© 2016 Nicolaus Copernicus University. All rights reserved. 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 DYNAMIC ECONOMETRIC MODELS 16 (2016) 133–144 141 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. 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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.