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http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2016.009  Vol. 16 (2016) 133−144 

Submitted November 11, 2016  ISSN (online) 2450-7067 

Accepted December 20, 2016 ISSN (print) 1234-3862 

Beata Szetela, Grzegorz Mentel, Stanisław Gędek
*
 

Dependency Analysis between Bitcoin  
and Selected Global Currencies 

A b s t r a c t. In this research we  have tried to identify the relationship between the exchange 

rate for bitcoin to the leading currencies such as Dollar, Euro, British Pound and Chinese 

Yuan and Polish zloty as well. We have applied ARMA and GARCH models to model and to 

analyze the conditional mean and variance. The appliance of GARCH models have identified 

some dependency in explanation conditional variance between bitcoin and US Dollar, Euro 

and Yuan, while ARMA analysis have shown no relations between bitcoin and other depend-

ent variables. 

K e y w o r d s: ARMA, Bitcoin, Dependency, GARCH, Variability. 

J E L Classification: F31; C32. 

Introduction  

 Progressive globalization and dynamic technological development has 

led to inequalities in sharing of resources and increase of distrust to govern-

ments, banks and other state and financial institutions. People began to look 

for new alternatives that would respond to the growing consumerism and 

surveillance. The creation of virtual currencies (VC) was such an answer. 

VC’s are developing in a dynamic way and are gaining more and more atten-

                                                 
* Correspondence to: Beata Szetela, Rzeszow Technical University, Faculty of Manage-

ment, Al. Powstańców Warszawy 8, 35-959 Rzeszów, Poland, e-mail: beata@prz.edu.pl; 

Grzegorz Mentel, Rzeszow Technical University, Faculty of Management, Al. Powstańców 

Warszawy 8, 35-959 Rzeszów, Poland, e-mail: gmentel@prz.edu.pl; Stanisław Gędek, 

Rzeszow Technical University, Faculty of Management, Al. Powstańców Warszawy 8,  

35-959 Rzeszów, Poland, e-mail: gedeks@prz.edu.pl. 



Beata Szetela, Grzegorz Mentel, Stanisław Gędek 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

134 

tion. The experts from PriceWaterhouseCoopers in their report have admit-

ted, that VC is a beginning of a new phase of technology-driven markets that 

have the potential to disrupt conventional market strategies, longstanding 

business practices, and established regulatory perspectives—all to the bene-

fit of consumers and broader macroeconomic efficiency. [VC] Carry 

groundbreaking potential to allow consumers access to a global payment 

system—anywhere, anytime. (PwC, 2015). Following information Available 

via: mapofcoins.com, currently there are ca. 600 different, active virtual 

currencies. Among them is bitcoin – one of the most prominent examples, 

with the biggest capitalization (ca. 10 bn USD). It is the most popular and 

most widely recognized VC. Many global companies are now accepting 

payments in bitcoin, e.g. WordPress.com, Amazon, Victoria’s Secret, Sub-

way, Bloomberg.com, Sears, GAP, Apple App Store, Miscrosoft, DELL, 

LOT Polish Airlines, T-Mobile Polska, Profident, etc. The father of bitcoin 

is considered Satoshi Nakamoto, who in 2008 published an article "Bitcoin: 

A Peer-to-Peer Electronic Cash System", in which he described the concept 

of virtual, decentralized and independent means of payment. He has based 

his solution on the information protocol (block chain), which aims to elimi-

nate all of the transactions factors, which are based on trust
1
. Bitcoin sup-

porters consider anonymity as its major advantage, as well as its broadest 

sense independence. 

 Therefore in this article we have used autoregressive models, such as 

ARMA and GARCH, to model and to analyze Bitcoin’s conditional mean 

and variance in relation to other world currencies like US Dollar (USD), 

Euro (EUR), British pound sterling (GBP) and Chinese Yuan (CNY). The 

currencies were selected based on the transaction volume, in which the most 

of the transactions were performed. The results would give us an answer, 

whether bitcoin is impervious to external influences. Such a feature would 

imply that this VC is impossible to control by third party agents, hence can 

be seen as a fully independent means of payment. 

