Forecasting the Bitcoin-USD exchange rate
using the Dornbusch model
Bachelor of Science
Puck Koot 10774157
Bsc. Economics & Finance: Economics
Supervised by C. Sahin
Januari 31, 2018
The University of Amsterdam
Abstract
This paper attempts to forecast the Bitcoin-USD exchange rate by means of the Dornbusch exchange rate model. This will be done by looking at the cointegration between the exchange rate and the economic fundamental variables of the model. Since many researchers state that the Dornbusch model is outperformed by the random walk model, this will be compared through the RMSE. It is found that the Dornbusch model has four cointegrated variables from which three are between the exchange rate and the economic fundamental variables. This implies they share a long-run equilibrium, resulting in the model to create a more meaningful forecast. When looking at the RMSE, the Dornbusch model outperforms the random walk model.Statement of Originality
This document is written by Student Puck Koot who declares to take full responsibility for the content of this document. I declare that the text and the work presented in this document is original and that no source other than those mentioned in the text and its references have been used in creating it. The faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the content.Table of Contents
1. Introduction
4
2. Literature Review
6
2.1. Bitcoin as a currency 6 2.2. Dorbusch 7 2.3. Cointegration 83. Methodology Bitcoin
9
3.1. Bitcoin 93.2. Dornbusch 10
3.3. Random walk 14
3.4. Cointegration 14
3.5. Hypothesis 15
4. Data
16
5. Results
19
5.1. Unit root test 19 5.2. Cointagration 21 5.3. RMSE 226. Conclusion
23
7. Reference list
24
8. Appendix
27
1. Introduction
In November 2008 Satoshi Nakamoto published a white paper called ‘Bitcoin: A Peer-to-Peer Electronic Cash System’ (Nakamoto, 2008). In this paper Nakamoto introduces and describes an electronic payment system called Bitcoin. It achieves what many had tried and failed to create before: a protocol for a peer-to-peer payment system with a solution to the infamous double spending problem.1 The article from Nakamoto is by many seen as the beginning of the Bitcoin.2 Bitcoin is a decentralized system, as the system works without a central bank or a single administrator. The network is peer-to-peer and transactions take place between users directly, without an intermediary. This cryptographic proof-of-work makes it practically impossible to interfere or disrupt the protocol (Nakamoto, 2008). Nakamoto referred to Bitcoin as an electronic payment system, but many now call it a cryptocurrency due to its currency like characteristics. Since its creation, many argued that Bitcoin could be the first global currency ever existed. However, Bitcoin was also labelled as a hype and a gimmick, which in the near future would have no value left (Krugman, 2015). These contradictory views lead to the question whether Bitcoin has the ability to become a currency. To answer this, first needs to be determined why the Bitcoin is not a currency yet. This will be done by looking at the conditions it should meet following the European Central Bank (ECB), De Nederlandse Bank (DNB) and the Austrian school of economics. As a theoretical concept, Bitcoin can find support for this in the Austrian School of economics. They describe money to have one function, which Bitcoin fulfils (Mises, 1912). However, according to the ECB (2012) and DNB (2014), it must meet a number of conditions to become a global ‘currency’. The ECB states that Bitcoin does not fulfil all of these conditions momentarily. At this moment, the Bitcoin is an unreliable store of value, its value can double or halve in a couple of hours As Bitcoin is the first cryptocurrency in the world, it received both positive and negative media attention. The media attention was quickly followed up by adoption from consumers and speculators. This adoption made the price of one Bitcoin very volatile, which was $1 dollar per Bitcoin on the 14th of April 2011 and $5808 on the 15th of October 2017 (Coindesk, 2017).1 The principle of not being able to ensure that files cannot be duplicated, so that the money can be copied over and over, is seen as the double-spend problem. If digital money can be counterfeited (copied), it becomes worthless. Nakamoto wat the first to develop a protocol that solves that problem. 2 Since it is suspected that the name Nakamoto is a pseudonym, the inventor of the Bitcoin is still unknown. 10 Years earlier, Wei Dai (1998) was the first who described the idea of an electronic monetary system without intermediation of central authorities. Despite not explaining the operation of the Bitcoin or giving the monetary system a name; the decentralized aspect he mentioned is one of the key points of today’s Bitcoin (Nakamoto, 2008). Wei Dai can therefore be seen as one of the creators of the base of the Bitcoin.
