lowest GARCH coefficients

are for Barclays US Aggregate Bond and Baltic Dry Index, which shows that the relation

between the current and previous volatilities is not strong, there is not that high possibility the

volatility movements to be related.


of weaker ARCH effect. As a result, Ethereum and indices shocks are influenced by past





Ethereum Baltic Dry Index, but also between Ethereum and the S&P 500 index, which shows

that previous information of the one variable will affect the other. The lowest ARCH effect in

absolute value is detected between Ethereum and Gold.



between Ethereum and indices.

The highest GARCH effect, and thus volatility clustering, is detected between Ethereum and

Gold, while the lowest between Ethereum and Barclays US Aggregate Bond.

83


presented in the Appendix of the thesis.

It is observed from the produced graphs that the covariance between the Ethereum and Dow

Jones Conventional Electricity, Dow Jones, S&P 500, Gold and S&P Goldman Sachs

Commodity index is mainly positive, while the covariance between the Ethereum and

Barclays US Aggregate Bond, Baltic Dry Index, Crude Oil WTI and Dow Jones Real Estate is

mainly negative.


84

9.2.3. Diagonal BEKK Model for Cardano and Indices

Below is the Table including the descriptive statistics of the price returns of Cardano and

indices.

Table 20: Cardano & Indices Descriptive Statistics

ADA_RE

TURN AGG_RE

TURN BDI_RETU

RN

CRUDEIL

WTI_RETURN

DJCONVEL

ECTR_RETURN

DJREALES

TATE_RETURN

DOWJON

ES_RETURN

GOLD_RETURN

SP500_RETURN

SPGOLD

MAN_RETURN

Mean 0.023096 0.000287 0.004311 0.001284 0.00119 0.001356 0.002192 0.001359 0.002747 0.001053

Median 0.0033 0.000758 0.008715 0.006141 0.002055 0.003465 0.004142 0.001939 0.005819 0.003297

Maximum 1.537266 0.049123 0.470774 0.275756 0.158057 0.20403 0.12084 0.101022 0.114237 0.080997

Minimum -0.569369 -0.052249 -0.295538 -0.346863 -0.182883 -0.283845 -0.189978 -0.09897 -0.162279 -0.145503

Std. Dev. 0.229829 0.006906 0.113917 0.065901 0.032391 0.037759 0.030507 0.020466 0.027979 0.031273

Skewness 2.161141 -0.499068 0.25444 -0.837029 -0.694556 -1.517551 -1.452282 -0.108072 -1.324199 -1.283299

Kurtosis 15.72893 30.26244 4.220636 9.813769 15.52255 23.59225 13.86799 8.484994 11.88188 7.189005

Jarque-Bera 1528.487 6295.002 14.79288 416.4031 1342.708 3664.594 1070.4 254.8655 726.5861 204.1435

Probability 0 0 0.000613 0 0 0 0 0 0 0

Sum 4.688442 0.058222 0.875136 0.26071 0.241615 0.275307 0.445021 0.275855 0.557661 0.213671

Sum Sq.Dev. 10.66991 0.009634 2.621363 0.877263 0.211933 0.287995 0.187991 0.084609 0.158125 0.197552

ADF p-value 0 0 0 0 0 0 0 0 0 0

LM

statistics 0.038003 10.31125 0.781849 52.61651 41.92877 5.37017 22.24396 9.016329 17.03763 14.67332

P-value 0.8454 0.0013 0.3766 0 0 0.0205 0 0.0027 0 0.0001

Observ.

Number 203

Note: This table presents the descriptive statistics for the weekly return series of ADA: Cardano,

AGG: Barclays US Aggregate Bond, BDI: Baltic Dry Index, Crude Oil WTI, Dow Jones

Conventional Electricity Index, Dow Jones Rea Estate Index, Dow Jones Index, Gold, S&P 500

index, S&P Goldman Sachs Index for the sample period from 24 September 2017 to 15 August




The average return is positive for the Cardano and all the studied indexes for the examined

period. The Cardano is the variable that suffered the highest weekly loss of -56.94%, while

the same is the one with the highest average return of 153.73%.

85

The highest volatility is found for the Cardano, with volatility of 22.98%, while the lowest is





The skewness of almost all assets (except for Cardano and Baltic Dry Index) is negative,

indicating that they have a longer left tail. The positive skewness of Cardano and Baltic Dry

Index indicated that than large positive price returns are more common than the large negative

returns.







It is observed that there is volatility clustering on almost all variables, except for the Cardano.

In the following table, there is the summary of the BEKK model results for Cardano-Index

pair.

86

Table 21: GARCH BEKK Model Results for Cardano and Indices


AGG BDI WTI

Coefficient stand error prob coefficient stand error Prob coefficient stand error Prob

CARDANO_RETURN 0.01583 0.01282 0.21690 0.01211 0.01354 0.37090 0.01443 0.01375 0.29370

INDEX_RETURN -0.00010 0.00032 0.75590 0.00429 0.00820 0.60100 0.00236 0.00353 0.50260

Log likelihood 830.94800 201.81950 365.27460

C(1,1) 0.00139 0.00049 0.00480 0.00145 0.00047 0.00200 0.00121 0.00044 0.00550

C(2,2) 0.00001 0.00000 0.00000 0.00402 0.00183 0.02790 0.00048 0.00026 0.06230

A1(1,1) -0.21115 0.02536 0.00000 0.21925 0.02599 0.00000 0.18272 0.02289 0.00000

A1(2,2) 0.73503 0.07710 0.00000 0.41271 0.08687 0.00000 0.52709 0.07608 0.00000

B1(1,1) 0.95341 0.00952 0.00000 0.95129 0.00951 0.00000 0.96084 0.00822 0.00000

B1(2,2) 0.41325 0.15036 0.00600 0.71394 0.12331 0.00000 0.74596 0.09785 0.00000

ADA ARCH COEF 0.04459 0.04807 0.03339

INDEX ARCH COEF. 0.54026 0.17033 0.27783

ADA GARH COEF. 0.90900 0.90495 0.92320

INDEX GARCH COEF. 0.17077 0.50971 0.55646

COV ARCH COEF. -0.15520 0.09048 0.09631

COV GARCH COEF. 0.39400 0.67916 0.71675

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Cardano with AGG: Barclays US Aggregate Bond

Index, BDI: Baltic Dry Index, WTI: Crude Oil WTI for the sample period from 24 September 2017 to 15 August 2021. C(1,1) and C(2,2) is the constant

term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Cardano and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of

Cardano and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Cardano and Indices respectively and GARCH coefficient

as B(1,1)2 and B(2,2)2 for Cardano and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while


