At this stage, it is common knowledge that cryptocurrency prices are indeed, a bubble. However, does modern-day finance have the tools to detect explosive behaviour in absence of a fundamental value? Glad to have worked with Shane Jose to release a paper in a bid to answer the aforementioned question!

1. Econometric Investigation into Cryptocurrency Price
Bubbles in Bitcoin and Ethereum
Siddharth Hitkari & Shane Jose
Trinity College, University of Dublin
April 2018
Abstract
This paper delves into testing bubble activity in Bitcoin and Ethereum prices as well as
linking these explosive responses to real-world events through a three-pronged model,
consisting of recursive, rolling and reverse recursive ADF tests. Rolling and reverse
recursive ADF models prove highly credible in detecting ‘irrational exuberance’. Ad-
ditionally, our results indicate a low degree of complementarity between Bitcoin and
Ethereum, which is often overstated due to sporadic events.
Keywords: Asset-price Bubbles; Cryptocurrency; Augmented Dickey-Fuller
1. Introduction
Nobel laureate in economics, Robert Shiller, remarked earlier this year that while Bitcoin
was a ‘really clever idea’, it would not become a permanent part of the financial world
(Monaghan, 2018). The wild swings in the prices of cryptocurrencies have been a matter
of much on-going academic deliberation and polarisation. Such incessant and intriguing
volatility forms the epicentre of the motivation for this paper wherein we aim to detect
cryptocurrency price bubbles in Bitcoin and Ethereum.
However, detecting bubbles in cryptocurrencies is particularly challenging given an ab-
sence of fundamental value which was empirically confirmed by Cheah and Fry (2015). This
implies that traditional asset-pricing approaches developed by West (1987), Diba and Gross-
man (1988) cannot be implemented. Furthermore, Evans’ (1991) critique highlights the
limitations of standard unit-root procedures in detecting periodically collapsing bubbles.
As a result, we follow right-tailed approaches of recursive and rolling ADF testing success-
fully developed by Phillips, Wu and Yu (2011) and Phillips, Shi and Yu (2013) respectively.
We supplement this approach with a right-tailed reverse recursive ADF test developed by

2. Andrews and Kim (2006) to test for end-of-sample bubble activity. The aforementioned tests
form the cornerstone of our model which is explained in Section 3, while Section 4 mentions
the empirical results which successfully detect explosiveness in Bitcoin and Ethereum. In
Section 5, we delve into the economic interpretation of our results and draw parallels with
real-world events to validate our findings, before concluding in Section 6.
2. Description of data set
Count Max Min Range Std.dev Variance Skewness Kurtosis
BTC 2807 9.87008 -2.99573 12.8658 2.92341 8.54634 -.694872 2.84611
ETH 962 7.23347 -.8675006 8.10097 2.245559 5.042536 .109551 1.878663
2011 2012 2013 2014 2015 2016 2017
0
5
10
Year
Price
(a) Bitcoin
2015 2016 2017 2018
0
2
4
6
8
Year
Price
(b) Ethereum
Fig. 1. Log-Price Trends
The sample period for Bitcoin price data is 19th July 2011 to 23rd March 2018, consist-
ing of 2807 daily observations. Similarly, the sample period for Ethereum price data is 6th
August 2015 to 23rd March 2018, consisting of 962 daily observations. Inconsistent pricing
across crypto-exchanges can be attributed to a highly speculative pricing mechanism cou-
pled with their non-stationary explosive behaviour. Bitcoin pricing varies across exchanges
on degrees of volatility, volumes of transactions as well as ‘infrastructural’ technicalities that
certain exchanges face (Pilani and Haselton, 2017). It is for this reason that the histori-
cal data extracted for both, Bitcoin and Ethereum, was a weighted average obtained from
BitcoinAverage.com.
In order to correct for skewness and accurately estimate results, we took the natural log
of the currency prices and linearized the time-series data. Doing so, additionally aided in
2