Yermack (2014) found the US Dollar/Bitcoin exchange rate volatility to be 142% in 2013. This is extremely high compared to volatilities of other currencies, falling between 7% and 12%. A possible remedy to take away the uncertainty of Bitcoins volatility, which enlarges the change of Bitcoin becoming a currency, is the ability to forecast the Bitcoin exchange rate. When the Bitcoin-USD exchange rate can be forecasted, it would be less risky to invest in it. Knowing what the price will do will results in less sudden acts causing stability, which then leads to Bitcoin becoming less volatile. So far, there has been little research about the effect of forecasting the exchange rates of cryptocurrencies. However, this would be interesting because the high volatility is one of the biggest problems of Bitcoin. Solving this problem might lead to Bitcoin becoming a global currency. This paper attempts to forecast the Bitcoin-USD exchange rate by using ideas developed by Rudiger Dornbusch (1976). The Dornbusch exchange rate model is a theoretical explanation for high levels of exchange rate volatility. According to the model, each market reacts to changes of the economic fundamental variables of the model leading to a new long-run equilibrium. However, there is some discussing about whether the Dornbusch exchange rate model is as good in forecasting as the random walk model (Hwang, 2003). In this paper, this will be tested by comparing them through looking at cointegration and comparing the RMSE. Cointegration methodology allows researchers to test for the presence of equilibrium relationships between economic variables. The cointegrated vectors can be interpreted as the long-run equilibrium relationship among the variables towards which the system will tend to be drawn. Thus, more cointegrated vectors mean a more complete long-run image of the model and therefore a more meaningful forecast (Hwang, 2003). Using the multivariate cointegration technique proposed by Johansen (1988) and Johansen and Juselius (1990), this paper will determine the long-run multivariate relationship between the variables. Due to fully capturing the underlying time series properties of data, providing estimates of all possible existing cointegrating vectors and offering a test statistic for the number of cointegrating vector, this method is superior to the Engle-Granger (1987) methodology (Hwang, 2001). The research question of this paper will be the following: Can the Dornbusch exchange rate model create a meaningful forecast of the Bitcoin-US Dollar exchange rate and outperform the random walk model on basis of the Root Mean Square Error? It is found that the Dornbusch model can outperform a random walk model when looking at the Root Mean Square Error. When looking at if the exchange rate shares a long-run equilibrium with the economic fundamental variables of the models, it is shown that there are 4 cointegrated vectors. This leads to a more complete long-run image of the model and therefore a more meaningful forecast. It can be concluded that the Dornbusch model does contain meaningful information in forecasting the Bitcoin exchange rate.
To get to this answer on the research question, first I will review literature on the Bitcoin as a currency, Dornbusch’ overshooting exchange rate model and cointegration. Next, a theoretical framework of the Dornbusch model, its extensions, the random walk model and cointegration will be given. Then, the data used by this paper is discussed and presented. Afterwards, the research is conducted and the results are presented. At last, a conclusion will be given as well as an answer to the research question.
2.
Literature Review
In the first part of this section an overview of the discussion whether Bitcoin can be seen as a currency is given. The second part of this section compares the Dornbusch model to the random walk model. Finally, a short section is assigned to cointegration.2.1 Bitcoin as a currency
Since the number of merchant accepting the Bitcoin as payment is getting substantial, the Bitcoin is considered competition for official currencies like the euro and the US dollar (ECB, 2012). However, within the literature there is some discussion about whether the Bitcoin is economically considered to be money. A currency is generally considered by economists to be an instrument that serves as a medium of exchange, a unit of account, and a store of value. In order to identify the Bitcoin as money from an economic perspective, the Bitcoin must meet the three above-mentioned characteristics (ECB, 2015). Money as medium of exchange means that it must serve as numerical unit for the determination of the value of goods, services, belongings and depts. The second characteristic of money is that is must be a unit of account, meaning that is should be used as an intermediary when trading. The last characteristic is a store of value, containing that it should be used as a means to bring purchasing power of the present to the future. Although the term “virtual currency” is commonly used – indeed, it often appears in the report – the ECB does not regard virtual currencies, such as Bitcoin, as full forms of money as defined in economic literature (ECB, 2015). They judge that the Bitcoin is only a limited medium of exchange because of its low degree of acceptance within society. Because of the volatility of the exchange rate of the Bitcoin it is unusable as a store of value and because of both the low degree of acceptance and the volatile exchange rate it is useless as unit of account. De Nederlandse Bank (DNB) is in her research more nuanced by saying that the Bitcoin is almost no unit of account, to a limited extend a medium of exchange and to a lesser extent a store of value
(DNB, 2014). Given these considerations, we can conclude that the ECB and DNB take a negative point of view what concerns the economic status of the Bitcoin. However, they might be forced to answer the question ‘if the Bitcoin economically can be seen as money’ negatively due to the risk it brings. However, in 2012 the European Central Bank (ECB, 2012) published a study of the decentralized cryptographic money, in which they say: “the theoretical roots of the Bitcoin can be found in de Austrian school of economics”. The Austrian school of economic thought describes money to have only one primary function, which is a medium of exchange (Mises, 1912). Mises argues that all other functions of money arise from this function. Satoshi Nakamoto (2008) designed Bitcoin to be a medium of exchange, where users could exchange directly with each other, which makes it qualified. Furthermore, the Austrian school claims that currency is only possible if governments monopolize the issuance of a fiat currency (Clegg, 2014). Nowadays, most fiat currencies rely on the belief of the people and people should be therefore be able to choose any currency they wish to use (Hayek, 1976). If the governments will not keep the inflation low, people would simply switch to a different currency.3 Given that the Bitcoin fully rests on the belief of the people, it would be a good alternative since it gives the possibility to rupture the monopolized issuance of fiat currencies.
2.2 Dornbusch
Besides the theoretical literature, Bitcoin consists some more problems withholding it from being and becoming a global currency. Currently, Bitcoin is an unreliable store of value with an extremely high volatility. A possible remedy to take away this uncertainty, is the ability to forecast the Bitcoin-USD exchange rate. In forecasting exchange rates, random walk models are most often used. Meese and Rogoff (1983) found that a random walk model performs as well as any estimated exchange rate model. The Dornbusch exchange rate model was one of the models tested. They compared the models using Root Mean Square Errors (RMSE). None of the models achieved a significantly
3 The vast majority of today’s world currencies are fiat, which means that they are not backed by a
physical commodity like gold. In de 19th and the beginning of the 20th century, most successful currencies
around the world were easily exchangeable for a fixed amount of gold or a precious metal. Until between the 1920’s and 1970’s, when the gold standard collapsed. The economic growth exceeded the production of gold and the government wasn’t able to keep their promise to exchange the currency for the fixed amount of gold. Nowadays, the belief of the public that central banks will not increase the supply to quickly and therefore overinflate the value of the currency to worthlessness is what the fiat currencies rely on (Yermack, 2014).
