Master’s Thesis
Correlation of
cryptocurrencies:
A dynamic investigation
Author: Annick Torres Stienissen
Master of Science in Finance and Banking
UPF Barcelona School of Management
Academic year 2020 - 2021
Advisor: José B. Olmo
Correlation of cryptocurrencies: a dynamic investigation
ABSTRACT
The research line of this paper aims to capture and detect contagion between Bitcoin
and the main factors that could have an impact on Bitcoin such as; Ethereum, Ripple, S&P 500,
MSCIWorld, MSCIEM50, Gold, VIX, FSI, and new daily cases and deaths due to Covid-19.
For such purpose, the paper has been structured in three parts. The first part, aimed to detect
the change points in variance from 01/11/2019 to 31/03/2021 using daily data. Main results
suggested that Bitcoin change points were: 07/03/2020, 11/03/2020, and 20/03/2020. For
Ethereum were: 07/03/2020 and 20/03/2020, and for Ripple were: 7/12/2020, 20/12/2020, and
08/01/2021. These dates coincide with the announcement of COVID-19 virus as a global
pandemic (11/03/2020) and the third wave (December 2020 to the 8 of January, 2021). In the
second part of the analysis a DCC-MGARCH model was implemented, in which the
persistence of volatility, co-movement, and conditional correlation were studied between the
different cryptocurrencies and, between Bitcoin, the Equity Indices and Gold. Main results
suggested that cryptocurrencies are positively correlated while no correlations were found
between Bitcoin and the Equity Indices, nor Bitcoin and Gold. Finally, a Johansen test was
done to identify co-integration relationships between Bitcoin, the S&P 500, VIX, FSI and the
Covid-19 variables. No co-integration relationships were found. Finally, a Granger causality
test was performed among the variables. Main results suggested that the S&P 500 was Granger
causing Bitcoin, as well as the VIX. Daily new deaths was Granger causing Bitcoin and new
daily cases due to Covid-19. Finally, a relationship was found between new daily cases and
new daily deaths.
Keywords: Change Point Detection analysis, COVID-19, DCC-MGARCH, Granger
Causality, VEC, VAR, Bitcoin
JEL Classification: C12, C22, C51, C52, C58, G00
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Correlation of cryptocurrencies: a dynamic investigation
TABLE OF CONTENT
1.
INTRODUCTION AND OBJECTIVES ........................................................................ 7
2.
DESCRIPTION OF VARIABLES................................................................................ 12
2.1 Cryptocurrencies .......................................................................................................... 12
2.1.1 Bitcoin (BTC) .................................................................................................................... 12
2.1.2 Ethereum (ETH)................................................................................................................ 14
2.1.3 Ripple (XRP) ..................................................................................................................... 15
2.2 Stock markets ............................................................................................................... 16
2.2.1 S&P500 .............................................................................................................................. 16
2.2.2 MSCI World....................................................................................................................... 17
2.2.3 MSCI EM50 ....................................................................................................................... 17
2.3 Gold .................................................................................................................................. 17
2.4 Fear indices ................................................................................................................... 18
2.4.1 VIX ...................................................................................................................................... 18
2.4.2 FSI ...................................................................................................................................... 19
2.5 COVID-19 variables..................................................................................................... 19
3.
LITERATURE REVIEW .............................................................................................. 20
3.1 First approach - Cryptocurrencies ............................................................................. 20
3.2 Second approach - Bitcoin, Equity Indices and Gold ............................................... 21
3.3 Third approach - Model 1: Bitcoin and S&P 500 and VIX/FSI .............................. 23
3.4 Third Approach - Model 2: Bitcoin and S&P 500 and Covid-19 ............................ 25
4.
METHODOLOGY ......................................................................................................... 26
4.1 First approach - Change Point Analysis .................................................................... 26
4.1.1 Change point detection in variance ............................................................................... 26
4.1.1.1 Model and Methodology ................................................................................... 27
4.1.1.1.1 PELT........................................................................................................... 27
4.1.1.1.2 AMOC......................................................................................................... 28
4.2 Second Approach - DCC-MGARCH ......................................................................... 29
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Correlation of cryptocurrencies: a dynamic investigation
4.3 Third approach - VEC and VAR - Granger Causality tests .................................... 30
5.
PRELIMINARY ANALYSIS ........................................................................................ 32
5.1 Data and Descriptive Statistics ................................................................................... 33
5.1.1 Bitcoin, Ethereum, and Ripple ........................................................................................ 33
5.1.2 Equity Indices, Gold, Fear Indices and Covid-19 variables ...................................... 35
6.
RESULTS ........................................................................................................................ 39
6.1 Change Point Detection Results .................................................................................. 39
6.1.1 Bitcoin ................................................................................................................................ 39
6.1.2 Ethereum ............................................................................................................................ 40
6.1.3 Ripple ................................................................................................................................. 41
6.1.4 The influence of COVID-19 pandemic........................................................................... 43
6.2 DCC MGARCH Model................................................................................................ 44
6.2.1 Stationarity ........................................................................................................................ 45
6.2.2 DCC MGARCH results .................................................................................................... 47
6.3 VEC, VAR and Granger Causality tests model ........................................................ 51
6.3.1 Co-integration analysis between Bitcoin and S&P 500, VXI, and FSI ..................... 51
6.3.1.1 Granger causality test results ............................................................................ 53
6.3.2 Co-integration analysis between Bitcoin and S&P 500, and Covid-19 variables ... 56
6.3.2.1 Granger causality test results ............................................................................ 58
7.
CONCLUSION AND FURTHER RESEARCH .......................................................... 62
REFERENCES ....................................................................................................................... 66
ANNEX 1. CRYPTOCURRENCIES COMPARISON ...................................................... 71
ANNEX 2. IMPULSE RESPONSE FUNCTION PLOTS BITCOIN, S&P 500 AND VIX
.................................................................................................................................................. 72
ANNEX 3. IMPULSE RESPONSE FUNCTION PLOTS BITCOIN, S&P 500 AND
COVID-19 VARIABLES ...................................................................................................... 73
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Correlation of cryptocurrencies: a dynamic investigation
LIST OF TABLES
Table 2. Logarithmic returns of Bitcoin, Ethereum, and Ripple – Summary statistics .......... 34
Table 3. Equity Indices – daily logarithmic returns. Descriptive Statistics ............................ 35
Table 4. Gold – daily logarithmic returns. Descriptive Statistics ........................................... 36
Table 5. Fear indices. Descriptive Statistics ........................................................................... 37
Table 6. New daily Covid-19 Cases and New Daily Deaths due to Covid-19. Descriptive
Statistics ................................................................................................................................... 38
Table 7. Bitcoin Change Point detection - Results.................................................................. 40
Table 8. Ethereum Change Point detection - Results .............................................................. 41
Table 9. Ripple Change Point detection - Results................................................................... 42
Table 10. Variance before and after change point’s location .................................................. 43
Table 11. Checking for stationarity – ADF test. Without trend .............................................. 46
Table 12. Checking for stationarity – Phillips–Perron test, for non-stationary variables. ...... 46
Table 13. Checking for stationarity – ADF test. Without trend .............................................. 47
Table 14. Checking for stationarity – Phillips–Perron test, for non-stationary variables. ...... 47
Table 15. DCC MGARCH Model Bitcoin, Ethereum and Ripple .......................................... 48
Table 16. Correlations Bitcoin, Ethereum and Ripple ............................................................ 49
Table 17. DCC MGARCH Model Bitcoin, Equity Indices and Gold ..................................... 50
Table 18. Correlations Bitcoin, Equity Indices and Gold ....................................................... 50
Table 19. Lag order identification ........................................................................................... 52
Table 20. Number of Co-integrated relationships ................................................................... 52
Table 21. Granger causality Wald tests ................................................................................... 55
Table 22. Lag order identification ........................................................................................... 56
Table 23. Number of Co-integrated relationships ................................................................... 58
Table 24. Granger causality Wald tests ................................................................................... 60
Table 1. Cryptocurrencies comparison.................................................................................... 71
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Correlation of cryptocurrencies: a dynamic investigation
LIST OF FIGURES
Figure 1. Daily log returns of Bitcoin, Ethereum, and Ripple prices ..................................... 34
Figure 2. Bitcoin change point detection using PELT and AMOC methods .......................... 39
Figure 3. Ethereum Change Point detection using PELT and AMOC methods. .................... 40
Figure 4. Ripple Change Point detection using PELT and AMOC methods .......................... 41
Figure 5. Bitcoin and S&P 500 daily log returns, and the VIX differentiated daily index ..... 55
Figure 6. Bitcoin daily log returns, S&P 500 daily log returns and the Covid-19 new daily
cases and deaths ....................................................................................................................... 61
Figure 7. Impulse response function plot: Bitcoin, S&P 500 and VIX .................................. 72
Figure 8. Impulse response function plot: Bitcoin, S&P 500 and Covid-19 variables ........... 73
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Correlation of cryptocurrencies: a dynamic investigation
1. INTRODUCTION AND OBJECTIVES
According to the Bitcoin Organization (2021), Bitcoin is a decentralized, peer-to-peer
version of electronic cash that allows transactions to be irreversible through the cryptographic
treatment of transaction information, thereby eliminating the need for trust between agents and
intermediaries, such as financial institutions. Bitcoin is an open source and a completely digital
currency that uses cryptography as a method to validate transactions and control the creation
of new money, which is why it has been categorized as a cryptocurrency. Since its beginning
in January 2009, Bitcoin has become more widely known and used while seeing a rapid growth,
since 2012, in its price, number of transactions, as well as, the size of the companies that are
currently using, or experimenting, with Bitcoins. In fact, Bitcoin’s popularity has increased in
such extension that it became the largest digital currency by both market capitalization and
number of daily transactions (Stephen Ozvatic, 2015).
Due to the strong growth that Bitcoin has had in recent years and its participation in the
money market, it has become a political, economic and financial challenge for its stakeholders;
For example, some countries such as China and Japan have already taken measures regarding
transactions with cryptocurrencies, as well as several analysts and communication media that
speculate about cryptocurrencies’ performance. Additionally, cryptocurrencies usually
experience significant changes in price movement in short and unexpected periods of time, and
some analysts have identified that these changes are due to important events or events around
the world, provoking structural breaks in the data. Also, uncertainty of the cryptocurrency
market attracts scientist to keep studying its movements, making predictions and studying and
modelling against market indices, commodities, fear indices, and other cryptocurrencies,
among others. Likewise, as the price of cryptocurrencies increases, so does the interest of
investors and the general public, which forms a potential for a developing bubble.
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Correlation of cryptocurrencies: a dynamic investigation
On the other hand, given the current situation we are facing due to COVID-19
pandemic, it has become without any doubt one of the most discussed topics among researchers
at the time of doing this paper. COVID-19 pandemic and its impact on several economic and
financial factors has become the centre of attention. In fact, COVID-19 will play an important
role in this thesis as it will help to find irregularities in the data set and to find any possible
long-term or causality relationship between Bitcoin, Equity Indices and COVID-19.
Thus, due to the possible relationship that may exist between the price behaviour of
Bitcoin and other cryptocurrencies, as well as, gold, equity indices, fear indices, and COVID19, the main goal of this thesis is to use daily data of Bitcoin and the corresponding selected
variables, including COVID-19, which are: Ethereum and Ripple as cryptocurrencies, S&P
500, MSCI World, and MSCI EM50 as equity indices, VIX and FSI as fear indices, and Gold
as a commodity, to explain and model the dynamic correlation of Bitcoin with the above
mentioned variables.
To do so, the thesis will be divided in three main approaches/objectives. The first
approach, is the so called Change Point Detection analysis where daily data from 01/11/2019
to 31/03/2021 to observe any structural break on Bitcoin, Ethereum, and Ripple, time series,
using daily logarithmic returns, through two main techniques: PELT and AMOC, will be used.
Results obtained suggested that there were two change points detected for Bitcoin using PELT
technique: 11/03/2020 and 20/03/2020, while using the AMOC technique, one change point
was detected: 07/03/2020. Ethereum’s first change point detected by both techniques was
07/03/2020, while the second change point detected by PELT was 20/03/2020. Lastly, the
Ripple results significantly differed from Bitcoin and Ethereum, being the two change points
detected using PELT: 07/12/20 and 08/01/2021, and using AMOC: 20/12/2020. Given the fact
that the COVID-19 virus was announced, officially, as a global pandemic the 11/03/2020 and
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Correlation of cryptocurrencies: a dynamic investigation
that as of today it has had three waves – the first wave took place between 15th March and 30th
June, the second wave took place between 1st July and 15th October, and the third wave took
place from the very end of December 2020 to 8th January, 2021 – it can be concluded that the
COVID-19 pandemic had an impact on the volatility and has created change points in the time
series of the cryptocurrencies.
After having identified several structural breaks in the data set in all the three
cryptocurrencies, the second approach was defined. The second approach focuses on
analysing the persistence of volatility, the co-movement, and the conditional correlation, from
an econometric perspective, between Bitcoin and the cryptocurrencies, as well as, between
Bitcoin, the Equity Indices and Gold. To do so, a DCC-MGARCH time series model was
applied. In the first DCC-MGARCH model were the cross correlation between
cryptocurrencies (Bitcoin, Ethereum and Ripple) was studied, main results suggested that
Bitcoin had no influence and effect on Ethereum nor Ripple. However, the persistence of
volatility was very high for Ethereum (0.853) and Bitcoin (0.92), while for Ripple it was very
low (0.11). Regarding the DCC conditional dynamic correlation equation, results suggested
that the co-movement of the currencies is changing over time. Finally, a high and positive
correlation between Ethereum and Bitcoin (0.831), followed by a positive correlation between
Ethereum and Ripple (0.559) and a very low positive correlation between Bitcoin and Ripple
(0.384) was observed, all of them statistically significant at 1% level.
