MASTER THESIS 
Title: Bitcoin: A long-run equilibrium with East Asian currencies 
Author: David Anaya Luque 
Advisor: Samer Ajour El Zein 
Academic year: 2019-2020
M
as
te
r 
in
 A
ct
u
ar
ia
l a
n
d
 F
in
an
ci
al
 S
ci
en
ce
s 
Faculty of Economics and Business 
Universitat de Barcelona 
Master thesis 
Master in Actuarial and Financial Sciences 
Bitcoin: A long-run 
equilibrium with East Asian 
currencies 
Author: David Anaya Luque 
Advisor: Samer Ajour El Zein 
“The content of this document is the sole responsibility of the author, who declares that he
has not incurred plagiarism and that all references to other authors have been expressed in
the text.”
iAbstract
The purpose of the study will be focused on developing and demonstrating
the existence of cointegration between the most important cryptocurrency in
the world, BTC-Bitcoin, paired with three different local currencies from
selected southeast Asian countries: China (BTC-CNY), South Korea
(BTC-KRW) and Japan (BTC-JPY).
In order to achieve the proposed objective, it is extensively developed the
entire process currently established by the most important econometric
literature, from the analysis of stationarity of the series and the detection of
unit roots to the validation of a VAR model to be contrasted with the Johansen
test, its resultant cointegration equation (long-run model) and the dynamic
adjustment for each of the target variables.
Finally, it has been proposed an ARDL bounds testing approach model to
determine and contrast the results obtained with the Johansen test and ensure
the existence of a real long-term relationship of the selected series, where the
coefficients obtained that determine the link between integrated processes
and steady state equilibrium (no growth steady state) using Johansen
approach are: 0.0056(lnBTCKRWt) and 0.0037(lnBTCJPYt); and through the
use of an ARDL bounds testing approach the following coefficients are
obtained: 0.00556(lnBTCKRWt) and 0.00426(lnBTCJPYt).
Keywords: cryptocurrency, bitcoin, econometric, financial , cointegration,
ii
Contents
Abstract i
1 Introduction 1
1.1 Thesis goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Theoretical and conceptual framework 3
3 Econometric methodology, modeling and estimation 7
3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Financial time series . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.1 Time series analysis . . . . . . . . . . . . . . . . . . . . . 8
3.2.2 Econometric model building . . . . . . . . . . . . . . . . 15
VAR(p) model . . . . . . . . . . . . . . . . . . . . . . . . . 15
Granger causality . . . . . . . . . . . . . . . . . . . . . . . 18
Impulse - Response function . . . . . . . . . . . . . . . . 20
3.2.3 Cointegration tests . . . . . . . . . . . . . . . . . . . . . . 22
Johansen test . . . . . . . . . . . . . . . . . . . . . . . . . 24
ARDL Bounds Testing approach . . . . . . . . . . . . . . 26
4 Conclusions and research limitations 30
4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Limitations of the study . . . . . . . . . . . . . . . . . . . . . . . 32
A Additional ADF, PP and KPSS tables 34
B Orthogonal Impulse Response 36
C Short-run response coefficients 39
D Plotting VECM model responses 42
E ARDL model coefficients 43
F RECM coefficient values 44
G ARDL - Cointegration equation 45
Bibliography 46
iii
List of Figures
2.1 Total Market Capitalization . . . . . . . . . . . . . . . . . . . . . 5
2.2 Percentage Total Market Capitalization . . . . . . . . . . . . . . 6
3.1 Bitcoin price series/returns against local currency . . . . . . . . 8
3.2 ACF - PACF for regular Bitcoin price series . . . . . . . . . . . . 11
3.3 ACF - PACF for Bitcoin return series . . . . . . . . . . . . . . . . 11
B.1 Orthogonal Impulse Response from BTCCNY . . . . . . . . . . 37
B.2 Orthogonal Impulse Response from BTCKRW . . . . . . . . . . 37
B.3 Orthogonal Impulse Response from BTCJPY . . . . . . . . . . . 38
C.1 Response BTCCNY coefficients . . . . . . . . . . . . . . . . . . . 40
C.2 Response BTCKRW coefficients . . . . . . . . . . . . . . . . . . . 40
C.3 Response BTCJPY coefficients . . . . . . . . . . . . . . . . . . . . 41
D.1 VECM model responses . . . . . . . . . . . . . . . . . . . . . . . 42
E.1 ARDL model coefficients . . . . . . . . . . . . . . . . . . . . . . . 43
F.1 Short-run dynamic and long-run coefficients . . . . . . . . . . . 44
G.1 Cointegration equation - Projected vs Observed . . . . . . . . . 45
iv
List of Tables
2.1 Percentage of Total Market Capitalization . . . . . . . . . . . . . 4
3.1 ADF, PP and KPSS tests of BTC-CNY series (prices) . . . . . . . 13
3.2 ADF, PP and KPSS tests of BTC-CNY series (returns) . . . . . . 13
3.3 Zivot-Andrews test of BTC-CNY series (prices) . . . . . . . . . . 14
3.4 Zivot-Andrews test of BTC-CNY series (returns) . . . . . . . . . 14
3.5 VAR(p) model lag selection . . . . . . . . . . . . . . . . . . . . . 17
3.6 Multivariate Portmanteau Test (adjusted) . . . . . . . . . . . . . 18
3.7 Granger causality VAR(1), m=3 . . . . . . . . . . . . . . . . . . . 19
3.8 Variance - covariance matrix VAR(1), m=3 . . . . . . . . . . . . . 20
3.9 Correlation matrix VAR(1), m=3 . . . . . . . . . . . . . . . . . . . 20
3.10 OR - Choleski decomposition VAR(1), m=3 . . . . . . . . . . . . 21
3.11 Trace test - VAR(p=1), k=2 . . . . . . . . . . . . . . . . . . . . . . 24
3.12 Maximum eigenvalue test - VAR(p=1), k=2 . . . . . . . . . . . . 25
3.13 Bounds F-test (Wald) . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.14 ARDL model selection . . . . . . . . . . . . . . . . . . . . . . . . 28
3.15 Durbin-Watson test . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.16 Breusch-Pagan test . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.17 Ramsey RESET test . . . . . . . . . . . . . . . . . . . . . . . . . . 29
A.1 ADF, PP and KPSS tests of BTC-KRW series (prices) . . . . . . . 35
A.2 ADF, PP and KPSS tests of BTC-KRW series (returns) . . . . . . 35
A.3 ADF, PP and KPSS tests of BTC-JPY series (prices) . . . . . . . . 35
A.4 ADF, PP and KPSS tests of BTC-JPY series (prices) . . . . . . . . 35
1Chapter 1
Introduction
Cryptocurrencies (Lam and Lee, 2015) are a subset of the class of digital
currencies that have gone from being a digital payment instrument to a great
source of interest when it comes to making large-volume investments and
speculating on their evolution in the markets. Taking advantage of the wide
range of econometric methods that are currently used to manage financial
asset portfolios, the aim of this study is to focus on verify the existence of
cointegration of the selected three series of Bitcoin in order to determine the
long-run model and the short-run dynamics adjustments that explain the
model.
In order to detail the progress that it has achieved and what is being done at
every stage, it has been decided to organize the project as follows:
• In Chapter 2 it is exposed a brief approximation of the theoretical and
conceptual framework that exists around Bitcoin and general
cryptocurrencies in the market, the volume of capitalization and the
level of dominance of each of them in the actual market.
• Once they are introduced, Chapter 3 introduces the econometric
methodology, modeling and estimation that is going to be established in
order to achieve the objective of this work, where there are exposed the
main tests and contrasts applied during the econometric process and the
methodological development behind each one.
• Finally, Chapter 4 introduces the final conclusions and research
limitations observed during the econometric process of determining our
main objectives, identifying literature gaps and the main problems that
could currently alter the results of our study.
Chapter 1. Introduction 2
1.1 Thesis goals
The goals of the thesis are dual. The thesis is organized to commence with an
extensive literature review of the time series analysis that comprises methods
for analyzing time series data in order to extract meaningful statistics and
other characteristics of the data. In addition, it also focuses on that literature
that involve the whole process of cointegration analysis, the evidence that
exists and alternative methods to study this relationship in the long-run state.
Everything followed by an explanatory description of the project objectives
and methodology.
To meet the second goal, all the literature analyzed is applied to schematically
develop the whole econometric process that includes the determination of
cointegration equations and the dynamic adjustments using the Johansen test
and, comparatively, deriving and modelling an ARDL model using the
bounds testing approach. This process, together with the selected literature
review, allows us to derive robust results about the hypotheses developed in
this study and also to confirm that the goals of the thesis have been met
opening the line for a further research.
3Chapter 2
Theoretical and conceptual
framework
Nowadays it is hard to imagine someone who has never heard before the
word Bitcoin or cryptocurrency in any media such as television or economic
newspapers. That is because in 2009 someone decided to create an
open-source software to break with the traditional banking system and let a
digital asset flow away through the internet. That open-source was called
Bitcoin and it was considered the first decentralized cryptocurrency and
distributed ledger technology, typically well-known as a blockchain. It
couldn’t be copied, it couldn’t be duplicated or pirated, so those primary
attributes gave its confidence and secure to the people that wanted to get in
this world.
Since then, many cryptocurrencies have been created as a medium of
exchange to do secure financial transactions in internet outside the existing
banking system. Even to use them as a stock exchange in the market, most of
them paired with the most important currencies in the world and they have
generated unimaginable profits to all those people who decided to go long on
these currencies in its first years of existence.
By definition, and according to (Narayanan and Goldfeder, 2016), digital
currencies are, unlike physical currencies, not controlled by economic systems
and centralized banks. It is also an anonymous ecosystem, which means that
no one can identify who is making transactions while still making the
transaction data public. This makes it protected from third parties and lobbies
that could influence the demand or control the prices, and that is why certain
politicians from some different countries have banned the use of this type of
digital currencies over the said territory. The rate with which new currency is
created is defined from the beginning and is known to the public.
Chapter 2. Theoretical and conceptual framework 4
At the same time, though, most of them have also been designed to
progressively decrease the production of new units -making it reach a cap on
the total amount of currency that exists in order to prevent high inflation.
At present, according to CoinMarketCap1 reports, there are more than 5.000
cryptocurrencies in the world and more than 21.000 markets, where these
digital currencies are paired with standard reference currencies and in most of
them the US dollar predominates. Total market capitalization average stands
around $ 250.000.000.000, and the market share or dominance of the most
important cryptocurrencies over the total market capitalization is shown in
the following table, where it can be clearly seen how Bitcoin accounts for
almost 64% of the total.
TABLE 2.1: Percentage of Total Market Capitalization,
“Source: own elaboration”
Cryptocurrency % of total
Bitcoin 63.77 %
Ethereum 9.87 %
XRP 3.91 %
Tether 2.87 %
Bitcoin Cash 2.04 %
Bitcoin SV 1.62 %
Litecoin 1.30 %
The following graphic (Figure 1.1) shows the time series as a whole with
