An Analysis of Bitcoin Exchange Rates
Jacob Smith∗
University of Houston
March 31, 2015

Abstract
Bitcoins are digital gold. They are a purely electronic commodity traded for speculative purposes as well as in exchange for goods and services. Just like physical gold,
the relative price of bitcoins denominated in different currencies implies a nominal exchange rate. This is a departure from previous literature which treats bitcoin prices
themselves as nominal exchange rates. I argue that treating prices as exchange rates
is inappropriate as one would not consider the price of physical gold to be an exchange
rate. Therefore, this paper characterizes the behavior of nominal exchange rates implied by relative bitcoin prices. I show that the implied nominal exchange rate is highly
cointegrated with the nominal exchange rate determined in conventional foreign currency exchange markets. I also show that the direction of causality flows from theconventional markets to the bitcoin market and not vice-versa which can explain muchof the volatility in bitcoin prices.

Keywords: Bitcoin, Cointegration, Cryptocurrency, Exchange Rates
JEL Classification: E42, E47, C3∗

Special thanks for the helpful comments from David Papell, Dietrich Vollrath, German Cubas, and Christian Murray as well as feedback from attendees of the UH Graduate Research Seminar and Hashers United International Bitcoin Conference. Please address correspondence to: Jacob Smith, Department of Economics,
University of Houston, McElhinney 248, Houston, TX 77204-5019. Email: Jacob.BL.Smith@gmail.com.
Copyright: Jacob B. Smith c 20141

1

Introduction

Bitcoins are digital gold, both in design and behavior. This paper will highlight the similarities between the traditional commodity and its new digital counterpart. In doing so it isevident that the bitcoin market is more mature than previously thought.
The contribution of this paper is to characterize the behavior of bitcoin-implied exchange
rates, i.e. the relative price of bitcoin denominated in different currencies. Whereas previous
authors have concerned themselves with top-level bitcoin prices, my analysis goes further
in studying relative bitcoin prices. I argue that the study of these relative prices is a more
appropriate endeavor when attempting to compare the behavior of bitcoin prices to conventional currencies. In the same way that one would not call the price of physical gold an
exchange rate, neither do I call the price of bitcoins an exchange rate.
Three nominal exchange rates are considered for this analysis: The dollar-euro rate, thedollar-pound rate, and the dollar-Australian dollar rate. These three rates were chosen basedon their economic importance and the availability of the data. The data are daily and span
the period from September 1st , 2011 to January 31st , 2014. While bitcoin prices themselves
are highly volatile and only weakly correlated with conventional nominal exchange rates, theexchange rates implied by relative bitcoin prices are less volatile and are highly cointegrated
with conventional nominal exchange rates.
I employ Vector Error Correction modeling to describe the relationship between impliedand market exchange rates. For each exchange rate I estimate one cointegrating vector andfind that shocks to the conventional foreign exchange market have significant and permanent
effects on implied exchange rates in the bitcoin market. However, shocks to bitcoin-implied
exchange rates do not have any noticeable effects on conventional market rates. This oneway causality implies that bitcoin prices must do all of the adjusting in order for implied andmarket rates to return to parity. In this way, conventional exchange rate volatility drives
much of the bitcoin price volatility.
Previous comparisons of bitcoin prices to conventional exchange rates have shown little2

similarity. This has led some to conclude that “bitcoin is not a real currency” Yermack(2013). However, in thinking of bitcoin as digital gold the similarities become apparent. I
extend my analysis to include gold-implied exchange rates (i.e. relative gold prices) andfind that gold and bitcoin behave almost identically. Gold-implied exchange rates are highly
cointegrated with market exchange rates and shocks to the conventional market have large
persistent effects on the gold market but not vice versa. The similarities between gold andbitcoin speak to the idea that bitcoin is not a currency in the sense that dollars and pounds
are currencies, but are much more akin to physical commodities liked gold, coffee, and oil.
The rest of the paper proceeds as follows: Section 2 provides a primer on the technologyof bitcoins and brief review of the literature. Section 3 presents the data and describes somebasic characteristics of bitcoin prices and exchange rates. Section 4 fits an in-sample modeland tests the predictability of bitcoin nominal exchange rates. Section 5 concludes.

2

What is a Bitcoin?