87





CARDANO _RETURN 0.01385 0.01275 0.27760 0.02084 0.01316 0.11320 0.02198 0.01369 0.10820

INDEX_RETURN 0.00072 0.00176 0.68440 0.00177 0.00138 0.19810 0.00619 0.00143 0.00000

Log likelihood 515.87760 502.59110 514.22830

C(1,1) 0.00127 0.00042 0.00280 0.00097 0.00030 0.00110 0.00118 0.00038 0.00200

C(2,2) 0.00011 0.00005 0.02600 0.00009 0.00005 0.05720 0.00016 0.00005 0.00080

A1(1,1) 0.18733 0.02482 0.00000 0.06932 0.04108 0.09150 0.17264 0.02946 0.00000

A1(2,2) 0.45784 0.08099 0.00000 0.61865 0.08326 0.00000 0.95414 0.08779 0.00000

B1(1,1) 0.95948 0.00851 0.00000 0.97878 0.00703 0.00000 0.96361 0.00812 0.00000

B1(2,2) 0.78367 0.07760 0.00000 0.74290 0.08687 0.00000 0.42931 0.11320 0.00010

ADA ARCH COEF 0.03509 0.00481 0.02981

INDEX ARCH COEF. 0.20962 0.38272 0.91039

ADA GARH COEF. 0.92061 0.95801 0.92854

INDEX GARCH COEF. 0.61414 0.55191 0.18431

COV ARCH COEF. 0.08577 0.04288 0.16473

COV GARCH COEF. 0.75192 0.72714 0.41369

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Cardano with Dow Jones Conventional Electricity

Index, Dow Jones Real Estate Index and DJ: Dow Jones Index for the sample period from 24 September 2017 to 15 August 2021. C(1,1) and C(2,2) is

the constant term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Cardano and Indices respectively and the B1(1,1) and B(2,2) is the GARCH

term of Cardano and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Cardano and Indices respectively and GARCH

coefficient as B(1,1)2 and B(2,2)2 for Cardano and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as


88




Coefficient stand error prob coefficient stand error prob coefficient stand error prob

CARDANO _RETURN 0.01347 0.01326 0.30980 0.02148 0.01323 0.10440 0.01476 0.01399 0.29150

INDEX_RETURN 0.00044 0.00116 0.70850 0.00434 0.00126 0.00060 0.00253 0.00201 0.20920

Log likelihood 568.32030 531.54420 481.57490

C(1,1) 0.00119 0.00037 0.00130 0.00101 0.00029 0.00050 0.00129 0.00046 0.00470

C(2,2) 0.00003 0.00001 0.01260 0.00004 0.00002 0.04640 0.00010 0.00005 0.04610

A1(1,1) 0.14811 0.02424 0.00000 0.09269 0.03389 0.00620 0.17244 0.02247 0.00000

A1(2,2) 0.36938 0.04804 0.00000 0.76884 0.09421 0.00000 0.44586 0.08522 0.00000

B1(1,1) 0.96686 0.00770 0.00000 0.97626 0.00680 0.00000 0.96181 0.00834 0.00000

B1(2,2) 0.88972 0.02409 0.00000 0.72137 0.06722 0.00000 0.83686 0.05529 0.00000

ADA ARCH COEF 0.02194 0.00859 0.02974

INDEX ARCH COEF. 0.13644 0.59111 0.19879

ADA GARH COEF. 0.93483 0.95309 0.92507

INDEX GARCH COEF. 0.79160 0.52038 0.70033

COV ARCH COEF. 0.05471 0.07126 0.07688

COV GARCH COEF. 0.86024 0.70425 0.80489

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Cardano with Gold price, S&P 500 Index and S&P

Goldman Sachs Commodity Index for the sample period from 24 September 2017 to 15 August 2021. C(1,1) and C(2,2) is the constant term of the

equation, A1(1,1) AND A(2,2) is the ARCH term of Cardano and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of Cardano and

Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Cardano and Indices respectively and GARCH coefficient as B(1,1)2 and

B(2,2)2 for Cardano and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the Covariance


89











Bond indices have the greatest ARCH effect. In contrast, gold has the lowest ARCH effect.





are for Barclays US Aggregate Bond and Dow Jones, which shows that the relation between

the current and previous volatilities is not strong, there is not that high possibility the

volatility movements to be related.


of weaker ARCH effect. As a result, Cardano and indices shocks are influenced by past





Cardano Dow Jones, but also between Cardano and the S&P 500 index, which shows that

previous information of the one variable will affect the other. The lowest ARCH effect in

absolute value is detected between Cardano and Dow Jones Real Estate Index.



between Cardano and indices.

The highest GARCH effect, and thus volatility clustering, is detected between Cardano and

Gold, while the lowest between Cardano and Barclays US Aggregate Bond.

90


presented in the Appendix of this thesis.

It is observed from the graphs that the covariance between the Cardano and Dow Jones, S&P

500, Crude Oil WTI, Dow Jones Real Estate, Gold and S&P Goldman Sachs Commodity

index is mainly positive, while the covariance between the Cardano and Dow Jones

Conventional Electricity, Barclays US Aggregate Bond and Baltic Dry Index is mainly

negative.


91

9.2.4. Diagonal BEKK Model for Litecoin and Indices

Below is the Table including the descriptive statistics of the price returns of Litecoin and

indices.

Table 24: Litecoin & Indices Descriptive Statistics

AGG_RE

TURN BDI_RET

URN

CRUDEIL

WTI_RETURN

DJCONV

ELECTR_RETURN

DJREALEST

ATE_RETURN

DOWJONES_RETURN

GOLD_RETURN

LTC_RETURN

SP500_RETURN

SPGOLD

MAN_RETURN

Mean 0.000184 0.003111 -0.000888 0.001277 0.001149 0.001952 0.001071 0.010475 0.00217 -0.000456

Median 0.000626 0 0.002294 0.002327 0.003301 0.003026 0.001241 0.008787 0.00377 0.000417

Maximum 0.049123 0.470774 0.275756 0.158057 0.20403 0.12084 0.112555 0.762648 0.114237 0.080997

Minimum -0.052249 -0.295538 -0.346863 -0.182883 -0.283845 -0.189978 -0.09897 -0.726006 -0.162279 -0.145503

Std. Dev. 0.006062 0.101366 0.059636 0.027427 0.030905 0.025345 0.020786 0.153277 0.023657 0.029695

Skewness -0.590343 0.313375 -0.633291 -0.708576 -1.558371 -1.473456 0.285249 0.527552 -1.288543 -0.859657

Kurtosis 29.22883 4.364868 8.799123 17.22773 29.57753 16.66891 7.924094 7.399112 13.41262 5.704763

Jarque-Bera 10368.91 33.92918 529.9789 3075.067 10771.03 2940.997 369.6063 307.8342 1730.755 154.5047

Probability 0 0 0 0 0 0 0 0 0 0

Sum 0.066561 1.123035 -0.320647 0.461071 0.414757 0.704774 0.386582 3.78147 0.783332 -0.16479

Sum Sq.Dev. 0.01323 3.699029 1.280312 0.270813 0.343853 0.231259 0.155547 8.457728 0.201468 0.317443

ADF p-value 0 0 0 0 0 0 0 0 0 0

LM

statistics 18.98375 2.41838 79.39372 93.40422 14.3708 41.15508 1.419976 2.909259 31.55692 19.16992

P-value 0 0.1199 0 0 0.0002 0 0.2334 0.0881 0 0

Observ.