1. Literature Overview  

Current achievements of scientists related to bitcoin, can be divided into four 

main streams of interest. Considerations of a general and theoretical form for 

example: (Dwyer, 2015), (Dopierała and Borodo, 2014), (Liu et.al., 2015), 

(Jagwani, 2015), (Rogojanu and Badea , 2015), etc. This group include 

                                                 
1 For more information about bitcoin and it’s technical details of creation, mining and 

functioning, please refer to (Nakamoto, 2008), (Nowakowski, 2013) and (Nielsen, 2013). 



Dependency analysis between Bitcoin and selected global currencies 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

135 

a number of studies on the classification of bitcoin as a commodity or a cur-

rency. The scientists in their work roll extensive discussion regarding the 

intrinsic value of bitcoin, the future and the potential it can bring. The Com-

mittee for Payments and Market Infrastructure (CPMI) acting at the Bank for 

International Settlements in its publication "Virtual currency” (CPMI, 2015), 

has stated, that bitcoin is similar in its concept to goods, such as gold, whose 

price is created by the power of supply and demand, with an except that it 

has no intrinsic value. (Haubno-Dyhrberg, 2016) raises other similarities 

between bitcoin and gold, e.g. limited amount and a similar way of its ex-

traction ( "mining"). Other similarities lists (Weber, 2016), who points out 

that both gold and bitcoin supply is not controlled by any state or institution, 

and both products have the same means of exchange. The second group of 

papers, for example: (Badev and Chen, 2014), (Taha, 2015), (Luther and 

Olson, 2015), (Campbell, 2014) (Tasca and De Roure, 2014), (Karama et.al., 

2015) and others, is focusing mainly on issues relating to acquisition (min-

ing), trade and broadly understood security. 

Articles concerning regulatory and tax issues are covered by the third group, 

for example. (Mandjee, 2014), (Bryans, 2014), (Plassaras, 2013), etc. This 

group also include the various types of reports and publications of banks, 

financial institutions and governments on bitcoin, i.e. European Central 

Bank (ECB, 2012), (ECB, 2015), (Draghi, 2015); Congressional Research 

Service acting on the needs of the US Congress (Murphy et.al. 2015), (Con-

gress, 2014); Canadian Central Bank (Gans and Halaburda, 2013), (BoC, 

2014), (Chiu and Wong, 2015), and others. The last group focuses on the 

application of quantitative methods in the study of bitcoin, for example: 

(Brandvold et.al., 2015) using the price discovering process studied which of 

the bitcoin’s trading platforms has the most impact on the bitcoin’s price and 

which one is following the general trend. The authors found that Mtgox and 

Btce are undisputed animators of the bitcoin’s prices. (Haubno-Dyhrberg, 

2016a) and (Haubno-Dyhrberg, 2016b) have used GARCH to prove that 

bitcoin bears similarities to both the US dollar and gold, and that it can serve 

as an instrument to minimize the risk by people characterized by a strong 

risk aversion. (Cheah and Fry, 2015) showed that bitcoin like other 

cryptocurrency tend to generate bubbles, and they do not have fundamental 

value. (Gronwald, 2015) using a generalized model of autoregressive condi-

tional heteroscedasticity said that the fluctuations in bitcoin are characterized 

by sudden surges and extremes pricing, which is characteristic for immature 

markets. (Szetela, 2016) has investigated bitcoin and dollar price variability 

using control charts. She has confirmed that price variability is strongly in-

fluenced by price jumps on the one hand, but on the other hand bitcoin’s 



Beata Szetela, Grzegorz Mentel, Stanisław Gędek 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