respect to the total market capitalization of each of the cryptocurrencies,
where the volume remains constant during the previous years to 2016 due to
the low opening of this type of currency to the world and the technological
environment.
Additionally, it must be taken into account that, for example, Bitcoin was
born in 2009 and was one of the youngest to appear in this environment, but
most of the 5,000 that currently exist have been born in mid-2013, which
makes them have a fairly tight margin to grow in a market where Bitcoin has
been made with practically all of it.
1https://coinmarketcap.com/all/views/all/
Chapter 2. Theoretical and conceptual framework 5
FIGURE 2.1: Total Market Capitalization,
“Source: CoinMarketCap”
Even so, the significant growth that Bitcoin showed did what it could be
said today as the starting point of everything that exists right now, and what
is to come.
New digital currencies began to emerge, new blockchain-related technologies
that further improved the anonymity and security of participants in the
network, new protocols that were created allowed more efficient mining, in
environments such as Tor, and therefore, more profit and more speculation.
Two of the cryptocurrencies that managed to move important whales2 and
private investment funds to break the uptrend of Bitcoin in 2017 were
Ethereum and XRP.
Both were born in 2013 and 2012 respectively and thanks to the development
of new technologies, they managed to be one of the currencies with the
highest gains in 2017 3, and large-scale displacement of an important market
volume towards them.
2https://btcmanager.com/what-are-whales-how-do-they-affect-cryptocurrencies/
3https://www.coindesk.com/900-20000-bitcoins-historic-2017-price-run-revisited
Chapter 2. Theoretical and conceptual framework 6
Because of that, as it is explained in (Fry and Cheah, 2016), the prices of
the currencies in the markets began to move very quickly, causing great
profits and at the same time great losses to large investors that didn’t know
why volatility had shot up so aggressively, as that large shifting of currencies
from portfolio to portfolio was displacing supply and demand out of control.
Fortunately, time has made it possible for these situations to be foreseen as far
as possible, and there are web pages that centralize the registration of all
transactions in real time (which are public as explained above) to detect
anomalies4 or important movements of currencies between wallets that can
shake the whole system. Although the origin or final destination of a
transaction can be unknown, with the data of each movement available to
anyone who wants information about how much money is flowing through
the network at a certain point in time, the system may be somewhat more
transparent against the excess of anonymity.
The next graphic shows how Bitcoin’s market dominance has been declining
since 2013 and, especially, the great debacle in late 2017 and early 2018 in a
very graphic way.
FIGURE 2.2: % of Total Market Capitalization,
“Source: CoinMarketCap”
4https://whale-alert.io/
7Chapter 3
Econometric methodology,
modeling and estimation
One of the aspects that is being very important in this field especially to
explain certain pairs of economic or financial variables is what is known as
the existence of cointegration, formalized by (Engle and Granjer, 1987), and
approached as a long-run economic relationship explained by a linear
combination I(0) of two or more variables I(1). In the following sections to be
developed - within the econometric environment with the aim of detecting
the movement pattern of each of the series analyzed and its behavior in the
long term, as well as the short-term shocks that may alter the steady state, a
study based on demonstrating the existence or not of cointegrated series and
the entire scaled process to develop the study will be considered, including
both theoretical and practical approaches.
First steps are focused on analyzing the time series data - a summary of the
main descriptive statistics for each series, verifying the adequacy of them
using techniques to ensure the existence of stationarity and non correlated
residuals. Following to the next steps, it is going to be focused on
determining an optimal VAR(p) model through the use of different
information criterion and verifying the adequacy of the model but this time
using multivariate tests. In addition, the Impulse-Response function and the
Granger causality test are applied to describe the evolution of the model’s
variables in reaction to a shock or to determine whether one time series is
useful in forecasting another.
Finally, it is covered the entire process derived from determining the
calculations related to the resultant cointegration equation. In this section, it
is first exposed the Johansen method, and later on, the determination of an
ARDL Bounds testing approach model as an alternative method to compare
the obtained results and verify that the results obtained are consistent and
may or may not justify the existence of co-integrated series.
Chapter 3. Econometric methodology, modeling and estimation 8
3.1 Data
To begin the analysis, the financial time series referring to the price of Bitcoin
will be used at par with the local currency of each country. So in total there
will be three series with the following labels: BTC-CNY for the Chinese case;
BTC-KRW in the case of South-Korean; and BTC-JPY for the Japanese case.
The sequence of discrete-time data will take place from (2014.9) to (2019.9),
with a daily observation frequency (base 365.25). The data has been extracted
from the Yahoo! Finance using the Quantmod package in R.
The reason why Bitcoin is paired with the local currency is because it is
desired to observe if there any macroeconomic implication or impact from the
current national currency from each of the countries in our estimations.
3.2 Financial time series
3.2.1 Time series analysis
The first step will consist of analyzing the data available for each of the series
and performing the first diagnostic tests to determine if they are series that
will be adapted to our objectives. To do this, it has to be identified if it is
necessary to stabilize mean and variance using logarithmic differentiation.
FIGURE 3.1: Bitcoin price series/returns against local currency,
“Source: own elaboration”
Chapter 3. Econometric methodology, modeling and estimation 9
The three plots on the left display the prices of each of the series and how
invariable they remain from late-2014 until mid-2016. From early 2017 they
rise exponentially from a global average of 1-3 dollars to an average of
400-1500 dollars, with average growth rates of 20,000% and 30,000%,
respectively.
Once they top the maximum resistance in mid-2018, they reverse immediately
to the initial average prices as a consequence of the massive closing long
positions.
Then, it can be affirmed that they are not stationary, because they present a
variability that increases with the level of the series and their behaviour keep
changing over time in an unpredictable way. This is what is known as
heteroskedasticity.
The other three plots on the right display the returns, where it can be
anticipated that these, just as it should be, will be stationary and integrated of
order 0, because it is known that all stationary processes are integrated I(0),
but not all I(0) processes are stationary.
It is clear the existence of relationship between each of the series: same
pattern in the evolution of the price and range of similar returns, with no
clearly defined volatility clusters over the analyzed period of time. The
following tables contain the descriptive statistics for each series.
Before starting to execute commands, it is necessary to make an analysis of
the most important statistics of our data to ensure that there are no errors, and
in the same way, to better know what type of observations are selected.
The following tables contain the descriptive statistics for each series:
basicStats BTC-CNY basicStats BTC-KRW basicStats BTC-JPY
NAs 0
Mean 520660.2
Stdev 16793.51
Skewness -0.096634
Kurtosis -0.496225
NAs 0
Mean 2966433
Stdev 4086366
Skewness 1.739114
Kurtosis 2.554908
NAs 0
Mean 298831
Stdev 418792.3
Skewness 1.824194
Kurtosis 3.000417
If the tables are analyzed, it can be seen that there is no NA values, therefore,
it can be ensured that we do not have incomplete data in our series. In the
same way, it can be seen that the skewness is not 0, it is greater(lower) than
this value, but with levels very close to 0, therefore, it will differ in terms of
data size in both queues, but a very small difference. It can be applied the
same criteria in the case of kurtosis.
Chapter 3. Econometric methodology, modeling and estimation 10
To continue with the process of verifying the stationarity of the time series
data, it will be proceeded to start informal methods to evaluate the existence
of unit root.
To do that, it is going to calculate the ACF and PACF, which is known as
Autocorrelation Function and Partial Autocorrelation Function that it is
introduced by (Dickey and Fuller, 1979), in order to detect the linear
dependence of a variable with itself at two points in time.
By definition, it is known that for stationary processes, autoccorrelation
between any two observations only depends on the time lag h between them.
Additionally, for testing the significance of a single lag-h autocorrelation it is
calculated the standard error.
So, mathematically:
ρh = Corr(yt, yt − h) = γhγ o
(3.1)
ρˆh =
∑Tt=h+1(yt − yˆ)(yt−h − yˆ)
∑Tt=1(yt − yˆ)2
(3.2)
SEp =
√
(1+
2∑h−1i=1 ρˆ
2
i
N
(3.3)
On the next page there are exposed the outputs obtained by applying this
procedure:
• Figure 2.2 shows how the price series do not look stationary in each of
the ACF plots and it can be seen how they remain constant and fall very
slowly over the lags period.
• At the same time, Figure 2.3 shows in the opposite way how the series
of returns fall sharply in the ACF and PACF plots, meaning that only
describe the direct relationship between an observation and its lag.
Chapter 3. Econometric methodology, modeling and estimation 11
FIGURE 3.2: ACF - PACF for regular Bitcoin price series
FIGURE 3.3: ACF - PACF for Bitcoin return series
Chapter 3. Econometric methodology, modeling and estimation 12
To make easier the interpretation, (Ljung and Box, 1978) introduced an
aproximation of what is known as Ljung-Box Q-test, that it is also applied to
check, not just visually, the null hypothesis that a series of residuals exhibits
no autocorrelation for a fixed number of lags, against the alternative
hypothesis that some autocorrelation coefficient is different from zero - they
should be near zero for any and all time-lag.
Q = T(T + 2)
L
∑
K=1
(
ρ(k)2
(T − k) (3.4)
As expected, once it is computed, it is clear that it is necessary to transform
the time series data in order to meet the first requirement, that is stationarity.
To achieve this, logarithms will be applied to correct the heteroscedasticity,
and it will be differentiated to eliminate the trend and the seasonal
component.
But, before carrying out these transformations, there are other ways to prove
that the series really have to be differentiated, and they are what are called as
formal methods.
• Augmented Dickey-Fuller test (ADF): to test the null hypothesis that
exist a unit root under the approximation of the critical value by (Dickey
and Fuller, 1979).
• Phillips-Perron test (PP): to test the null hypothesis that a time series is
I(1), introduced later than the previous methodology and developed by
(Phillips and Perron, 1988) in their publication.
• Kwiatkowski-Phillips-Schmidt-Shin test (KPSS): to test the same but
as H1. This one is the last tests introduced by (Kwiatkowski and Shin,
1992) with some additions that differs from the previous ones.