Number 361


US Aggregate Bond, BDI: Baltic Dry Index, Crude Oil WTI, Dow Jones Conventional

Electricity Index, Dow Jones Rea Estate Index, Dow Jones Index, Gold, LTC: Litecoin, S&P

500 index, S&P Goldman Sachs Index for the sample period from 14 September 2014 to 15

August 2021. Std. dev refers to the values of standard deviation. J-B is the Jarque-Bera test

statistic of normality. LM statistics and corresponding p-values, refer to the ARCH test. ADF

is the Augmented Dickey-Fuller test for unit root. The number of observations is 361.

The average return is positive for the Litecoin and most of the studied indexes for the

examined period, except for the Crude Oil WTI and S&P Goldman Sachs Commodity

Indices, which have negative average returns. The Litecoin is the variable that suffered the

92

highest weekly loss of -72.60%, while the same is the one with the highest average return of

72.63%.

The highest volatility is found for the Litecoin, with volatility of 15.33%, while the lowest is





The skewness of almost all assets (except for Litecoin, Gold and Baltic Dry Index) is

negative, indicating that they have a longer left tail. The positive skewness of Litecoin, Gold

and Baltic Dry Index indicated that than large positive price returns are more common than

the large negative returns for the studied period.







It is observed that there is volatility clustering on almost all variables, except for the Litecoin,

Gold and Baltic Dry Index.

In the following table, there is the summary of the BEKK model results for Cardano-Index

pair.

93

Table 25: GARCH BEKK Model Results for Litecoin and Indices


AGG BDI WTI

Coefficient stand error Prob coefficient stand error Prob coefficient stand error Prob

LITECOIN_RETURN 0.00056 0.00832 0.94620 0.00173 0.00837 0.83660 0.00529 0.00790 0.50340

INDEX_RETURN -0.00008 0.00026 0.74630 0.00530 0.00522 0.30950 0.00173 0.00261 0.50660

Log likelihood 1574.69700 513.25670 750.84310

C(1,1) 0.00323 0.00133 0.01500 0.00421 0.00109 0.00010 0.00468 0.00136 0.00060

C(2,2) 0.00001 0.00000 0.00000 0.00288 0.00104 0.00570 0.00053 0.00022 0.01400

A1(1,1) 0.28302 0.03878 0.00000 0.41498 0.04420 0.00000 0.36557 0.03728 0.00000

A1(2,2) 0.66942 0.04055 0.00000 0.39675 0.06567 0.00000 0.47716 0.05158 0.00000

B1(1,1) 0.88636 0.04520 0.00000 0.81396 0.04784 0.00000 0.81633 0.05254 0.00000

B1(2,2) 0.35878 0.07974 0.00000 0.74367 0.08725 0.00000 0.77236 0.07347 0.00000

LTC ARCH COEF 0.08010 0.17221 0.13364

INDEX ARCH COEF. 0.44812 0.15741 0.22768

LTC GARH COEF. 0.78563 0.66254 0.66639

INDEX GARCH COEF. 0.12872 0.55305 0.59654

COV ARCH COEF. 0.18946 0.16464 0.17443

COV GARCH COEF. 0.31801 0.60532 0.63050

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Litecoin with AGG: Barclays US Aggregate Bond

Index, BDI: Baltic Dry Index, WTI: Crude Oil WTI for the sample period from 14 September 2014 to 15 August 2021. C(1,1) and C(2,2) is the constant

term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Litecoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of

Litecoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Litecoin and Indices respectively and GARCH coefficient

as B(1,1)2 and B(2,2)2 for Litecoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while


94




Coefficient stand error Prob coefficient stand error prob coefficient stand error Prob

LITECOIN _RETURN 0.00178 0.00796 0.82330 0.00360 0.00827 0.66360 0.00284 0.00834 0.73380

INDEX_RETURN 0.00087 0.00119 0.46400 0.00123 0.00097 0.20430 0.00286 0.00097 0.00320

Log likelihood 1048.79200 1040.10300 1072.37400

C(1,1) 0.00375 0.00127 0.00320 0.00345 0.00147 0.01930 0.00365 0.00147 0.01340

C(2,2) 0.00011 0.00005 0.01680 0.00009 0.00004 0.01530 0.00006 0.00002 0.00290

A1(1,1) 0.32669 0.03763 0.00000 0.28425 0.04017 0.00000 0.28165 0.03811 0.00000

A1(2,2) 0.40793 0.05293 0.00000 0.54271 0.05486 0.00000 0.58729 0.06258 0.00000

B1(1,1) 0.85767 0.04687 0.00000 0.87893 0.05066 0.00000 0.87502 0.04935 0.00000

B1(2,2) 0.79055 0.07294 0.00000 0.76102 0.06878 0.00000 0.78067 0.04817 0.00000

LTC ARCH COEF 0.10673 0.08080 0.07933

INDEX ARCH COEF. 0.16641 0.29454 0.34490

LTC GARH COEF. 0.73560 0.77252 0.76566

INDEX GARCH COEF. 0.62496 0.57915 0.60945

COV ARCH COEF. 0.13327 0.15427 0.16541

COV GARCH COEF. 0.67803 0.66889 0.68310

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Litecoin with Dow Jones Conventional Electricity

Index, Dow Jones Real Estate Index and DJ: Dow Jones Index for the sample period from 14 September 2014 to 15 August 2021. C(1,1) and C(2,2) is

the constant term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Litecoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH

term of Litecoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Litecoin and Indices respectively and GARCH

coefficient as B(1,1)2 and B(2,2)2 for Litecoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as


95




Coefficient stand error Prob coefficient stand error prob coefficient stand error prob