136 

price variability is tending to decrease. (Macdonell, 2014) using ARMA 

model and LPPL showed that the price of bitcoin depends on the CBOE 

Volatility Index, which indicates the great potential of this speculative cur-

rency. (Vockathaler, 2015) confirmed that fluctuations in the price of bitcoin 

are positively correlated with the amount of users BTC and determined by 

the endogenous shocks of unknown source, origin and are not generated by 

the impact of specific variables, such as indexes S&P 500, gold rate against 

the US dollar (XAU) and the Shanghai Stock exchange index (SSE). Ac-

cording to (Chu et.al., 2015) bitcoin’s price fluctuations are best character-

ized by generalized hyperbolic distribution. (Bouoiyour and Selmi, 2015) is 

their current research investigated the variability of Bitcoin against the dollar 

in the period before and after the year 2015. They have found that despite the 

fact that the volatility of bitcoin in 2015 significantly decreased compared 

with the preceding period, still it cannot be said that the bitcoin can be re-

garded as mature currency.  

3. Methodology 

To analyze the relationship between given time series, we have used vector-

autoregressive models formulated by Sims in 1980.The analysis of the rela-

tionship between time series is complex and complicated. Often, the depend-

ent variables are modeled not only by a set of explanatory variables, but may 

also depend on their own historical observations and/or historical values of 

the independent variables. Vector-autoregressive models (VAR) allow for 

simultaneous analysis of this type of relationship. An important advantage of 

these models is the lack of restrictions concerning the division of variables 

into endogenous and exogenous. This model also allows to analyze the bidi-

rectional relation, that is when two variables interact with each other. This 

property is used e.g. (Matuszewska and Witkowska, 2006) to examine the 

interdependence between the exchange rate EUR/USD and selected eight 

independent variables. VAR method is based on ARMA and GARCH mod-

els with all their generalizations and modifications. It is often use to model 

rates of return on various types of assets. ARMA allows the modeling of 

conditional mean, while the GARCH – conditional variance. In the literature, 

the combination of these two models to analyze the exchange rates applied, 

among others, (Nakatsuma and Tsurumi, 1999) (Quaicoe et.al., 2015), 

(Marreh et.al., 2014), (Doman and Doman, 2014), etc. 

ARMA (p, q) process is a combination of two types of processes, ie. an auto-

regressive of order p – AR (p) and the moving average of order q – MA (q). 



Dependency analysis between Bitcoin and selected global currencies 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

137 

Following (Montgomery et.al., 2008), the autoregressive process of an order 

p, can be written the formula: 

                              , (1) 

where:    is a white noise;            – parameters. 

The realization of a process at time t depends on p prior observations and the 

white noise at point t. If however the realization of a process at time t de-

pends on q previous random terms, then such a process is a moving average 

of order q (Pesaran, 2015). Such dependency has the form: 

                                  
 
    , (2) 

where:    is a white noise;            – parameters. 

The ARMA model is a combination of the autoregressive process of order p 

and a moving average process of order and can be written in the following 

form: 

            
 
              

 
   , (3) 

The ARCH models (Auto Regressive Conditional Heteroskedasticity) were 

introduced to the literature by Eagle in 1982. They are based on the assump-

tion that variance of the process residuals is not constant over time. (Man-

delbrot, 1963) has proven, that in particular the financial data are affected by 

the presence of outliers and volatility clustering, which impacts the distortion 

of the distribution of the exanimated variables. 

ARCH models assume that the conditional variance of the error term at point 

t is dependent on the p previous error terms:  

             
  

   , (4) 

where    – variance,    – error term,            – parameters,    – con-

stant. 

The generalization of the ARCH model, a GARCH models
2
, was introduced 

in 1986 by Boleslev, and has a form: 

  
           

  
           

  
   , (5) 

where:    is a white noise;   
  – variance at time t;           ,            

– parameters. 

                                                 
2 For further details considered applied models please refer to Hamilton (1994), Mont-

gomery et.al. (2008) and Alberg et.al. (2008). 