The recent surge in the popularity of bitcoins has garnered the attention of business professionals and academics unlike any other cryptocurrency. However, most of the discussion on
bitcoins has been confined to the computer science literature or blogosphere with almost no
academic research devoted to the economics of bitcoin.
The first question one might ask is “What is a Bitcoin?” Ron and Shamir (2012) provide
a concise definition:“Bitcoins are digital coins which are not issued by any government, bank, ororganization and rely on cryptographic protocols and a distributed network ofusers to mint, store, and transfer.”
In other words, bitcoins are a purely digital, highly-liquid asset that are traded for speculative purposes and used as a currency. They are not backed by any tangible asset nor are
they sanctioned as legal tender. Thus, bitcoins are non-commodity and non-fiat.
The technology behind bitcoins was first introduced by the pseudonymous Satoshi Nakamoto3

in 2009. Fundamentally, the bitcoin network is centered around a massive but decentralized
public ledger called the “Block Chain.” This ledger records every bitcoin transaction that’s
ever occurred since the network’s genesis. Each bitcoin is given a unique serial number. The
Block Chain publicly displays which bitcoins are associated with which accounts. When atransaction occurs the Block Chain is appended to show that a particular bitcoin has moved
from one account to another. In this way, the bitcoin market is not unlike the stone wheelsof Yap.
The novelty of bitcoin is in the way in which transactions are processed. Suppose Jake
wishes to to send a bitcoin to Josie. Jake announces his intent to the entire network.
The proposed transaction spreads through the network until it finds a “Miner.” A Mineris someone who verifies transactions. The Miner first verifies that Jake has the right tosend that particular bitcoin by checking Jake’s “Private Key” which is analogous to an
ATM PIN. This lets the Miner know that Jake is the rightful owner of the account. Once
Jake’s private key is verified the Miner sets about verifying the authenticity of the particular
bitcoin that Jake is trying to send. The Miner does this through a difficult “proof-of-work”
algorithm which scrolls through the Block Chain to check the bitcoin’s history. Proof-ofwork algorithms are hard to complete but easy to check. Any tampering with the bitcoin’s
code or any attempt to spend the same bitcoin twice will generate large deviations fromthe algorithm’s expected result. For example, I know 2+2=4 but if I get sneaky and try to
add 2.0000001+2 the proof-work-algorithm will report an answer something like 1,729. If
the algorithm generates the correct result (or “hash”) then the transaction is deemed to bevalid. The Miner then bundles this transaction along with other verified transactions into a“Block.” This new Block is broadcast to the network and appended to the Block Chain. At
this point the bitcoin that was in Jake’s account is now publicly recognized to be in Josie’s
account and the transaction is complete. The entire process takes approximately ten minutes
to complete.
Verifying these transactions is quite costly, requiring specialized equipment, technical4expertise, and a significant amount of electricity. To compensate Miners for their efforts
they are awarded newly minted bitcoins every time they complete a Block. It is important
to note that the bitcoin supply is not governed by a central bank or any type of monetary
policy rule but by the efforts of Miners. This is by design and intended to make the supplyof bitcoin similar (at least in theory) to the supply of gold.
As time goes on, the reward for verifying transactions is automatically reduced until the
maximum number of 21 million bitcoins have been mined. From then on Miners will no longer
be rewarded by the system for processing transactions. At its current pace, the network is
expected to mine its last bitcoin in the year 2140. This hard limit on the total supply of
bitcoins is feared to put deflationary pressure on prices and to thus encourage hoarding.
Indeed, Ron and Shamir (2012) find that 78% of all bitcoins are held in accounts which
receive coins but never spend them. Questions about the long-run incentives in the bitcoin
market lead Kroll, Davey, and Felten (2013) to conclude that some form of governance will
be required for long-run sustainability of the market.
Others have also leveled criticisms against the economic viability of bitcoin. Meiklejohn
et al. (2013) show that interactions in the bitcoin market aren’t that anonymous as theBlock Chain provides a public record of all transactions. This is an issue for bitcoin as
the anonymity of illicit transactions was and is one of the major draws for its utility as acurrency (Christin 2012). Pirrong (2013) attacks the efficiency of the market arguing that
too few actors and small market cap leave the market vulnerable to manipulation. Even
more concerning, Eyal and Sirer (2014) describe a strategy in which bitcoin Miners may
unilaterally cheat the system to receive more than their fair share of mining proceeds.
Yermack (2013) questions the maturity of the market by asking “Is Bitcoin a Real Currency?” The author shows that there is almost no correlation between the price of bitcoinsand conventional exchange rates. Moreover, the volatility of bitcoin prices is an order ofmagnitude higher than the volatility observed in conventional nominal exchange rates. This
leads Yermack to conclude that “bitcoin behaves more like a speculative investment than a5currency.”
This paper looks deeper into the behavior of the bitcoin market and endeavors to explain
the extraordinarily high price volatility. Understanding the technology and construction of
bitcoins is crucial to understanding the behavior of agents in the bitcoin market. Like othercommodities (such as gold), bitcoins are bought and sold primarily for investment purposes
and assume a secondary role as a currency. And like these other commodities, the relative
price of bitcoins denominated in different currencies implies an exchange rate. That is, theC/B) implies a dollar-euro
dollar price of bitcoins ($/B) divided by the Euro price of bitcoins (Aexchange rate ($/A
C). Focusing on this bitcoin-implied exchange rate gives insight into the
behavior of the bitcoin market. Contrasting the behavior of bitcoin-implied exchange rates
with similar gold-implied exchange rates allows a more direct comparison between traditionalcommodity money and the burgeoning market for cryptocurrency. I find that bitcoins are
subject to the same motivations and dynamics as physical gold which leads me to conclude
that the bitcoin market, though small, is relatively mature.