In the second DCC-MGARCH model a cross correlation between Bitcoin, the Equity
Indices (S&P 500, MSCIWorld, and MSCIEM50) and Gold was studied. Results suggested
that Bitcoin has an influence and effect on the MSCIEM50 index at a 5% significance level.
The persistence of volatility was on average 0.94 for each of the specifications, the comovement of the variables was changing over time, and only two statistically significant
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Correlation of cryptocurrencies: a dynamic investigation
correlations at 1% level were observed which were: Gold and S&P 500 (-0.108), and Gold and
the MSCIWorld (-0.149).
To end up, a third approach was proposed which focused on studying long-term
relationship and causal relationship between Bitcoin and the selected variables as correlation
does not imply causality, nor co-integration. Being the selected variables: S&P 500, VIX, FSI,
and COVID-19 variables (new daily cases and new daily deaths due to COVID-19). This third
part answers mainly to two questions: (1) Are there any co-integrated relationships between
Bitcoin and the selected variables? In order to answer this question a VEC model was applied.
(2) Does any of the selected variables help to predict Bitcoin daily logarithmic returns, and vice
versa? In order to be able to answer this question a VAR model and, consequently, a Granger
Causality Wald test was implemented.
To answer these two questions, the analysis was split in two different models. The first
model was studying the existence of co-integration relationships and causality between Bitcoin,
the S&P 500 and the two Fear Indices, and the second model was used to study the same but
between Bitcoin, the S&P 500, and the two COVID-19 variables.
Main results suggested that no co-integration relationships were found in any model.
However, in the first model, results suggested that a rise of the S&P 500 implies a negative
effect of -0.36 on Bitcoin. On the contrary, Bitcoin was not Granger causing S&P 500.
Additionally, results suggested that the VIX index was not Granger causing Bitcoin daily log
returns nor the S&P 500 daily log returns. However, it was found that the S&P 500 was Granger
causing the VIX index. Indeed, a rise of the S&P 500 implied a negative effect of -83.28 on
the first difference of the VIX index.
In relation the second model, it was observed that new daily deaths due to Covid-19
Granger causes Bitcoin. Indeed, an increase in the number of deaths due to Covid-19 implied
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Correlation of cryptocurrencies: a dynamic investigation
a positive effect of 7.19e-06. On the contrary, Covid-19 variables were not Granger causing
S&P 500. Finally, it was also observable that new daily deaths due to Covid-19 Granger cause
new daily cases due to Covid-19, meaning that an increase in new daily deaths due to Covid19 implies a negative effect of -0.53 on new daily cases. To end up, new daily cases of Covid19 also Granger cause new daily deaths due to Covid-19. Indeed, a rise (fall) in new daily cases
implies a negative (positive) effect on new daily deaths.
These objectives are of interest as I will contribute to the fast-growing literature of the
correlation between Bitcoin and other assets, the value added of sentiment variables on return
and volatility predictions, and the impact of COVID-19 on Bitcoin dynamics.
The structure of this paper is organized as follows. Section 2 presents the definitions of
the variables. Section 3 presents the literature review. Section 4 presents the methodology of
the study. Section 5 introduces the preliminary analysis. Section 6 presents the results. Finally,
the paper concludes in Section 7.
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Correlation of cryptocurrencies: a dynamic investigation
2. DESCRIPTION OF VARIABLES
2.1 Cryptocurrencies
2.1.1 Bitcoin (BTC)
This section discusses the definition of Bitcoin, how it operates, the risks it entails and its
market history. Based on a paper by Satoshi Nakamoto1 in 2008, Bitcoin was the first
cryptocurrency created. However, other digital currencies already existed before Bitcoin, such
as E-gold, but those were mostly able to last very short time before being shut down, or actually
collapsing, see, e.g. Ben-Sasson et al. (2014); Reid and Harrigan (2013). Nevertheless, other
cryptocurrencies have emerged after the appearance of Bitcoin with different levels of success,
such as peer-coin and/or lite-coin.
The main difference between cryptocurrencies such as Bitcoin and previous digital
currencies is the existence of a third party that validates the transaction (mining process) and
stops the so called double-spending. However, bitcoin does not need a third party to validate
transactions as it is designed to be decentralised (i.e. no central bank needed) which allows a
peer-to-peer network to do this validation cryptographically through a proof-of-work system
which can be checked, trusted and is irreversible.
Bitcoins operate through a current owner of Bitcoins who can transfer them to another
owner by creating a digital history of the previous transaction of those bitcoins along with the
public key of the next owner. Each transaction is recorded in a public ledger through
blockchain2, making sure that the transfer of Bitcoins is controlled through a chain of
transactions. When a transaction ends, they are grouped into blocks that are validated by nodes
(anyone with the software and hardware can become a node) on the peer-to-peer network
Satoshi Nakamoto’s identity is not confirmed. There are many who believe that the name is a pseudonym, however it is not
known whether Nakamoto is a person or a group.
2
Blockchain: digital record of transactions
1
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Correlation of cryptocurrencies: a dynamic investigation
software which is designed to solve a cryptographic problem that appears as part of the proofof-work system. It is a system in which the answer to the problem is very difficult to find and
it is very time-consuming. However, it is easily verified as correct, see Karame, Androulaki,
and Capkun (2012) or Ben-Sasson et al. (2014). Thus, every time a node solves the problem,
the block is added to the blockchain, which is the only chain where you can find all the
validated blocks and it is the only source of truth which solves the double-spending problem.
The block confirmation time is of 10 minutes and the size of the reward has a limit of 21
million3 Bitcoins in circulation. An interesting observation in the Bitcoin market is that Bitcoin
is known for its price volatility, hence large spot price increases followed by smaller (still large)
declines in magnitude.
When Bitcoin went public, people started to mine new currency units by running the so
called mining nodes4. According to the Gold Price Organization (2021), during 2010, Bitcoin
was traded for the first time, on a Bitcoin forum through peer-to-peer, and during 2011 and
2012 Bitcoin was able to arrive and exceed parity with the US dollar, reaching a value of $31
per Bitcoin (approx.) in June 2011 before falling to less than 10 percent of that value, remaining
so during the following year. In March 2013, the international bailout of Cyprus pushed the
price of bitcoin up to 500 percent to $238 from February 2013, which declined substantially in
April 2013 after a possible attack to the Bitcoin exchange Mt. Gox5 by a hacker. A consequently
fall was seen at the end of 2014 and at the beginning of 2015, bringing some rumours about
the possible collapse of the coin. However, the price continued to falter while big tech
companies like Microsoft started to accept BTC as a payment method. In 2017, bitcoin started
to be more known and the demand increased so much that the price increased from $1,000 to
3
Based on 50 bitcoins as the starting reward, 50 · 210000 · (1 + 1 2 + 1 4 + · · · ) = 21000000 (Stephen Ozvatic, 2015)
Mining nodes; special network nodes
5 Mt. Gox; The world’s largest bitcoin exchange
4
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Correlation of cryptocurrencies: a dynamic investigation
$20,000. However, at the end of 2017, the price of Bitcoin was continuously decreasing and
this gave room to rumours about a bubble burst. However, during the first half of 2019 the
price started to recover and stabilized around $10,000.
To end up, main risks are defined: (1) the main external threat that Bitcoin has, which
is potentially the largest, are the hackers, fraud and malware. The above mentioned incident
with Mt. Gox is an example, as they were forced to close (went bankrupt) due to hackers. Also,
the lack of local and global regulation on Bitcoin provides opportunities for fraud and other
illegal practices6. (2) Another important risk is the maximum amount of 21 million Bitcoins.
The accumulation of Bitcoins would imply a downward pressure on the amount of transactions,
hence a downward pressure on the fees created for miners, decreasing their incentive and
increasing the possibility of having what is called as: attacks from history-revision. (3)
According to Barber et al. (2012), another important problem is the fact of losing your Bitcoins
and not being able to recover them. This can happen if you lose a Bitcoin e-wallets, or it can
also happen when companies that manage e-wallets make errors. (4) According to the same
author, the fourth risk is the history-revision attack. This directs people to the question of
whether the network of miners can be trusted and if Bitcoins are feasible.
2.1.2 Ethereum (ETH)
The ETH project was created in July 2015 to allow flexibility and increase its
functionality, on a blockchain, to provide the capability to program different types of smart
contracts in the ETH system in a decentralized way. This flexibility that ETH smart contracts
offered, attracted many developers, users and investors, which led ETH to become the second
largest cryptocurrency. ETH is not intended to be a currency but a by-product. When a contract
is executed, it is verified by all the updated participants of the blockchain. It is done like this
6
(e.g. black market purchases)
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Correlation of cryptocurrencies: a dynamic investigation
to guarantee the correct execution of the blockchain by consensus. What mainly differentiates
Ethereum from Bitcoin is the Turing programming language that allows anyone to create
contracts for any use (Kim et al., 2018)
Rouhani and Deters (2018) state that ETH has two types of accounts: (1) Contract
accounts and (2) Externally Owned Accounts (EOA) which are those accounts that are used by
users to directly send transactions. Furthermore, certain smart contracts use some mechanisms
that allow the exchange, or sharing digital assets, which are known as crypto-tokens, on the
blockchain. Contracts have a permanent storage and a set of functions that can be requested by
users or by other contracts. Current users have the possibility to send transactions to the ETH
network for 3 purposes: (1) create a new contract, (2) call on a function from a contract, and
(3) send an ETH to other users or to contracts. Finally, those ETH transactions start with a first
block (genesis block) and then the other transactions that are created, process and develop new
blocks (Rouhani and Deters, 2018).
2.1.3 Ripple (XRP)
The Ripple cryptocurrency, also known as XRP is the fourth largest cryptocurrency in
the market in terms of market capitalization. According to Armknecht et al. (2015), the display
of Ripple, nowadays, is just managed by Ripple Labs, and Rosner and Kang (2016) suggest
that XRP is an open-source Internet software that allows (in a very easy way) its users to make
payments beyond national boundaries in different currencies. Thus, an advantage of XRP is
that it offers other currencies to make transactions, not just its cryptocurrency.
Furthermore, Rosner and Kang (2016) keep insisting that XRP’s protocol is using a
distributed ledger, which is collection of updated financial accounts through which XRP’s users
can make payments across borders which are faster, cheaper and more efficient than traditional
payments. In fact, that is why the XRP cryptocurrency is of bank’s interest. According to
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Correlation of cryptocurrencies: a dynamic investigation
Moreno-Sanchez, Zafar, and Kate (2016), the bank Santander stated that embracing XRP could
make them save around $20 billion a year.
A reason why we include XRP in this paper is because it offers two payment types: (1)
Direct XRP payment, which according to Moreno-Sanchez et al. (2016), it is basically the
transfer of Ripple between two wallets that do not need a credit track between them. (2) Pathbased settlement transactions. Those are transactions that transfer any kind of credit between
two wallets. The XRP network, is a replicated public database (XRP ledger). Meaning that
everyone can see the historical activity of all their transactions, however those transactions are
transparent as they are done under pseudonyms (Armknecht et al., 2015).
Finally, the main difference between Bitcoin and Ripple is that in Bitcoin transactions
can be done from different accounts while in Ripple payments are done from a unique account
as input. Thus, to make the XRP protocol more secure, what they require is to have a small
reserve of 20 XRP to, at least, be able to create transactions. In Annex 1 you will find a
comparison table between the three cryptocurrencies.
2.2 Stock markets
2.2.1 S&P500
The S&P 500 refers to the Standard & Poor's company and is a stock index that is
composed of the 500 largest companies in the United States that are publicly traded, and it is
weighted according to the market capitalization of each of the companies. The way in which it
is weighted is by using a float weight which means that the market cap of each company is
adjusted according to the number of shares that are available for public trading. Also, the index
is considered to be the best indicator of large capitalization equities in the United States and,
as a result of that, there are many funds which are dedicated to track the behavior or, in other
words: the performance of the S&P 500 (Investopedia, 2021).
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Correlation of cryptocurrencies: a dynamic investigation
2.2.2 MSCI World
The MSCI World Index is composed by 23 different developed countries and 1,562
companies that have large and mid-capitalization. This index covers around 85 percent of the
total free float which is adjusted with respect to the market capitalization of each company in
each country. During 2020 the annual performance of the MSCIWorld, in percentage, was of
15.90, compared to 18.31 of the MSCI Emerging Markets. Finally, the three main key
exposures that may drive risk and return to the index are: (1) momentum, (2) low volatility,
and (3) quality (MSCI, 2021).
2.2.3 MSCI EM50
The MSCI EM 50 Index is a tradable version of the market leading MSCI Emerging
Markets Index. It comprises 50 of the most important components of the leading MSCIWorld
Index. The index excludes some of the smallest emerging market (EM) countries and uses a
certificate which is negotiable and issued by banks which represent the shares in a foreign
company which is traded on a local stock exchange for those markets that do not have easy
access to foreign investors. The index experienced an annual performance rate of 29.87 percent
compared to an 18.31 of the MSCI Emerging Markets. In terms of risk and return exposure,
the order differs a bit compared to the MSCIWorld: (1) low size, (2) quality, and (3) low
volatility (MSCI.1, 2021).
