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 18: GARCH BEKK Model Results for Ethereum and Indices .................................... 80
Table 19: GARCH BEKK Model Results for Ethereum and Indices .................................... 81
Table 20: Cardano & Indices Descriptive Statistics .............................................................. 84
Table 21: GARCH BEKK Model Results for Cardano and Indices ...................................... 86
Table 22: GARCH BEKK Model Results for Cardano and Indices ...................................... 87
Table 23: GARCH BEKK Model Results for Cardano and Indices ...................................... 88
Table 24: Litecoin & Indices Descriptive Statistics .............................................................. 91
Table 25: GARCH BEKK Model Results for Litecoin and Indices ...................................... 93
Table 26: GARCH BEKK Model Results for Litecoin and Indices ...................................... 94
Table 27: GARCH BEKK Model Results for Litecoin and Indices ...................................... 95
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
Table 31: GARCH BEKK Model Results for Bitcoin and Indices for the pro-COVID 19
period................................................................................................................................ 102
Table 32: GARCH BEKK Model Results for Bitcoin and Indices for the pro-COVID 19
period................................................................................................................................ 103
Table 33: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period
......................................................................................................................................... 104
Table 34: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period
......................................................................................................................................... 105
Table 35: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period
......................................................................................................................................... 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)
19
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
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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.
∗∗∗ Indicates statistical significance at the 1% level.
∗∗ Indicates statistical significance at the 5% level.
∗ Indicates statistical significance at the 10% level.
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.
∗∗∗ Indicates statistical significance at the 1% level.
∗∗ Indicates statistical significance at the 5% level.
∗ Indicates statistical significance at the 10% level.
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.
∗∗∗ Indicates statistical significance at the 1% level.
∗∗ Indicates statistical significance at the 5% level.
∗ Indicates statistical significance at the 10% level.
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.
∗∗∗ Indicates statistical significance at the 1% level.
∗∗ Indicates statistical significance at the 5% level.
∗ Indicates statistical significance at the 10% level.
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.
∗∗∗ 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.
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
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
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
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
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
Note: This table presents the descriptive statistics for the weekly return series of 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, 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.
The Jarque-Bera normality test values are all greater than the critical value, so the null
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.
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.
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
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
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
the Covariance GARCH coefficient as B(1,1)*B(2,2).
80
Table 18: GARCH BEKK Model Results for Ethereum and Indices
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
DJ CONV ELEC DJ REAL EST DJ
Coefficient stand error prob coefficient stand error prob coefficient stand error Prob
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
Table 19: GARCH BEKK Model Results for Ethereum and Indices
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
GOLD SP500 SP GOLDMAN
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
The BEKK models above, include both index (or cryptocurrency) specific volatility and
index-bitcoin pair volatility spill over effects.
The log likelihood for all developed models is higher than 405 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 ARCH coefficients, meaning the coefficients A, measure the impact of previous
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.
The GARCH coefficients, meaning the coefficients B, examine the persistence of the return
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.
From these results we conclude that there is strong evidence of GARCH effect and existence
of weaker ARCH effect. As a result, Ethereum 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
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.
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 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
The conditional variances and covariances graphs which were calculated by BEKK model are
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.
The conditional covariance graphs are plotting the magnitude of volatility.
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
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 203.
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
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.
The Jarque-Bera normality test values are all greater than the critical value, so the null
hypothesis for the normal distribution is rejected.
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.
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.
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
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
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
the Covariance GARCH coefficient as B(1,1)*B(2,2).
87
Table 22: GARCH BEKK Model Results for Cardano and Indices
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
DJ CONV ELEC DJ REAL EST DJ
Coefficient stand error prob coefficient stand error prob coefficient stand error Prob
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
A(1,1)*A(2,2), while the Covariance GARCH coefficient as B(1,1)*B(2,2).
88
Table 23: GARCH BEKK Model Results for Cardano and Indices
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
GOLD SP500 SP GOLDMAN
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
GARCH coefficient as B(1,1)*B(2,2).
89
The BEKK models above, include both index (or cryptocurrency) specific volatility and
index-bitcoin pair volatility spill over effects.