3

Data

Daily data on bitcoin prices were downloaded from bitcoincharts.com. The prices are thosewhich prevailed on the Mt. Gox1 exchange at 12:00 a.m. GMT (UTC). There are any
number of bitcoin exchanges with freely floating prices, however, Mt. Gox was by far thelargest and most active (Meiklejohn et al. 2013). Data on bitcoin prices denominated in
US Dollars, British Pounds, Euros, and Australian Dollars were collected for the period
September 1st , 2011 to January 31st , 20142 . Implied exchange rates were then calculatedby dividing the Dollar price of bitcoins ($/B) by the foreign price of bitcoins (£/B, A
C/B,
1

Not only was Mt. Gox the largest and most active bitcoin exchange, it was also one of the most active
bitcoin market places in general (Meiklejohn et al. 2013).
2
Data are available after January 31st , 2014, however, in early February 2014 news broke that Mt. Gox’s
security protocols had been compromised allowing hackers to seize bitcoins stored in customers’ accounts.
Thus, there was a subsequent run on Mt. Gox bitcoins which caused the price to plummet and forced Mt.
Gox into bankruptcy.

6

A$/B). Conventional market exchange rates as well as gold prices were obtained from theSt. Louis Fed’s FRED database. The nominal exchange rates are the “noon buying rates in
New York City for cable transfers payable in foreign currencies.” While gold prices are 3:00
p.m. (London Time) fixing prices which prevailed in the London Bullion Market. Figures
1-5 illustrate the data and Table 1 provides some descriptive statistics.
The descriptive statistics in Table 1 reiterate the findings of Yermack (2013): bitcoin
prices are indeed extraordinarily volatile when compared with conventional market exchange
rates. A fact illustrated in Figure 1. However, if one considers the bitcoin implied exchange
rates one notices that the mean and standard deviation are almost identical to the market
exchange rates. This fact is born out in Figures 3-5. While it is true that bitcoin implied
rates remain slightly more volatile than conventional market rates, it is clear the the impliedand market rates follow each other very closely. This is despite the fact that bitcoin prices
themselves are almost totally uncorrelated with market exchange rates.
One sees a similar pattern with physical gold. Gold prices themselves are highly volatileand seemingly uncorrelated with market exchange rates, however, gold-implied exchange
rates track closely with market rates.
The relationship between the implied and market exchange rate is motivated by arbitrage. When the bitcoin (or gold) implied exchange rate finds itself out of parity with theconventional market rate there exists an opportunity for arbitrage. As agents take advantageof these arbitrage opportunities the exchange rates are driven back together. For example,
suppose the implied dollar-euro rate was greater in the bitcoin market than in the conventional market. This means the euro is relatively stronger in the bitcoin market than in theconventional market. Thus, a trader could exchange dollars for euros in the conventional
market (where euros are relatively cheap) and take those euros to buy bitcoins in the bitcoin
market (where euros are relatively valuable). The purchase of bitcoins with euros drives up
the euro price of bitcoin thereby driving down the implied exchange rate.
Such arbitrage is common whenever commodities are traded in different currencies (Sjaas-