 



Beata Szetela, Grzegorz Mentel, Stanisław Gędek 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

138 

Vector autoregressive models can be applied to stationary time series. 

A time series is assumed to be strict stationarity, if the probability distribu-

tion function remains unchanged at each point of time, (Markellos & Mills, 

2008): 

                                            (6) 

The process can be described as stationary in a broader sense, if it has a con-

stant mean and variance, and covariance of the observation depends solely 

on the distance between them. Formally this conditions are described as 

follows: 

                             (7) 

                            
  (8) 

                 (9) 

In order to investigate, whether a considered time series follows a stationary 

process, a Dickey Fuller test is applied, which verifies a presence of a unit 

root under the Null hypothesis (       ) versus an alternative hypothesis, 

which assumes process stationarity (        .   

4. Empirical Results 

In our research we have taken into the consideration daily logarithmic rate of 

return for bitcoin to Polish zloty (rBTC/PLN) and compared it with the loga-

rithmic rate of return for euro to zloty (rEUR/PLN), US Dollar to zloty 

(rUSD/PLN), British pound to zloty (rGBP/PLN) and Chinese Yuan to zloty 

(rCNY/PLN). The mentioned currencies were chosen based on the volume 

of transactions performed on bitcoin
3
. We analyzed the period from January 

2014 to June 2016. The Data were collected from www.quandl.com and 

contained in total 641 full observations. 

The assumption of stationarity was verified by the Dickey Fuller Unit Root 

test (DF). The test results presented in table 1, show that in all cases the DF 

test was able to reject the null hypothesis of appearance of a unit root, prov-

ing stationarity for all of included variables.  
  

                                                 
3 Following information Available via: bitcoinity.org, the majority of transactions in terms 

of volume was performed in yuan, which accounted for approx. 84% of all transactions. The 

US dollar covered approximately 13% of all transactions performed on bitcoin, euro ca. 1%, 

British pound ca. 0.3% and Polish zloty ca. 0.2%. 



Dependency analysis between Bitcoin and selected global currencies 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

139 

Table 1.  Testing for the unit roots and stationarity results for the logarithmic rate of 

return for bitcoin to zloty (rBTC/PLN), euro to zloty (rEUR/PLN), dollar to 

zloty (rUSD/PLN), pound to zloty (rGBP/PLN), daily observations from 

the period Jan. 2014–June, 2016 

Dickey-Fuller Unit Root Tests 

Variable Zero Mean p-Value Single Mean p-Value Trend p-Value 

rBTC/PLN –17.54 <.0001 –17.53 <.0001 –17.74 <.0001 
rEUR/PLN –16.68 <.0001 –16.68 <.0001 –16.68 <.0001 
rUSD/PLN –16.25 <.0001 –16.32 <.0001 –16.32 <.0001 
rGBP/PLN –15.73 <.0001 –15.75 <.0001 –15.75 <.0001 
rCNY/PLN –16.42 <.0001 –16.44 <.0001 –16.44 <.0001 

Note: ADF computes a test statistic for the time series with a Zero Mean, a Single Mean, which includes 
a constant term and a Trend, which includes constant and a deterministic trend. 

For the adopted order of the model p and q, selected by the smallest value of 

Corrected Akaike Inrofmation Criterion, ARMA model parameters were 

estimated and the results are presented in table 2. None of the lagged de-

pendent variables were statistically significant in explaining dependency 

between bitcoin and other currencies. The p-Values are clearly above the 5% 

significance level. The results indicate, that no dependency exists between 

logarithmic rate of return of bitcoin to zloty and all other exchange rates in 

modeling conditional mean.  