The first of the tests, the ADF, tests the null hypothesis from the following
expression:
yt = c+ δt +∅yt−1 + βp∂yt−p + et, (3.5)
where the null hypothesis is empty set is equal to 1 (exist a unit root), and the
alternative hypothesis is that empty set is strictly less than 1.
The second of the tests, the PP, tests the null hypothesis from the following
expression:
yt = c+ δt +∅yt−1 + e(t), (3.6)
where the alternative hypothesis is that a time series is not integrated of order
Chapter 3. Econometric methodology, modeling and estimation 13
1, and the null hypothesis is that the time series is integrated of order 1, I(1),
under the equation:
(1− L)dXt, d = 1 (3.7)
Finally, the third of the tests, the KPSS, test the inverse null hypothesis of the
PP test following expression:
yt = ct + δt + u1t (3.8)
ct = ct−1 + u2t (3.9)
The following tables collect the results of applying each of the tests in each of
the series, with the corresponding p-value (t-value) and the t-statistic value.
TABLE 3.1: ADF, PP and KPSS tests of BTC-CNY series (prices),
“Source: own elaboration”
Test ADF PP KPSS
Null hypothesis Non stationary Non stationary Stationary
P-value/T-value 0.1699 -1.2246 2.1867
T-statistic value - -2.863999 0.146
Results NRH0 NRH0 RH0
TABLE 3.2: ADF, PP and KPSS tests of BTC-CNY series (returns),
“Source: own elaboration”
Test ADF PP KPSS
Null hypothesis Non stationary Non stationary Stationary
P-value/T-value 0.01 -36.8573 0.0056
T-statistic value - -2.864016 0.146
Results RH0 RH0 NRH0
It is only shown for BTC-CNY time series, but the pattern is the same for all
series. Once the series are differentiated, they are stationary, and if it is plotted
all three differentiated series in just one plot it can be observed that they are
practically identical. Even so, the tables with the results for the remaining two
variables are attached in Appendix B, for both price series and returns.
Chapter 3. Econometric methodology, modeling and estimation 14
Additionally, it can be applied the Zivot-Andrews test to contrast the null
hypothesis of unit root with structural break in the intercept. Although this
test is not usually included in the group of formal methods to identify the
existence of unit root, it is usually used especially in economic and financial
series to detect potential break points at a certain point in time.
This one developed by (Zivot and Andrews, 1992) in their publication related
with the Great Crash and the Oil-Price Shock.
TABLE 3.3: Zivot-Andrews test of BTC-CNY series (prices),
“Source: own elaboration”
Test Zivot-Andrews
Null hypothesis Non stationary
P-value/T-value -3.2656
T-statistic value -4.8
Results NRH0
TABLE 3.4: Zivot-Andrews test of BTC-CNY series (returns),
“Source: own elaboration”
Test Zivot-Andrews
Null hypothesis Non stationary
P-value/T-value -31.1698
T-statistic value -4.8
Results RH0
As it is rejected the null hypothesis of unit root in favour of a one time break
in the intercept, then it is assumed the alternative hypothesis of a trend
stationary process with a break in the intercept at position 1186 ["2017-09-15"].
The same process can be applied for the other series and it will be achieved the same
result, which return series are stationary and the same potential break point at the
same position.
Chapter 3. Econometric methodology, modeling and estimation 15
3.2.2 Econometric model building
VAR(p) model
The main objective of carrying out this analysis through the use of
multivariate models is to determine if in the long term these three series,
BTC-CNY, BTC-KRW and BTC-JPY, will tend to move in the same way over
the time and if they will converge each other at the same price level. This
statistical property is known as cointegration, and reflects the presence of a
long-run equilibrium towards which the economic system converges over
time in order to maximize forecasting methods. By definition, if there exists a
stationary linear combination of the three non stationary series, the series
combined are said to be cointegrated.
Once it is assured that the price series are integrated of order 1 and returns
integrated of order 0, the next step would be focused on determinate a Vector
Autoregressive Model, which is a multidimensional extension well detailed
in (Hamilton, 1994) of the classical autoregressive models.
It is a model that contains a system of n equations of n distinct, stationary
response variables as linear functions of lagged responses. It is characterized
by their degree p, VAR(p), that represents the number of lags of all variables
in the system.
A general Vector Autoregressive Model is similar to the AR(p) model except
that each quantity is vector valued and matrices are used as the coefficients.
Taking into consider this idea, the general form of a VAR(p) model is:
Xt = α+ β1Xt−1 + β2Xt−2 + ...+ βpXt−pet (3.10)
where every X multiplying the beta parameter is called the p-th lag of Xt;
α parameter is a k-vector of constants (intercepts); β parameter is a
time-invariant (k x k)-matrix, and the last component is a k-vector of error
terms.
For example: VAR(1) with three variables/series (m=3)
X1,t = α1 + β1,1X1,t−1 + β1,2X2,t−1 + β1,3X3,t−1e1t
X2,t = α1 + β2,1X1,t−1 + β2,2X2,t−1 + β2,3X3,t−1e2t
X3,t = α1 + β3,1X1,t−1 + β3,2X2,t−1 + β3,3X3,t−1e3t
Chapter 3. Econometric methodology, modeling and estimation 16
For example: VAR(1) with (m=3) can be written in matrix form as:X1,tX2,t
X3,t
 =
β1,1 β1,2 β1,3β2,1 β2,2 β2,3
β3,1 β3,2 β3,3
 x
X1,t−1X2,t−1
X3,t−1
+
e1,te2,t
e3,t
 (3.11)
Then, in order to select the optimal number of lags to use to estimate the model,
it is going to be estimated a VAR(p) by OLS per equation using the function
VARselect() in R programming language. This reports the appropiate number
of lags using an information criteria, such as:
• Akaike’s Information Criterion, (Akaike, 1974) :
AIC(n) = −2max[log(L)] + 2K, (3.12)
where max[log(L)] is the maximized log likelihood function and K the number
of parameters.
• Hannan-Quinn Information Criterion, (Quinn, 1980) :
HQ(n) = Klog(n)− 2max[log(L)], (3.13)
where max[log(L)] is the maximized log likelihood function, K the number of
parameters and n the number of observations.
• Schwarz Information Criterion, (Schwarz, 1978) :
SC(n) = −2max[L] + 2K ∗ log(log(n)), (3.14)
where max[L] is the maximized likelihood function, K the number of
parameters and n the number of observations.
Computing the previous function taking into account the structure and its
components:
The previous table details, by minimizing the result obtained by each
information criterion, that the four possible alternatives suggest that the
VAR(p=1) model is the most appropriate. (The following figures refer to the
differentiated log variables)
BTCCNYt = α1 + β1,1BTCCNYt−1 + β1,2BTCKRWt−1 + β1,3BTCJPYt−1e1t
Chapter 3. Econometric methodology, modeling and estimation 17
TABLE 3.5: VAR(p) model lag selection,
“Source: own elaboration”
VAR(p) AIC(n) HQ(n) SC(n) FPE(n)
1 -27.82954 -27.81309 -27.78548 8.199405e-13
2 -27.82031 -27.79153 -27.74321 8.275439e-13
3 -27.81302 -27.77190 -27.70287 8.335987e-13
4 -27.80608 -27.75262 -27.66289 8.394053e-13
5 -27.79716 -27.73136 -27.62092 8.469318e-13
... ... ... ... ...
10 -27.78380 -27.65631 -27.44233 8.583392e-13
Selection 1 1 1 1
BTCKRWt = α2 + β2,1BTCCNYt−1 + β2,2BTCKRWt−1 + β2,3BTCJPYt−1e2t
BTCJPYt = α3 + β3,1BTCCNYt−1 + β3,2BTCKRWt−1 + β3,3BTCJPYt−1e3t
And the coefficients for the estimated model are, in the previous order:
̂BTCCNY = (0.0020)+ (0.0094)BTCCNYt−1+(0.4200)BTCKRWt−1− (0.4090)BTCJPYt−1
̂BTCKRW = (0.0020)+ (0.062)BTCCNYt−1+(0.3328)BTCKRWt−1− (0.3772)BTCJPYt−1̂BTCJPY = (0.0019)+ (0.0698)BTCCNYt−1+(0.3672)BTCKRWt−1− (0.4167)BTCJPYt−1
High values of p (overestimation or over-parametrization) will reduce the
forecast precision of the corresponding estimated VAR(p) model, the
estimation precision of the impulse response, and the approximated mean
square error matrix with each additional lag level.
For that reason, it is decided to accept p = 1 as the ideal number of lags that
minimize the value of the each of the information criteria method to carry out
the study with a VAR at this level/order.
But before consolidating this lag order as the appropriate value for the model,
it has to be diagnosed in order to detect undesirable situations like the
existence of autocorrelation of the residuals.
To do that, (Edgerton and Shukur, 1999) expose clearly what is and how to
deal with the Portmanteau Autocorrelation Test, that computes the
multivariate Box-Pierce/Ljung-Box Q-statistics for residual serial correlation
up to the specified order. The diagnosis tests used in VAR models are similar
Chapter 3. Econometric methodology, modeling and estimation 18
to those used in AR models, but amplified to a dimension of k correlation
matrix of the residuals.
To check if the VAR residual are white noise, the hypothesis to test is:
H0 : ρ1 = ρ2 = ... = ρm = 0 (3.15)
H1 : ρi 6= 0
where the multivariate Portmanteau adjusted approximation has the following
form:
Qk(m) = T2
m
∑
k=1
(
1
(T − k) tr(ρˆ
′
kρˆ
−1
0 ∗ ρˆkρˆ−10 ) (3.16)
ρˆk = ρˆij(k)
Computationally, the null hypothesis of no residual autocorrelation up to lag
m is tested to check how many of them pass the test. In this case, as the series
are daily observed over the analyzed period, it is contrasted the VAR(p=1)
until the lag 14 (2 weeks seems enough) to test for serially correlated errors.
H0 : ρ2 = ρ3 = ρ4 = ρ5 = ρ6 = ... = ρ14 = 0 (3.17)
The following table shows that as the p-value is in the acceptance region, it
cannot be rejected with a significance of 5% the null hypothesis of no residual
autocorrelation of VAR(1) up to lag 14.
TABLE 3.6: Multivariate Portmanteau Test (adjusted),
“Source: own elaboration”
Portmanteau Test (adjusted) Residuals of VAR
Chi-squared 117.72
df 117
p-value 0.4639
Result NRH0
Granger causality
The (Granger, 1969) causality test is a statistical hypothesis test for
determining whether one time series is useful in forecasting another. That
means that, in Granger’s sense, it shows if the past values of one of the time
series analyzed Granger cause the other two series; or in other words, if the
Chapter 3. Econometric methodology, modeling and estimation 19
first time series helps to forecast one of the other two series.
The idea of this test is that there may exist co-movements between the three
time series (variables) - they will trend together in finding a cointegrating
relationship, then there is causality between these variables at least in one
direction. It can exists unidirectional causality relationships, a bidirectional
causality relationships between them, or lack of any causal relationships.
For example, assuming the estimated VAR(1), the null hypothesis is such as:
TABLE 3.7: Granger causality VAR(1), m=3,
“Source: own elaboration”
Granger causality Coefficient p-value
BTCCNYt does 6 Granger−−−−→ BTCKRWt β2,1 = 0 0.9486
BTCKRWt does 6 Granger−−−−→ BTCCNYt β1,2 = 0 0.06315
BTCJPYt does 6 Granger−−−−→ BTCKRWt β2,3 = 0 0.0525
BTCKRWt does 6 Granger−−−−→ BTCJPYt β3,2 = 0 0.06315
BTCJPYt does 6 Granger−−−−→ BTCCNYt β1,3 = 0 0.0525
BTCCNYt does 6 Granger−−−−→ BTCJPYt β3,1 = 0 0.9486
Using a robust heteroscedasticity-consistent variance-covariance matrix for
the Granger test, in every resultant p-value from the previous table of each
relationships it can be said with a significance of 5% that:
BTCCNYt does 6 Granger−−−−→ BTCKRWt
BTCKRWt does 6 Granger−−−−→ BTCCNYt
BTCJPYt does 6 Granger−−−−→ BTCKRWt
BTCKRWt does 6 Granger−−−−→ BTCJPYt
BTCJPYt does 6 Granger−−−−→ BTCCNYt
BTCCNYt does 6 Granger−−−−→ BTCJPYt
Therefore, it can be affirmed that none of the paired variables can affect the
other variables of the same vector and does not provide anything significant
when it comes to predicting at some stage in the future.