7

tad and Scacciavillani 1996). These arbitrage trades often result in the series being cointegrated. Two non-stationary series are said to be cointegrated if a linear combination of
the series is stationary. If one considers the spread between implied and market rates (i.e.
$/A
Cmarket − $/A
Cimplied ) in Figure 6 it seems clear that the linear combination of the series is stationary about 0. This is mirrored in Figures 7 and 8 for the dollar-pound anddollar-Australian dollar rates respectively.
One interesting comparison that can be made between the gold and bitcoin implied rates
is the relative size of the so-called arbitrage bands. The bands are much larger far the bitcoin
implied rates than for the gold implied rates. The width of the arbitrage band is dictated bytransaction costs, i.e. arbitrage trades are only made once the profits exceed the transactions
costs. Thus, higher transaction costs allow for higher disparity. Because the arbitrage bands
are much larger for bitcoins than for gold it must be that transactions costs are higher in the
bitcoin market than in the gold market. The exact reasoning for this is outside the scope of
this paper but further research into the size and dynamics of the exchange rate spread could
prove an interesting topic for further discussion.

4

Empirical Analysis

Engle and Granger (1987) show that if two series are cointegrated then a simple regressionof ∆yt on ∆xt is misspecified. A system of I(1) unit root processes is said to be cointegrated
if there exists some linear combination of the series which is stationary, or I(0). If we let yt
be a vector of time series then the system is cointegrated if there exists some non-zero vector
β such that β 0 yt is stationary. The system is said to be in “equilibrium” when β 0 yt = 0 andout of equilibrium when β 0 yt 6= 0. We can denote this “equilibrium error” as zt = β 0 yt .

8Consider the bivariate system,

yt + αxt = t ,

t = t−1 + ξt

(1)

yt + βxt = νt ,

νt = ρνt−1 + ζt , |ρ| < 1(2)

where ξt and ζt are white noise. Note that the reduced form of yt and xt will be functionsof both  and ν. Because t is I(1) it must be the case that yt and xt are also I(1). Now
consider (2) which is a linear combination of yt and xt . Because νt is stationary it must be
the case that yt + βxt is also stationary. Therefore, yt and xt are cointegrated with a vector
β 0 = (1, β).
Engle and Granger (1987) show that we can rewrite the system as,∆yt = αδzt−1 + η1t(3)∆xt = −δzt−1 + η2t(4)

where δ = (1 − ρ)/(β − α) and the η’s are linear combinations of t and νt . Recall thatzt = yt + βxt . Thus, equations (3) and (4) illustrate the vector error correction (VEC)
representation of the system which describes how the series respond to disequilibrium.
If, however, yt is a VAR(p) the VEC representation can be expanded and written as,∆yt = αβ 0 yt−1 +p−1XΓi ∆yt−i + t

i=1

For this analysis, ∆yt is a 2x1 vector

 ∆yt ∆xt

where the ∆yt is the first difference in thenatural log of the implied exchange rate and ∆xt is the first difference in the natural log ofthe market exchange rate. The vectors α and β are also 2x1 and contain the adjustment
parameters and cointegrating vector respectively. A simple VAR of ∆yt and ∆xt would omit
αβ 0 yt−1 and therefore be misspecified if indeed the series are cointegrated.
The matrix Γp is a 2x2 matrix of lag parameters
9





γyy,p γyx,p 
Γp = 

γxy,p γxx,pApplying the VEC method to implied and market exchange rates follows Sjaastad andScacciavillani (1996). In their paper, Sjaastad and Scacciavillani estimate a cointegrating
relationship between gold-implied exchange rates and the market rate then attempt to use
this vector to forecast the nominal market rate. The authors show that the adoption of
floating exchange rate regimes has contributed significantly to the volatility of gold prices. I
will show that this is also a major factor in the extreme volatility in bitcoin prices. Sjaastad
and Scacciavillani (1996) also find that that the direction of causality flows from the nominal
exchange rate market to the gold market, but not vice versa. We will see that the bitcoin
market exhibits the same behavior. At the end of this section, I replicate portions of Sjaastad
and Scacciavillani (1996) using my data and find similar results.
The contribution of this paper is to apply Sjaastad and Scacciavillani’s methodology
to the bitcoin market. Doing so will shed light on the dynamics of the market as well as
illustrate the similarities between purely digital bitcoins and other physical commodities.
Determining the order of the VAR means considering five lag-selection methodologies.
The five methodologies all involve minimizing some criterion, either the Likelihood-Ratio, theFinal Prediction Error (FPE), the Akaike Information Criterion (AIC), the Hannan-Quinn
Information Criterion (HQIC), or the Schwartz Bayesian Information Criterion (SBIC). The
lag length which minimizes the majority of these five criteria is taken to be the most appropriate. For the dollar-euro and the dollar-Australian dollar rates a lag length of 2 is selected.
For the dollar-pound rates I use a lag-length of 5.
From there I apply Johansen’s (1988, 1991, 1995) methodology to formally tests for the
existence of cointegration. The results of the Johansen tests show that for all three currencies
I strongly reject the null of no cointegrating vector but fail to reject the null of at most 1
cointegrating vector. The results of these tests can be found in Table 2.