lower RMSE than the random walk model at any horizon, even though forecasts were based on actual realised values of explanatory variables. Dornbusch' (1976) exchange rate overshooting hypothesis is a central building block in international macroeconomics, stating that an increase in the interest rate should cause the nominal exchange rate to appreciate instantaneously, and then depreciate in line with uncovered interest parity (UIP). However, when confronted with data few empirical studies that analyse the effects of monetary policy have found support for Dornbusch’ model (Bjørnland, 2009). But there are exceptions. Hoffman and Schlagenhauf (1983) state that the monetary model of the exchange rate behaviour does have empirical content. They estimated a monetary model of exchange rate determination under the assumptions that expectations are formed ‘rationally’. All restrictions typically associated with the monetarist approach were consistent with the data. Moreover, the structural estimated where extremely plausible. Bjørnland (2009) also counters it, stating many researchers make mistakes when evaluating the Dornbusch model. To establish the quantitative effects of monetary policy in the above-mentioned study, the common approach had been via structural vector autoregressive (VAR). However, there is a major challenge when analysing the open economy through structural VARs; namely how to properly address the simultaneity problem between monetary policy and the exchange rate. When analysing the open economy through VAR models, most studies restrict either the exchange rate from reacting instantaneously to a change (shock) in the monetary policy of monetary policy from reacting instantaneously to a shock in the exchange rate. However, this is not consistent with established theory on either monetary policy of on exchange rate determination. By not allowing for potential simultaneity effects in the identification of monetary policy shock, they may have produced a numerically important bias in the estimate of the degree of interdependence. Bjørnland (2009) found that when one allows full simultaneity between the exchange rate and the monetary policy, but restricts monetary policy from having any long run effect on the exchange rate, the results are precisely as Dornbusch hypothesized: a contractionary monetary policy shock results in an immediate (within 1-2 quarters) appreciation of the exchange rate, after which it gradually depreciates back to baseline.
2.3 Cointegration
The Dornbusch model creates a more meaningul forecast when there is a more complete long-run image of the model and the image will be more complete when there are more cointegrated vectors (Hwang, 2003). This will be tested using the multivariate cointegration techniqueproposed by Johansen (1988) and Johansen and Juselius (1990). The Johansen and Juselius (1990) method is superior to other methods because it fully captures the underlying time series properties of data, provides estimates of all possible existing cointegrating vectors and offers a test statistic for the number of cointegrating vector. With the Engle-Granger (1987) methodology, null hypothesis of no cointegration were not rejected while there were signs of cointegration using the Johansen maximum likelihood method (Hwang, 2001). According to these researchers (MacDonald and Taylor, 1993 and Moosa, 1994), the Johansen and Juselius (1990) method fully captures the underlying time series properties of the data and is therefore preferred to the Engle and Granger (1987) method. The Johansen and Juselius method provides test statistics for the total number of cointegrating vectors, and permits direct hypothesis testing on the coefficients of the cointegrating vectors. Furthermore, its results are invariant with respect to the direction of normalization because it makes all of the variables explicitly endogenous. There are two different approaches to forecasting the exchange rate, a technical and a fundamental analysis (Hwang, 2001). In this paper, the fundamental analysis is used. The fundamental analysis is based on the belief that there are some economic variables (or fundamentals) that determine the exchange rate. The variables typically include money supply, income, interest rates and price level changes. Its success depends on whether the forecaster correctly specifies the underlying economic relationship among the macroeconomic variables that influence the exchange rate (Hwang, 2001).
3. Methodology
The technology behind Bitcoin is a quite new and complex phenomenon. This will be explained in the first section of this part. Second, the methodology of the Dornbusch model will be explained. Then a short section is assigned to the methodology of the random walk model. This is followed by the cointegration methodology. Finally, the hypotheses will be given.3.1 Bitcoin
Bitcoin is a decentralized, private digital coin (Grinberg, 2012). It is used as an electronic payment method and also is commonly a cryptocurrency (Dwyer, 2015). This is because the techniques behind Bitcoin are related to digital cryptographic algorithms.4 All transactions happen by means of crypto graphical calculations, which are verified by all fellow users. All transaction a being kept in the so-called ‘blockchain’, which serves as public administration4 This means that every bitcoin is represented by a unique code consisting numbers and letters.
(Dwyer, 2015). The blockchain is linked with the bitcoin-network, at which all computers from all users are connected. This network of all connected computers is called a peer-to-peer network (Grinberg, 2012). When certain software is downloaded on the computer, this computer is connected to the Bitcoin’s peer-to-peer network. Miners can use a computers mathematic power to ‘mine’ the bitcoins (Yermack, 2014). Miners are anonymous persons of private institutions who have access to powerful computers. Mining is the process of computers who perform or carry out complex crypto graphical calculations which results in the creation of new bitcoins (Dwyer, 2015). The integrity of the bitcoin is protected by the complexity of these calculations (Yermack, 2014). The estimation is that there is a maximum of 21 million bitcoins, which will be in circulation in 2040 (Grinberg, 2012). Bitcoin has no physical appearance, meaning that there are no coins or bills in circulation (Yermack, 2014). The quantity of bitcoins, which are brought in circulation, are not determined by a central bank or any other financial institution because of the decentralized character of the Bitcoin (ECB, 2012). The bitcoin is public, there is no administrator or owner and everyone who has access to a computer with Internet is able to buy bitcoins (Grinberg, 2012). Every user of the Bitcoin owns a bitcoin address that is not linked to his or her identity, which makes the use of bitcoin anonymous (Nakamoto, 2008). In comparison with regular giro transactions (between different currencies), the transaction velocity is significantly much higher with bitcoins (EBA, 2014). This is due to the crypto graphical technique behind the Bitcoin (Dwyer, 2015). With transactions, no fees are withheld (EBA, 2014), which makes the transaction costs low (Nakamoto, 2008). The reason there are no fees withheld is because of the absence of a bank, which guarantees the safety of the Bitcoin (EBA, 2014).