LITECOIN _RETURN 0.00256 0.00830 0.75800 0.00181 0.00820 0.82540 0.00494 0.00801 0.53760

INDEX_RETURN 0.00098 0.00098 0.31630 0.00289 0.00090 0.00130 0.00124 0.00155 0.42190

Log likelihood 1080.50600 1096.34100 962.58340

C(1,1) 0.00498 0.00134 0.00020 0.00415 0.00154 0.00700 0.00525 0.00140 0.00020

C(2,2) 0.00002 0.00001 0.04260 0.00003 0.00001 0.00560 0.00012 0.00006 0.03180

A1(1,1) 0.39150 0.04436 0.00000 0.29906 0.03821 0.00000 0.39947 0.04251 0.00000

A1(2,2) 0.25056 0.02426 0.00000 0.62515 0.05166 0.00000 0.35768 0.05179 0.00000

B1(1,1) 0.80265 0.05427 0.00000 0.85675 0.05266 0.00000 0.78831 0.05747 0.00000

B1(2,2) 0.95030 0.01181 0.00000 0.79375 0.03668 0.00000 0.85601 0.05492 0.00000

LTC ARCH COEF 0.15327 0.08943 0.15958

INDEX ARCH COEF. 0.06278 0.39082 0.12793

LTC GARH COEF. 0.64425 0.73402 0.62144

INDEX GARCH COEF. 0.90307 0.63004 0.73275

COV ARCH COEF. 0.09809 0.18696 0.14288

COV GARCH COEF. 0.76276 0.68005 0.67480

Note: This table presents the Diagonal BEKK model results for the weekly return series of pairs of Litecoin with Gold price, S&P 500 Index and S&P

Goldman Sachs Commodity Index for the sample period from 14 September 2014 to 15 August 2021. C(1,1) and C(2,2) is the constant term of the

equation, A1(1,1) AND A(2,2) is the ARCH term of Litecoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of Litecoin and

Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Litecoin and Indices respectively and GARCH coefficient as B(1,1)2 and

B(2,2)2 for Litecoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the Covariance


96











Bond indices have the greatest ARCH effect. In contrast, Gold has the lowest ARCH effect.





are for Barclays US Aggregate Bond, which shows that the relation between the current and

previous volatilities is not strong, there is not that high possibility the volatility movements to

be related.


of weaker ARCH effect. As a result, Litecoin and indices shocks are influenced by past


With regards to the covariance coefficients, the strongest ARCH effect is detected between

the Litecoin and Barclays US Aggregate Bond index, but also between Litecoin and S&P 500

index, which shows that previous information of the one variable will affect the other. The

lowest ARCH effect is detected between Litecoin and Gold.



between Litecoin and indices.

The highest GARCH effect, and thus volatility clustering, is detected between Litecoin and

Gold, while the lowest between Litecoin and Barclays US Aggregate Bond.


presented in the Appendix.

97

It is observed from the graphs that the covariance between the Litecoin and Dow Jones, S&P

500, Crude Oil WTI, Dow Jones Real Estate and Barclays US Aggregate Bond index is

mainly positive, while the covariance between the Cardano and Dow Jones Conventional

Electricity, Baltic Dry Index, Gold and S&P Goldman Sachs Commodity is mainly negative.


98

9.3. Diagonal BEKK Model for Bitcoin and Indices: Pre and During COVID-19

period

The last study refers to the Bitcoin and Indices volatility dynamics prior and during COVID-

19 crisis.

The prior COVID-19 period is considered up to 31 December 2019, while the during COVID-

19 period starts from 01 January 2020 up to 15 August 2021. Weekly prices for Bitcoin and

Indices are used for this approach as well.

The descriptive statistics for the two periods are presented in the following tables.

Table 28: Bitcoin & Indices Descriptive Statistics for the Pro-COVID 19 period

AGG_PR

O BDI_PRO BTC_PRO

CRUDEOI

L_PRO

DJCONVEL

ECTR_PRO

DJREALES

TATE_PRO

DOWJON

ES_PRO

GOLD_P

RO

SP500_PR

O

SPGOLDMAN_PR

O

Mean 0.000103 -0.001163 0.022736 -0.000457 0.001434 0.001255 0.00205 0.0006 0.002183 -0.000307

Median 0.000626 -0.002265 0.008658 0.001516 0.002451 0.003563 0.003167 0.00146 0.003078 0.000941

Maximum 0.010415 0.27813 0.822906 0.127072 0.045667 0.064156 0.067776 0.112555 0.071284 0.061453

Minimum -0.02015 -0.335448 -0.71562 -0.159019 -0.053319 -0.120237 -0.071149 -0.101316 -0.074603 -0.118266

Std. Dev. 0.004647 0.088262 0.166469 0.041063 0.017529 0.021197 0.01832 0.021094 0.018732 0.024591

Skewness -0.861777 -0.050133 0.874515 -0.344907 -0.350526 -0.808126 -0.517084 -0.159976 -0.5744 -0.497079

Kurtosis 4.331188 3.701755 9.084112 4.058811 3.21009 5.94282 4.900243 6.042764 5.217297 4.239523

J-B 97.42291 10.32248 823.218 32.80348 11.00237 231.555 96.14376 192.286 128.1009 51.86298

Probability 0 0.005735 0 0 0.004082 0 0 0 0 0

Sum 0.050637 -0.57357 11.20868 -0.225267 0.706839 0.618711 1.010455 0.295732 1.076257 -0.151117

Sum Sq.

Dev. 0.010626 3.832728 13.63435 0.829576 0.151183 0.221063 0.165132 0.218928 0.172638 0.29752

ADF p-value 0 0 0 0 0 0 0 0 0 0

LM-statistics 0.327729 0.496088 30.59666 0.035973 3.304456 4.120197 18.92534 0.962614 18.46675 134166

P-value 0.567 0.4218 0 0.8496 0.0691 0.0424 0 0.3265 0 0.7142

Observations 493

Note: This table presents the descriptive statistics for the weekly return series of BTC: Bitcoin,



index, S&P Goldman Sachs Index for the sample period from 18 July 2010 to 31 December




99

Table 29: Bitcoin & Indices Descriptive Statistics for the COVID 19 period

AGG_AF

T BDI_AFT BTC_AFT

CRUDEOI

L_AFT

DJCONVEL

ECTR_AFT

DJREALES

TATE_AFT

DOWJON

ES_AFT

GOLD_A

FT

SP50_AF

T

SPGOLD

MANO_

AFT

Mean 0.000359 0.013774 0.022075 0.000965 0.000514 0.001527 0.002318 0.001583 0.003559 0.001349

Median 0.001091 0.022741 0.019457 0.006776 0.002008 0.003635 0.005159 0.003926 0.007126 0.007542