2.3 Gold
Over the years, Gold has become a tradable asset and it maintains its value in turbulent
periods which makes it to be a refuge value. Some of the key factors that affect gold’s price
are: (1) national interest rates as when this increases, gold prices tend to decrease. This is
mainly due to the fact that investors move to government bonds and other assets whose yield
is related to the interest rate, (2) geopolitical events also affect the price of gold as in times of
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Correlation of cryptocurrencies: a dynamic investigation
international tension, the price of gold tends to increase because investors buy the product to
have a high degree of security in times of uncertainty, (3) levels of supply and demand also
affect as the production of gold increases and decreases as time goes by, and so does the supply.
Finally, (4) the industrial production also affect gold prices as when the production increases,
the demand for gold increases, and the other way around (IG.com, 2020).
2.4 Fear indices
2.4.1 VIX
The Cboe Volatility Index (VIX) is a real time index that represents the expectations of
the market for the volatility over the coming thirty days and it is a measure of the level of fear,
stress, or risk in the market. It is necessary to understand that it measures volatility over the
next 30 days. That is, it does not measure past volatility, but future volatility (Investopedia.1,
2021). It is understood that low values of the indicator give moments of market tranquility and
sustained upward trends. On the contrary, high values correspond to moments of panic in which
a sustained downtrend or decline is exacerbated. It is more a barometer of investors' fear of
possible falls, than of their complacency in a market rally. Typically, the VIX has an inverse
relationship to the stock market. When stocks go down, the VIX goes up and vice versa.
Therefore, an increase in stocks will be considered a lower risk factor. Whereas, if it is bearish
and stocks fall, it carries a higher risk. The greater the perceived risk, the greater the volatility.
So this volatility is more susceptible to the direction of the market. A turn or fall to the
downside causes an increase in volatility. A normal reading of the VIX is between 20 and 30,
below 20 investors are not worried, there is complacency. Above 30 indicates that there is
nervousness, that is, fear in the market.
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2.4.2 FSI
The Financial Stress Index (FSI) is an index that measures the level of financial stress
in different financial markets and it is built from 33 financial market variables: yield spreads,
valuation measures, interest rates, among others. This index is interpreted in the following way:
the index will be positive if the contribution of the weighted average stress of the indicator is
positive and, the other way around, the index will be negative if the contribution of the weighted
average stress of the indicator is negative and, the index is zero if the average is zero which
suggests that stress levels are normal.
2.5 COVID-19 variables
Given the current situation with the new occurred pandemic of coronavirus, also known
as COVID-19, and that this topic became the recent research topic among scientists who started
to measure the impact of the pandemic on financial markets, cryptocurrencies and stocks, in
this study I will use two main variables related to Covid-19. The first variable is the ‘New daily
cases of Covid-19’, which are new daily cases meaning that the values are not cumulative, and
the ‘New daily deaths due to Covid-19’, which are new daily deaths meaning that there are no
cumulative values involved.
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3. LITERATURE REVIEW
Many scientist, during the past years, have been analysing which were the factors that were
influencing Bitcoin’s prices and volatility. However, very recent studies have shown that using
the same study method but with data of two months later, results were completely different.
This might be due to the fact that Bitcoin’s market is very unstable but also very unpredictable.
Therefore, there is a need to study whether Bitcoin and the selected variables are correlated. In
the following sub-sections the literature review is being presented for each approach/objective.
3.1 First approach - Cryptocurrencies
The detection of change or structural change in the parameters of a given model has
been an extensive area of investigation, within time series models, when interpreting the results
change points are found by giving a representation to structural changes that are represented
when there are instantaneous or permanent, invariable and unexpected modifications in one or
more components, due to specific events. A structural change or point of change in a time series
occurs when there are modifications in one or more components. Therefore, if a structure that
represents it in a time series is included in the model, a more complete model is reached in
order to arrive at a more precise forecast. When structural changes are present, the
autocorrelation of the series is affected, so the estimation of the simple and partial
autocorrelation function is not effective, since its identification makes it difficult.
By the time of writing this thesis, there were not many studies that were implementing
the Change Point Analysis on cryptocurrencies. However, James, Menzies, and Chan (2021)
– who used a two-phase change point detection algorithm to obtain the change points, which
are also known as structural breaks – found that the cryptocurrency market, during the COVID19 pandemic, was being disrupted compared to the cryptocurrency market before COVID-19.
Additionally, Thies and Molnár (2018) – who used the Bayesian Change Point on Bitcoin
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returns and volatility to study any possible structural break – found out that structural breaks
are very common on Bitcoin returns and volatility. Thus, it can be concluded that Change Point
analysis can be used to detect irregularities in time series.
Aside from the fact the cryptocurrencies are prompt to have structural breaks, it has
been said that Bitcoin and Ethereum maintain a very high correlation, while Bitcoin and Ripple
do not. In fact, according to Juan Sebastián et. al (2020), Bitcoin and Ethereum have very
similar behaviours in terms of volatility. The use of this and other cryptocurrencies began to
increase due to the growth of Bitcoin, it is for this reason that their behaviour was so similar.
Additionally, he found out that Ripple was a bit different from Bitcoin and Ethereum, given
that its rise in 2017 was much more noticeable, this may be because, unlike the other two
cryptocurrencies, Ripple is a centralized currency and in 2017 it was accepted by different
banks such as BBVA, Santander and Bank of America. Finally, his results suggested that the
highest correlation was between Bitcoin and Ethereum with 0.92, followed by Ethereum and
Ripple with 0.85, and Bitcoin with Ripple with 0.83.
To conclude, a Change Point Detection analysis will be implemented in this study as it
helps to check whether there are irregularities in the data set and, as there have been evidences
of the existence of a correlation between cryptocurrencies, a DCC-MGARCH model will be
implemented to study the volatility, co-movement, and conditional correlation of
cryptocurrencies.
3.2 Second approach - Bitcoin, Equity Indices and Gold
Some empirical findings show that there is a relationship between Bitcoin and the S&P
500. In fact, they indicate that the price of Bitcoin can be altered by the S&P 500 returns. Klein
et al. (2018), found that when Bitcoin engages with the S&P 500 during bearish times, it causes
correlations to rapidly turn into positive values. Also, during 2017 and 2018 they found an
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inverse movement of correlation between Bitcoin and the S&P 500. Kjærland et al. (2018)
found that there was a positive relationship between the Bitcoin price and the S&P 500. This
means that when the S&P 500 was increasing by 1 percent, the Bitcoin’s price was increasing
by 1.77 percent. The other way around, authors such as Sovbetov (2018) found that the S&P
500 had a positive (but weak) long-term relationship on Bitcoin, implying that if the S&P 500
was increasing by 1 percent then the Bitcoin’s price was increasing by 0.8 percent. However,
in the short run there was a negative (but weak) relationship, meaning that by 1 percent increase
in the S&P 500, the Bitcoin price was decreasing by -0.2 percent. The same results were also
found by Georgoula et al. (2015).
Going further, Klein et al. (2018) found that by only using the average correlation over
the whole sample Bitcoin was slightly positively correlated to other assets and the MSCIWorld
was the assets with the highest correlation, which was between 0.045 and 0.05. Additionally,
they observed that Bitcoin and the MSCIWorld were negatively correlated when the market
was suffering downturns. However, the correlation was mainly positive when Gold was very
volatile. Thus, as soon as the correlation between Gold and MSCI World becomes positive, the
correlation between Bitcoin and the MSCI World index becomes negative, and the other way
around. Finally, Klein et al. (2018), also found that when Bitcoin engages with the MSCIWorld
during bearish times, this also provoke correlations to be positive very fast which is the case
between Bitcoin and the S&P 500 index.
Furthermore, we have the MSCI EM50 index which, according to Klein et al. (2018),
is the one that has the lowest correlation with Bitcoin compared to S&P 500 and MSCIWorld.
In fact, the same author observed that Bitcoin was negatively correlated to the MSCIEM50
index when the market of this index was in distress. Summing up, there is very little literature
on the correlation between the MSCI EM50 and Bitcoin. However, from the literature observed
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we can identify a low correlation among both variables. Thus, given the little amount of
literature available, it seems reasonable to provide new results including the most recent data
which will provide us with more updated results.
Finally, we have the Gold, which according Kristoufek (2015), who studied the
relationship between Bitcoin and Gold prices, found that BTC was not linked to the dynamics
of gold. Hence, no significant impact was found. Other studies such as the one done by
Sovbetov (2018) and Kjærland et al. (2018), who studied the connection between Bitcoin and
dynamics of Gold also found that gold price was not a significant variable in their studie.
However, there was one author (Poyser, 2017) whose results showed that Bitcoin’s price was
negatively correlated to gold’s price. In fact, he stated that if the gold price was increasing by
1 percent, then Bitcoin price was decreasing by -0.6 percent. Other authors like Conrad et al.
(2018), studied long-term volatility of Bitcoin compared to Gold, and found that Bitcoin’s
volatility was different from that of Gold. Thus, it is observable that most of the researchers’
results conclude that the change in price of cryptocurrencies is not influenced by the price of
gold. Thus, as most of the studies were done before the COVID-19 pandemic, it can be
interpreted that by including the most recent data I will be able to provide better and more
updated results.
For such purpose, a DCC-MGARCH model will be used to study the conditional on
past history covariance matrix of the dependent variable (Bitcoin) to follow a dynamic
conditional correlation study to obtain the persistence of volatility, co-movement and the
conditional correlation between Equity Indices, Gold and Bitcoin.
3.3 Third approach - Model 1: Bitcoin and S&P 500 and VIX/FSI
The VIX and FSI are two other variables that were included in the model since several
authors included one of the two in their studies. According to Soldevilla Estrada (2017) – who
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used a Granger causation test to study the relationship between Bitcoin and the S&P 500,
Bitcoin’s price and the VIX, Bitcoin’s volatility and the S&P 500, and Bitcoin’s volatility and
the VIX as previously mentioned – found out that Bitcoin was not Granger causing the VIX,
nor the VIX was Granger causing Bitcoin. Furthermore, the same author found out that Bitcoin
was not Granger causing the S&P 500, nor the S&P 500 was Granger causing Bitcoin. These
results were coinciding with other findings such as the one by Ciaian, Rajcaniova, and Kancs
(2016) who, using a similar approach, did not find evidence that financial variables had an
impact on Bitcoin’s prices. Finally, Chung et al. (2011) who studied the volatility, behavior on
returns, and prediction of the S&P 500 and the VIX, found out that both indices behave very
similarly but are not identical.
Additionally, according to Bouri et al. (2018), who studied the conditional dependence
between the FSI and Bitcoin returns, found out that there was a strong dependence between the
FSI and the Bitcoin. In other words, the FSI was strongly Granger-causing Bitcoin returns.
Other similar studies such as the one done by Kristoufek (2015), found out that the FSI was
Granger-causing Bitcoin price but just in one period. Finally, according to Caporin, Corazzini,
and Costola (2019), who used the Bayesian method to study whether the FSI was Granger
causing the S&P 500, found out that the S&P 500 was Granger causing the FSI.
To end up, as there is a lack of studies that demonstrate whether one variable Granger
causes the other variable, any statement specified in this paper might be limited by the bounded
amount of findings. However, according to the above mentioned findings, it can be said that
the VIX does not Granger cause Bitcoin prices, nor the other around, FSI was strongly Grangercausing Bitcoin, Bitcoin does not Granger cause the S&P500, nor the other way around, and
the S&P 500 Granger causes the FSI. For that reason, a VEC, VAR and Granger causality test
to study the relationship between the four variables will be used.
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3.4 Third Approach - Model 2: Bitcoin and S&P 500 and Covid-19
When we talk about financial and macroeconomic developments, we talk about the
existing literature which studied the relationship between Bitcoin prices and macroeconomic
indicators. In fact, according to Panagiotidis et al. (2019) what seems to have a greater impact
on Bitcoin’s prices are the external shocks. The new occurred pandemic of coronavirus, also
known as COVID-19, became the recent research topic among scientists who started to
measure the impact of the pandemic on financial markets, cryptocurrencies and stocks. Some
of them modelled cryptocurrencies’ behavior during the pandemic to analyze if it could behave
as a safe haven, while other authors studied the impact of the pandemic on cryptocurrencies’
prices and/or volatility. However, results showed that Bitcoin was not acting as a safe haven
but rather decreased in price at the same time as the S&P 500 (Chen et al., 2020).
During the outbreak of the coronavirus, Bitcoin was very affected and lost half its value
in a few days. However, there is no literature that explains the fall in the price of Bitcoin.
Nevertheless, according to Ali, Alam, and Rizvi (2020); Apergis and Apergis (2020) and GilAlana and Monge (2020), who studied the impact of COVID-19 on financial markets, found
that they are associated as there was a decline in asset prices and an increase in the volatility
of the market. Additionally, Baig et al. (2020) found that there is an impact of negative
sentiment which leads to an increase of volatility in the market, but also to a decrease of
liquidity. A study done by Goodell and Goutte (2021), who used the wavelet method of
Grinsted, Moore, and Jevrejeva (2004) to daily data of COVID-19 (daily Bitcoin prices but
also number of deaths in the world), found that COVID-19 was causing an increase in Bitcoin
prices. In fact, the author found that during the 5th of April of 2020 to the 29th of April of 2021,
the number of deaths due to Covid-19 were causing an increase in Bitcoin prices, as well as an
increase in S&P 500 prices.