Table 2. Results for the significance test for the estimated ARMA(1,0) model for the 

BTC/PLN as an dependent variable 

ARMA 

 
rUSD/PLN rEUR/PLN rGBP/PLN rCNY/PLN 

Const 0.01410 0.02764 0.01374 0.02496 
BTC/PLN (t–1) 0.04343 0.04758 0.04593 0.04629 

rx/PLN (t–1) 0.19033 –0.50527 0.42876 –0.09269 

P rUSD/PLN rEUR/PLN rGBP/PLN rCNY/PLN 

Const 0.9321 0.8671 0.9337 0.8800 
BTC/PLN (t–1) 0.04015 0.2360 0.2516 0.2502 

rx/PLN (t–1) 0.26859 0.2721 0.1531 0.7314 

std.err. rUSD/PLN rEUR/PLN rGBP/PLN rCNY/PLN 

Const 0.16549 0.16506 0.16499 0.16532 
BTC/PLN (t–1) 0.2798 0.04011 0.04003 0.04022 

rx/PLN (t–1) 0.4788 0.45963 0.29974 0.26988 

Note: BTC/PLN(t–1) is a lagged by one period BTC/PLN, rx/PLN(t–1) – lagged by one period independ-
ent variable, i.e. rUSD/PLN, rEUR/PLN, rGBP/PLN, rCNY/PLN.  

We have applied the Lagrange multiplier test to detect the ARCH effects in 

residuals. The highly significant p-value (<.0001) points at rejection of the 

null hypothesis, what indicates the existence of autocorrelation in residuals. 

To model the conditional variance, we have estimated forty different 



Beata Szetela, Grzegorz Mentel, Stanisław Gędek 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

140 

GARCH models. Based on the smallest value of the information criterion, 

we have chosen an exponential GARCH(1,1), as a model best fitted to the 

data. The results of a significance test, presented in table 3 show, that all 

EGARCH parameters are statistically significant. Moreover BTC/PLN loga-

rithmic exchange rate is influenced by logarithmic exchange rate of 

USD/PLN, EUR/PLN and CNY/PLN.  

Table 3.  Results for the significance test for the estimated EGARCH(1,1) model for 

the BTC/PLN as an dependent variable 

EGARCH 

 rUSD/PLN rEUR/PLN rGBP/PLN rCNY/PLN 

rx/PLN 0.4053 0.9990 –0.1348 0.3742 
EARCH0 0.2423 0.2472 0.2528 0.2477 
EARCH1 0.2958 0.2994 0.3057 0.2952 

EGARCH1 0.9245 0.9225 0.9208 0.9224 
THETA –0.0668 –0.0609 –0.0563 –0.0633 
Restrict –10.1788  1.2521 –9.7345 

P rUSD/PLN rEUR/PLN rGBP/PLN rCNY/PLN 

rx/PLN 0.0406 0.0026 0.5392 0.0746 
EARCH0 <.0001 <.0001 <.0001 <.0001 
EARCH1 <.0001 <.0001 <.0001 <.0001 

EGARCH1 <.0001 <.0001 <.0001 <.0001 
THETA 0.2145 0.2570 0.3068 <.0001 
Restrict 0.0210  0.7469 0.0301 

Note: rx/PLN–independent variable, i.e. rUSD/PLN, rEUR/PLN, rGBP/PLN, rCNY/PLN.  

To confirm the quality of the model fit we performed the BDS test (table 4) 

and stability of the parameters were verified by the Chow test (table 5).  