Maximum 0.049123 0.470774 0.237202 0.275756 0.158057 0.20403 0.12084 0.101022 0.114237 0.080997

Minimum -0.052249 -0.289055 -0.539353 -0.346863 -0.182883 -0.283845 -0.189978 -0.09897 -0.162279 -0.145503

Std. Dev. 0.009222 0.133291 0.115244 0.090362 0.045823 0.05396 0.040383 0.027759 0.036223 0.040196

Skewness -0.407629 0.437267 -1.349465 -0.708764 -0.512268 -1.214681 -1.34731 -0.141366 -1.265819 -1.326743

Kurtosis 22.43414 3.900097 8.415353 6.454618 9.159843 13.47764 10.3994 5.809201 9.557235 5.849609

J-B 1355.755 5.643693 131.1868 49.96518 139.7261 414.5299 222.2102 28.56472 177.04 54.32786

Probability 0 0.059496 0 0 0 0 0 0.000001 0 0

Sum 0.03086 1.184565 1.898467 0.08298 0.044241 0.131292 0.199321 0.136118 0.306086 0.116011

Sum Sq.

Dev. 0.007229 1.510157 1.128899 0.694057 0.178479 0.247492 0.138619 0.065498 0.111531 0.137338

ADF p-value 0 0 0 0 0 0.0202 0.0001 0.0001 0.0001 0

LM-statistics 3.84109 0.016312 0.180308 20.62602 14.50816 1.418709 8.417714 3.12503 6.668636 4.555993

P-value 0.05 0.8984 0.6711 0 0.0001 0.2336 0.0037 0.0771 0.0098 0.0328

Observations 86

Note: This table presents the descriptive statistics for the weekly return series of BTC: Bitcoin,



index, S&P Goldman Sachs Index for the sample period from 01 January 2021 to 15 August




For the period prior to the COVID-19 crisis, it is observed that the average return of all the

studied variables is positive, except for the Baltic Dry index, the Crude Oil and the S&P

Goldman Sachs Commodity Index. In contrast, for the period of COVID-19, all the studied

variables have positive average return.

For the pro COVID -19 period, Bitcoin has both the highest average return but also suffered

the greatest lost. It also has the greatest volatility of 16.65%. For the COVID-19 period, Baltic

Dry Index has the greatest maximum return of 47.08% and highest volatility of 13.33%, while

the Bitcoin remains the variable with the greatest loss of -53.94%.

For both periods the lowest volatility is observed for the Barclays US Aggregate Bond index,

which also has the lowest average return, which means that is the most stable index.

100

The Jarque-Bera normality test values are almost all greater than the critical value (except for

the Baltic Dry Index at both periods), so the null hypothesis for the normal distribution is

rejected. This is due to the leptokurtic kurtosis, with kurtosis more than 3, in the return

distributions.

For both periods, the skewness of almost all assets (except for Bitcoin) is negative, indicating

that they have a longer left tail. In contrast, the positive skewness of Bitcoin indicates that

than large positive price returns are more common than the large negative returns.







From the results presented in the tables above, it is observed that for the period prior to

COVID-19 crisis, only the following four variables have ARCH effects: Bitcoin, Dow Jones

Real Estate Index, Dow Jones, and S&P 500.

For the COVID-19 period, the following four variables have ARCH effects: Barclays US

Aggregate Bond Inex, Crue Oil WTI, Dow Jones Conventional Electricity Index, Dow Jones,

S&P 500, and S&P Goldman Sachs Commodity Index.

The next step is the use of a multivariate GARCH model to examine for the variables’

conditional variances and covariances and so to test their volatility co-movements.

Below are the tables with the GARCH BEKK model results for both periods, before and

during COVID-19 for Bitcoin-Indices pairs.

101

Table 30: GARCH BEKK Model Results for Bitcoin and Indices for the pro-COVID 19 period


AGG BDI WTI

coefficient stand error prob coefficient stand error Prob Coefficient stand error Prob

BITCOIN_RETURN 0.01653 0.00721 0.02180 0.01653 0.00721 0.02180 0.01653 0.00721 0.02180

INDEX_RETURN 0.00015 0.00023 0.51330 0.00015 0.00023 0.51330 0.00015 0.00023 0.51330

Log likelihood 2235.43000 788.96990 1166.52100

C(1,1) 0.00172 0.00018 0.00000 0.00171 0.00019 0.00000 0.00176 0.00018 0.00000

C(2,2) 0.00000 0.00000 0.17060 0.00523 0.00136 0.00010 0.00004 0.00002 0.03010

A1(1,1) 0.42089 0.03383 0.00000 0.41511 0.03415 0.00000 0.42282 0.03304 0.00000

A1(2,2) -0.17881 0.05588 0.00140 0.40377 0.07678 0.00000 0.19318 0.03284 0.00000

B1(1,1) 0.87098 0.01275 0.00000 0.87330 0.01304 0.00000 0.87046 0.01231 0.00000

B1(2,2) 0.93308 0.04558 0.00000 0.40647 0.23305 0.08110 0.96847 0.01107 0.00000

BTC ARCH COEF 0.17715 0.17231 0.17877

INDEX ARCH COEF. 0.03197 0.16303 0.03732

BTC GARH COEF. 0.75861 0.76265 0.75770

INDEX GARCH COEF. 0.87063 0.16521 0.93794

COV ARCH COEF. -0.07526 0.16761 0.08168

COV GARCH COEF. 0.81269 0.35497 0.84302


Index, BDI: Baltic Dry Index, WTI: Crude Oil WTI for the sample period from 18 July 2010 to 31 December 2019. C(1,1) and C(2,2) is the constant

term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of

Bitcoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH coefficient as

B(1,1)2 and B(2,2)2 for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the


102





BITCOIN_RETURN 0.01582 0.00718 0.02750 0.01634 0.00673 0.01520 0.01609 0.00680 0.01790

INDEX_RETURN 0.00169 0.00080 0.03450 0.00128 0.00093 0.16800 0.00279 0.00067 0.00000

Log likelihood 1579.69100 1499.37200 1581.72000

C(1,1) 0.00184 0.00019 0.00000 0.00182 0.00022 0.00000 0.00170 0.00019 0.00000

C(2,2) 0.00001 0.00001 0.23230 0.00004 0.00002 0.02230 0.00004 0.00001 0.00160

A1(1,1) 0.43139 0.03394 0.00000 0.43442 0.03884 0.00000 0.40098 0.03327 0.00000

A1(2,2) -0.12166 0.05140 0.01790 0.29757 0.04356 0.00000 0.42802 0.05454 0.00000

B1(1,1) 0.86372 0.01296 0.00000 0.86714 0.01623 0.00000 0.88092 0.01337 0.00000

B1(2,2) 0.97433 0.02031 0.00000 0.90729 0.03120 0.00000 0.83744 0.03961 0.00000

BTC ARCH COEF 0.18609 0.18872 0.16078

INDEX ARCH COEF. 0.01480 0.08855 0.18320

BTC GARH COEF. 0.74600 0.75193 0.77602

INDEX GARCH COEF. 0.94932 0.82318 0.70131

COV ARCH COEF. -0.05248 0.12927 0.17163

COV GARCH COEF. 0.84154 0.78675 0.73772


Index, Dow Jones Real Estate Index and DJ: Dow Jones Index, for the sample period from 18 July 2010 to 31 December 2019. C(1,1) and C(2,2) is the

constant term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term

of Bitcoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH coefficient

as B(1,1)2 and B(2,2)2 for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while