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Summing up, it can be observed that the COVID-19 pandemic has become a widely
analysed topic among researchers and results differ from each other. Thus, it suggests that other
approaches are necessary to give another point of view to the impact of COVID-19 towards
Bitcoin and the S&P 500. For that reason, I will study whether co-integration relationships
between Bitcoin, the S&P 500 and the Covid-19 variables exist, as well as if any variable
Granger causes another variable. To do so, a VEC model and the Granger causality Wald test
will be applied, respectively. Additionally, I will follow the study done by Goodell and Goutte
(2021) and I will take into account new daily deaths and, additionally, new daily cases due to
Covid-19 as Covid-19 variables, to contribute to the existing literature.
4. METHODOLOGY
4.1 First approach - Change Point Analysis
For the detection of the change points, the change point analysis (CPA) is implemented,
this is a statistical tool used to detect changes in the parameters of the distribution in the data
set. Depending on the statistical tests used, models of change points are obtained that can detect
changes in location parameters, scale, combinations of both, or more general changes.
4.1.1 Change point detection in variance
According to Christian Rohrbeck (2013) the change point detection in variance is a
method that has been well studied in the past years and there are several methods in which this
can be performed. The first one is the cumulative sums of square by Inclán and Tiao (1994),
the second method is the penalized likelihood by Yao (1988), and the third one is the Bayesian
posterior odds by Fearnhead (2006). The change point detection is what we call the problem
of estimating the point in which the statistical properties of a sequence of observations change
and it is very important to detect those changes in many areas. Some recent examples include
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medical imaging, finance and climatology among others. Thus, as there is an increasing
collection of signal streams but also time series, there is an increasing need to accurately and
efficiently identify and estimate the location of multiple change points.
Log returns will be used as it is well documented by financial empirical studies as prices
are believed to be unit roots, hence non-stationary, which is solved using log returns.
Additionally, another positive advantage of using log returns is the fact that data is n.n.d
(normalized and normally distributed), and are defined as the first difference of the natural
algorithm.
4.1.1.1 Model and Methodology
In order to study the change point detection of each daily logarithmic returns of Bitcoin,
Ethereum and Ripple time series, the PELT (Pruned Exact Linear Time) and the AMOC (At
most on change point) methods were used. Their assumptions and specifications are expressed
below:
4.1.1.1.1 PELT
The Pruned Exact Linear Time is a method to detect the change point which considers
that the data is sequential and looks for the solution space in a very exhaustively way. With
this method, the ccomputational efficiency is reached by detaching solution paths which are
known to not lead to an optimal point. In this context, pruning will be used by eliminating those
values of T which will never be minima from the minimization that is performed at each
iteration in the following equation:
𝑚+1
∑ [𝐶(𝑦(𝑇𝑖−1 +1):𝑇𝑖 )] + 𝛽𝑓(𝑚),
𝑖=1
According to Dorcas Wambui (2015), the main assumption of the PELT algorithm is
that the change point values increase linearly together with the increase of the dataset. In other
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words, the change points are distributed through all the data and are not restricted to one part
of the data. Additionally, as it is mentioned by the same author, the PELT method uses pruning
to modify the optimal partitioning. This method combines pruning with optimal partitioning to
get an efficient and precise computational cost which is linear in n. The optimal segmentation
is 𝐹(𝑛) where:
𝑚+1
𝐹(𝑛) = min τ { ∑ [𝐶(𝑦𝑇𝑖−1 +1,…, 𝑦𝑇𝑖 ) + 𝛽]}
𝑖=1
If we condition on the last change point which is 𝑇𝑚 and we calculate the optimal segmentation
of the data until we arrive to that change point, we will have the following:
𝑚
𝐹(𝑛) = min τ𝑚 {min τ|τ𝑚 ∑[𝐶(𝑦𝑇𝑖−1 +1,…, 𝑦𝑇𝑖 ) + 𝛽] + 𝐶(𝑦𝑇𝑚+1,…, 𝑦𝑛 )}
𝑖=1
As mentioned by Dorcas Wambui (2015), the model starts with the calculation of F(1),
then it continues to F(2) until it arrives at F(n) . Therefore, at each part, the optimal
segmentation which is up to 𝑇𝑚+1 is saved and when the model arrives to F(n) it basically
means that the optimal segmentation for the whole data has been found and completed, and the
number and location of change points is being saved and every minimization step over 𝑇𝑚
covers all the previous values. Hence, the computational efficiency of the model is achieved
by eliminating the candidate values of 𝑇𝑚 from the minimization in each step.
4.1.1.1.2 AMOC
AMOC which means ‘At most one change point’ is a technique used to detect a single
hypothesised change point. In this case, the null hypothesis is not having a change point, and
its maximum logarithmic likelihood is given by 𝑙𝑜𝑔 𝑝 (𝑦1:𝑛 |𝜃̂1 ), where 𝑝 (. ) is the probability
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density function which is related to the distribution of the data and theta hat (𝜃̂) is the maximum
likelihood estimate of the parameters (Dorcas Wambui, 2015).
The other way around, the alternative hypothesis takes into account the change point at
τ1 where τ1 can be any discrete number, that is: τ1 ∈ {1, 2, . . . , n − 1} . the maximum
logarithmic likelihood is expressed in the following way: τ1 𝑖𝑠 𝑀𝐿(τ1 ) = log 𝑝 (𝑦1:𝑛 |𝜃̂1 ) +
log 𝑝 (𝑦(𝑇1+1):𝑛 |𝜃̂2 ). Thus, it is assume that the alternative hypothesis is = τ1 𝑀𝐿(τ1 ) is the
maximum log-likelihood value and this maximum is gathered from all possible change point
locations given the fact that the change point location is a discrete variable. Hence, the t-statistic
is λ = 2 [ max τ1 ML (τ1 ) − log p ( 𝑦1:𝑛 | 𝜃̂ )] and the null hypothesis is rejected if lambda
is bigger than 𝐶, and the value of 𝐶 is the determined threshold.
4.2 Second Approach - DCC-MGARCH
In order to analyze the correlation between Bitcoin, Ethereum, and Ripple, and the
relationship between Bitcoin and the three Equity Indices and Gold, a DCC-GARCH model
introduced by Engle and Sheppard in 2001 will be used. This type of model is a parsimonious
option to model portfolios with a large number of assets since with the conditional correlations
and the conditional volatility, the entire conditional matrix of variance and covariance of a
particular portfolio can be estimated. The DCC-GARCH model is expressed in the following
1/2
way: 𝑟𝑡 = µ𝑡 + 𝑎𝑡 , 𝑎𝑡 = 𝐻𝑡
𝑧𝑡 , 𝑯𝒕 = 𝑫𝒕 𝑹𝒕 𝑫𝒕 , where, rt is the n x 1 vector of log returns
of n assets at time t, 𝒂𝒕 the k x1 vector of mean-corrected returns of n assets at time t, i.e.
E[𝒂𝒕 ] = 0. Cov[𝒂𝒕 ] = 𝐇t, µ𝐭 the k × 1 vector of the expected value of the conditional rt,
1/2
Ht the k × k matrix of conditional variances of 𝒂𝒕 at time t, Ht
any k × k matrix at time
t such that Ht is the conditional variance matrix of 𝒂𝒕 , 𝐃𝐭 the k × k, diagonal matrix of
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conditional standard deviations of 𝒂𝒕 at time t, 𝐑 𝐭 the k × k conditional correlation matrix
of 𝒂𝒕 at time t, and zt the k × 1 vector of IID errors such that E[𝐳𝐭 ] = 0 and E[𝐳𝐭 𝐳𝐭𝐓 ] = I.
It is worth making a point, and remembering that the GARCH models have different
orders, generally the simplest model is the GARCH (1,1) and it is usually the most appropriate.
The 𝛼𝑖𝑞 represents the ARCH effect (short-term persistence of the “shock” in the profitability
of asset 𝑖) and 𝛽𝑖𝑝 represents the GARCH effect (contribution of the “shock” of the profitability
𝑄
𝑃
𝑖
𝑖
of the asset 𝑖 to the long-term persistence [∑𝑞=1
𝛼𝑖𝑞 + ∑𝑝=1
𝛽𝑖𝑝 ]). 𝑅𝑡 is the matrix of
conditional correlations of the standardized residuals 𝜖𝑡 , where: 𝜖𝑡 = 𝐷𝑡−1 𝑎𝑡 ~𝑁(0, 𝑅𝑡 ), being
𝑅𝑡 a symmetric matrix.
4.3 Third approach - VEC and VAR - Granger Causality tests
First of all, in order to be able to apply any of the two models, I need to study whether the
variables are stationary, or not. To do so, I will apply an Adjusted Dickey-Fuller test and the
Phillips Pherron test. In the Adjusted Dickey-Fuller test, I can exclude the constant and include
a linear trend. The ADF test consists of estimating the following model:
𝑚
△ 𝑥𝑡 = 𝛽0 + 𝛽1 𝑡 + 𝛿𝑥𝑡−1 + 𝛼𝑖 ∑
△ 𝑥𝑡−1 + 𝜔𝑡
𝑖=1
The contrast is similar to the Dickey-Fuller test case: 𝐻0: 𝛿 = 0 → There is a unit root, 𝑥𝑡 is
not stationary. 𝐻1: 𝛿 ≠ 0 → There is no unit root, 𝑥𝑡 is stationary. If;
𝜏 -calculated in absolute value> 𝜏 -critical in absolute value: 𝐻0 is rejected.
𝜏 -calculated in absolute value <𝜏 -critical in absolute value: 𝐻0 is accepted.
The Phillips Pherron test estimates a regression by correcting for the matrix of variances
and covariances of the residuals. The correction is by means of a non-parametric method. In
this the following regression is estimated: △ 𝑥𝑡 = △ 𝛽 + 𝑝𝑥𝑡 − 1 + 𝑤𝑡
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Unlike the ADF test, there are no delayed difference terms. The null hypothesis 𝐻0 of the
Phillips-Perron test is the path of unit root with trend and the alternative is stationarity with
trend, if the value 𝑡 associated with the coefficient of 𝑥𝑡−1 is greater in absolute value than the
MacKinnon statistic, the null hypothesis of the existence of a unit root.
If variables are unit roots, I will apply first differences to the variables to turn them into
stationary variables. Once this is done, I will identify the number of lags that I will use in the
regression. Thus, to find the order of the model, I will examine the so-called Information
criteria, which are certain corrections on the sample value of the logarithm of Likelihood
function. The best known are: Akaike (AIC), Schwarz (SBC or BIC), and Hannan-Quinn.
Having done so, I will proceed to study the existence or non-existence of co-integrated
relationships. Thus, to study whether there is any co-integration relationship between Bitcoin
and the S&P 500, VIX and the Covid-19 variables, will use a VEC model (or vector error
correction models). The VEC model is a model that belongs to the context of multivariate time
series, but is characterized by containing co-integrated variables; that is, variables that maintain
a long-term equilibrium relationship between them. It includes both: the dynamics of
adjustment of the variables in the short term, when an unexpected shock occurs that causes
them to temporarily move away from their long-term equilibrium relationship, and the
reestablishment of the relationship of equilibrium in the long term, the information it provides
on the speed of adjustment towards such equilibrium is especially useful.
Cointegration refers to linear combinations of stationary variables. Nonlinear combinations
may exist, but these cannot be found using econometric methods at present. The cointegration
vector is not unique. If (β1, β2,…, βn)’ is a cointegration vector then (λβ1, λβ2,…, λβn)' is also a
cointegration vector. Generally, one of the variables is normalized to set its coefficient to unity,
(λ = 1 / β1). All variables must be integrated in the same order. If xt has n components there
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can be at most n-1 cointegration vectors. Clearly, if xt contains only 2 variables, there can be
at most 1 cointegration vector. The existence of cointegration between the variables of a VAR
model implies a long-term relationship between them.
In order to build the model I will use Johansen's methodology which estimates in just one
stage and deals with more than one co-integration relationship. Johansen's (1988) procedure is
nothing more than a multivariate generalization of unit root tests. The equation to estimate is:
∆𝑥𝑡 = 𝜋𝑥𝑡−1 + 𝜀𝑡 . By analogy with the univariate case, if the range of π is zero all the variables
in the system have unit roots. If the range of π is n, then all variables are stationary. In
intermediate cases the range of π determines the number of co-integration relationships in the
system. The number of co-integration relations can be contrasted by seeing what is the rank of
the matrix.
If no co-integrated relationships are found, I will proceed with a VAR model, using the
functions varsoc to identify the lag order, based on the information criterion previously
explained, followed by a varbasic function to estimate the VAR model with the selected
amount of lags. Once the model is estimated a varlmar function, which is a diagnostic test to
know whether the model is correct or not, whether there is autocorrelation or not, will be
applied. Finally, the vargranger function, which is typically used to do the Granger causality
Wald test to check whether one variable is explained by another variable, will be used. It is
said that a variable z does not cause the variable y if adding the past of z to the previous equation
does not add explanatory power.
5. PRELIMINARY ANALYSIS
This section is composed of two sub-sections. In the first sub-section the data and the
necessary descriptive statistics to proceed with the change point analysis results will be
introduced. This change point analysis will be applied to Bitcoin, Ethereum, and Ripple. The
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second part of this section will be dedicate to introduce some descriptive statistics of Equity
Indices, Gold, Fear Indices and Covid-19 variables to be able to further proceed with the
relationship among the variables.
5.1 Data and Descriptive Statistics
5.1.1 Bitcoin, Ethereum, and Ripple
As previously mentioned, in this research daily log returns for cryptocurrencies from
01/11/2019 to 31/03/2021, will be used. There are several ways of calculating returns.