Chapter 3. Econometric methodology, modeling and estimation 20
Impulse - Response function
Since all variables in a VAR model depend on each other, individual
coefficient estimates only provide limited information on the reaction of the
system to a shock. Then, here appears the IR function, popularized in
econometrics by Sims (1980), which the main purpose of using it is to describe
the evolution of a model’s variables in reaction to a shock in one or more
variables estimating standard deviation of the disturbance term to determine
the impulse. This allows to trace the effects of an innovation shock to one
variable on the response of all variables in the VAR model system for several
periods in future, starting in t = 0 on the own variable to the others in t = 1 .
TABLE 3.8: Variance - covariance matrix VAR(1), m=3,
“Source: own elaboration”
Variance - covariance matrix BTCCNY BTCKRW BTCJPY
BTCCNY 0.001564549 0.001556504 0.001563982
BTCKRW 0.001556504 0.001569861 0.001561939
BTCJPY 0.001563982 0.001561939 0.001589296
In the previous table it can be seen that the off-diagonal elements of the
estimated variance-covariance matrix are not zero, so it can be assumed that
there is contemporaneous correlation between the variables in the VAR (1)
model estimated.
The correlation matrix can be expressed as follows:
TABLE 3.9: Correlation matrix VAR(1), m=3,
“Source: own elaboration”
Correlation matrix BTCCNY BTCKRW BTCJPY
BTCCNY 1.0000000 0.9931735 0.9918244
BTCKRW 0.9931735 1.0000000 0.9888514
BTCJPY 0.9918244 0.9888514 1.0000000
But the main goal of using the IR function is to determine in which direction
the impact will affect the behavior of the rest of the variables of the model,
and the correlation matrix do not express this information at all.
Chapter 3. Econometric methodology, modeling and estimation 21
An approach to identify how the shocks impact on the VAR(1) model
estimated is computing Orthogonalized Responses using Cholesky
decomposition.
The basic idea, well-developed in (Lütkepohl, 2006) is to decompose the
previous estimated variance-covariance matrix to obtain the lower triangular
matrix with positive diagonal elements through the use of Choleski
decomposition. Doing this, it can be observed the sensitivity to a
contemporaneous shock on each of the variables.
TABLE 3.10: OR - Choleski decomposition VAR(1), m=3,
“Source: own elaboration”
Choleski decomposition BTCCNY BTCKRW BTCJPY
BTCCNY 0.03955438 0.000000000 0.00000000
BTCKRW 0.03935100 0.004621719 0.00000000
BTCJPY 0.03954005 0.001297926 0.00491894
Note that the obtained matrix using Choleski decomposition is a lower
triangular matrix, where the first equation does not have any other
endogenous variables from the system, the second equation has the first two
and the third has the first three.
Then, the first has only its own innovation, while the second has implicitly
the first coming from the inclusion of the contemporaneous first variable. In
other words, the variable in the first row (BTCCNY) will never be sensitive to
a contemporaneous shock of any other variable, and the last variable
(BTCJPY) in the system will be sensitive to shocks of all other variables.
Graphically, the orthogonal impulse response from each of the variables has
been moved to the Appendix C, but it can be deduced in the following points:
• The first plot shows the estimated standard deviation of the disturbance
term as the response of BTCCNY to BTCKRW, which it has a minimum
positive impact at t = 1 but then it gradually decreases until the effect of
the shock disappears at t = 2.
• The second plot (Figure 2.5) shows the estimated standard deviation of
the disturbance term as the response of BTCKRW to BTCCNY, which it
has a no impact (not sensitive to a contemporaneous shock) at t = 0 but
then it gradually increases at t = 1, where it hits its steady value, and
then it decreases to the a value close to initial one at t = 3.
Chapter 3. Econometric methodology, modeling and estimation 22
• The third plot (Figure 2.6) shows the estimated standard deviation of the
disturbance term as the response of BTCJPY to BTCCNY, which it has a
no impact at t = 0 but then it decreases at t = 1, where it hits its steady
value, and then it increases to the a value close to initial one at t = 3.
3.2.3 Cointegration tests
It is normally assumed that when two or more series follow the same trend or
pattern, that is, they tied together in the long term, these are said to be
cointegrated.
Before start working on this concept, let’s introduce some important rules of
linear combinations of integrated series:
Single combination [x1t]:
• If x1,t ∼ I(0), then ω + φ x1,t ∼ I(0)
• If x1,t ∼ I(1), then ω + φ x1,t ∼ I(1)
Linear combination [x1t, x2t]:
• If x1,t ∼ I(0) and x2,t ∼ I(0), then ω x1,t + φ x2,t ∼ I(0)
• If x1,t ∼ I(0) and x2,t ∼ I(1), then ω x1,t + φ x2,t ∼ I(1)
• If x1,t ∼ I(1) and x2,t ∼ I(1), then ω x1,t + φ x2,t ∼ I(1)
But there is an exception that breaks this last rule and let to be ω x1t + φ x2t ∼
I(0), and Granger (1987) pointed out that situation:
"A linear combination of two or more non stationary series can be stationary. If this
linear stationary combination or I(0) exist, it is said that the series are cointegrated."
There is also one important case in which the previous statistical result is not
applied, and it is when [x1t, x2t] ∼ I(1), but they are not cointegrated. This
means that there is no linear combination of the variables that ∼ I(0). This
situation was pointed out by Granger and Newbold (1974), and it is known as
the spurious regression problem.
It take place when is regressed one random walk onto another independent
random walk and the estimated coefficient is significant, despite that the true
value of the coefficient β = 0. So the residual is non stationary and the
apparently significant relationship between two series just because they move
together along the time are, in fact, unrelated series.
Mathematically, a linear combination using three time series (true related)
Chapter 3. Econometric methodology, modeling and estimation 23
that are independently non stationary, [x1t, x2t, x3t], and grouped by a single
vector Xt, such that:
Xt = [x1 = (x1,1, x1,2, ..., x1,t), x2 = (x2,1, x2,2, ..., x2,t), x3 = (x3,1, x3,2, ..., x3,t)]
where they can be combined in a way that their linear combination is a
stationary equilibrium relationship.
In the following expression, β represents what is known as the cointegrating
vector:
βXt = β1x1,t + β2x2,t + β3x3,t ∼ I(0) (3.18)
Because of the fact that there can be multiple cointegrating vectors that fit the
same model, it has to be imposed a restriction to normalize the cointegrating
vector for estimation, such as β1 = 1.
βYt = y1,t − β2y2,t − β3y3,t ∼ I(0) (3.19)
y1,t = β2y2,t + β3y3,t + µt,
By taking the residuals from a linear combination, such as:
yt = β1 + β2xt + et (3.20)
eˆt = yt − βˆ1 − βˆ2xt
If this eˆ ∼ I(0) are stationary, then xt and yt are cointegrated.
The Engle and Granjer (1992) cointegration test considers the case that there is
a single cointegrating vector and a procedure method in two stages, r = 1 and
k = 2, based on estimated residuals. It follows the the previous idea that if
variables are cointegrated, then the residual of the cointegrated regression
should be stationary.
x1,t = α+ β1x2,t + e1,t, (3.21)
x2,t = α+ β1x1,t + e2,t,
Regress and test for no-cointegration by testing for a unit root for each
equations (ADF test):
∆e1,t = φ1e2,t−1 + ρ1,t, (3.22)
∆e2,t = φ2e2,t−1 + ρ2,t,
Chapter 3. Econometric methodology, modeling and estimation 24
If it is not possible to reject the null hypotheses that φ1, φ2 = 0, then it cannot
be rejected the hypothesis that this linear combination of variables are not
cointegrated.
Johansen test
In this situation, it is gonna be applied the (Johansen, 1988) maximum
eigenvalue and trace tests for cointegration under the empirically relevant
situation of near-integrated variables. One of the main features in front of the
Engle-Granger test is that this one permits more than one cointegrating
relationship, so it can be applied more generally when working in situations
with more than two variables or financial data, for example.
Trace tests and Maximum eigenvalue tests examine the number of linear
combinations (r, rank) to be equal to a given value (r∗) under the following
statistic and hypothesis:
λtrace(r) = −T
n
∑
i=r∗+1
ln(1− λˆi) (3.23)
λmax(r, r+ 1) = −Tln(1− λˆr+1)
This means that for the trace test statistic, λtrace(r), it is contrasted the null
hypothesis that the number of cointegrating vectors is r = r∗ < k against the
alternative hypothesis that the number of cointegrating vectors is r = k,
where k = p(lags) + 1.
In the case of the maximum eigenvalue test statistic, λmax(r, r + 1), it is
contrasted the null hypothesis that the number of cointegrating vectors is
r = r∗ < k against the alternative hypothesis that the number of cointegrating
vectors is r = r ∗+1.
TABLE 3.11: Trace test - VAR(p=1), k=2,
“Source: own elaboration”
Rank Test 10 pct 5 pct 1 pct
r <= 2 1.50 7.52 9.24 12.97
r <= 1 6.70 17.85 19.96 24.60
r = 0 37.06 32.00 34.91 41.07
Chapter 3. Econometric methodology, modeling and estimation 25
TABLE 3.12: Maximum eigenvalue test - VAR(p=1), k=2,
“Source: own elaboration”
Rank Test 10 pct 5 pct 1 pct
r <= 2 1.50 7.52 9.24 12.97
r <= 1 5.20 13.75 15.67 20.20
r = 0 30.37 19.77 22.00 26.81
As it can be observed in the following tables, trace test identified at least
one cointegrating vector while the maximum eigenvalue test found no
cointegration for a model with a constant in the cointegrating equation. So it
is has some conflict with cointegration rank tests, and that is because they test
different alternative hypothesis, (Enders, 2014)
In this situation, it is more consistent (preferred) the output that reports the
trace test, (r = 1) in order to estimate the potential Vector Error correction
model (VECM) and calculate its coefficients (β) as the long run cointegrating
vector equation.
Cointegration equation (long-run model)
lnBTCCNYt = − 105.4364+ 0.0056(lnBTCKRWt)
+ 0.0037(lnBTCJPYt)
(3.24)
Estimated VECM with BTCCNY as target variable
∆lnBTCCNYt−1 = − 0.0588ECTt−1 − 0.7022∆(lnBTCCNYt−1)+
0.0057∆(lnBTCKRWt−1)− 0.0132∆(lnBTCJPYt−1) (3.25)
Estimated VECM with BTCKRW as target variable
∆lnBTCKRWt−1 = − 7.6521ECTt−1 − 109.0729∆(lnBTCCNYt−1)+
0.8747∆(lnBTCKRWt−1)− 1.9267∆(lnBTCJPYt−1) (3.26)
Estimated VECM with BTCJPY as target variable
∆lnBTCJPYt−1 = − 0.4829ECTt−1 − 8.5105∆(lnBTCCNYt−1)+
0.0852∆(lnBTCKRWt−1)− 0.3006∆(lnBTCJPYt−1) (3.27)
In the Appendix C there are attached three tables of the R outputs obtained
with the significance level and any other important information for each of the
short-run equations [∆BTCCNYt, ∆BTCKRWt, ∆BTCJPYt] and Appendix D
contains the plotted forecast of the same model responses.
Chapter 3. Econometric methodology, modeling and estimation 26
ARDL Bounds Testing approach
The non-derivation of a possible consistent solution to our financial series,
and especially when it has been trying to demonstrate the existence of a
correlation between them, has led to search for alternative literature about the
models that are currently trying to measure the existence of cointegration. It
is taken for granted that the series that have been exposed in this work show
a strong pattern of movement and long-term trend that clearly demonstrate
this phenomenon, but for some reason derived from the analysis of the
variables, it has not been possible to reach a definitive solution even assuring
the requirements for the application of the Johansen’s methodology.
The Autoregressive Distributed Lags (ARDL) cointegration methodology,
(Pesaran and Shin, 1999), is based on determining the long run relationship
between series following Eagle and Granger method of cointegration with