103




coefficient stand error prob coefficient stand error Prob coefficient stand error prob

BITCOIN_RETURN 0.01562 0.00694 0.02430 0.01610 0.00677 0.01740 0.01687 0.00724 0.01990

INDEX_RETURN 0.00113 0.00085 0.18520 0.00320 0.00069 0.00000 -0.00010 0.00112 0.93030

Log likelihood 1501.02200 1580.30000 1412.31600

C(1,1) 0.00180 0.00018 0.00000 0.00172 0.00020 0.00000 0.00172 0.00018 0.00000

C(2,2) 0.00001 0.00001 0.08330 0.00004 0.00001 0.00140 0.00002 0.00001 0.13300

A1(1,1) 0.42773 0.03408 0.00000 0.39880 0.03533 0.00000 0.41962 0.03332 0.00000

A1(2,2) 0.23651 0.03400 0.00000 0.47190 0.05358 0.00000 0.16016 0.04186 0.00010

B1(1,1) 0.86820 0.01267 0.00000 0.88183 0.01440 0.00000 0.87223 0.01233 0.00000

B1(2,2) 0.95572 0.01621 0.00000 0.82252 0.03917 0.00000 0.97224 0.01538 0.00000

BTC ARCH COEF 0.18295 0.15904 0.17608

INDEX ARCH COEF. 0.05594 0.22269 0.02565

BTC GARH COEF. 0.75377 0.77763 0.76079

INDEX GARCH COEF. 0.91340 0.67654 0.94525

COV ARCH COEF. 0.10116 0.18820 0.06721

COV GARCH COEF. 0.82976 0.72533 0.84802


Goldman Sachs Commodity Index for the sample period from 18 July 2010 to 31 December 2019. C(1,1) and C(2,2) is the constant term of the equation,





104



AGG BDI WTI

coefficient stand error prob coefficient stand error Prob coefficient stand error Prob

BITCOIN_RETURN 0.02711 0.01366 0.04710 0.01855 0.01525 0.22390 0.02126 0.01524 0.16280

INDEX_RETURN -0.00019 0.00050 0.69740 0.01987 0.01376 0.14860 0.00555 0.00769 0.47010

Log likelihood 394.79430 120.93190 177.59000

C(1,1) 0.00593 0.00928 0.52280 0.00368 0.00736 0.61680 0.00473 0.02577 0.85430

C(2,2) 0.00001 0.00000 0.04150 0.00294 0.00275 0.28540 0.00068 0.00064 0.29040

A1(1,1) -0.29870 0.23213 0.19820 0.19663 0.25004 0.43160 -0.04116 0.13010 0.75170

A1(2,2) 0.84923 0.17701 0.00000 0.43377 0.13212 0.00100 0.56058 0.20455 0.00610

B1(1,1) 0.69092 0.52916 0.19170 0.82930 0.38080 0.02940 0.80130 1.22640 0.51350

B1(2,2) 0.46854 0.22607 0.03820 0.79399 0.14240 0.00000 0.74537 0.16972 0.00000

BTC ARCH COEF 0.08922 0.03866 0.00169

INDEX ARCH COEF. 0.72118 0.18815 0.31425

BTC GARH COEF. 0.47736 0.68774 0.64208

INDEX GARCH COEF. 0.21953 0.63043 0.55557

COV ARCH COEF. -0.25366 0.08529 -0.02307

COV GARCH COEF. 0.32372 0.65846 0.59726


Index, BDI: Baltic Dry Index, WTI: Crude Oil WTI for the sample period from 01 January 2020 to 15 August 2021. C(1,1) and C(2,2) is the constant

term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH term of

Bitcoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH coefficient as

B(1,1)2 and B(2,2)2 for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as A(1,1)*A(2,2), while the


105





BITCOIN_RETURN 0.02219 0.01520 0.14430 0.02027 0.01541 0.18850 0.02356 0.01540 0.12600

INDEX_RETURN -0.00003 0.00434 0.99390 0.00192 0.00118 0.10530 0.00761 0.00302 0.01180

Log likelihood 236.97060 233.92130 243.08940

C(1,1) 0.00393 0.01472 0.78940 0.00482 0.01880 0.79750 0.00389 0.01482 0.79280

C(2,2) 0.00032 0.00028 0.24710 -0.00001 0.00003 0.68480 0.00021 0.00011 0.04890

A1(1,1) -0.03335 0.12279 0.78590 -0.08689 0.14239 0.54170 0.04083 0.08377 0.62600

A1(2,2) 0.55862 0.28501 0.05000 0.98743 0.15645 0.00000 1.17475 0.17664 0.00000

B1(1,1) 0.83884 0.67530 0.21420 0.79203 0.92922 0.39400 0.84072 0.68169 0.21750

B1(2,2) 0.65889 0.31149 0.03440 0.69365 0.09774 0.00000 0.34075 0.18694 0.06830

BTC ARCH COEF 0.00111 0.00755 0.00167

INDEX ARCH COEF. 0.31205 0.97502 1.38004

BTC GARH COEF. 0.70365 0.62730 0.70680

INDEX GARCH COEF. 0.43413 0.48115 0.11611

COV ARCH COEF. -0.01863 -0.08580 0.04797

COV GARCH COEF. 0.55270 0.54939 0.28648


Index, Dow Jones Real Estate Index and DJ: Dow Jones Index, for the sample period from 01 January 2020 to 15 August 2021. C(1,1) and C(2,2) is

the constant term of the equation, A1(1,1) AND A(2,2) is the ARCH term of Bitcoin and Indices respectively and the B1(1,1) and B(2,2) is the GARCH

term of Bitcoin and Indices respectively. ARCH coefficient is calculated as A(1,1)2 and A(2,2)2 for Bitcoin and Indices respectively and GARCH

coefficient as B(1,1)2 and B(2,2)2 for Bitcoin and Indices respectively. The Covariance ARCH coefficient of the model/ pair is calculated as