However, one of the most common ways to do so when analyzing financial data is by applying
the continuous compounding (Ruppert, 2014). The following equation shows the model used
for converting daily prices into logarithmic returns for each cryptocurrency: 𝑟𝑡 = ln(𝑃𝑡 ) −
ln(𝑃𝑡−1 ), where 𝑟𝑡 are the log returns at time t and 𝑃𝑡 is the price of cryptocurrency in USD at
time t. thus, we are following an independent and identically distributed (i.i.d.) and normally
distributed log returns as suggested by Ruppert (2014).
In Figure 1, the logarithmic returns time series of the three cryptocurrencies: Bitcoin,
Ethereum and Ripple can be seen. From these three graphs, it is observed that the trend does
not differ a lot, they have similar patterns and from a first sight, it is seen that all of them have
a spike at the beginning of 2020 in March. However, this spike is less noticeable in the Ripple
graph, which shows a higher spike at the end of 2020 in December. Bitcoin and Ethereum seem
to have more movement over the period from 11/1/2019 to 11/1/2020 while Ripple returns
have shown more stability. However, from November on, Ripple had more movement than
Bitcoin and Ethereum. The spike in March for Bitcoin and Ethereum and the spike in December
for Ripple, which is when the Covid-19 pandemic was announced and the third wave of the
pandemic was about to start, respectively, is a first indicator that suggests that there will, or
might, be a possible result once having implemented the change point analysis on this data.
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Master of Science in Finance and Banking
0
-.1
-.3
-.2
log returns BTC
.1
.2
Correlation of cryptocurrencies: a dynamic investigation
11/1/2019 12/1/2019 1/1/2020
2/1/2020 3/1/2020
4/1/2020
5/1/2020
6/1/2020
7/1/2020
8/1/2020
9/1/2020 10/1/2020 11/1/2020 12/1/2020 1/1/2021
2/1/2021 3/1/2021
9/1/2020 10/1/2020 11/1/2020 12/1/2020 1/1/2021
2/1/2021 3/1/2021
-.2
-.4
log returns ETH
0
.2
Date
2/1/2020 3/1/2020
4/1/2020
5/1/2020
6/1/2020
7/1/2020
2/1/2020 3/1/2020
4/1/2020
5/1/2020
6/1/2020
7/1/2020
8/1/2020
Date
-.4
-.2
0
log returns XRP
.2
.4
11/1/2019 12/1/2019 1/1/2020
11/1/2019 12/1/2019 1/1/2020
8/1/2020
9/1/2020 10/1/2020 11/1/2020 12/1/2020 1/1/2021
2/1/2021 3/1/2021
Date
Figure 1. Daily log returns of Bitcoin, Ethereum, and Ripple prices
In Table 2 you will be able to identify the descriptive statistics of the logarithmic returns
of the three cryptocurrencies. It can be observed that Ripple reached the largest minimum value
of -.4289956 and the greatest maximum value of .2474847 over the period. However, the
largest mean was of Ethereum (.005144) and the mean of Ripple was very close to 0 (.0004).
Table 2. Logarithmic returns of Bitcoin, Ethereum, and Ripple – Summary statistics
Cryptocurrency
Obs
Mean
Std. Dev.
Min
Max
Bitcoin (BTC)
Ethereum (ETH)
Ripple (XRP)
511
511
511
.0042564
.005144
.000399
.0378391
.0501878
.0558353
-.3159456
-.423604
-.4289956
.1510413
.1862117
.2474847
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Correlation of cryptocurrencies: a dynamic investigation
To summarize, it can clearly be observed that there abnormalities (outliers) in the data,
especially during March of 2020 for Bitcoin and Ethereum, and December 2020 for Ripple,
where all the cryptocurrencies faced minimum values. Thus, the usage of the change point
analysis will help to identify whether this assumption is correct and if it can be related to Covid19 pandemic.
5.1.2 Equity Indices, Gold, Fear Indices and Covid-19 variables
Equity indices play an important role in this paper as they will be included in the analysis.
The Equity Indices were chosen taking into account findings of previous studies and the three
ones selected were the S&P 500, MSCIWorld, and MSCIEM50. In order to be able to calculate
the daily logarithmic return of each of the indices, the initial data on daily prices was taken
from Yahoo Finance (2021). In Table 3 you will be able to observe the descriptive statistics
results for each Equity Index.
Table 3. Equity Indices – daily logarithmic returns. Descriptive Statistics
Variable
Observations
Mean
Std. Dev.
Min
Max
Skewness
Kurtosis
S&P500
MSCIWorld
MSCIEM50
511
.0005951
.0158372
-.1276522
.0896831
-1.128256
20.62947
511
.0012932
.0242616
-.1399031
.1658111
-.0148362
13.81482
511
.0005279
.0115443
-.0694251
.0557375
-1.262255
12.64785
Table 3 is showing the amount of observations that are considered and it is showing
that, within those observations, there were fluctuations during the selected period as the
minimum and maximum values of each of the equity indices significantly vary. However, the
mean of the series and of each of the indices is close to zero and the standard deviation is low
which indicates that most of the data in the sample tends to be clustered close to its mean.
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Correlation of cryptocurrencies: a dynamic investigation
Something else to look at is the Skewness and Kurtosis. Skewness measures whether
the tail of the distribution is longer to the right or left. For the three indices the values are
negative which indicates a distribution with an asymmetric tail extended toward negative
values. However, the skewness of the MSCIWorld is (-0.015) which is very close to 0. If the
skewness would be 0, this would imply a normally distributed data, which by definition exhibit
relatively little skewness. Kurtosis on the other side, indicates how the tails of a distribution
differ from the normal distribution. All of the three equity indices experience a large and
positive kurtosis this means that for all of them the distribution has heavier tails than the normal
distribution, hence we will have a leptokurtic distribution which is when it’s more pointed and
with tails less wide than the usual curve.
Another important variable to take into account is the Gold daily logarithmic returns.
The price used to calculate the log returns was obtained from Gold.org (Gold Hub, 2021). In
this website, the gold price is normally calculated, by default, as a unit per troy ounce in
USD. In Table 4 you can find the results obtained from the descriptive statistics.
Table 4. Gold – daily logarithmic returns. Descriptive Statistics
Variable
Gold
Observations
Mean
Std. Dev.
Min
Max
Skewness
Kurtosis
511
.0002286
.0096942
-.0526457
.0513344
-.5522391
9.964163
It is observable that during the selected period gold daily log returns were very close to
0 as of: mean and variance. Also, the minimum and maximum differ slightly from each other
indicating a possible period of fluctuations. Skewness was negative indicating that the
distribution with an asymmetric tail is extended toward negative values, and a positive Kurtosis
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Correlation of cryptocurrencies: a dynamic investigation
which shows that there is, again, a leptokurtic distribution. It is observable that the behaviour
of gold is very similar to the behaviour of the Equity Indices.
Furthermore, the Cboe Volatility Index (VIX) and the Financial Stress daily index (FSI)
were also included. The data for the VIX index was obtained from Investing.com (Indices –
Volatility – S&P500 historical data) and the data for the FSI index was obtained from financial
research page of the Government (financialresearch.gov).
In terms of interpretation, a normal reading of the VIX is between 20 and 30, below 20
investors are not worried, there is complacency. Above 30 indicates that there is nervousness,
that is, fear in the market. Similarly, in the case of the FSI, the index will be positive if the
contribution of the weighted average stress of the indicator is positive, and the other way
around, the index will be negative if such average is negative and, the index is zero if the
average is zero which suggests that stress levels are normal. Table 5 shows the results.
Table 5. Fear indices. Descriptive Statistics
Variable
Observations
Mean
Std. Dev.
Min
Max
Skewness
Kurtosis
VIX
FSI
511
26.2264
11.5148
11.54
82.69
1.874813
7.98666
511
-1.426869
3.221726
-4.36
10.27
1.738653
5.382647
Table 5 is indicating that during the selected period both indices were experiencing
fluctuations, hence not being stable, since the minimum and maximum values significantly
differ from each other. Also, it is observable that the mean of the VIX is around 26. Taking
into account that this is the mean, which means that it is doing an average of all the values
during the selected period, we can notice that to obtain this mean many huge values had to be
encountered in the data set, thus we can deduce that during many days, the VIX value was
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Correlation of cryptocurrencies: a dynamic investigation
above 30, indicating that there was nervousness, that there was fear in the market, and given
the selected period, this could be linked to the Covid-19 pandemic. Additionally, from the
obtained results, we can also say that during the period under analysis, the average stress
contribution of the indicators was negative as we encountered a negative mean of -1.43.
Finally, two social factors were included in the model which were: new daily cases of
Covid-19 (not cumulative) and new daily deaths due to Covid-19 pandemic (not cumulative),
which will be used to study the long term relationship between these two variables, the S&P
500, and Bitcoin as we found out, using the change point detection analysis, that Covid-19
could possibly impact the volatility on cryptocurrencies, in this case Bitcoin. In Table 6 the
results are shown. The data was obtained from CovidTracking.com (2021).
Table 6 is indicating that during the selected period, the maximum number of cases due to
Covid-19 was 248,724.9 a day and the maximum number of deaths a day was 4.409. The mean
is indicating that, on average, there were 58,456.46 new daily cases of Covid-19 and 1,156.4
new daily deaths which are very huge numbers.
Table 6. New daily Covid-19 Cases and New Daily Deaths due to Covid-19. Descriptive
Statistics
Variable
Observations
Mean
Std. Dev.
Min
Max
Variance
Skewness
Kurtosis
New daily cases
New daily deaths
511
58456.46
65480.63
0
248724.9
4.32e+09
1.32937
3.705002
511
1156.372
1066.478
0
4409
1124254
.9749288
3.393615
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Correlation of cryptocurrencies: a dynamic investigation
6. RESULTS
6.1 Change Point Detection Results
In the following sub-section, the results of the implementation of the change point
detection on the variance using PELT and AMOC on Bitcoin, Ethereum and Ripple, and the
presented data, which is presented separately, will be presented, analyzed, and compared. The
default model penalty and the assumption of normal distribution were used to detect the
changing points in both methods.
6.1.1 Bitcoin
As it is observable in Figure 2, different methods lead to different outcomes. While
using the PELT method, two change points were found, and while using the AMOC method,
-.3
-.2
-.1
0
log returns BTC
.1
.2
one unique change point was found.
500
700
800
Date
900
1000
-.3
-.2
-.1
0
log returns BTC
.1
.2
600
500
600
700
800
Date
900
1000
Figure 2. Bitcoin change point detection using PELT and AMOC methods
In Table 7 you will find the exact dates that were detected using both methods and, as
it can be observed, the dates are between the 7th and 20th of March.
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
Table 7. Bitcoin Change Point detection - Results
Change Point method
PELT
AMOC
Change Point - Day number
Change Point – Calendar date
649
11/03/2020
658
20/03/2020
645
07/03/2020
6.1.2 Ethereum
Going further, the same change point detection procedure was applied. The PELT and
AMOC technique were used, and it was found out that the PELT method was detecting two
change points, while AMOC was just detecting one change point as previously happened. It is
interesting to mention that the second change point found with PELT method is the same as
600
700
800
Date
900
1000
-.4
-.2
0
log returns ETH
.2
500
.4
-.4
-.2
0
log returns ETH
.2
.4
the one found with AMOC method (07/03/2020). Results are observed in Figure 3.
500
600
700
800
Date
900
1000
Figure 3. Ethereum Change Point detection using PELT and AMOC methods.
From Table 8 it can be observed that 07/03/2020 day was detected by both methods.
However, the most interesting part is that both dates coincide with the dates that were found
for Bitcoin. Hence, comparing Ethereum results to Bitcoin results, it can be observed that the
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
dates are almost equal with both methods, except for the first change point on day 649 found
for Bitcoin using PELT method.
Table 8. Ethereum Change Point detection - Results
Change Point method
PELT
AMOC
Change Point - Day number
Change Point – Calendar date
645
07/03/2020
658
20/03/2020
645
07/03/2020
6.1.3 Ripple
Ripple seems to be the exception. The same change point detection procedure was
applied and, interestingly, the results were different compared to those of Bitcoin and
Ethereum. Using the PELT method two change points were found, while using the AMOC
-.4
-.2
0
log returns XRP
.2
method just one change point was observed. The change points can be observed in Figure 4.
600
700
600
700
800
Date
900
1000
-.4
-.2
0
log returns XRP
.2
500
500
800
Date
900
1000
Figure 4. Ripple Change Point detection using PELT and AMOC methods
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
In the following table, the results of the Ripple were identified. It can be seen that both
methods, PELT and AMOC, found a change point that was not detected on Bitcoin and
Ethereum. We refer to date 17/12/2020, 20/12/2020 and 08/01/2020. However, as happened
with Bitcoin, the result using AMOC method is between the dates found when using PELT
method.
Table 9. Ripple Change Point detection - Results
Change Point method
Change Point - Day number
Change Point – Calendar date
PELT
930
952
17/12/2020
08/01/2021
AMOC
933
20/12/2020
Once the change point analysis was done and we were able to identify at which point
in time this was happening, we proceeded with the analysis of change in variance of each
cryptocurrency using the logarithmic returns on prices. First of all, we excluded the period
from 645 to 658 when analysing the variance for Bitcoin and Ethereum to be able to compare
the variance difference before and after this excluded period, as it was the most common change
point location of both models. And, to analyse the variance difference for Ripple, we excluded
the period between 930 and 952. The results are shown in Table 10. It can be observed from
Table 10 that the variance of the three cryptocurrencies’ logarithmic returns from period
01/11/2019 to 06/03/2020 and from period 21/03/2020 to 31/03/2021 were smaller compared
to the variance of the whole dataset which goes from period 01/11/2019 to 31/03/2021. This
result is telling us that the periods that we excluded to perform the variance analysis had a great
impact on the overall variance, meaning that if we include the ‘excluded’ period, the variance
increases, while the contrary was happening if we were not including them. Therefore, it can
be concluded that these excluded periods are significant in the change point analysis as larger
variance is indicating that the values in the set are far from the mean and from each other.