3.2 The Dornbusch model
The overshooting model, first developed by economist Rudi Dornbusch, is a theoretical explanation for high levels of exchange rate volatility (Dornbusch, 1976). It assumes the following (Copeland, 2008): • The standard open economy IS-LM mechanism determines aggregate demand. • Financial markets react and adjust instantaneously to shocks. Uncovered interest rate parity holds at all times, because investors are risk neutral. • Prices are sticky, meaning that they are fixed in the short run and flexible in the long run. The aggregate supply curve is horizontal in the short run, getting steeper in the adjustment phase, and vertical in the long run. The model can be specified throughout different markets (Copeland,2008).Expectations In theory, the exchange rate is always adjusting towards its long-run value. The equation shows that exchange rate expectations are formed by multiplying the difference between the long-term exchange rate and the short exchange term rate with a parameter value. This parameter value indicates how fast the adjustment towards the long-term exchange rate is. ∆se = Θ (ŝt - st) Expected appreciation/depreciation (3.1) where Δst denotes the expectation of the exchange rate; ŝt denotes the log of the long-term exchange rate at time t; st denotes the actual exchange rate at time t; Θ denotes a parameter for the elasticity of the expectation. Note that the higher is Θ, the faster the exchange rate is expected to return to its equilibrium value. In a world in which financial markets are expected to adjust immediately, the UIRP condition represents a no-arbitrage state in which investors are indifferent between investing with domestic banks as opposed to investing at a foreign bank. rt = r*t + ∆se Uncovered interest rate parity (3.2) Where rt is the domestic interest rate at time t; r*t is the foreign interest rate at time t; ∆se is the expected movement of the exchange rate at time t. Financial Markets This equation is a basic log-linear formulation with the assumption that the financial demand is equal to the financial supply(md=ms), therefore both money demand and money supply are written as m. mt – pt = kyt – lrt real money demand (3.3) where mt is the log of the nominal financial supply at time t; pt denotes the log of the price index at time t; yt denotes the log of GDP at time t; rt denotes the domestic interest rate at time t; k and
y represents parameters for respectively the GDP and domestic interest rate elasticity. Goods market Here we see demand for Bitcoins treated as a function of the real exchange rate. The real exchange rate is set by the difference between price level and the short-term exchange rate. At a higher real exchange rate Bitcoins are more competitive, thus creating greater demand. ydt = h(s - p)t Demand for domestic output (3.4) where ydt denotes the demand for GDP in the domestic country at time t; st denotes the log of the exchange rate; pt denotes the log of the price index; h denotes a parameter for the elasticity of demand.
If demand deviates from its level in an economy with assumed full employment and fixed income, the result is a drawn-out adjustment in the level of prices in the economy. Inflation
increases when the level of demand deviates away from the level of output.
þt = π(ydt – ŷt) De-/inflation (3.5) where þt denotes the inflation at time t; ydt denotes the log of the demand for domestic GDP at time t; ŷt denotes the log of the long term exchange rate at time t; π denotes a parameter for adjustment elasticity.
3.2.1 Long Run
The Dornbusch Overshooting Model consists of two equal important parts used for estimation. The first part is the long-run, where we can claim that in the long-term prices can be adjusted freely as the divergence of demand and supply is clear. According to Dornbusch, when the economy is at its long-run equilibrium, it follows three conditions. First, the aggregate demand is equal to the aggregate supply, meaning there is no pressure on the price level. Second, domestic interest rates equal foreign rates, resulting in a static exchange rate with no depreciation or appreciation expectations. At last, the real exchange rate is at its long-run level, following in a delta of zero in the current account of the balance of payments (Copeland, 2008). This looks as follows, reflected in the equations (Dornbusch, 1976):1. The inflation rate is zero. So pt =pydt - yt = 0, implying ydt = yt and thus:
ŷt / h = ŝt - þVt 2. The expected rate of depreciation is zero. So Dset=0, and therefore rt = r*t. Reformulating money demand gives: þVt = mt - kŷt + lr*t Combining these two results in the expression for the long-run exchange rate: ŝt = (h-1 – k) ŷt + mt + lr*t
3.2.2 Short Run
The short-term part of the Dornbusch perspective is vulnerable to shocks and can term off the model’s equilibrium. In the short-term price levels are ‘sticky’, which means that in a short time span the demand and supply of goods do not affect its price. Only after a certain amount of time trends are being noted and prices are adjusted accordingly (Copeland, 2008). In the short-run, demand for domestic currency changes, when expectations change. Because of the prospect of low purchasing power, currency’s demand decreases, in case of an expected depreciation of domestic currency (Dornbusch, 1976). As can been seen in de equations, an expected depreciation is reflected by an increase in ∆𝑠𝑒. In his turn, this results in an increase in domestic interest rate, as shown in equation 3.1. Following equation 3.3, an increase in domesticinterest rate leads to a lower real money demand. Which at last results in an actual depreciation of the