stationary series, integrated of the same order and cointegrated. The model
contains the lagged value of the dependent variable, the current value and the
lagged value of regressors as explanatory variables, combining endogenous
and exogenous variables - unlike a VAR model that is strictly for endogenous
variables.
ARDL test equation:
ARDL(p, q) : yt = y0 + ρ1yt−1 + ρ2yt−2 + ...+ ρpyt−p
+ γ1xt + γ2xt−1 + ...+ γqxt−q + et
(3.28)
Then, summarizing lagged values of y and x (values of the regressors and its
lagged values) it is obtained the following equation:
ARDL(p, q) : yt = y0 +
p
∑
i=1
ρiyt−i +
q
∑
i=0
γixt−i + et
where the variables in xt are allowed to be I(0), I(1) or cointegrated variables.
The ARDL bound test, (Pesaran and Smith, 2001), is based on determining
the long run relationship between series with different order of integration -
independent variables are allowed to be individually either I(0) or I(1) but
lower than I(2), to obtain the short-run dynamics and long-run relationship.
ARDL bound test equation:
ARDL : ∆yt = y0 +
p
∑
i=1
ρi∆yt−i +
q
∑
i=0
γi∆xt−i + φ1yt−1 + φ2xt−1et,
Chapter 3. Econometric methodology, modeling and estimation 27
where the new adjusted model, respect the previous one, contemplates the
components of:
• Short-run (ρi,γi) such as: ∑pi=1 ρi∆yt−i +∑
q
i=0 γi∆xt−i.
• Long-run (φ1, φ2) such as: φ1yt−1 + φ2xt−1.
• Error/disturbance such as: et.
The steps to follow to develop the estimation process of this adjusted model
by Pesaran is structured quite similar as the Johansen method, but with certain
alterations in the structure.
1. Identify serial correlation.
2. If there are serially correlated errors, then Least Square Estimation - HAC
standard errors.
3. Estimate an AR(p) Error Model, Non linear Least Squares Estimation.
4. Determinate ARDL bound test model.
5. Determinate the optimal lag structure.
6. Apply cointegration test on the series.
7. Estimate long-run levels model to obtain ECT (residuals).
8. Estimate VECM with the estimated residuals.
9. Estimate short-run model (short-run dynamics).
From our data, to perform the ARDL bound test for cointegration, the
conditional model with three variables is specified as:
∆BTCCNYt = α0 +
p
∑
i=1
ρi∆BTCCNYt−i +
q1
∑
i=0
γi∆BTCKRWt−i+
q2
∑
i=0
ωi∆BTCJPYt−i + φ0BTCCNYt−1+
φ1BTCKRWt−1 + φ2BTCJPYt−1 + et
(3.29)
Under the F-bound test null and alternative hypothesis:
H0 : φ0 = φ1 = φ2 = 0
H1 : φ0 6= φ1 6= φ2 (3.30)
where the null hypothesis exposes that "a long-run relationship does not
exists" and the alternative hypothesis that "a long-run relationship exists".
Chapter 3. Econometric methodology, modeling and estimation 28
TABLE 3.13: Bounds F-test (Wald), “Source: own elaboration”
statistic Lower-bound I(0) Upper-bound I(1) p-value
-3.8422 -2.86 -3.53 0.02142
According to Pesaran (2001), there are 5 differents cases for testing the
cointegration bound test. In our case, it is considered the case 3 where there
is/are unrestricted/s intercept/s and no trends. As it is shown in the table, the
Wald statistic (-3.84) falls above the upper-bound (-3.53) and then it can be
safely said that exists cointegration. If it falls between the lower-bound and
the upper-bound critical value, then it is considered inconclusive. And if it
falls below the lower-bound, then it is said that there is no cointegration.
All of this under the selected ARDL model (4,5,5) - (limited to a max order of
5 to avoid overparameterization):
TABLE 3.14: ARDL model selection, “Source: own elaboration”
BTCCNY BTCKRW BTCJPY AIC
4 5 5 21651.76
5 5 5 21653.56
3 5 1 21657.27
2 5 1 21658.98
3 5 2 21659.05
... ... ... ...
2 2 2 21697.05
In the Appendix E there are attached three tables of the R outputs obtained
with the significance level and any other important information for each of
the coefficients of BTCCNY, BTCKRW, BTCJPY and their different orders.
Additionally, it is applied the Durbin-Watson test to ensure that there is no
autocorrelation of residuals on this linear regression fit, (Durbin and Watson,
1950; Durbin and Watson, 1951). The obtained output shows that:
TABLE 3.15: Durbin-Watson test, “Source: own elaboration”
Durbin-Watson statistic p-value
1.9888 0.7963
Chapter 3. Econometric methodology, modeling and estimation 29
Then, it cannot be rejected the null hypothesis that there is no autocorrelation
of residuals. The same criteria can be applied to ensure the existence of
homoscedasticity (variance does not depend on auxiliary regressors) using
the studentized Breusch-Pagan test, (Breusch and Pagan, 1979), as it follows:
TABLE 3.16: Breusch-Pagan test, “Source: own elaboration”
Breusch-Pagan statistic p-value
78.962 0.0821
In this case the p-value is slightly higher than 0.05, but again it cannot be
rejected the null hypothesis of the existence of homoscedasticity.
In order to determine that the model is well-specified and there is no
misspecification functional form, such as the model does not account for
some important nonlinearities or omit important variables, it can be applied
the Ramsey RESET test, introduced by (Ramsey, 1969).
TABLE 3.17: Ramsey RESET test, “Source: own elaboration”
Ramsey RESET statistic p-value
0.045941 0.9551
Finally, it is estimated the Restricted Error Correction Model (RECM) under
the case 3 (unrestricted intercepts and no trend, c1 = 0), where applying our
data it is obtained the following equations:
∆lnBTCCNYt = α0 +
3
∑
i=1
ρi∆lnBTCCNYt−i +
4
∑
i=0
γi∆lnBTCKRWt−i+
4
∑
i=0
ωi∆lnBTCJPYt−i + λ3ECTt−1 + et
(3.31)
where this equation represents the short-run dynamic model and the
coefficient of the parameter error correction term, ECTt−1, represents the
speed of adjustment towards equilibrium. In the Appendix F there are
attached R outputs with the significance level and important information for
each of the parameters.
The long-run model can be written taking into consider the next multipliers
(elasticities):
lnBTCCNYt = −115.1117+ 0.005564(lnBTCKRWt) + 0.004261(lnBTCJPYt)
(3.32)
30
Chapter 4
Conclusions and research
limitations
4.1 Conclusions
The whole process to get to the formulation of the cointegration model has
been adjusted well during the process. It has been shown that the temporary
series of the prices are non stationary and integrated of order 1, I(1),
something needed to evaluate our goal. Parallelly, it has been observed that
the returns of each of the variables are stationary and integrated of order 0,
I(0), something predictable after looking at the series of returns in the first
figure in the previous chapter. Once reviewed, it has been proposed a VAR(1)
after determining the optimal number of lags to use to compute the Johansen
test.
Then it has been evaluated the Granger causality test, where with a VAR(1) it
has not been observed any variable that was Granger-cause to the other ones.
Strictly speaking, one variable is Granger causal for another one if the first
one helps predict the second one at some stage in the future and non of the
selected variables seem to be showing this behaviour.
In the same way, it has been observed that the Impulse response function
(IRF) tracks the impact of any variable on others in the system but as a very
short term impact, because after period 2 or 3 (depending on the variable) the
response is not significantly different from zero (at the 5% level) and the idea
of a % change in something of the first variable produces a % change in
something of the second variable and its effect disappears.
Finally, according to the literature and the reasoning established by the
authors, if the long-run equilibrium shows the relationship between the
variables without any short-run shock or the relationship from which
variables deviate but always return to, then it can be shown that the
estimated long-run model equation seems to be accurate. Applying the
Chapter 4. Conclusions and research limitations 31
Johansen test, it has been derived an equation to explain the equilibrium in
the future, but the results of the coefficients, especially the error-correction
coefficient for every dynamic short-run model estimated, does not seem to be
statistically significant at any confidence level.
For example, if it is selected the dynamic short-run model for the target
variable BTC-KRW, it is observed that the value of this error-correction
coefficient is -7.6521, which is too high to be a value delimited between [-1,0].
This interval has to be imposed because this coefficient captures the speed of
adjustment such that when the dependent variable exceeds the long-run
relationship, this has to be corrected to revert to the steady state. Considering
this value, this coefficient means that about 765.25 % of this disequilibrium is
corrected between 1 year (annualized data) - and this does not make sense.
A plausible argument that can justify this high value is probably explained by
the fact that prices are national prices, not standardized, and therefore, there
is a large difference in volume in them. So the correction factor may need to
be scaled to higher volumes as it is not standardized to a single currency.
From the cointegration equation (long-run) , it has been shown that an
increment of 1% of the prices of BTC-CNY increases the prices of BTC-KRW
in about +0.56%, but what happens to BTC-JPY, +0.37%, seems to be not
significant in order to get this equilibrium in the future. These values seem to
be quite accurate, but as it has been said before, what differs in the integrity of
the estimated model are the coefficient values obtained in dynamic short-run
equations and increasing the lag order established in the VAR model is not an
option, since it has been tested and has not turned out to be the solution.
Because of that, an Autoregressive Distributed Lag (ARDL) bounds testing
approach has been proposed to determine the existence of cointegration and
thus be able to contrast the results obtained with the Johansen test. The
selected model, an ARDL (4,5,5), has passed the F-bound test to prove the
existance of long-run relationship and it has been ensured that there is no
autocorrelation of residuals applying the use of the Durbin-Watson test. At
the same time, it has been proved using the Breusch-Pagan test that the
variables have the same finite variance (homoscedasticity). Finally, with the
Ramsey RESET test, it has been determined that the structure of the model
has as an adequate functional form.
From the cointegration equation (long-run), it has been shown that an
increment of 1% of the prices of BTC-CNY increases the prices of BTC-KRW
in about +0.56%, and, unlike what it has obtained with the Johansen test,
what happens to BTC-JPY, +0.43%, seems to be significant in order to get this
equilibrium in the future.
Chapter 4. Conclusions and research limitations 32
These values are quite similar in both resultant equations, and this allows us
to ensure that there is a long-term relationship of Bitcoin series and that the
dynamic adjustment equations have a significant impact when determining
the steady state in each of the observed variables.
4.2 Limitations of the study
The results obtained are subject to the limitations existing in the applied
methodology and the following literature gaps that has been identified.
Considering the Johansen test to obtain the cointegration equation and its
respective dynamics adjustment, it is known that is an improvement over the
"initial" cointegration test exposed by Engle-Granger test, as (Bilgili, 1998;
Gonzalo and Lee, 1998; Haug, 1996) point out in their respective research. It
can detect multiple cointegrating vectors and it is more consistent for
multivariate analysis, which is one of the objectives of the study. Another
robust property is that Johansen test deals with the variables as endogenous
variables once it is computed the model, as it has been observed when