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Correlation of cryptocurrencies: a dynamic investigation
Table 10. Variance before and after change point’s location
Variance until
07/03/02020
Variance after
20/03/2020
Variance using the
whole dataset
Bitcoin
0.0007271
0.0012258
0.0014318
Ethereum
0.0014604
0.0021804
0.0025188
Variance until
17/12/02020
Variance after
08/01/2021
Variance using the
whole dataset
0.0024939
0.0032104
0.0034884
Cryptocurrency
Ripple
6.1.4 The influence of COVID-19 pandemic
One of the objectives of detecting the change point and studying the variance was to
investigate whether the fluctuation in logarithmic returns of cryptocurrencies was being
influenced by Covid-19. That was one of the reasons why, for this analysis, it was decided to
take into account the periods from 01/11/2019 to 31/03/2021 as they were covering the Covid19 period. According to the World Health Organization (WHO, 2021), the first Covid case was
detected in December 2019 in Wuhan, China. However, the worldwide pandemic was not
declared until the 11th of March of 2020. This pandemic resulted in three waves. The first wave
took place between 15th March and 30th June, the second wave took place between 1st July
and 15th October, and the third wave took place from the very end of December 2020 to 8th
January, 2021.
Therefore, the objective was to search whether the World Wide pandemic could have
had an influence on cryptocurrencies. To do so, the change point detection analysis was applied
to Bitcoin, Ethereum and Ripple using the daily logarithmic returns on prices. After analysing
the change point detection results, it was observed that the change point location dates were:
07/03/2020, 11/03/2020, 20/03/2020, 17/12/2020, 20/12/2020, and 08/01/2021, which indicate
that at the beginning and at the middle-end of March the log returns of the studied
cryptocurrencies, had a significant change on the variance as it was detected by both methods.
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For that reason, we could possibly consider that the announcement of the pandemic could have
had an impact on the change in variance and the specific dates in which both methods detected
the change points.
To conclude, both methods: PELT and AMOC, performed similarly and detected
almost the same change points for Bitcoin and Ethereum, with the same change point for
AMOC and only differing in the first point when using PELT method. It is worth mentioning
that AMOC only detects one change point, while PELT detects several change points.
However, in the analysis the cryptocurrency Ripple had a different result as it was the only
cryptocurrency that had the change point location at the end of December 2020 and beginning
of January 2021, this dates coincide with the third Covid-19 wave. Finally, after having
analyzed the change point detection, I considered that it was worth to check the variance of the
data. Once having checked and compared the variance of each cryptocurrency, before and after
the location change points, it was found out that the variance, before and after the selected
change point locations were smaller compared to the variance of the whole dataset. Meaning
that the change points that both methods detected were correctly identified as the excluded
periods were significantly increasing the overall variance of each cryptocurrency.
6.2 DCC MGARCH Model
In the first part we detected the change points on Bitcoin, Ethereum, and Ripple. In this
second part, the main objective is to further study the relationship between cryptocurrencies
and the relationship between Bitcoin, Equity Indices and Gold. To do so, a new variable called
time that goes from 1 to n was generated, as well as a time series framework. As previously
mentioned, a DCC-MGARCH model introduced by Engle and Sheppard in 2001 will be used,
as this type of model will allow us to study the conditional on past history covariance matrix
of the dependent variable which, in this case, is Bitcoin, to follow a dynamic conditional
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Correlation of cryptocurrencies: a dynamic investigation
correlation study. Daily logarithmic returns of cryptocurrencies (Bitcoin, Ethereum, and
Ripple), Equity Indices (SP&500, MSCIWorld, and MSCIEM50), and Gold were used from
November 1st, 2019 to March 31st, 2021. I will model the conditional mean of the logarithmic
returns as a first order vector autoregressive process and the conditional covariance as a DCC
MGARCH process, following a GARCH(1,1) process for the variance of each disturbance.
6.2.1 Stationarity
In order to be able to explain the Granger causality, their predictability, the presence of
dynamic correlations and the conditional volatility, the first step is to study whether the
variables are stationary or not. Once this is done, the model will be built and tested.
A time series is said to be stationary when its distribution and its parameters do not vary
with time. In more concrete terms, the mean and variance of a stationary series do not change
over time, nor do they follow a trend. This is something very important because a stationary
series is much easier to predict. If it behaved in a way in the past (say with a certain mean and
variance), we can assume that it will continue to behave in the same way in the future, or that
it has a high probability of continuing to behave in the same way. Most models that describe
and attempt to predict the behavior of time series work under the assumption that the series is
stationary. However, there is a problem which is that when we talk about the stock market this
almost never happens. Quotes, that is, the prices of financial assets, do not have a stationary
behavior. In order to deal with this problem, we can transform and convert non-stationary series
into stationary series. Nevertheless, since we were aware of this fact we calculated the log of
returns of cryptocurrencies and equity indices before testing for stationarity. In the following
table you will be able to identify the test for stationarity following and Adjusted Dickey-Fuller
test. If any variable appears to be non-stationary a Phillips-Perron test will be done to double
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
check the outcome. If variables keep being non-stationary then we will apply first differences
to the variable to convert it into a stationary variable.
Table 11 shows the results using an Adjusted Dickey-Fuller test, under which the
hypothesis is as follows: 𝐻0: 𝛿 = 0 → There is a unit root, 𝑥𝑡 is not stationary, 𝐻1: 𝛿 ≠ 0 →
There is no unit root, 𝑥𝑡 is stationary. Similarly, the Phillips-Perron tests for the null hypothesis
that 𝑥 has a unit root. From the results obtained we can observe that most of the variables are
stationary at 5% of significance level by the ADF test. The variables that were not significant
at 5% were the VIX, FSI and New daily cases of Covid-19. For these three variables we decided
to double check if they were non-stationary using a Phillips Perron test and we can conclude,
looking at Table 12, that the variables are non-stationary. In this case, first differences to
convert them into stationary variables, will be applied.
Table 11. Checking for stationarity – ADF test. Without trend
Variable
Tstatistic
1%
Critical
Value
5%
Critical
Value
10%
Critical
Value
p-value
Stationarity
rBTC
rXRP
rETH
rSP500
rMSCIWorld
rMSCIEM50
VIX
FSI
New daily cases
New daily deaths
-22.76
-22.12
-22.05
-32.91
-28.34
-21.32
-2.74
-1.15
-0.95
-4.91
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
0.00
0.00
0.00
0.00
0.00
0.00
0.06
0.69
0.77
0.00
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Non-Stationary
Non-Stationary
Non-Stationary
Stationary
Table 12. Checking for stationarity – Phillips–Perron test, for non-stationary variables.
Variable
Z(rho)
Z(t)
p-value
Stationarity
VIX
FSI
New daily cases
-11.822
-3.624
-1.117
-2.489
-1.319
-1.136
0.1182
0.6203
0.7083
Non-Stationary
Non-Stationary
Non-Stationary
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
After doing first differences to non-stationary variables, that is: VIS, FSI and New daily
cases, the stationarity test was implemented and the results were the following:
Table 13. Checking for stationarity – ADF test. Without trend
Variable
rBTC
rXRP
rETH
rSP500
rMSCIWorld
rMSCIEM50
dVIX
dFSI
dNew daily cases
New daily deaths
TStatistic
1%
Critical
Value
5%
Critical
Value
10%
Critical
Value
pvalue
Stationarity
-22.76
-22.12
-22.05
-32.91
-28.34
-21.32
-29.56
-24.74
-18.80
-4.95
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.43
-3.44
-3.44
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.86
-2.87
-2.87
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
-2.57
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Table 14. Checking for stationarity – Phillips–Perron test, for non-stationary variables.
Variable
Z(rho)
Z(t)
p-value
Stationarity
dVIX
dFSI
dNew daily cases
-684.80
-679.42
-545.39
-29.040
-24.77
-19.86
0.00
0.00
0.00
Stationary
Stationary
Stationary
As it can be observed from Table 13 and 14, after having applied first differences (dVIX,
dFSI, and dNew daily cases) to the variables and running the stationarity test, it can be
concluded that the variables are stationary at a 5% of significance level with both: ADF and
PP tests.
6.2.2 DCC MGARCH results
Table 15 shows the results of the estimation of the multivariate DCC MGARCH model
using cryptocurrencies. The parameter 𝛾2 of the equation of the mean is not significant and
represents the zero influence and effect that the Bitcoin has on the other two cryptocurrencies.
Regarding the lag order, there are multiple information criteria, including LL, LR, FPE, AIC,
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
HQIC and SBIC, which confirm the inclusion of a maximum of 1 to 2 lags for the mean
equation. The ARCH and GARCH coefficients of the variance equation are highly significant,
which corroborates the specification of the model in which the variance of each error follows
a GARCH (1,1) type process. This last equation yields the coefficients 𝛼 and 𝛽, which show
that the persistence of volatility is 0.853 for Ethereum, 0.11 for Ripple and 0.92 for Bitcoin.
Being the persistence of volatility high for Ethereum and Bitcoin. Regarding the DCC
conditional dynamic correlation equation, both coefficients 𝜆1 𝑦 𝜆2 are statistically significant
at 1%, so it is concluded that the co-movement of the currencies is changing over time. In fact,
these coefficients show that they are different from zero, but their sum is less than unity, which
rules out the presence of unit roots, exhibiting a high persistence in the correlations with values
for the sum of the coefficients that oscillate between 0.853 and 0.956.
Table 15. DCC MGARCH Model Bitcoin, Ethereum and Ripple
Ethereum
Ripple
Bitcoin
𝛾2
-0.098
(-0.09)
-0.04
(-0.35)
𝜇
0.004
(1.86)
-0.000
(-0.03)
0.002
(1.54)
𝛼
0.096
(4.01)**
0.188
(3.11)**
0.086
(4.07)**
𝛽
0.757
(10.97)**
-0.078
(-2.31)*
0.834
(19.11)**
𝜔
0.000
(2.82)**
0.000
(5.71)**
0.000
(2.84)**
𝜆1
0.055
(4.09)**
𝜆2
0.901
(36.28)**
Equation of the mean
Equation of the variance
DCC equation
∗ 𝑝 < 0.05; ∗∗ 𝑝 < 0.01
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
Table 16 summarizes the conditional correlations between the standardized residuals of
the Bitcoin and the Ethereum and Ripple, observing a high and positive correlation between
Ethereum and Bitcoin with a value of 0.831, followed by a positive correlation between
Ethereum and Ripple of 0.559 and a very low positive correlation between Bitcoin and Ripple
of 0.384, all of them statistically significant at 1% level.
Table 16. Correlations Bitcoin, Ethereum and Ripple
Ethereum
Ripple
Bitcoin
1
0.559
0.831
Ripple
0.559
1
0.384
Bitcoin
0.831
0.384
1
Ethereum
Following the same methodology described above, the results of the DCC MGARCH
model for the Bitcoin, the three Equity Indices (S&P 500, MSCIWorld, and MSCIEM50) and
the Gold are summarized in Table 17. The results of the following model significantly differ
from the previous model. The model presents one significant value of the parameter 𝛾2, that is:
MSCIEM50, being statistically significant at 5%, which represents the influence and effect that
the Bitcoin has on the MSCIEM50. Additionally, the equation of the variance yields the
coefficients 𝛼 and 𝛽, which, added together, show that the persistence of volatility is on average
0.94 for each of the specifications. Regarding the DCC conditional dynamic correlation
equation, both coefficients 𝜆1 𝑦 𝜆2 are statistically significant at 1%, so it is concluded that the
co-movement of the variables is changing over time.
Additionally, in Table 18, when analyzing the conditional correlations between the
standardized residuals of Bitcoin, the Equity Indices and Gold, a lower correlation is observed
between them. In fact, we only observed two statistically significant correlations at 1% level
which were Gold and S&P 500, and Gold and the MSCIWorld, both having a statistically
significant negative correlation with Gold at 1% level. Also, it can be deduced that the degree
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Correlation of cryptocurrencies: a dynamic investigation
of influence of the Bitcoin on these variables is less than that observed with cryptocurrencies
(Table 16).
Table 17. DCC MGARCH Model Bitcoin, Equity Indices and Gold
S&P 500
MSCIWorld MSCIEM50
Gold
Bitcoin
Equation of the mean
𝛾2
-0.001
(-0.18)
𝜇
0.001
(3.72)**
Equation of the variance
-0.020
(-0.95)
0.021
(2.16)*
-0.002
(-0.25)
0.001
(2.47)*
0.000
(2.36)*
0.000
(0.48)
0.002
(1.94)*
𝛼
0.292
(5.32)**
0.280
(4.56)**
0.211
(4.90)**
0.148
(3.58)**
0.139
(4.15)**
𝛽
0.676
(14.68)**
0.626
(9.29)**
0.717
(14.14)**
0.799
(16.44)**
0.809
(19.17)**
𝜔
0.000
(3.54)**
0.000
(3.53)**
0.000
(3.19)**
0.000
(2.78)**
0.000
(2.88)**
𝜆1
0.182
(6.55)**
𝜆2
0.240
(2.29)**
DCC equation
∗ 𝑝 < 0.05; ∗∗ 𝑝 < 0.01
Table 18. Correlations Bitcoin, Equity Indices and Gold
S&P 500
MSCIWorld MSCIEM50
Gold
Bitcoin
1
0.014
-0.020
-0.108
0.018
MSCIWorld
0.014
1
-0.071
-0.149
0.008
MSCIEM50
-0.020
-0.071
1
0.080
-0.039
Gold
-0.108
-0.149
0.080
1
0.042
Bitcoin
0.018
0.008
-0.039
0.042
1
S&P 500
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
6.3 VEC, VAR and Granger Causality tests model
Once having studied the effects, persistence, volatility and correlation between Bitcoin,
cryptocurrencies, Equity Indices, and Gold. We could observe that Bitcoin, Ethereum and
Ripple were correlated, while Bitcoin and the Equity Indices were not correlated, nor Gold was
correlated to Bitcoin, we rather observed a negative correlation between Gold and S&P500,
and Gold and MSCIWorld. In this third part, we will study whether a co-integration between
Bitcoin and the different selected variables: Equity Indices, Fear Indices and Covid-19, exists.