Table 4.  Results for the BDS test for the estimated EGARCH(1,1) model for the 

BTC/PLN as an dependent variable 

BDS Test 

 EUR/PLN USD/PLN GBP/PLN CNY/PLN 
Dimen-

sion 
BDS Pr > 

|BDS| 
BDS Pr > 

|BDS| 
BDS Pr > 

|BDS| 
BDS Pr > 

|BDS| 

2 0.7035 0.4817 0.7654 0.4440 0.6031 0.5465 0.8044 0.4211 
3 0.3521 0.7247 0.4092 0.6824 0.2908 0.7712 0.4463 0.6554 
4 0.4129 0.6797 0.5354 0.5924 0.3920 0.6951 0.5691 0.5693 
5 0.5807 0.5614 0.7242 0.4690 0.5796 0.5622 0.7769 0.4372 
6 0.7871 0.4312 0.9528 0.3407 0.7898 0.4297 1.0111 0.3120 
7 0.9275 0.3537 1.0873 0.2769 0.9216 0.3568 1.1630 0.2448 
8 0.9925 0.3210 1.1470 0.2514 0.9824 0.3259 1.2311 0.2183 
9 0.9798 0.3272 1.1308 0.2582 0.9781 0.3280 1.2127 0.2252 
10 0.8498 0.3954 0.9882 0.3231 0.8494 0.3956 1.0701 0.2846 

Note: Test prints the results for the correlation between residuals up to 10 lags. 



Dependency analysis between Bitcoin and selected global currencies 

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141 

The results of BDS test indicate, that there exists no serial non-linear rela-

tionship between residual values, therefore it is assumed that all linear rela-

tionships have previously been removed from the model, what points at good 

model fit (Brabazon and O’Neill, 2008).  

The results of the Chow test show that at the 5% significance level the test 

has failed to reject the null hypothesis, which assumes that the structural 

parameters of the estimated model are stable over time which provides 

a good fit to the data model. 

Table 5.  Results of the Chow testing the structural stability of the parameters 

Chow Test 

Variable F Value Pr > F 

rEUR/PLN 1.20 0.2733 
rUSD/PLN 0.37 0.5415 
rGBP/PLN 0.51 0.4765 
rCNY/PLN 0.18 0.6714 

The above generated results lead to the conclusion that in terms of condi-

tional mean bitcoin to polish zloty may be consider as independent from the 

impact of dollar, euro, pound and yuan. In term of conditional variance 

bitcoin seems to be dependent from USD, EUR and CNY, therefore it cannot 

be said, that bitcoin is fully independent currencies.  

Conclusions 

  One of the most important question which appear regarding bitcoin is its 

independency. This is one of its the main advantages, which is underlined 

strongly by its supporters. Our research has focused on the analysis of cer-

tain relationships between Bitcoin expressed in Polish Zloty and selected 

global currencies. 

 In terms of conditional mean, modelled by ARMA process, we can say 

that bitcoin is independent from the influence of all of analyzed currencies. 

However bitcoin’s conditional variance, modelled by GARCH process, is 

influenced by the logarithmic rate of return of EUR, USD and CNY to PLN 

and is independent from GBP/PLN.  

 Additionally we have indicated exponential GARCH, as the most suita-

ble to model bitcoin’s conditional variance.  

 The results show that BTC/PLN is not fully independent from the exter-

nal influences VC, thus it can be control by third party agents. Such feature 

can be used, for example, by speculators to achieve abnormal gains.  



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DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 

142 

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Analiza zależności pomiędzy bitcoinem a wybranymi walutami 

Z a r y s  t r e ś c i. W badaniach staraliśmy się przeanalizować i określić zależność pomiędzy 

kursem bitcoina do polskiego złotego, a innymi czołowymi walutami, takimi jak dolar, euro, 

funt brytyjski i chiński Yuan. Waluty zostały wybrane w oparciu o wielkość wolumenu trans-

akcji do bitcoina. Zastosowaliśmy modele ARMA do modelowania warunkowej średniej oraz 

modele GARCH do analizy warunkowej wariancji. Wyniki nie wykazały związku pomiędzy 

logarytmiczną stopą zwrotu z bitcoina do złotówki, a pozostałymi kursami walut w zakresie 

warunkowej średniej. Natomiast zastosowanie modeli GACH wykazało pewną zależność 

pomiędzy bitcoinem a innymi walutami, w kontekście modelowania warunkowej wariancji.  

S ł o w a  k l u c z o w e: ARMA, Bitcoin, GARCH,  VAR, Zależność. 