determining the dynamic adjustment equations for each one of the analyzed
variables (Johansen, 1995).
But, as it has been reported by (Gonzalo and Lee, 1998), in some scenarios
Engle-Granger test can be more robust than Johansen likelihood ratio test.
This can be pointed out in some research papers that compare performance of
both tests, (Wee and Tan, 1997; Agoraki and Kouretas, 2019; Pascual, 2003),
and this is why it has been selected Johansen as a main methodology to
follow and compare with alternative models to study the cointegration effect
than Engle-Granger, where in that case it is selected an ARDL Bounds Testing
approach.
One of the arguments that had to be justified has been the high values
obtained in the dynamic equations of BTC-KRW, specially the speed of
adjustment: (-7.6521), and why the coefficients obtained are not significant at
any confidence level, except two or three. This situation is observed and
suggested in other levels by different authors, such as (Maggiora and
Skerman, 2009).
In that case, once the ARDL bounds testing approach model is computed, it is
obtained that most of the parameters are significant and in the case of
BTC-KRW they are very much lower than obtained with the Johansen
methodology. The main empirical argument that justifies this situation is
marked by the ARDL model being less restrictive to raise the order of each of
the variables to adequate levels to adjust the dynamic adjustment equation.
Chapter 4. Conclusions and research limitations 33
This is observed by (Haq and Larsson, 2016; Musa and Zoramawa, 2014;
Bouri and Shahbaz, 2018) in their studies and the methodology used to
diagnose the value of the parameters obtained in their series. Another reason
can be explained, as it has been exposed before in the conclusions section, by
the fact that prices are not standardized, and therefore, there is a large
difference in volume. This makes that the correction factor may need to be
scaled to higher volumes as it is not standardized to a single currency (Biggeri
and Ferrari, 2012).
Another limitation or hypothesis adopted in the formulation of the ARDL
model is that it is limited to a maximum order of 5. This ensures that there is
no excess of parameters in the model, but also restricts that for values greater
than 5 it can be found a model that better fits our data. Many authors agree
that this situation may effect the results and the optimal model selection,
(Wang and Cao, 2020; Tursoy, 2019), but in our data it seems that the selected
ARDL (4,5,5) model order fits quite well.
Finally, the time period chosen for the analysis of the time series has
coincided with a period of price increased and optimal economic situation.
The impact derived from COVID-19 could affect the projections that can be
made based on this established model or the cointegration equation and the
long-term equilibrium for projections beyond this year, which would cause
the model and the calculations made with the Johansen test had to be
calibrated again. Many authors have already tried to capture the impact
derived from the pandemic and have established correction hypotheses using
ARDL models to quantify the impact, for example, in crude oil prices or
tourism, (Albulescu, 2020; Ghosh, 2020).
But even so, the goals of the thesis have been met, opening the line for a
further research with alternative approaches such as the use of machine
learning algorithms or bayesian neuronal models.
34
Appendix A
Additional ADF, PP and KPSS
tables
Appendix A. Additional ADF, PP and KPSS tables 35
TABLE A.1: ADF, PP and KPSS tests of BTC-KRW series (prices),
“Source: own elaboration”
Test ADF PP KPSS
Null hypothesis Non stationary Non stationary Stationary
P-value/T-value -1.173 -1.1915 4.79
T-statistic value -2.86 -2.863615 0.463
Results NRH0 NRH0 RH0
TABLE A.2: ADF, PP and KPSS tests of BTC-KRW series (returns),
“Source: own elaboration”
Test ADF PP KPSS
Null hypothesis Non stationary Non stationary Stationary
P-value/T-value -30.6073 -42.5698 0.1653
T-statistic value -2.86 -2.863616 0.463
Results RH0 RH0 NRH0
TABLE A.3: ADF, PP and KPSS tests of BTC-JPY series (prices),
“Source: own elaboration”
Test ADF PP KPSS
Null hypothesis Non stationary Non stationary Stationary
P-value/T-value -1.3436 -1.4288 4.6408
T-statistic value -2.86 -2.863615 0.463
Results NRH0 NRH0 RH0
TABLE A.4: ADF, PP and KPSS tests of BTC-JPY series (returns),
“Source: own elaboration”
Test ADF PP KPSS
Null hypothesis Non stationary Non stationary Stationary
P-value/T-value -30.6235 -42.718 0.1677
T-statistic value -2.86 -2.863616 0.463
Results RH0 RH0 NRH0
36
Appendix B
Orthogonal Impulse Response
Appendix B. Orthogonal Impulse Response 37
FIGURE B.1: Orthogonal Impulse Response from BTCCNY,
“Source: own elaboration”
FIGURE B.2: Orthogonal Impulse Response from BTCKRW,
“Source: own elaboration”
Appendix B. Orthogonal Impulse Response 38
FIGURE B.3: Orthogonal Impulse Response from BTCJPY,
“Source: own elaboration”
39
Appendix C
Short-run response coefficients
Appendix C. Short-run response coefficients 40
FIGURE C.1: Response BTCCNY coefficients,
“Source: own elaboration”
FIGURE C.2: Response BTCKRW coefficients,
“Source: own elaboration”
Appendix C. Short-run response coefficients 41
FIGURE C.3: Response BTCJPY coefficients,
“Source: own elaboration”
42
Appendix D
Plotting VECM model responses
FIGURE D.1: VECM model respones, “Source: own elaboration”
43
Appendix E
ARDL model coefficients
FIGURE E.1: ARDL model coefficients,
“Source: own elaboration”
44
Appendix F
RECM coefficient values
FIGURE F.1: Short-run dynamic and long-run coefficients,
“Source: own elaboration”
45
Appendix G
ARDL - Cointegration equation
FIGURE G.1: Cointegration equation - Projected vs Observed,
“Source: own elaboration”
46
Bibliography
Agoraki M-E., Georgoutsos D. and G. Kouretas (2019). “Capital markets
integration and cointegration: Testing for the correct specification of stock 
market indices.” In: Journal of Risk and Financial Management 12.
Akaike, H. (1974). “A new look at the statistical model identification.” In: IEEE
Transactions on Automatic Control AC-19, pp. 716–723.
Albulescu, C. (2020). “Do COVID-19 and crude oil prices drive the US
economic policy uncertainty?.” In: HAL Id: hal-02509450. URL: 
https://hal.archives-ouvertes.fr/hal-02509450.
Biggeri, L. and G. Ferrari (2012). “Price indexes in time and space: methods
and practice.” In: Physica Contributions to Statistics.
Bilgili, F. (1998). “Stationarity and cointegration tests: Comparison of Engle -
Granger and Johansen methodologies.” In: Journal of Faculty of Economics 
and Administrative Sciences, Erciyes University 13, pp. 131–141.
Bouri E., Grupta R. Lahiani A. and M. Shahbaz (2018). “Testing for
asymmetric nonlinear short- and long-run relationships between bitcoin,
aggregate commodity and gold prices.” In: Resources Policy 57,
pp. 224–235.
Breusch, T. and R. Pagan (1979). “A simple test for heteroscedasticity and
random coefficient variation.” In: Econometrica 47, 1287–1294.
Dickey, D. A. and W. A. Fuller (1979). “Distributions of the estimators for
autoregressive time series with a unit root.” In: Journal of the American
Statistical Association. 75, pp. 427–431.
Durbin, J. and G. Watson (1950). “Testing for serial correlation in least squares
regression I.” In: Biometrika 37, pp. 409–428.
— (1951). “Testing for serial correlation in least squares regression II.” In:
Biometrika 38, pp. 159–178.
Edgerton, D. and G. Shukur (1999). “Testing autocorrelation in a system
perspective.” In: Econometric Reviews 18, pp. 43–386.
Enders, W. (2014). “Applied Econometric Time Series”. In: Wiley 3.
Engle, R.F. and C. W. J. Granjer (1987). “Co-integration and error correction:
Representation, estimation and testing.” In: Econometrica 55, 251–276.
Fry, J. and E. T. Cheah (2016). “Negative bubbles and shocks in cryptocurrency
markets.” In: International Review of Financial Analysis 47, pp. 343–352.
Bibliography 47
Ghosh, S. (2020). “Asymmetric impact of COVID-19 induced uncertainty on
inbound Chinese tourists in Australia: insights from nonlinear ARDL
model.” In: Quantitative Finance and Economics 4, pp. 343–364.
Gonzalo, J. and T.H. Lee (1998). “Pitfalls in testing for long run relationships.”
In: Journal of Econometrics 86, pp. 129–154.
Granger, C. W. J. (1969). “Investigating causal relations by econometric models
and cross-spectral methods.” In: Econometrica 37, pp. 424–438.
Hamilton, J. (1994). “Time series analysis.” In: Princeton University Press 0.
Haq, S. and R. Larsson (2016). “The dynamics of stock market returns and
macroeconomic indicators: An ARDL approach with cointegration.” In:
Master of Science Thesis INDEK 2016:59 KTH Industrial Engineering and
Management, Stockholm.
Haug (1996). “Tests for cointegration A Monte Carlo comparison.” In: Journal
of Econometrics 71, pp. 89–115.
Johansen, S. (1988). “Statistical analysis of cointegration vectors.” In: Journal of
Economic Dynamics and Control. 12, pp. 231–254.
— (1995). “Identifying restrictions of linear equations with applications to
simultaneous equations and cointegration.” In: Journal of Econometrics 69,
111–132.
Kwiatkowski D., Phillips P. C. B. Schmidt P. and Y. Shin (1992). “Testing the
null hypothesis of stationarity against the alternative of a unit root: How
sure are we that economic time series have a unit root?.” In: Journal of
Econometrics 54, pp. 159–178.
Ljung, G.M. and G.E.P. Box (1978). “On a measure of lack of fit in time series
models.” In: Biometrika 65, pp. 297–303.
Lütkepohl, H. (2006). “New Introduction to Multiple Time Series Analysis.”
In: Springer.
Maggiora, D. and R. Skerman (2009). “Johansen cointegration analysis of
american and european stock market indices.” In: Master Thesis Lund
University.
Musa Y., Usman U. and A. Zoramawa (2014). “Relationship between money
supply and government revenues in Nigeria.” In: CBN Journal of Applied
Statistics 5, pp. 117–136.
Narayanan A., Bonneau J. Felten E. Miller A. and S. Goldfeder (2016). “Bitcoin
and cryptocurrency technologies: a comprehensive introduction.” In:
Princeton University Press. 336.
Pascual, A. (2003). “Assessing European stock markets (co)integration.” In:
Economics Letters 78, 197–203.
Pesaran, M. and Y. Shin (1999). “An Autoregressive Distributed-Lag
Modelling Approach to Cointegration Analysis.” In: Cambridge University
Press Econometrics and Economic Theory in the 20th Century: The Ragnar
Frisch Centennial Symposium.371-413.
Bibliography 48
Pesaran M., Shin Y. and R. Smith (2001). “Bounds testing approaches to the
analysis of level relationships.” In: Journal of Applied Econometrics 16.289-
326.
Phillips, P.C.B. and P. Perron (1988). “Testing for a unit root in time series
regression.” In: Biometrika 75(2), pp. 335–346.
Quinn, B. (1980). “Order determination for a multivariate autoregression.” In:
Journal of the Royal Statistical Society B42, pp. 182–185.
Ramsey, J. (1969). “Tests for specification error in classical linear least squares
regression analysis.” In: Journal of the Royal Statistical Society 31, 350–371.
Schwarz, G. (1978). “Estimating the dimension of a model.” In: Annals of
Statistics 6, pp. 461–464.
Tursoy, T. (2019). “The interaction between stock prices and interest rates in
Turkey: empirical evidence from ARDL bounds test cointegration.” In:
Financial Innovation 5.
Wang Q., Hao Y. and J. Cao (2020). “ADRL: An attention-based deep
reinforcement learning framework for knowledge graph reasoning.” In:
Knowledge-Based Systems 197.
Wee, C. and R. Tan (1997). “Performance of Johansen’s cointegration test.” In:
East Asian Economic Issues, pp. 402–414.
Zivot, E. and Donald W.K. Andrews (1992). “Further Evidence on the Great
Crash, the Oil-Price Shock, and the Unit-Root Hypothesis.” In: Journal of
Business and Economic Statistics 10(3), pp. 251–270.
###########Packages and libraries########### 
 