10

The Johansen tests confirm the presence of cointegration for all three currencies making
VEC modeling the most appropriate technique. Table 3 reports the parameter estimates of(5) when considering bitcoin implied rates and conventional market rates.
In examining to the results from Panel 3a we see that the estimates of α1 are negativeand highly significant for each of the exchange rates. Recall that α1 is the way in which thefirst series responds to disequilibrium, i.e. the implied rate’s “error correction.” The fact that
α1 is negative and significant for all three currencies indicates that when the bitcoin-implied
exchange rate finds itself out of equilibrium with the conventional market exchange rate,
the implied rate quickly adjusts to restore balance. Moreover, this error correction dynamic
explains between 36% and 49% of the movement in the implied rates meaning that shocks
to market exchange rates have enormous knock-on effects in the bitcoin market. However,
if one considers the output in Panel 3b one sees that the same error correction dynamic is
not present when the market rate is the response variable. The estimated coefficients on α2
are insignificant for each of the currencies, thus, the conventional market does not respondto disequilibrium with the bitcoin market.
A further illustration of this one-way causality can be found in the plots of the impulseresponse functions. Figures 9-11 show the response of the implied rate to a shock in themarket rate for the $/A
C, $/£, and $/A$ respectively. One can see that shocks to the implied
rate are significant and persistent, i.e. shocks to the market rate have positive effects on
the implied rate which never wear off. For all three currencies, the shocks are almost fully
realized within the first few days indicating an active and alert pool of traders.
However, if one considers the IRF’s in Figures 12-14, shocks to the implied rate yield
almost no effects on the conventional market rate. In other words, the conventional market
does not react to changes in the bitcoin market. This is intuitive as the total capitalizationof the entire bitcoin market was less than $5 billion at the time of this writing whereas
M2 was nearly $11.5 trillion. The bitcoin market is simply not large enough to move theglobal currency markets but global currency markets are more than large enough to move11

the bitcoin market.
The parameters β define the cointegrating vector. Following Johansen (1988, 1991, 1995)
β1 is normalized to 1 for identification. In the bivariate system, this means β2 defines
the equilibrium condition. Recall that equilibrium occurs where 0 = yt + β2 xt , or whenyt = −β2 xt . Thus, βˆ2 gives an estimate of the relationship between yt and xt in equilibrium.
In Panel 3b we see that the estimated β2 is very precise but not statistically different from
-1 for the $/A
C rate. Therefore, we fail to reject the null that the equilibrium condition is forparity between the implied and market rates. yt = xt . The β2 estimates for the $/£ and$/A$ are are also very precise and close to -1, however, they do remain statistically different
from -1. Thus, I cannot make the same claim that the equilibrium condition is parity. Thatbeing said, the absolute difference from -1 is small in both cases.
Taken together, this tells us that when the bitcoin market is out of equilibrium with theconventional exchange market bitcoin prices are forced to make the entire adjustment back toparity. This is the same source of volatility described by Sjaastad and Scacciavillani (1996)
within the gold market. To fully illustrate the similarities between bitcoins and physical gold
I replicate the above analysis using gold-implied exchange rates rather than bitcoin-implied
exchange rates.
As with bitcoin-implied rates, Johansen tests for cointegration in Table 4 indicate thepresence of one cointegrating vector for both the $/A
C and $/£ rate. Table 5 presents theestimates from the VEC model. The estimates on α1 are negative and highly significant for
both currencies in Panel 5a where the gold-implied exchange rate is the response variable.
Once again, this indicates that as the gold-implied rate finds itself out of equilibrium withthe market rate the implied rate adjusts to restore parity. This error-correction dynamic
accounts for 14% and 8% of the variation in the implied $/A
C and $/£ rates respectively.
While market exchange rate volatility is a contributing factor to gold price volatility it is notas significant a factor as in the bitcoin market. Moreover, in considering the magnitudes of ofαˆ1 we see that the gold-implied rate adjusts much more rapidly than do bitcoin-implied rate.