exchange rate. When looking at equation 3.4, the actual depreciation of the exchange rate is followed by an increase in demand for domestic goods, which follows in an increase in price. However, since prices are sticky in the short run this only happens in the long run. The increased prices decrease real money supply but increase the domestic real interest rate, which increases demand for domestic currency. Finally, this results in an appreciation of the domestic currency. By combining equation 3.1, 3.2 and 3.3 we are able to derive an equation which describes the short-run dynamics of the exchange rate (Copeland, 2008 & Dornbusch, 1976). Combining 3.1 and 3.3 gives us: 𝑚 t− 𝑝t = 𝑘𝑦t − 𝑙(𝑟∗t + ∆𝑠𝑒t) When adding 3.2 to this equation, we obtain: 𝑚t − 𝑝t = 𝑘𝑦t − 𝑙(𝑟∗t + 𝜃(ŝt-st)) Solving for 𝑠 results in: ŝt = 𝑠t +(1/𝑙𝜃)𝑚t − (1/𝑙𝜃)𝑝t − (1/𝑙𝜃)𝑦t + (1/𝜃)𝑟∗t
3.2.3 Model specification
The equation to be estimated for the Dornbusch model is derived from combining the equations 3.1-3.5. It is formulated as follows (Dornbusch, 1976): st = β0+β1(mt-mt*)+β2(yt-yt*)+β3(rt-rt*)+β4(pet-pet*)+ εt (3.6) Where: s = Bitcoin-US Dollar exchange rate m = Log of the United States money supply m* = Log of the Bitcoin money supply y = Log of the United States income y* = Log of the world income r = United States interest rate r * = World interest rate Pe = Log of the United States expected inflation Pe* = Log of the Bitcoin expected inflation εt = The error term (white noise) However, as described in the data section, expected inflation will be omitted of the equation.Simplifying equation 3.6 gives us the following:
St =β0+β1m̂t +β2ŷt +β3rt̂ + εt (3.7) Where:
m̂t =mt–mt* Net log Money supply (US – Bitcoin)
ŷt =yt–yt* Net log income series (Us– world)
rt̂ = rt – rt* Net interest rate series εt Error term Note that a circumflex above a variable denotes the differential of that variable between domestic and foreign values here.
3.3 Random walk model
The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk and thus cannot be predicted (Fama 1965). Stock price changes have the same distribution and are independent of each other, so the past movements or trend of a stock price cannot be used to predict the future movement. The model can be shown though the following equation: st = st-1 + et In other words, we predict that this period's value will equal last period's value plus an error term representing the average change between periods.3.4 Cointegration
Cointegration methodology allows researchers to test for the presence of equilibrium relationships between economic variables. In other words, the cointegrating vectors can be interpreted as the long-run equilibrium relationships among the variables towards which the system will tend to be drawn. If the separate economic time series are stationary after differencing but a linear combination of their levels is stationary, then the series are said to be cointegrated. This paper implements a cointegration technique to detect whether stable long-run relationships exist between exchange rates and fundamental variables. The more cointegrated variables, the more long-run equilibrium relationships and the better the model makes a long-run forecast (Hwang, 2003). We start modelling the VAR model based on the following equation (Johansen, 1988): Xt = AtXt-1 + … + AkXt-k + εt Letting ∆ represent the first-difference operator, the above mentioned equation can be reformulated into a vector error correction form: ∆Xt = 𝛤1Xt-1 + .. + 𝛤kXt-k+1 + ΠXt-k + εtWhere:
Xt = (n x 1) vector I(1) variable
Ai = (n x n) matrix of parameters 𝛤1 = -(I – A1 - ... - Ai), (I = 1, …, k – 1) Π = - (I – A1 - … - Ak) εt = a n x 1 vector of residual Cointegration can be determined by examining the (n x n) matrix: Π = aß Where: a = the speed of adjustment to disequilibrium ß = a matrix of long-run coefficients If the Π matrix has rank zero, implying no linear combination of Xt that are I(0), then the appropriate model will be a VAR first difference. If Π is of full rank n, implying that all the variables in Xt are stationary, then the appropriate strategy is to estimate the standard Sims-type VAR in levels. If Π has reduced rank (0 < r ≤ n -1), there are r ≤ (n-1) cointegration vectors present in ß. So, testing for cointegration amounts to finding the number of r linearly independent columns in Π. To determine the cointegration rank, the maximum likelihood estimator of ß (cointegrating vector) can be obtained by the likelihood ratio test. The Johansen cointegration test is testing hypothesis about the cointegrating rank r of the long-run matrix Π. λtrace is tested formulating the null hypothesis of r cointegrating vectors against the alternative hypothesis of n cointegrating vectors.
λtrace = - 2 log (Q) = - T ∑ni = r+1 log (1 – λ^i)
Where Q = (restricted maximized likelihood / unrestricted maximized likelihood) and λ^I, …, λ^n are the n-r smallest eigenvalues. Alternatively, the likelihood ratio statistics to test the null hypothesis or r cointegrating vectors against the alternative of r + 1 cointegrating vectors is called the maximum eigenvalue test, shows as: λmax = - T log (1 - λ^r+1) (Johansen, 1988)
3.5 Hypothesis
Two hypotheses are formulated in accordance with the research question of this paper:• H0: It is not possible for the Dornbusch exchange rate model to create a meaningful forecast of the Bitcoin-USD exchange rate and outperform the random walk model on the basis of the Root Mean Square Error. • H1: It is possible for the Dornbusch exchange rate model to create a meaningful forecast of the Bitcoin-USD exchange rate and outperform the random walk model on the basis of the Root Mean Square Error.