install.packages("car")   
install.packages("urca")   
install.packages("vars")   
install.packages("tseries")   
install.packages("ggplot2")   
install.packages("tsDyn") 
install.packages("forecast")   
install.packages("fGarch") 
install.packages("quantmod") 
library(car) 
library(urca) 
library(vars) 
library(tseries) 
library(ggplot2) 
library(tsDyn) 
library(forecast) 
library(fGarch) 
library(quantmod) 
library(ARDL) 
 
###########Data########### 
 
getSymbols("BTC-CNY",from="2014-09-01", to="2019-09-01", periodicity="daily")  
getSymbols("BTC-KRW",from="2014-09-01", to="2019-09-01", periodicity="daily")  
getSymbols("BTC-JPY",from="2014-09-01", to="2019-09-01", periodicity="daily") 
 
#Price selection - adjusted prices and format dates# 
 
 
btccny=`BTC-CNY`[,6] 
btckrw=`BTC-KRW`[,6] 
btcjpy=`BTC-JPY`[,6] 
Dates<-as.Date(rownames(zoo(`BTC-CNY`))) 
#Dates<-as.Date(rownames(zoo(`BTC-KRW`)))# Same length on each case. 
#Dates<-as.Date(rownames(zoo(`BTC-JPY`)))# Same length on each case. 
btccny<-as.numeric(`BTC-CNY`$`BTC-CNY.Adjusted`) 
btckrw<-as.numeric(`BTC-KRW`$`BTC-KRW.Adjusted`) 
btcjpy<-as.numeric(`BTC-JPY`$`BTC-JPY.Adjusted`) 
 
#Derive returns each series and alter [-1] date structure# 
 
rendbtccny <- diff(log(btccny)) 
rendbtckrw <- diff(log(btckrw)) 
rendbtcjpy <- diff(log(btcjpy)) 
diffDates<-as.Date(Dates[-1],"%d/%m/%Y")  
 
#basicStats() function to statistic summary# 
 
basicStats(btccny) 
basicStats(btckrw) 
basicStats(btcjpy) 
basicStats(rendbtccny) 
basicStats(rendbtckrw) 
basicStats(rendbtcjpy) 
 
#Plot grouped data series# 
 
win.graph(width=8,height=5) 
par(mfrow=c(3,2),font=3,font.lab=4,font.axis=2,las=1) 
plot(Dates,btccny,type="l",col="black",main="BTC - CNY Prices", ann=FALSE) 
plot(diffDates,rendbtccny,type="l",col="black",main="BTC - CNY Returns", ann=FALSE) 
plot(Dates,btckrw,type="l",col="black",main="BTC - KRW Prices", ann=FALSE) 
plot(diffDates,rendbtckrw,type="l",col="black",main="BTC - KRW Returns", ann=FALSE) 
plot(Dates,btcjpy,type="l",col="black",main="BTC - JPY Prices", ann=FALSE) 
plot(diffDates,rendbtcjpy,type="l",col="black",main="BTC - JPY Returns", ann=FALSE) 
 