12This gives us some perspective as to the relative size and maturity of the bitcoin market.
In turning to Panel 5b there is little evidence to suggest that the market exchange rates
respond to disparities with the implied rate. The estimate of α2 for the $/£ rate is notstatistically different from 0 while the estimate for the $/A
C rate is only marginally significant.
This is the same one-way, market-to-implied causality that we saw in the bitcoin analysis.
Again, this one-way causality is born out in the graphs of the impulse-response functions.
Figures 15-18 show the permanent effects of a market exchange rate shock to the gold-implied
exchange rate while illustrating the negligible, transitory effects of a gold-implied rate shockto the currency market.
As in the bitcoin analysis, β1 is normalized to 1 for identification while β2 describes theequilibrium condition. We see that for both the $/A
C and $/£ rates βˆ2 is exactly -1 meaning
the equilibrium condition is for the implied rate to exactly equal the market rate. Again,
the fact that βˆ2 is precisely -1 in Panel 5b while somewhat less precise in Panel 5a may be
indicative of a smaller, more immature bitcoin market. That is, traders in the bitcoin market
might not be as sophisticated or motivated to act on arbitrage opportunities. It could also
simply be a function of the presumably larger transactions costs in the bitcoin market which
prevent arbitrage trades from occurring and therefore allow for larger disparities between
the implied and market rates.

5

Conclusion

I show that the most appropriate way to think about bitcoins is as digital gold. Whilenominal bitcoin prices are extremely volatile and seemingly uncorrelated with other nominal
exchange rates, relative bitcoin prices or implied nominal exchange rates are indeed highly
cointegrated with conventional market exchange rates. This mirrors the relationship between
physical gold and conventional nominal exchange rates. Driven by arbitrage opportunities,
relative bitcoin prices adjust rapidly to restore parity with market exchange rates, however13market exchange rates seem unaffected by changes in bitcoin prices. Indeed, almost half ofthe movement in relative bitcoin prices can be explained by this cointegrating vector. Thus,
floating nominal exchange rates are a major source of price volatility in the bitcoin market
just as they are in conventional commodity markets. While the bitcoin market remains
relatively small and highly volatile it appears as though there is a deep and active pool of
traders which maintain a level of market efficiency. However, floating exchange rates canonly explain part of bitcoin’s volatility. It is clear that the market is still small enoughto be manipulated by a small handful of investors and questions are being raised about
the robustness of bitcoin’s underlying technology. It remains to be seen whether bitcoin’s
innovations as a currency can withstand these pressures.