4. Data and variables
The analysis requires data on the exchange rate (St), the domestic money supply (Mt), the
foreign money supply (Mt*), the domestic GDP (Yt), the foreign GDP (Yt*), the domestic interest
rate (It), the foreign interest rate (It*), the domestic inflation (Pet) and the foreign inflation (Pet*).
To obtain enough observations for the analysis, the data is converted into weekly data. Conversion from daily to weekly data has been done through geometric averaging. Conversion from yearly or monthly to weekly data had been done through linear interpolation. The time span used is from Monday 19 July 2010, which is the first date Bitcoin Supply data is available, till Monday 27 November 2017. This gives 385 observations. The Bitcoin Price Index (the unified price of a Bitcoin in US Dollars) historical data (St) is available from CoinDesk with a daily frequency (2017). For this data, weekly averages are used. In this paper, the exchange rate (St) is defined as 1/BPI, so Bitcoin/US Dollar. However, there is no so-called Bitcoin Bank, so Bitcoins can be obtained by exchanging conventional currency at Bitcoin exchanges. But these exchanges do not necessarily have equal prices. The attempt to create a unified price, carried out by the Bitcoin Price Index (BPI), is done by calculating the average of several large exchanges, like Bitfinex, Bitstamp and BTC-e. Coindesk calculates this live and expresses it at the bid/ask spread. For the US Dollar, it is defined as US Dollar/Bitcoin. Daily Bitcoin supply data (Mt*) is available from Blockchain.info (2017). To convert the series to a weekly frequency, every Monday value is used. Weekly US money supply data (Mt) is available from the Board of Governors of the Federal Reserve System (2017). The United States' money supply is chosen to include cash and checking deposits (M1) as well as saving deposits, money market mutual fund and other time deposits. Thus, this can be defined as the M2 money stock. Gross Domestic Product data is available from OECD.StatExtracts (2018a) on a yearly basis. For United States GDP (Yt), no national averaging is done. The yearly data points have been assumed to be that year's last month value. The remaining months are filled in via linear interpolation. The conversion from a monthly to a weekly series is done in the same manner assuming the monthly data is that month’s last week value. However, GDP data is only available
until 2016 from OECD.StatExtract (2018a). Remaining GDP data of 2017 is yearly available from the International Monetary Fund (2018) and converted in the same manner. For United States' overall demand, US Gross Domestic Product data is used. Since the Bitcoin is a global currency, and no country exists with the Bitcoin as its national currency, the foreign demand should be a global variable. Foreign GDP data (Yt*) is available on a yearly basis from OECD.StatExtract (2018a). For a global currency, an average of the top 20 countries with the biggest Bitcoin network is created. This list is available from Bitnodes.earn.com (2018).5 The averaged, yearly data points have been assumed to be that month's last week value. The remaining months are filled in via linear interpolation. The conversion from a monthly to a weekly series is done in the same manner assuming the monthly data is that month’s last week value, giving a weekly foreign GDP series (Yt*). Again, GDP data is only available until 2016 from OECD.StatExtract (2018a). Remaining GDP data of 2017 is yearly available from the International Monetary Fund (2018), where data from top 20 countries is averaged and converted in the same manner. Short-term interest rate data (It*) is available on a monthly basis from OECD.StatExtracts (2018b). Foreign interest rate should be a global variable, for the same reason foreign demand should be. Thus, for foreign interest rate, an average of the above mentioned top 20 bitcoin countries data is used, collected from Bitnodes.earn.com (2018). Since there was no data available for Bulgaria, Norwegian data is used for the foreign interest rate. Each year’s data point is assumed to be that years last week value. Remaining weeks are filled in linearly via linear interpolation to obtain a weekly foreign interest rate series (It*). For the United States interest rate (It), no averaging is needed. Weekly historical data on the 3 month Eurodollar rate is available from the Board of Governors of the Federal Reserve System (2018). US Consumer Price Index data has been used for the unobserved United States expected inflation (Pet), which is available from US Bureau of Labor Statistics (2017) on a monthly basis. Converting to weekly frequency has been done through linear interpolation, by assuming the monthly observations to be the month's first week observation. To construct the unobserved expected inflation, the CPI series percentage change over the past 12 months is taken. Seasonally adjusted monthly data, is used. For the unobserved Bitcoin expected inflation (Pet*), the US CPI data has been converted to Bitcoins using the exchange rate series. CPI percentage change over the past month is used as
5The top 20 countries are; United States, Germany, France, China, the Netherlands, Canada, United Kingdom, Russia, Australia, Japan, Sweden, Swiss, the Republic of Korea, Spain, Lithuania, Italy, Ireland, Czech Republic and Bulgaria. This list consists the first 20 countries for which data was available.