The log likelihood for all developed models is higher than 201 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 ARCH coefficients, meaning the coefficients A, measure the impact of previous
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.
The GARCH coefficients, meaning the coefficients B, examine the persistence of the return
volatility. Most GARCH coefficients are above 0.5204, 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 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.
From these results we conclude that there is strong evidence of GARCH effect and existence
of weaker ARCH effect. As a result, Cardano 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
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.
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 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
The conditional variances and covariances graphs which were calculated by BEKK model are
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.
The conditional covariance graphs are plotting the magnitude of volatility.
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
Note: This table presents the descriptive statistics for the weekly return series of 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, 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
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.
The Jarque-Bera normality test values are all greater than the critical value, so the null
hypothesis for the normal distribution is rejected.
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.
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.
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
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
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
the Covariance GARCH coefficient as B(1,1)*B(2,2).
94
Table 26: GARCH BEKK Model Results for Litecoin and Indices
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
DJ CONV ELEC DJ REAL EST DJ
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
A(1,1)*A(2,2), while the Covariance GARCH coefficient as B(1,1)*B(2,2).
95
Table 27: GARCH BEKK Model Results for Litecoin and Indices
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
GOLD SP500 SP GOLDMAN
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
GARCH coefficient as B(1,1)*B(2,2).
96
The BEKK models above, include both index (or cryptocurrency) specific volatility and
index-bitcoin pair volatility spill over effects.
The log likelihood for all developed models is higher than 513 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 ARCH coefficients, meaning the coefficients A, measure the impact of previous
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.
The GARCH coefficients, meaning the coefficients B, examine the persistence of the return
volatility. Most GARCH coefficients are above 0.5792, 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, 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, Litecoin and indices shocks are influenced by past
information which is common to the respective assets.
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.
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 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.
The conditional variances and covariances graphs which were calculated by BEKK model are
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.
The conditional covariance graphs are plotting the magnitude of volatility.
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,
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 18 July 2010 to 31 December
2019. 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 493.
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,
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 01 January 2021 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 86.
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.
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 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
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 0.01653 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
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 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
Covariance GARCH coefficient as B(1,1)*B(2,2).
102
Table 31: GARCH BEKK Model Results for Bitcoin and Indices for the pro-COVID 19 period
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
DJ CONV ELEC DJ REAL EST DJ
Coefficient stand error prob coefficient stand error Prob coefficient stand error Prob
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
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 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 Covariance GARCH coefficient as B(1,1)*B(2,2).
103
Table 32: GARCH BEKK Model Results for Bitcoin and Indices for the pro-COVID 19 period
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
GOLD SP500 SP GOLDMAN
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
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 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 Covariance GARCH
coefficient as B(1,1)*B(2,2).
104
Table 33: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period
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 0.02711 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
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 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
Covariance GARCH coefficient as B(1,1)*B(2,2).
105
Table 34: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
DJ CONV ELEC DJ REAL EST DJ
Coefficient stand error prob coefficient stand error Prob coefficient stand error Prob
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
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 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 Covariance GARCH coefficient as B(1,1)*B(2,2).
106
Table 35: GARCH BEKK Model Results for Bitcoin and Indices for the COVID-19 period
GARCH = C + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
GOLD SP500 SP GOLDMAN
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
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 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 Covariance GARCH
coefficient as B(1,1)*B(2,2).
107
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 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
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.
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.
The highest GARCH effect, and thus volatility clustering, is detected between Bitcoin and
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.
The highest GARCH effect, and thus volatility clustering, is detected between Bitcoin and
S&P Goldman Sachs Commodity Index, same with the pre-COVID period, while the lowest
between Bitcoin and Down Jones Index.
The conditional variances and covariances graphs which were calculated by BEKK model are
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
common information of the variables is less significant than the persistence of covariance
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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119
120
121
B. Ethereum and Indices
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123
124
125
C. Cardano and Indices
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D. Litecoin and Indices
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E. Bitcoin and Indices for the Pre-COVID-19 and COVID-19 periods
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