106




coefficient stand error prob coefficient stand error Prob coefficient stand error prob

BITCOIN_RETURN 0.02212 0.01483 0.13570 0.02249 0.01577 0.15380 0.02384 0.01584 0.13240

INDEX_RETURN 0.00086 0.00279 0.75950 0.00797 0.00303 0.00860 0.00625 0.00422 0.13840

Log likelihood 258.96170 248.11210 230.76320

C(1,1) 0.00405 0.01806 0.82260 0.00401 0.01507 0.79030 0.00358 0.01343 0.78990

C(2,2) 0.00024 0.00018 0.18800 0.00023 0.00011 0.03550 0.00011 0.00010 0.28650

A1(1,1) 0.01256 0.25191 0.96020 -0.00279 0.09137 0.97560 0.04494 0.08840 0.61120

A1(2,2) 0.49644 0.20852 0.01730 0.98942 0.18998 0.00000 0.45327 0.20072 0.02390

B1(1,1) 0.83453 0.82881 0.31400 0.83640 0.69946 0.23180 0.85463 0.60287 0.15630

B1(2,2) 0.64800 0.29246 0.02670 0.42033 0.17158 0.01430 0.85553 0.08281 0.00000

BTC ARCH COEF 0.00016 0.00001 0.00202

INDEX ARCH COEF. 0.24645 0.97895 0.20545

BTC GARH COEF. 0.69643 0.69956 0.73040

INDEX GARCH COEF. 0.41990 0.17668 0.73193

COV ARCH COEF. 0.00623 -0.00276 0.02037

COV GARCH COEF. 0.54077 0.35156 0.73117


Goldman Sachs Commodity Index for the sample period from 01 January 2020 to 15 August 2021. C(1,1) and C(2,2) is the constant term of the equation,





107


index-bitcoin pair volatility spillover effects.

The log likelihood for all developed models is higher than 788 for the pre-COVID period and

higher than 120 for the post COVID period, which makes the null hypothesis to be rejected.

For the pre-COVID 19 period the constant of the conditional variance of the Baltic Dry Index

(BDI) is the largest among the rest conditional variances of the indexes, which suggests

greater risk in this index. The constant of the conditional variance for all the other indices is

close to zero and they are quite similar which suggests that information is quickly shared

between them.

For the COVID 19 period the constant of the conditional variance of the Baltic Dry Index

(BDI) is the largest among the rest conditional variances of the indexes, which suggests

greater risk in this index for this period as well. The constants of the rest indices are quite

similar in this case as well which suggests that information is quickly shared between them.

The individual variables ARCH coefficients for pre-COVID-19 period, measure the impact of

previous innovation. Among all indices, the Dow Jones and the S&P 500 indices have the

greatest ARCH effect. Dow Jones Conventional Electricity Index has the lowest ARCH

effect.

The individual variables ARCH coefficients for COVID-19 period, measure the impact of

previous innovation. Among all indices, the Dow Jones and the S&P 500 indices have the

greatest ARCH effect. Baltic Dry Index has the lowest ARCH effect.

Comparing the individual variables ARCH coefficient values between the two periods, it is

observed that the pre-COVID period has maximum ARCH coefficient close to 0.22, however

the ARCH coefficient for the post-COVID period exceeds the 1.30. The fact that the ARCH

coefficient is greater than one, indicates that the variance is not stationary, according to Perez

(2007) and a more detailed study for subject variable is required.

The individual variables GARCH coefficients for pre-COVID-19 period, examine the

persistence of the return volatility. Most GARCH coefficients are above 0.70, which proves

the existence of volatility clustering. The lowest GARCH coefficients are for S&P 500 and

Baltic Dry Index, which shows that the relation between the current and previous volatilities

is not strong, there is not that high possibility the volatility movements to be related.

The individual variables GARCH coefficients for COVID-19 period are significantly lower

than the pre-COVID calculated ones, varying mainly from 0.11 to 0.48, which shows that

there is no significant volatility clustering and that the relation between the current and

previous volatilities is not strong so there is not that high possibility the volatility movements

to be related. The highest GARCH coefficients are Baltic Dry Index, S&P Goldman Sachs

Commodity Index and the Crude Oil WTI, which shows the existence of volatility clustering

for these indices only.

From these results we conclude that for the pre-COVID period there is strong evidence of

GARCH effect in individual variables and existence of weaker ARCH effect. As a result,

indices shocks are influenced by past information which is common to the respective assets.

Nevertheless, for the COVID period, it is observed that the opposite happens. There is strong

evidence of individual variables ARCH effect and existence of weaker individual variables

GARCH effect.

108



return volatility regarding the covariance.

For the pre-COVID period, the strongest ARCH effect is detected between the Bitcoin and

S&P 500 index, but also between Bitcoin and the Dow Jones Index, which shows that

previous information of the one variable will affect the other. The lowest ARCH effect in

absolute value is detected between Bitcoin and Dow Jones Conventional Electricity Index.


S&P Goldman Sachs Commodity Index, while the lowest between Bitcoin and Baltic Dry

Index.

The covariance ARCH effect is lower than the GARCH effect, which shows that the influence

of the past common information of the variables is less significant than the persistence of

covariance between Bitcoin and indices for the period prior to COVID-19.

For the COVID-19 period, the strongest covariance ARCH effect in absolute value is detected

between the Bitcoin and Barclays US Aggregate Bond, which shows that previous

information of the one variable will affect the other. The lowest ARCH effect is detected

between Bitcoin and S&P 500 Index.

The covariance ARCH effect is lower than the GARCH effect, which shows that the influence

of the past common information of the variables is less significant than the persistence of

covariance between Bitcoin and indices for the COVID-19 period.


S&P Goldman Sachs Commodity Index, same with the pre-COVID period, while the lowest

between Bitcoin and Down Jones Index.


presented in the Appendix of this thesis. The left column includes the pre-COVID-19 period

results while the right column includes the COVID-19 period results.

From the produced graphs, for the pre-COVID-19 period, it is observed from the covariance

diagrams that the covariance between Bitcoin and indices shows both positive and negative

relationship, with positive and negative movements to be almost equally presented. For the

COVID-19 period, this is not the case. It seems that the covariance between Bitcoin and

indices is mainly negative, with only the covariance between Bitcoin and Gold, Dow Jones

and S&P Golman Sachs Commodity Index pairs to present positive covariance.