expected inflation. However, due to collinearity, foreign expected inflation and therefore domestic inflation, will be omitted.6
Descriptive statistics of the variables are as follows:
St Mt Mt* Yt Yt* It It*
Unit Bitcoin/US dollar Billions of US Dollars Bitcoins Millions of US Dollars Millions of US Dollars Percentag e Percentag e Mean 591.0605 11203.27039 10139765.2 5 16989299.5 7 2601113.60 3 0.51481 1.43386 Median 238.0614 11217.8 10764425.6 2 16863271.8 2633054.23 0.39 1.402 Maximum 10315.1849 13809.6 14319872.3 3 20078805.7 7 2801520.27 4 1.55 2.35 Minimum 0.0600 8599.7 2964090.52 9 14725657.5 7 2301806.39 1 0.24 0.34 Std. Dev. 1233.4585 1524.520018 1.03445E+1 3 2.0242E+12 137135.423 9 0.32286 0.42440 Skewness 4.2496 0.00131 -0.55891 0.23841 -0.43599 1.50046 0.27671 Kurtosis 21.4410 -1.11829 -0.80992 -0.96525 -1.01218 1.16570 -0.59900 Sum 227558.285 8 4313259.1 390380962 1 654088033 6 100142873 7 198.2 552.0375 Observatio ns 385 385 385 385 385 385 385 Table 1 | Descriptive statistics of variables If we look at the statistics of the variables, this could differ from regular foreign values due to the fact that Bitcoin has no country. An average of 20 countries is created for the foreign GDP and
6The conversion done, violates the Ordinary Least Squares assumption stating independent variables should not be linearly correlated (i.e. no collinearity). With collinearity, the estimated coefficient from the OLS regression cannot be interpreted, as the t- statistics and accompanying probabilities are likely to be incorrect. This would result in failure to reject the null hypothesis that the coefficients are zero, when it might entirely be possible that the coefficients are different from zero. Using the exchange rate series in the process to construct the Bitcoin expected inflation series, which in turn was used to construct the net expected inflation series causes that nearly all exchange rate information is already available in the net exchange series. To ensure a correct interpretation of the estimated coefficient, it is best to omit the net expected inflation from the Dornbusch equation.
interest rate. For the foreign money supply and the exchange rate, Bitcoin information is used. This could cause incoherence between the foreign variables and therefore lead to odd results. Note that natural logs are taken of the original variables, in order to be able to interpret the coefficients as elasticities This only counts for the variables a log is necessary, thus the money supply (Mt and Mt*) and the GDP (Yt and Yt*).
5. Results
To answer the hypothesis if the Dornbusch overshooting model creates a meaningful forecast of the Bitcoin-Us Dollar exchange rate, one should look at the cointegration and compare the RMSE with the random walk model.
5.1 Unit Root Test
However, prior to testing for cointegration, one needs to examine the time series of properties of the variables. They should be integrated of the same order to be cointegrated. To do so, one should conduct a unit root test, whose null hypothesis is that a unit root is present. If the presence of a unit root is not rejected, a common remedy is to difference the series (Hwang, 2001). A stationary process has the property that the mean, variance and autocorrelation structure do not change over time. The most common cause of violation of stationarity is a trend in the mean, which can be due the presence of a unit root. A unit root is a feature that can cause issues in statistical inference in an autoregressive model. The Bitcoin-US Dollar exchange rate time series is tested for stationarity by using an Augmented Dickey-Fuller test. The Dickey-Fuller test tests the null hypothesis stating the series has a unit root (Hill, Griffiths, & Lim, 2012). The alternative hypothesis is that the time series is stationary. Only when the null hypothesis can be rejected for all levels of significance (1%,5%,10%) there is no evidence of the series begin non-stationary. Variables should be stationary after differencing each time series the same number of times. Most macroeconomic variables have been found to be nonstationary in their levels and stationary in the first differences (Hwang, 2001). To implement the augmented Dickey-Fuller test, we estimate the regression as: ∆Xt = α + ßXt-1 + ∑kj=1 γj Xt-j + εt Where ∆ is the difference operator, Xt is the series being tested, k is the number of lagged differences, and ε is an error term. It is necessary to determine the appropriate lag length (k). Too few lags may result in over-rejecting the null when it is true, while too many tags may reduce the power of the test. To determine the number of lags the so called Schwert criterion isused (Hwang, 2001).
Test results of determining the number of lags with the Schwert criterion for the net money supply (mt), the net income series (yt) and the net interest rate (it) are presented in table 2.
Mt Yt It SBIC Criteria -11.3426 (1) -17.4229 (2) -4.14266 (2) Table 2 | Number of lags Testing the net money supply series (mt) for existence of a unit root, which is estimated with k is 1, shows strong evidence of unit root. The null hypothesis cannot be rejected. Taking the first difference, the null hypothesis can be rejected. The net money supply series is integrated of order 1. Results are shown in table 3. More information about the Dickey-fuller test about the net money supply series can be found in the appendix under table 7. Testing the net income series (yt) for existence of a unit root, which is estimated with k is 2, also shows strong evidence of unit root. The null hypothesis cannot be rejected. When taking the first difference, the null hypothesis can only be rejected at the 10% level. When taking the second difference, the null hypothesis can be rejected for all levels and there is no evidence of being non-stationary. The net income series is integrated of order 2. Results are shown in table 3. More information about the Dickey-fuller test for the net income series can be found in the appendix under table 8. Testing the net interest rate series (rt) for existence of a unit root, which is also estimated with k is 2, shows strong evidence of unit root. The null hypothesis cannot be rejected. When taking the first difference, the null hypothesis can be rejected. The net interest rate series is integrated of order 1. Results are shown in table 3. More information about the Dickey-fuller test for the net interest rate series can be found in the appendix under table 9.