14

References[1] Nicolas Christin. “Traveling the Silk Road: A Measurement Analysis of a Large Anonymous Online Marketplace”. Unpublished. 2012. url: https://www.cylab.cmu.edu/
files/pdfs/tech_reports/CMUCyLab12018.pdf.
[2] Robert F Engle and Clive W J Granger. “Co-integration and Error Correction: Representation, Estimation, and Testing”. Econometrica 55.2 (1987), pp. 251–76.
[3] Ittay Eyal and Emin Gün Sirer. “Majority is not Enough: Bitcoin Mining is Vulnerable”. Unpublished. 2014. url: http://www.cs.cornell.edu/~ie53/publications/btcProcArXiv.pdf.
[4] Soren Johansen. “Statistical Analysis of Cointegration Vectors”. Journal of Economic
Dynamics and Control 12.2-3 (1988), pp. 231–254.
[5] Soren Johansen. “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models”. Econometrica 59.6 (1991), pp. 1551–80.
[6] Soren Johansen. “A Statistical Analysis for Cointegration for I(2) Variables”. Econometric Theory 11.1 (1995), pp. 25–59.
[7] Lutz Kilian. “Exchange Rates and Monetary Fundamentals: What Do We Learn from
Long-Horizon Regressions?” Journal of Applied Econometrics 14.5 (1999), pp. 491–
510.
[8] Joshua Kroll, Ian Davey, and Edward Felten. “The Economics of Bitcoin Mining, orBitcoin in the Presence of Adversaries”. Unpublished. 2013. url: https://www.cs.
princeton.edu/~kroll/papers/weis13_bitcoin.pdf.
[9] Sarah Meiklejohn et al. “A Fistful of Bitcoins: Characterizing Payments Among Men
with No Names”. Unpublished. 2013. url: https://cseweb.ucsd.edu/~smeiklejohn/
files/login13.pdf.
[10] Tyler Moore and Nicolas Christin. “Beware the Middleman: Empirical Analysis of
Bitcoin-Exchange Risk”. Unpublished. 2013. url: http://fc13.ifca.ai/proc/12.pdf.
[11] Satoshi Nakamoto. “Bitcoin: A Peer-to-Peer Electronic Cash System”. Unpublished.
2009. url: https://bitcoin.org/bitcoin.pdf.
[12] Craig Pirrong. “Was the Bitcoin Flock Just Sheared?” Streetwise Professor Blog (2013).
url: http://streetwiseprofessor.com/index.php?s=bitcoin.
[13] Dorit Ron and Adi Shamir. “Quantitative Analysis of the Full Bitcoin Transaction
Graph”. Unpublished. 2012. url: https://eprint.iacr.org/2012/584.pdf.
[14] Larry A Sjaastad and Fabio Scacciavillani. “The Price of Gold and the Exchange Rate”.
Journal of International Money and Finance 15.6 (1996), pp. 879–897.
[15] VEC Intro. 13th ed. Stata Corp. 2014. url: http://www.stata.com/manuals13/
tsvecintro.pdf.
[16] David Yermack. “Is Bitcoin a Real Currency?” NBER Working Paper Series Number
19747. 2013. url: http://www.nber.org/papers/w19747.
15

16* Indicates failure to reject H0 : max rank ≤ x

Table 2: Johansen Tests for Cointegration
$/A
C
$/£
Maximum
Log
Trace
Log
Trace
Rank
Likelihood Statistic 5% CV
Likelihood Statistic 5% CV
04003.69
158.4712.53
3977.25
36.45
12.53
1
4082.790.29*
3.84
3995.44
0.07*
3.84
Lags
2
5
N
602
598

Mean
Std. Dev
High
Low
N

Bitcoin Prices
$/B
A
C/B
£/B
A$/B
118.41 87.31 80.86
145.53245.86 178.67 156.16
289.01
1,230.00 885.10 738.00 1,333.64
2.05
1.54
1.40
2.07
884
882
877
859

Gold
Implied Rates
$/A
C
$/£
1.34
1.59
0.06
0.04
1.49
1.67
1.21
1.49
774
774

Market Rates$/A
C $/£ $/A$
1.32 1.58 1.00
0.04 0.04 0.06
1.43 1.66 1.08
1.21 1.48 0.87
605 605
605

$/A$
Log
Trace
Likelihood Statistic 5% CV
3484.72
145.4412.53
3557.19
0.51*
3.84
2
595

Table 1: Descriptive Statistics
Bitcoin
Implied Rates
Gold Prices$/A
C $/£ $/A$
$/oz
A
C/oz
£/oz
1.31 1.57 0.99
1,541.31 1,156.61
972.540.05 0.05 0.05
170.81143.04109.06
1.58 1.87 1.21
1,895.00 1,382.27 1,182.82
1.19 1.45 0.87
1,192.00
873.14730.10882 877
859
774
774
774

Table 3a: Vector Error Correction Model
Response Variable: Natural Log of Bitcoin-Implied Exchange Rates

α1
β1

$/A
C
Estimate R2 N
-0.500*** 0.36 602(0.037)1na

γyy,1 -0.144***(0.036)
γyy,2
γyy,3
γyy,4

γxy,1 -0.067(0.092)
γxy,2
γxy,3
γxy,4

$/£
Estimate R2 N
-0.345*** 0.49 598(0.056)1
na$/A$
Estimate R2 N
-0.631*** 0.46 595(0.049)1na-0.541***(0.057)-0.385***(0.055)-0.392***(0.049)-0.152***(0.037)-0.233***(0.038)-0.078(0.149)0.201(0.148)0.472***(0.146)
0.369***(0.144)0.231*(0.126)* Significant at 10% level, ** Significant at 5% level, *** Significant at 1% level.