Covariance graphs also show that during the COVID-19 period, the absolute value of

covariance for the pairs of Bitcoin and Baltic Dry Index, Gold, S&P 500, is significantly

reduced in relation to the pre-COVI-19 period. For the rest pairs the absolute value of

covariance have almost the same order of magnitude.

109

It can also be pointed out that there is an either positive or negative peak in March 2020 for

all the Bitcoin-Indices pairs, which is the result of the COVID-19 outbreak and quarantine

measures’ introduction all over the world.

110

10. Conclusions

In this part of the thesis a summary of the empirical results will be presented.

For the first part of the study, where the interconnection of four major cryptocurrencies namely

Bitcoin, Ethereum, Cardano and Litecoin is examined with the use of the Ordinary Least

Squares (OLS) method, the conclusion is that there is positive relationship between the Bitcoin

and other cryptocurrencies returns. It is also shown that there exists heteroskedasticity in the

model, there is no autocorrelation and that ARCH effects are present. Moreover, from the

Engle-Granger Cointegration Test that has been performed among the pairs of the four

cryptocurrencies, it is determined that all the series/ pairs except for the Ethereum – Cardano,

are not cointegrated.

For the second part of the study, where the volatility dynamics of the four cryptocurrencies

namely Bitcoin, Ethereum, Litecoin and Cardano in relation to the nine indices namely S&P

500, Dow Jones, Gold Price, Crude Oil Price WTI, Dow Jones Conventional Electricity, Dow

Jones Real Estate, Baltic Dry index (BDI), Barclays US Aggregate Bond Index, S&P Goldman

Sachs Commodity Index is examined with the employment of the Diagonal BEKK model, the

conclusions vary and depend on the studied parameters. More specifically, it is observed that

the average return is positive for all parameters, except for the S&P Goldman Sahs Commodity

Index and Crue Oil WTI in some studied periods, while the studied cryptocurrencies have the

highest weekly loss, highest average return, and highest volatility. The lowest volatility for all

cases is observed for the Barclays US Aggregate Bond, which also presents the lowest average

return, that means that subject index is the most stable index for the studied periods. The Jarque-

Bera normality test values are all greater than the critical value, so the null hypothesis for the

normal distribution is rejected for all the cryptocurrency cases and also from the Augmented

Dockey-Fuler tests for unit it is concluded that all studied variables are stationary. Moreover,

Engle’s test for ARCH effects shows that volatility clustering exists only on Bitcoin among

cryptocurrencies and for the study of Litecoin and indices Gold and Baltic Dry Index do not

also appear to have volatility clustering for the specific studied period.

As for the Diagonal BEKK model results, the Baltic Dry Index and the Crude Oil WTI have

the greatest constants of the conditional variance for all cases that shows a high risk in these

indices. The Barclays US Aggregate Bond and S&P 500 indices have the greatest ARCH

coefficients for all the cases, while the lowest ARCH coefficient belongs mainly to Gold, which

indicates that new information related to Barclays US Aggregate Bond and S&P 500 is of the

attention of the community, something that does not apply for the case of Gold. The lowest

111

GARCH coefficient is for Barclays US Aggregate Bond. Thus, Barclays US Aggregate Bond

has a strong impact of previous innovation and shocks in this market persist the most.

The highest covariance ARCH coefficient is observed for the pairs of studied cryptocurrencies

and S&P 500 index, while the lowest for the pairs of studied cryptocurrencies and Gold, except

for the Cardano case where the lowest is for the Real Estate pair index. The greatest covariance

GARCH coefficient is for the pairs of studied cryptocurrencies and Gold showing strong

volatility clustering, while the lowest for the pairs with Barclays US Aggregate Bond. The

ARCH effect is lower than the GARCH effect, which shows that the influence of the past


between all cryptocurrencies and indices. So, it can be concluded that the S&P 500 index

previous information strongly affects the cryptocurrencies’ returns and vice versa, while for the

Gold and cryptocurrencies case, previous information has the least impact on their returns

comparing to the other indices.

The last part of the study refers to the comparison of the BEKK model results for the pre-

COVID-19 period and the COVID-19 period. It is observed that the highest covariance ARCH

coefficient is observed for the pairs of Bitcoin and S&P 500 and Down Jones indices for the

pre-COVID-19 period while for the COVID-19 period the highest is for the Bitcoin and

Barclays US Aggregate Bond Index. The lowest covariance ARCH coefficient is observed for

the pairs of Bitcoin and Down Jones Conventional Electricity index for the pre-COVID-19

period while for the COVID-19 period the lowest is for the Bitcoin and S&P 500 Index. The

greatest covariance GARCH coefficient is for both periods between Bitcoin and S&P Goldman

Sachs Commodity Index, showing strong volatility clustering, while the lowest is observed for

the Bitcoin and Baltic Dry Index for the pre-COVID-19 period, while for the Bitcoin and Dow

Jones for the COVID-19 period.

An important implication of the above results is that investors who select one of the studied

cryptocurrencies for their portfolio, should be aware that their returns are interconnected, so

they move together. Thus, investing only in these cryptocurrencies will not diversify away

portfolio risk.

Also, investors shall keep in mind that for all studied cryptocurrencies, the previous

information of S&P 500 index and Cryptocurrencies will strongly affect the other, which does

not apply for the case of Gold and Cryptocurrencies. Moreover, Cryptocurrency and Gold

pairs present the highest volatility clustering, while Cryptocurrency and Barclays US

Aggregate Bond the lowest. In this regard, investing only on

112

The results imply that own transmissions are always much larger than the cross-market

spillovers.

The study outcome has considerable implications for portfolio managers and institutional

investors in the evaluation of investment and asset allocation decisions. The cryptocurrency

and asset markets participants should focus on the assessment of the worth of across linkages

among these markets as well as their volatility transmissions. Owning a high level of

volatility creates anxiety and so investors might become more risk averse. So, diversification

of investment portfolio targets to the maximization of returns and minimization of the risk.

The findings also have relevant implications for policymakers in the context of the

cryptocurrency markets and their use in parallel with other financial assets and indices.

Particularly, international portfolio managers and hedgers could better understand the

volatility linkage between cryptocurrency and asset market overtime. This may be useful for

the forecasting of the behaviour of the cryptocurrency market by capturing the available

information for the indices and asset markets. In the same way, governments could also be

guided from these results, and understand in a higher level the cryptocurrency linkage with

the financial assets and global indices.

113

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Appendix – BEKK Model Conditional Variances and Covariances Graphs

A. Bitcoin and Indices

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B. Ethereum and Indices

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C. Cardano and Indices

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D. Litecoin and Indices

133

E. Bitcoin and Indices for the Pre-COVID-19 and COVID-19 periods

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