We will do so by using the Johansen VEC model as it finds several co-integration relationships.
6.3.1 Co-integration analysis between Bitcoin and S&P 500, VXI, and FSI
One of the important parts of the model is to make sure that variables are stationary. If
recall from the previous stationary test, where we performed an Adjusted Dickey Fuller test
and a Phillips’ Perron test, we observed that the logarithmic return of Bitcoin and the S&P 500
was stationary, while the VIX and the FSI were non stationary. As it is important that we
consider stationary variables to avoid any spurious result, we applied first difference to both
variables to turn them into stationary variables. After having turned all necessary variables into
stationary variables, we studied how many lags had to be included in the model. To do so, we
used the varsoc function as it is the most common function used to determine the number of
lags to be used in this type of models. As mentioned in Section 3 we will decide the lag order
based on the information criterion (AIC, HQIC and SBIC). As you can observe in Table 19,
results suggested 1 lag using the HQIC and SBIC information criterion method as both have
an (*).
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Correlation of cryptocurrencies: a dynamic investigation
Table 19. Lag order identification
Selection-order criteria
Included observations: 511
Lags
LL
LR
df
0
949.864
1
1194.1
488.47
16
2
1220.65
53.107
3
1236.18
4
1277.11
p
FPE
AIC
HQIC
SBIC
2.8e-07
-3.73
-3.72
-3.70
0.000
1.1e-07
-4.64
-4.57*
-4.47*
16
0.000
1.1e-07
-4.68
-4.56
-4.38
31.048
16
0.013
1.1e-07
-4.68
-4.51
-4.24
81.864*
16
0.000
9.9e-08*
-4.77*
-4.55
-4.21
The next step before being able to build the model is to determine if there is any cointegrating relationship. If there are no co-integrated relationships we will not be able to reject
the null hypothesis of no co-integration and we will proceed to study if at least, one of the
variables helps to predict the other variable using the Egranger model. To do so, we will
implemented the vecrank function which can be observed in the following table.
Table 20. Number of Co-integrated relationships
Johansen tests for cointegration
Included observations: 511
Trend assumption: Linear deterministic trend
Series: Bitcoin, S&P 500, VIX and FSI
Lags interval: 1
Unrestricted Co-integration Rank Test (Trace)
Maximum
rank
Parms
LL
Eigenvalue
Trace
statistic
5% critical
value
0
4
397.71
.
1618.56
47.21
1
11
854.42
0.833
705.14
29.68
2
16
1004.76
0.446
404.45
15.41
3
19
1119.22
0.362
175.54
3.76
4
20
1206.99
0.291
Trace test indicates no co-integration at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
As it is observable from Table 20, the Johansen test in levels was carried out for Bitcoin
and the S&P500, the VIX and FSI index, in order to identify any possible presence of cointegration between the different variables. The results rule out the presence of co-integration
between the variables used in this analysis. From this results we could suggest that during the
selected period, which covers the Covid-19 pandemic, the variables were not co-integrated,
meaning that there is not a strong long-term relationship between the variables. This implies
that whether one variable falls or grows, the other variable does not behave in a synchronized
way. Thus, we can conclude that Bitcoin, S&P 500, the VIX and the FSI do not maintain any
relationship over time. Only after this correction could the Granger test be implemented to
establish possible causal relationships between Bitcoin and the S&P 500, VIX and FSI.
6.3.1.1 Granger causality test results
Since we found out that there are no co-integration relationships among the variables,
we will further study whether one variable helps to predict another variable. To do so, we will
apply the Granger-causality Wald test. It is important to mention that for this part of the paper,
the FSI variable will be excluded as we tested whether the model was correct, or not, and we
found out that the model with FSI was not correct. Thus, we will only consider the following
variables: Bitcoin, S&P 500 and VIX.
First of all, we identified the number of lags to be included in the model using the
information criterion (AIC, HQIC and SBIC). According to the results it has been determined
that the optimal lag is 2, therefore a restriction is established and the command (varbasic) is
used. That is to say that it is a model of VAR (2) it is interpreted as a model of autoregressive
vectors of three variables and two lags. Furthermore, the varlmar function was used to identify
whether the model was correct, or not. According to the results our model was correct at lag 2
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Correlation of cryptocurrencies: a dynamic investigation
(p-value>0.05; 0.171), meaning that there was no presence of autocorrelation at lag 2.
Therefore, the null hypothesis of no autocorrelation at lag order two is accepted.
Finally, we performed the Granger causality Wald test using the Vargranger function.
In Table 21 you can find the results. As we can observe in Table 21 the p-value is below 0.05
in the row ‘rSP500 excluded’ for the ‘rBTC equation’. This strongly rejects the null hypothesis
that the S&P 500 does not cause Bitcoin. In the jargon: ‘S&P 500 Granger causes Bitcoin’, that
is: the S&P 500 daily log returns Granger cause the Bitcoin daily log returns as we have rejected
the null hypothesis of no Granger cause relationship. This findings, of the relationship between
Bitcoin and the S&P 500, was also found by other authors such as: Georgoula et al. (2015);
Kjærland et al. (2018); and Sovbetov and Sovbetov (2018). To see how a one-time-only oneunit increase in rSP500 (the ‘impulse’) affects the log return of Bitcoin (rBTC) (‘the response’),
consult the lower-left panel of the impulse response function plots in Annex 2.
On the contrary, we could identify that the VIX index, which was the one that had to
be differentiated in order to turn the variable into a stationary variable, does not Granger cause
the Bitcoin daily log returns as the p-value is bigger than 0.05, meaning that we do not reject
the null hypothesis, hence VIX does not Granger causes Bitcoin.
Something interesting to observe is that the S&P 500 Granger causes Bitcoin, however
Bitcoin does not Granger cause the S&P 500 as it can be observed by its high p-value,
indicating that we do not reject the null hypothesis, hence Bitcoin does not Granger cause the
S&P 500. Another interesting observation is the fact that the VIX index does not Granger
causes the S&P 500 as it can, again, be observed by its high p-value, meaning that we do not
reject the null hypothesis. However, it can be observed that the S&P 500 does Granger causes
the VIX index as the p-value is equal to 0.000 which shows that we strongly reject the null
hypothesis that S&P 500 does not cause VIX. In other words, S&P 500 Granger causes the
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Correlation of cryptocurrencies: a dynamic investigation
VIX. In Figure 5 you will find the daily log returns of Bitcoin plotted together with the daily
log returns of the S&P500, as well as, a two-way plot of the Bitcoin and VIX, and the S&P 500
and the VIX.
Table 21. Granger causality Wald tests
Equation
Excluded
chi2
df
Prob > chi2
rBTC
rSP500
10.58
2
0.005*
rBTC
dVIX
1.62
2
0.445
rBTC
ALL
10.59
4
0.031
rSP500
rBTC
1.23
2
0.540
rSP500
dVIX
4.66
2
0.097
rSP500
ALL
5.60
4
0.231
dVIX
rBTC
1.62
2
0.443
dVIX
rSP500
210.47
2
0.000*
dVIX
ALL
213.26
4
0.000
500
600
700
800
900
1000
Date
log returns BTC
log returns SP500
30
20
10
0
-10
-20
-20
-.3
-10
-.2
0
-.1
10
0
20
.1
.2
30
Figure 5. Bitcoin and S&P 500 daily log returns, and the VIX differentiated daily index
500
600
700
800
Date
log returns BTC
1000
dVIX
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Master of Science in Finance and Banking
900
500
600
700
800
Date
log returns SP500
900
1000
dVIX
Correlation of cryptocurrencies: a dynamic investigation
6.3.2 Co-integration analysis between Bitcoin and S&P 500, and Covid-19 variables
Lastly, the Bitcoin and the S&P 500 daily log returns are paired with two COVID-19
variables which are: new daily deaths due to Covid-19 (not accumulative) and new daily cases
of Covid-19 (not cumulative). As previously mentioned, all variables need to be stationary.
Applying the Adjusted Dickey Fuller test and a Phillips’ Perron test, results suggested that the
Covid-19 variable: new daily cases was non-stationary. To turn this variable into a stationary
variable we applied first differences. After having turned all necessary variables into stationary
variables, we studied how many lags had to be included in the model. To do so, we used the
varsoc function and, based on the information criterion (AIC, HQIC and SBIC), we selected
the lag order. As you can observe in Table 22, results suggested 2 lags using the SBIC, 3 lags
using HQIC, and 4 lags using AIC, this can be observed as the corresponding value has an
asterisk on the right side (*). Since every information criterion has a different outcome, we
decided to go for the AIC method as it coincides with the LR and FPE.
Table 22. Lag order identification
Selection-order criteria
Included observations: 511
Lags
LL
LR
df
0
-6995
1
-6483
1024.6
16
2
-6431
104.32
3
-6399
4
-6373
p
FPE
AIC
HQIC
SBIC
1.2e+07
27.61
27.62
27.64
0.000
1.4e+06
25.65
25.72
25.82
16
0.000
1.4e+06
25.51
25.63
25.81*
63.833
16
0.000
1.3e+06
25.45
25.62*
25.88
51.792*
16
0.000
1.3e+06* 25.41*
25.63
25.97
The next step before being able to build the model is to determine if there is any cointegrating relationship. If there are no co-integrated relationships we will not be able to reject
the null hypothesis of no co-integration and we will proceed to study if at least, one of the
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Correlation of cryptocurrencies: a dynamic investigation
variables helps to predict the other variable using the Egranger model. To do so, we will
implemented the vecrank function which can be observed in the following table.
As it is observable from Table 23, the Johansen test was carried out for Bitcoin, the
S&P500, and two Covid-19 variables, in order to identify any possible presence of cointegration between the different variables. The results rule out the presence of co-integration
between the variables used in this analysis. No rank is being selected and the trace statistics is
indicating that there are no co-integration relationships at 5% significance level. For that
reason, from these results we could suggest that during the selected period, which covers the
Covid-19 pandemic, the variables were not co-integrated, meaning that there is not a strong
long-term relationship between the variables. This implies that whether one variable falls or
grows, the other variable does not behave in a synchronized way. Thus, we can conclude that
Bitcoin, the S&P 500, and the Covid-19 variables do not maintain any relationship over time.
It is important to mention that given the proximity of rank 3 to being statistically
significant, the number of lags were increased to test whether any co-integration relationship
could be find and, indeed, we found at lag order 6 three co-integration relationships. However,
at the time of building the model and check if the model was correct, it was found out that the
model was not correct, proving the previous assumption of no co-integration relationships.
Only after this correction could the Granger test be implemented to establish possible causal
relationships between Bitcoin, the S&P 500, and the two Covid-19 variables.
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Correlation of cryptocurrencies: a dynamic investigation
Table 23. Number of Co-integrated relationships
Johansen tests for cointegration
Included observations: 507
Trend assumption: Linear deterministic trend
Series: Bitcoin, S&P 500, New daily deaths due to Covid-19 and new daily cases of Covid19
Lags interval: 4
Unrestricted Co-integration Rank Test (Trace)
Maximum
rank
Parms
LL
Eigenvalue
Trace
statistic
5% critical
value
0
52
-6494.3
.
241.2
47.21
1
59
-6446.2
0.172
145.0
29.68
2
64
-6401.9
0.160
56.5
15.41
3
67
-6376.8
0.094
6.37
3.76
4
68
-6373.6
0.012
Trace test indicates no co-integration at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
6.3.2.1 Granger causality test results
Since it was found out that there are no co-integration relationships among the variables,
I will further study whether one variable helps to predict another variable. To do so, we will
apply the Granger-causality Wald test. First of all, the number of lags to be included in the
model were identified using the information criterion: AIC, HQIC and SBIC. According to the
results, it has been determined that the optimal lag is 3, therefore a restriction is established
and the command (varbasic) is used. That is to say that it is a model of VAR (3) which is
interpreted as a model of autoregressive vectors of four variables and three lags. Finally, the
Granger causality test was performed using the vargranger function. In Table 24 you can find
the results.
As we can observe in Table 24 the p-value is below 0.05 in the row ‘COVIDnewdeaths
excluded’ for the ‘rBTC equation’. This strongly rejects the null hypothesis that new daily
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
Covid-19 deaths does not cause a rise in Bitcoin log returns. In the jargon: ‘New daily Covid19 deaths Granger causes Bitcoin’. This findings, of the relationship between New daily deaths
due to Covid-19 and Bitcoin, was also found by Goodell and Goutte (2021). The author found
that for the period that oscillates between April 5th, 2020 and April 20th, 2020 the deaths levels
due to Covid-19 caused an increase in Bitcoin prices. To see how a one-time-only one-unit
increase in COVIDnewdeaths (the ‘impulse’) affects the log return of Bitcoin (rBTC) (‘the
response’), consult the lower-left panel of the impulse response function plots in Annex 3.