17

Table 3b: Vector Error Correction Model
Response Variable: Natural Log of Conventional Market Exchange Rates

α2
β2

$/A
C
Estimate R2 N
-0.0110.01 602(0.018)-0.996***(0.003)

γyx,1 0.011(0.017)
γyx,2
γyx,3
γyx,4

γxx,1 -0.047(0.044)
γxx,2
γxx,3
γxx,4

$/£
Estimate R2 N
-0.0140.00 598(0.016)-0.989***(0.004)$/A$
Estimate R2 N
-0.0170.01 595(0.017)-0.897***(0.024)0.024(0.017)0.020(0.016)0.024*(0.014)0.028***(0.011)-0.017(0.013)-0.061(0.044)-0.016(0.044)-0.051(0.043)-0.047(0.043)-0.033(0.044)Significance of the Cointegrating Equation

χ2

$/A
C

$/£

$/A$

81047.29***57583.73***1419.54**** Significant at 10% level, ** Significant at 5% level, *** Significant at 1% level.

18

Table 4: Johansen Tests for Cointegration
$/A
C
$/£
Maximum
Log
Trace
Log
Trace
Rank
Likelihood Statistic 5% CV
Likelihood Statistic 5% CV
04003.69
158.4712.53
5193.23109.5112.53
14082.790.29*
3.84
5247.97
0.03*
3.84
Lags
2
5
N
602
598
* Indicates failure to reject H0 : max rank ≤ x

Table 5a: Vector Error Correction Model
Response Variable:
Natural Log of Gold-Implied Exchange Rates
$/A
C
$/£
2
Estimate R
N
Estimate R2 N
α1 -1.459*** 0.14 582-0.745*** 0.08 582(0.238)(0.218)
β1 1
1
na
na
γyy,1 0.410**(0.211)
γyy,2 0.392**(0.162)γyy,3 0.192*(0.109)-0.004(0.193)0.181(0.159)0.122(0.273)

γxy,1 -0.473**(0.211)
γxy,2 -0.431***(0.167)
γxy,3 -0.238**(0.116)-0.053(0.194)-0.169(0.159)-0.173(0.114)* Significant at 10% level** Significant at 5% level*** Significant at 1% level

19

Table 5b: Vector Error Correction Model
Response Variable:
Natural Log of Conventional Market Exchange Rates

α2
β2

$/A
C
Estimate R2 N
-0.448*0.02 582(0.254)-1.000***(0.000)$/£Estimate R2 N
0.078
0.02 582(0.218)-1.000***(0.000)

γyx,1 0.498**(0.219)γyx,2 0.528***(0.173)γyx,3 0.233**(0.116)0.214(0.209)0.385**(0.171)0.268**(0.120)

γxx,1 -0.526**(0.225)
γxx,2 -0.543***(0.178)
γxx,3 -0.256**(0.124)-0.236(0.209)-0.358**(0.172)-0.288**(0.123)Significance of the Cointegrating Equation

χ2

$/A
C

$/£

1.40x107 ***3.51x107 **** Significant at 10% level** Significant at 5% level*** Significant at 1% level20Figure 1: Bitcoin Prices

Figure 2: Gold Prices

21

Figure 3: Dollar-Euro Nominal Exchange Rate ($/A
C)

Figure 4: Dollar-Pound Nominal Exchange Rate ($/£)

22

Figure 5: Dollar-Australian Dollar Nominal Exchange Rate ($/A$)

Figure 6: Implied vs. Market Rate Spread for $/A
C Rate23

Figure 7: Implied vs. Market Rate Spread for $/£ Rate

Figure 8: Implied vs. Market Rate Spread for $/A$ Rate24

Figure 9: Bitcoin Market-to-Implied IRF for $/A
C RateFigure 10: Bitcoin Market-to-Implied IRF for $/£ Rate25

Figure 11: Bitcoin Market-to-Implied IRF for $/A$ Rate

Figure 12: Bitcoin Implied-to-Market IRF for $/A
C Rate26

Figure 13: Bitcoin Implied-to-Market IRF for $/£ Rate

Figure 14: Bitcoin Implied-to-Market IRF for $/A$ Rate27

Figure 15: Gold Market-to-Implied IRF for $/A
C RateFigure 16: Gold Market-to-Implied IRF for $/£ Rate28

Figure 17: Gold Implied-to-Market IRF for $/A
C RateFigure 18: Gold Implied-to-Market IRF for $/£ Rate29