Variables Levels First Differences Second Differences
Test-statistics Probability Test-statistics Probability Test-statistics Probability Mt (1) -1.578 0.8010 -14.709 0.0000 - - Yt (2) 1.567 1.0000 -3.418 0.0490 -8.084 0.0000 It (2) 1.328 0.9968 -11.276 0.0000 - - Table 3 | Dickey-Fuller Unit Root Test
5.2 Cointegration
Testing the cointegration between the different variables will be done through the maximum likelihood test of Johansen. However, to proceed with this test, the number of lags for the 4-variable model should be determined first. This will be done by looking at the likelihood-ratio test. Results of the likelihood ratio point out the correct number of lags is 2. Table 4 shows the results of the Johansen maximum likelihood estimation. A null hypothesis of no cointegration between the variables will be tested against an alternative hypothesis of variables with cointegration. The null hypothesis will be rejected when the trace statistics exceeds the 5% critical value. When looking at the multivariate model, the trace statistics exceeds its critical value for r=0, we strongly reject the null hypothesis of no cointegration. Similarly, we reject the null hypothesis that there is one of fewer cointegrated variables. This also counts for the null hypothesis there are two or fewer and that there are three of fewer cointegrated variables. Because Johansen’s method for estimating the rank is to accept the first rank for which the null hypothesis is not rejected, we accept r=4 as our estimate of the number of cointegrating equations between these four variables. As we know, a cointegration vector implies a long-run relationship among jointly endogenous variables arising from constraints implied by the economic structure on the long-run relationship. It means that the more number of cointegrating vectors, the more stable the system of non-stationary cointegrated vector.Maximum rank (r) LL Eigenvalue Trace statistics 5% critical value
0 3788.0114 . 335.6500 47.21 1 3875.3693 0.36706 160.9337 29.68 2 3922.7413 0.21966 66.1903 15.51 3 3953.4769 0.14864 4.7190 3.76 4 3955.8364 0.01228 Table 4 | Results from multivariate Johansen Maximum Likelihood Estimation However, the multivariate model only indicates the number of cointegrated vectors without specifying which variables are cointegrated. By performing the same test on the bivariate model, this information will be given. Results are shown in table 5. This corresponds to the 4 cointegrated variables from the multivariate test. The results indicate the existence of one
cointegrating vector between the exchange rate and the net money series, the exchange rate and the income series, the exchange rate and the Interest rate series and between the net money supply series and the interest rate series.
Pairs (r=1) Trace statistic 5% critical value Results
St - Mt 78.6109 3.76 Cointegrated St - Yt 6.1294 3.76 Cointegrated St - It 53.3014 3.76 Cointegrated Mt - Yt 3.1093 3.76 Not cointegrated Mt - It 63.6497 3.76 Cointegrated Yt - It 3.2497 3.76 Not cointegrated Table 5 | Results from bivariate Johanson Maximum Likelihood Estimation The existence of the cointegrating vectors indicate the above mentioned cointegrated variables share a long-run equilibrium. It is shown that there are up to 3 cointegrated vectors between the exchange rate and economic fundamental variables.
5.3 RMSE
Table 6 reports the RMSE statistic of the Dornbusch model and the random walk model. RMSE is the principal criterion to test the out-of-sample forecast performance. When comparing the Dornbusch model with the random walk model, one model outperforms the other model by showing a lower value of the RMSE statistics. Looking at the results from table 6, the Dornbusch model has a lower RMSE that the random walk model and performs therefore better. Models RMSE Dornbusch 902.3 Random walk 1233.5 Table 6 | Out-of-Sample Forecast: RMSE6. Conclusion
The research question of this paper is of the Dornbusch exchange rate model is able to create a meaningful forecast of the Bitcoin-USD exchange rate and outperform the random walk model. The following hypothesis were formulated: • H0: It is not possible for the Dornbusch exchange rate model to create a meaningful forecast of the Bitcoin-USD exchange rate and outperform the random walk model on the basis of the Root Mean Square Error. • H1: It is possible for the Dornbusch exchange rate model to create a good meaningful of the Bitcoin-USD exchange rate and outperform the random walk model on the basis of the Root Mean Square Error. It is found that the Dornbusch exchange model itself was able to create a meaningful forecast of the Bitcoin-USD exchange rate regarding the cointegration test results. It was shown that the exchange rate was cointegrated with all of the economic fundamental variables, namely the net money supply, the net income and the net interest rates. This implies they share a long-run equilibrium. More cointegrated vectors, mean a more complete long-run image of the model and therefore a more meaningful forecast. So, the Dornbusch exchange rate model is able to realize a meaningful forecast of the Bitcoin-USD exchange rate when looking at the number of cointegrating vectors. When looking at the RMSE the Dornbusch model outperforms the random walk model, due to having a lower value as the RMSE. In conclusion, we can say that the Dornbusch model is able to create a meaningful forecast of the Bitcoin-USD exchange rate and that it outperforms the random walk model when looking at the RMSE. It contains meaningful information which helps to improve the models forecasting performance. I note that more research may be necessary to really confirm this conclusion, as the amount of data used is limited. Using longer time series and more cryptocurrencies may result in a different conclusion.7. References
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Appendix
Augmented Dickey-Fuller tests:
Levels First difference
t-Statistic Prob. t-Statistic Prob. Augmented Dickey-fuller test statistics -1.578 0.8010 -14.709 0.0000 Test critical values: 1% level 5% level 10% level -3.985 -3.425 -3.130 -3.985 -3.425 -3.130 Table 7 | Augmented Dickey-Duller test results on net money supply
Levels First difference Second difference
t-Statistic Prob. t-Statistic Prob. t-Statistic Prob.
Augmented Dickey-fuller test statistics 1.567 1.0000 -3.418 0.0490 -11.276 0.0000 Test critical values: 1% level 5% level 10% level -3.985 -3.425 -3.130 -3.985 -3.425 -3.130 -3.985 -3.425 -3.130 Table 8 | Augmented Dickey-Duller test results on net income Levels First difference
t-Statistic Prob. t-Statistic Prob. Augmented Dickey-fuller test statistics 1.328 0.9968 -8.084 0.0000 Test critical values: 1% level 5% level 10% level -3.449 -2.875 -2.570 -3.450 -2.875 -2.570 Table 9 | Augmented Dickey-Duller test results on net money interest rate