On the contrary, it was observable that any of the Covid-19 variables was Granger
causing a change in the S&P 500. Finally, it was also observable that new daily deaths due to
Covid-19 Granger cause new daily cases due to Covid-19, as we strongly reject the null
hypothesis since the p-value is 0.000, meaning that an increase in new daily deaths due to
Covid-19 implies a negative effect of -0.53 on new daily cases. To end up, new daily cases of
Covid-19 also Granger cause new daily deaths due to Covid-19. Indeed, a rise (fall) in new
daily cases implies a negative (positive) effect on new daily deaths.
In Figure 6 you will find the daily log returns of Bitcoin plotted together with the daily
log returns of the S&P500, as well as, a two-way plot of the Bitcoin and VIX, and the S&P 500
and the VIX.
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
Table 24. Granger causality Wald tests
Equation
Excluded
chi2
df
Prob > chi2
rBTC
rSP500
0.01
1
0.916
rBTC
dCOVIDnewcases
2.32
1
0.127
rBTC
COVIDnewdeaths
13.05
1
0.000*
rBTC
ALL
14.79
3
0.002
rSP500
rBTC
0.58
1
0.443
rSP500
dCOVIDnewcases
0.32
1
0.571
rSP500
COVIDnewdeaths
2.17
1
0.140
rSP500
ALL
2.67
3
0.444
dCOVIDnewcases
rBTC
1.65
1
0.198
dCOVIDnewcases
rSP500
0.05
1
0.821
dCOVIDnewcases
COVIDnewdeaths
12.27
1
0.000*
dCOVIDnewcases
ALL
12.88
3
0.005
COVIDnewdeaths
rBTC
5.04
1
0.055
COVIDnewdeaths
rSP500
0.09
1
0.760
COVIDnewdeaths
dCOVIDnewcases
7.38
1
0.007*
COVIDnewdeaths
ALL
12.73
3
0.005
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Master of Science in Finance and Banking
Correlation of cryptocurrencies: a dynamic investigation
Figure 6. Bitcoin daily log returns, S&P 500 daily log returns and the Covid-19 new daily
.1
-.05
0
log returns SP500
-.15
-.3
-.1
-.2
-.1
0
log returns BTC
.1
.05
.2
cases and deaths
500
600
700
800
900
500
1000
600
700
900
1000
20000
0
-20000
-10000
0
dCOVIDnewcases
10000
20000
10000
-10000
-20000
dCOVIDnewcases
500
600
700
800
900
500
1000
600
700
800
900
1000
800
900
1000
Date
4000
3000
2000
0
1000
0
1000
2000
COVIDnewdeaths
3000
4000
Date
COVIDnewdeaths
800
Date
Date
500
600
700
800
900
1000
500
Date
700
Date
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Master of Science in Finance and Banking
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Correlation of cryptocurrencies: a dynamic investigation
7. CONCLUSION AND FURTHER RESEARCH
The aim of the paper was to use daily data from Bitcoin and selected variables including:
Ethereum, Rippler, Equity Indices (S&P 500, MSCIWorld, and MSCIEM50), Gold and Covid19 variables to explain and model the dynamics of Bitcoin. For such purpose we divided the
study into three parts. The first part, consisted of studying the Change Point Detection analysis
in variance of daily logarithmic returns of Bitcoin together with two other cryptocurrencies:
Ethereum and Ripple. To do so, we applied the PELT and AMOC techniques from which we
obtained similar results. In the case of the AMOC method, one unique change point was
detected and it was the same change point for Bitcoin and Ethereum: March 7th, 2020, and
PELT found two change points which were between the 07/03/2020 and the 20/03/2020,
however just one coincide for both: Bitcoin and Ethereum (March 20th, 2020). Ripple was the
exception as using both techniques, different change points were detected. Using the PELT
technique, we found two changing points: 7/12/2020 and 08/01/202, while using AMOC we
just detected one: 20/12/2020. As it is observable Ethereum is the one that behaves very
similarly to Bitcoin as of change point, while Ripple significantly differ from both: Bitcoin and
Ethereum.
To sum up, the change point detection analysis in variance results suggest that there is a
possibility that the change point in March and the change point in December/January, occurred
due to the current Covid-19 pandemic situation as it was announced as the 11th of March of
2020, the World Health Organization declared the Covid-19 virus as a pandemic, and according
to the WHO the third wave of Covid-19 pandemic appeared during December 2020 and
January 2021, which coincides with the change point detected for Ripple. Once the change
point was detected, we studied the change in variance by excluding the periods from
07/03/2020 to 20/03/2020 and studying the change before and after this period. The obtained
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Correlation of cryptocurrencies: a dynamic investigation
results were showing that the variance was smaller before and after the excluded dates while
the variance for the whole data set was significantly higher. This is telling us that the periods
that were excluded to perform the variance analysis had a great impact on the overall variance,
meaning that by including the ‘excluded’ period the variance was increasing, while the contrary
was happening if we were not included the dates from the 07/03/2020 to the 20/03/2020.
Therefore, we can conclude that these excluded periods are significant in the change point
analysis as larger variance is indicating that the values in the set are far from the mean and
from each other.
The second part of the analysis consisted of the implementation of a DCC-MGARCH
model. By applying the DCC MGARCH methodology, the cross correlations between
cryptocurrencies (Bitcoin, Ethereum and Ripple), and between Bitcoin, Gold and different
financial assets were estimated. In the first DCC-MGARCH model we studied the cross
correlation between cryptocurrencies (Bitcoin, Ethereum and Ripple). The results of the model
suggested that the Bitcoin had no influence and effect on Ethereum and Ripple. However, the
persistence of volatility was very high for Ethereum (0.853) and Bitcoin (0.92), while for
Ripple it was very low (0.11). Regarding the DCC conditional dynamic correlation equation,
we observed that both coefficients 𝜆1 𝑦 𝜆2 were statistically significant at 1% level, which
implies that the co-movement of the currencies is changing over time. Finally, we observed a
high and positive correlation between Ethereum and Bitcoin with a value of 0.831, followed
by a positive correlation between Ethereum and Ripple of 0.559 and a very low positive
correlation between Bitcoin and Ripple of 0.384, all of them statistically significant at 1% level.
In the second DCC-MGARCH model we studied the cross correlation between Bitcoin
and the Equity Indices (S&P 500, MSCIWorld, and MSCIEM50) and Gold. In this case we
could observe that the model was presenting one significant value of the parameter 𝛾2, that is:
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Correlation of cryptocurrencies: a dynamic investigation
MSCIEM50, being statistically significant at 5%, which represents the influence and effect that
the Bitcoin has on the MSCIEM50. Additionally, the equation of the variance yielded the
coefficients 𝛼 and 𝛽, which, added together, show that the persistence of volatility is on average
0.94 for each of the specifications. Regarding the DCC conditional dynamic correlation
equation, both coefficients 𝜆1 𝑦 𝜆2 were statistically significant at 1%, so it is concluded that
the co-movement of the variables is changing over time. When analyzing the conditional
correlations between the standardized residuals of Bitcoin, the Equity Indices and Gold, a lower
correlation is observed between them. In fact, we only observed two statistically significant
correlations at 1% level which were: Gold and S&P 500 (-0.108), and Gold and the
MSCIWorld (-0.149), both having a statistically significant negative correlation with Gold at
1% level. Also, it can be deduced that the degree of influence of the Bitcoin on these variables
is less than that observed with cryptocurrencies.
Regarding the third part, the possible causal relationships, and through the use of Vector
Autoregressive (VAR) models and the application of the Granger test, the existence of causal
relationships between the movements of the Bitcoin and the S&P 500 and VIX index was
determined. Indeed, a rise of the S&P 500 implies a negative effect of -0.36 on Bitcoin. This
finding, which shows a relationship between Bitcoin and the S&P 500, was also found by other
authors such as: Georgoula et al. (2015); Kjærland et al. (2018); and Sovbetov and Sovbetov
(2018). On the contrary, Bitcoin was not Granger causing S&P 500. Additionally, we observed
that the VIX index was not Granger causing Bitcoin daily log returns nor the S&P 500 daily
log returns. However, we found out that the S&P 500 does Granger causes the VIX index.
Indeed, a rise of the S&P 500 implies a negative effect of -83.28 on the first difference of the
VIX index.
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Correlation of cryptocurrencies: a dynamic investigation
In relation to Bitcoin, the S&P 500 and the Covid-19 variables, it was observed that
new daily deaths due to Covid-19 Granger causes Bitcoin. This finding was also found by
Goodell and Goutte (2021). The author found that for the period that oscillates between April
5th, 2020 and April 20th, 2020 the deaths levels due to Covid-19 caused an increase in Bitcoin
prices. Indeed, in our analysis, an increase in the number of deaths due to Covid-19 implies a
positive effect of 7.19e-06.
On the contrary, we could observe that any of the Covid-19
variables Granger causes S&P 500. Finally, it was also observable that new daily deaths due to
Covid-19 Granger cause new daily cases due to Covid-19, meaning that an increase in new
daily deaths due to Covid-19 implies a negative effect of -0.53 on new daily cases. To end up,
new daily cases of Covid-19 also Granger cause new daily deaths due to Covid-19. Indeed, a
rise (fall) in new daily cases implies a negative (positive) effect on new daily deaths.
It is worth adding that before the implementation and use of VAR models in this study,
the Johansen test in levels was carried out for all the models, in order to study the possible
presence of co-integration between the different assets. The results rule out the presence of cointegration between the assets used in this analysis. Only after this correction could the Granger
test be implemented to establish possible causal relationships between Bitcoin and the S&P500,
VIX index, and Covid-19 variables.
To conclude, since the empirical section has not given very significant results, whose
reason for the lack of significance may be due to the presence of different regimes (structural
breaks), some further research could be devoted to study each regimen, separately, using the
same method as used in this paper. Additionally, this further research could contain a longer
data set by including the first half of 2021, as the vaccination is more advanced, hence more
data is available.
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Correlation of cryptocurrencies: a dynamic investigation
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ANNEX 1. CRYPTOCURRENCIES COMPARISON
Table 1. Cryptocurrencies comparison
Launch date
Circulation supply
Bitcoin (BTC)
2009
18,693,393 BTC
Ethereum (ETH)
2015
115,653,049 ETH
Max supply
21,000,000 BTC
-
$54,334.53
$1,016,842,275,217
49.4%
$46,598,770,068
$2,622.24
$302,566,753,372
14.7%
$31,638,864,762
Ripple (XRP)
2017
45,404,028,640 XRP
100,000,000,000
XRP
$1.33
$60,516,181,416
2.53%
$10,845,052,610
0.04602
0.1045
0.1793
7
20
1500
10-30 minutes
15 seconds
4 seconds
PoW
Low
Yes
PoW
Low
Yes
Strength
Largest and most
popular
Smart contracts
Downside
Slow and expensive
transactions
Slow and energyhungry
Consensus
Low
No
Fast and multicurrency
transactions
It is not
decentralized
enough
Price
Market cap
Dominance
24h volume trade
Volume/Market
cap
Max transactions
per second
Approx.
transaction time
Proof type
Anonymity
Mineable
Table 1. Cryptocurrencies comparison. Data retrieved from https://coinmarketcap.com. Latest
review: April 28th, 2021.
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ANNEX 2. IMPULSE RESPONSE FUNCTION PLOTS BITCOIN, S&P 500 AND VIX
Figure 7. Impulse response function plot: Bitcoin, S&P 500 and VIX
varbasic, dVIX, dVIX
varbasic, dVIX, rBTC
varbasic, dVIX, rSP500
varbasic, rBTC, dVIX
varbasic, rBTC, rBTC
varbasic, rBTC, rSP500
varbasic, rSP500, dVIX
varbasic, rSP500, rBTC
varbasic, rSP500, rSP500
50
0
-50
-100
50
0
-50
-100
50
0
-50
-100
0
2
4
6
8
0
2
4
6
8
0
2
4
step
95% CI
impulse-response function (irf)
Graphs by irfname, impulse variable, and response variable
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Correlation of cryptocurrencies: a dynamic investigation
ANNEX 3. IMPULSE RESPONSE FUNCTION PLOTS BITCOIN, S&P 500 AND
COVID-19 VARIABLES
Figure 8. Impulse response function plot: Bitcoin, S&P 500 and Covid-19 variables
varbasic, COVIDnewdeaths, COVIDnewdeaths
varbasic, COVIDnewdeaths, dCOVIDnewcasesvarbasic, COVIDnewdeaths, rBTC
varbasic, COVIDnewdeaths, rSP500
20000
10000
0
-10000
-20000
varbasic, dCOVIDnewcases, COVIDnewdeaths
varbasic, dCOVIDnewcases, dCOVIDnewcasesvarbasic, dCOVIDnewcases, rBTC
varbasic, dCOVIDnewcases, rSP500
20000
10000
0
-10000
-20000
varbasic, rBTC, COVIDnewdeaths
varbasic, rBTC, dCOVIDnewcases
varbasic, rBTC, rBTC
varbasic, rBTC, rSP500
varbasic, rSP500, COVIDnewdeaths
varbasic, rSP500, dCOVIDnewcases
varbasic, rSP500, rBTC
varbasic, rSP500, rSP500
20000
10000
0
-10000
-20000
20000
10000
0
-10000
-20000
0
2
4
6
8
0
2
4
6
8
0
2
4
6
8
0
2
step
95% CI
impulse-response function (irf)
Graphs by irfname, impulse variable, and response variable
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4
6
8