###########Stantionary analysis of the series - Informal########### 
 
#Grouped ACF-PACF functions# 
 
win.graph(width=8,height=5) 
par(mfrow=c(2,3),font=2,font.lab=4,font.axis=2,las=1) 
acf(btccny,ylim=c(-1,1),main="BTC-CNY") 
pacf(btccny,ylim=c(-1,1),main="BTC-CNY") 
acf(btckrw,ylim=c(-1,1),main="BTC-KRW") 
pacf(btckrw,ylim=c(-1,1),main="BTC-KRW") 
acf(btcjpy,ylim=c(-1,1),main="BTC-JPY") 
pacf(btcjpy,ylim=c(-1,1),main="BTC-JPY") 
win.graph(width=8,height=5) 
par(mfrow=c(2,3),font=2,font.lab=4,font.axis=2,las=1) 
acf(rendbtccny,ylim=c(-1,1),main="BTC-CNY") 
pacf(rendbtccny,ylim=c(-1,1),main="BTC-CNY") 
acf(rendbtckrw,ylim=c(-1,1),main="BTC-KRW") 
pacf(rendbtckrw,ylim=c(-1,1),main="BTC-KRW") 
acf(rendbtcjpy,ylim=c(-1,1),main="BTC-JPY") 
pacf(rendbtcjpy,ylim=c(-1,1),main="BTC-JPY") 
 
#Box-Pierce and Ljung-Box tests# 
 
Box.test(btccny, lag = 1, type = c("Ljung-Box")) #replace for returns series 
Box.test(btccny, lag = 5, type = c("Ljung-Box")) 
Box.test(btccny, lag = 10, type = c("Ljung-Box")) 
Box.test(btccny, lag = 15, type = c("Ljung-Box")) 
Box.test(btccny, lag = 20, type = c("Ljung-Box")) 
Box.test(btckrw, lag = 1, type = c("Ljung-Box")) 
Box.test(btckrw, lag = 5, type = c("Ljung-Box")) 
Box.test(btckrw, lag = 10, type = c("Ljung-Box")) 
Box.test(btckrw, lag = 15, type = c("Ljung-Box")) 
Box.test(btckrw, lag = 20, type = c("Ljung-Box")) 
Box.test(btcjpy, lag = 1, type = c("Ljung-Box")) 
Box.test(btcjpy, lag = 5, type = c("Ljung-Box")) 
Box.test(btcjpy, lag = 10, type = c("Ljung-Box")) 
Box.test(btcjpy, lag = 15, type = c("Ljung-Box")) 
Box.test(btcjpy, lag = 20, type = c("Ljung-Box")) 
 
###########Stantionary analysis of the series - Formal########### 
 
###Prices series### 
 
#Augmented Dickey-Fuller test# 
 
btccny.df<-ur.df(btccny, type = c("drift"), selectlags = c("BIC")) 
summary(btccny.df)  
btckrw.df<-ur.df(btckrw, type = c("drift"), selectlags = c("BIC")) 
summary(btckrw.df) 
btcjpy.df<-ur.df(btcjpy, type = c("drift"), selectlags = c("BIC")) 
summary(btcjpy.df) 
 
#Phillips-Perron test# 
 
btccny.pp<-ur.pp(btccny, type = c("Z-tau"), model = c("constant"), lags = c("long")) 
summary(btccny.pp)  
btckrw.pp<-ur.pp(btckrw, type = c("Z-tau"), model = c("constant"), lags = c("long")) 
summary(btckrw.pp) 
btcjpy.pp<-ur.pp(btcjpy, type = c("Z-tau"), model = c("constant"), lags = c("long")) 
summary(btcjpy.pp) 
 
#Kwiatkowski-Phillips-Schmidt-Shin test# 
 
btccny.kpss<-ur.kpss(btccny, type = c("mu"), lags = c("long")) 
summary(btccny.kpss) 
btckrw.kpss<-ur.kpss(btckrw, type = c("mu"), lags = c("long")) 
summary(btckrw.kpss) 
btcjpy.kpss<-ur.kpss(btcjpy, type = c("mu"), lags = c("long")) 
summary(btcjpy.kpss) 
 
#Zivot-Andrews test# 
 
btccny.za<-ur.za(btccny, model = c("intercept"), lag=1) 
summary(btccny.za)  
plot(btccny.za) 
Dates[1186] 
btckrw.za<-ur.za(btckrw, model = c("intercept"), lag=1) 
summary(btckrw.za)  
plot(btckrw.za) 
btcjpy.za<-ur.za(btcjpy, model = c("intercept"), lag=1) 
summary(btcjpy.za)  
plot(btcjpy.za) 
 
###Retuns series### 
 
#Augmented Dickey-Fuller test# 
 
rendbtccny.df<-ur.df(rendbtccny, type = c("drift"), selectlags = c("BIC")) 
summary(rendbtccny.df)  
rendbtckrw.df<-ur.df(rendbtckrw, type = c("drift"), selectlags = c("BIC")) 
summary(rendbtckrw.df) 
rendbtcjpy.df<-ur.df(rendbtcjpy, type = c("drift"), selectlags = c("BIC")) 
summary(rendbtcjpy.df) 
 
#Phillips-Perron test# 
 
rendbtccny.pp<-ur.pp(rendbtccny, type = c("Z-tau"), model = c("constant"), lags = c("long")) 
summary(rendbtccny.pp)  
rendbtckrw.pp<-ur.pp(rendbtckrw, type = c("Z-tau"), model = c("constant"), lags = c("long")) 
summary(rendbtckrw.pp) 
rendbtcjpy.pp<-ur.pp(rendbtcjpy, type = c("Z-tau"), model = c("constant"), lags = c("long")) 
summary(rendbtcjpy.pp) 
 
#Kwiatkowski-Phillips-Schmidt-Shin test# 
 
rendbtccny.kpss<-ur.kpss(rendbtccny, type = c("mu"), lags = c("long")) 
summary(rendbtccny.kpss) 
rendbtckrw.kpss<-ur.kpss(rendbtckrw, type = c("mu"), lags = c("long")) 
summary(rendbtckrw.kpss) 
rendbtcjpy.kpss<-ur.kpss(rendbtcjpy, type = c("mu"), lags = c("long")) 
summary(rendbtcjpy.kpss) 
 
#Zivot-Andrews test# 
 
rendbtccny.za<-ur.za(rendbtccny, model = c("intercept"), lag=1) 
summary(rendbtccny.za)  
plot(rendbtccny.za) 
diffDates[1094] 
rendbtckrw.za<-ur.za(rendbtckrw, model = c("intercept"), lag=1) 
summary(rendbtckrw.za)  
plot(rendbtckrw.za) 
rendbtcjpy.za<-ur.za(rendbtcjpy, model = c("intercept"), lag=1) 
summary(rendbtcjpy.za)  
plot(rendbtcjpy.za) 
 
###########Estimating VAR model########### 
 
#Optimal lag selection# 
 
data_var<-cbind(rendbtccny, rendbtckrw, rendbtcjpy) 
VARselect(data_var,lag.max=20) 
VAR_1<-VAR(data_var, p=1) 
 
 
#Portmanteau Q and Lagrange Multiplier tests# 
 
archtest<-arch.test(VAR_1) 
plot(archtest) 
predict_VAR<-predict(VAR_1, n.ahead=50) 
plot(predict_VAR) 
 
#Multivariate Portmanteau adjusted test# 
 
mptad<-serial.test(VAR_1, lags.pt = 14, type = "PT.adjusted") 
 
#Granger causality test - Robust HC variance-covariance matrix# 
 
causality(VAR_1, cause = "rendbtccny", vcov.=vcovHC(VAR_1)) 
causality(VAR_1, cause = "rendbtckrw", vcov.=vcovHC(VAR_1)) 
causality(VAR_1, cause = "rendbtcjpy", vcov.=vcovHC(VAR_1)) 
 
#Impulse response function (IRF)# 
 
irf(VAR_1, n.ahead=10) #Change order variables inside vector to check impacts 
plot(irf(VAR_1, n.ahead=10)) 
VAR_summary<-summary(VAR_1) 
VAR_summary$covres 
VAR_summary$corres 
t(chol(VAR_summary$covres)) 
 
###########Cointegration########### 
 
###Johansen test### 
 
#Maximum Eigenvalue and Trace tests# 
 
data_coint<-cbind(btccny, btckrw, btcjpy) 
max_eigen_test<-ca.jo(data_coint, type="eigen", ecdet="const", K=2) 
summary(max_eigen_test) 
trace_test<-ca.jo(data_coint, type="trace", ecdet="const", K=2) 
summary(trace_test) 
 
#VECM estimation# 
 
VECM<-cajorls(max_eigen_test, r =1) 
summary(VECM$rlm) 
summary(vecm$rlm) 
VECM$beta 
 
 
###ARDL test### 
 
data_coint_ardl<-cbind(btccny, btckrw, btcjpy) 
 
#Model - order selection# 
 
models_ardl<- auto_ardl(btccny ~ btckrw + btcjpy, data = data_coint_ardl, max_order = 5) 
models_ardl$top_orders 
ardl_model<- models_ardl$best_model 
summary(ardl_model) 
 
#UEM-RECM estimation# 
 
uecm_ardl_model<- uecm(ardl_model) 
summary(uecm_ardl_model) 
recm_ardl_model<- recm(uecm_ardl_model, case = 3) #Case 3: unrestricted intercepts and no 
trend. 
summary(recm_ardl_model) 
 
#RECM coefficients# 
 
tail(recm_ardl_model$coefficients, 1) 
recm_ardl_model$coefficients[2] 
 
#Ramsey RESET test# 
 
resettest(ardl_model) 
 
#Durbin-Watson test# 
 
dwtest(recm_ardl_model, alternative = "two.sided") 
 #Breusch-Pagan test# 
 
bptest(recm_ardl_model) 
 
#Bounds F test# 
 
bounds_f_test(uecm_ardl_model, case = 3) #Case 3: unrestricted intercepts and no trend. 
t_bounds <- bounds_t_test(uecm_ardl_model, case = 3, alpha = 0.05) 
t_bounds$tab 
 
#Multipliers (sensibility) and Cointegration equation# 
 
multipliers(ardl_model) 
ce_ardl_model <- coint_eq(ardl_model, case = 3) #Case 3: unrestricted intercepts and no 
trend. 
 
#Plot projected vs observed CE# 
 
projvsobs <- cbind.zoo(btccny = data_coint_ardl[,"btccny"], ce_ardl_model) 
projvsobs_xts <- xts(projvsobs) 
plot(projvsobs_xts, legend.loc = "topleft")