3. reducing the scaling dimension of these time-series trends through providing stricter con-
ditions. Intuitively, despite having reduced the range-scale, a high degree of volatility can
be observed in the short time intervals in both cases and this will be verified empirically,
subsequently in this paper.
3. Empirical Approach
3.1. Optimal Lag Selection
While too few lags would inadequately eliminate residual serial correlation, too many lags
could cause biases in the model coefficients through higher standard errors (Brooks, 2014).
Optimal lag-order selection was executed by extrapolating the number of optimal lags
for each of the ‘maximum-lag values’ chosen. The maximum-lag values arbitrarily belonged
to the set {5, 8, 12, 15, 20, 24}. For each of these 6 statistics, the optimal lags (lag∗
) were
compared across each information criterion (IC), AIC, HQIC and SBIC. Most ICs agreed on
the optimal-lag value for the log-prices of both cryptocurrencies.
The Hannah-Quinn Information Criterion (HQIC) remained consistent at lag∗
= 8 for
Bitcoin, and the Schwarz Bayesian Information Criterion (SBIC) did not deviate from lag∗
=
1 for Ethereum, regardless of the maximum-lag chosen. Furthermore, Ivanov and Kilian
(2005), provide evidence for the HQIC rendering the most accurate results (in comparison
to other ICs) for large sample sizes; while SBIC performs better for smaller sample sizes.
This conjecture is consistent with our datasets – 2807 time-series observations for Bitcoin
and 962 for Ethereum.
3.2. The Model
In order to detect explosive behaviour in Bitcoin and Ethereum prices, we adopt the
models previously successfully deployed by Phillips, Wu and Yu (2011) and Phillips, Shi and
Yu (2013). For each time series pt (log of Bitcoin/Ethereum prices), we apply the Augmented
Dickey-Fuller test for a unit-root (H0 : δ = 1) against the alternative of an explosive root
(H0 : δ > 1). We, then, estimate the following regression by OLS for expanding and rolling
windows:
pt = αp + δpt−1 +
Q
q=1
ωq∆pt−q + p,t, p,t ≈ iid(0, σ2
p) (1)
where Q is the number of optimal lags chosen and the error term, , is normally and
independently distributed.
3

4. 3.2.1. Unit-Root Testing
Taking into account the endogeneity of the regressor pt−1, standard unit root-root tests
like Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) allow for lags ∆pt to be
included as regressors1
.
The null hypothesis without drift is chosen as the true process by treating the time-
variable as exogenous (drift = 0) and by obtaining non-zero means of cryptocurrency log-
prices (signifying the presence of a constant in the regression).
3.2.2. Forward Recursive Augmented Dickey-Fuller Test
Phillips, Wu and Yu (2011) proposed a forward recursive method to detect exuberance in
asset price series during an inflationary phase which is especially effective in cases of a single
bubble episode in the sample data2
. They develop a reduced form right-tailed approach to
bubble detection which focuses on the mildly explosive (submartingale) behaviour captured
by the alternative hypothesis.
Phillips et al. (2011) develop a test that entails a forward expanding sequence of samples
with a pre-decided initial window size (r0). The starting point (r1) is fixed at 0 while the
endpoint of the sample (r2) increases (from r0) recursively by 1 observation until r2 is equal
to the entire sample size. The supremum ADF (SADF) statistic is the largest ADF statistic
obtained by conducting a forward recursive ADF test across all subsamples and is defined
as:
SADF(r0) = supr2∈[r0,1][ADFr2
0 ] (2)
Our selection of the initial window size (r0) at 10% of the sample size (2807) for BTC was
consistent with that of Phillips et al. (2011). Given the smaller sample set for ETH (962),
we decided to take a larger value (15%) for r0 in order to reliably detect explosive behaviour.
3.2.3. Rolling Augmented Dickey-Fuller Test
Phillips, Shi and Yu (2013) postulate that in cases where the sample period includes mul-
tiple episodes of exuberance and collapse, the SADF test may have low explanatory power.
The complex nonlinear structure of a bubble with multiple collapsing episodes coupled with
SADFs reliance on the initial sample size selected (Shi, 2010), increases the risk of a false
positive.
1
The PP test additionally redresses unidentified heteroskedasticity, but is less accurate than the ADF
test in finite datasets (Davidson and MacKinnon, 2004).
2
Phillips, Wu and Yu (2011) successfully applied the SADF approach on to NASDAQ prices in the 1990s
to detect explosive behaviour
4

5. Phillips et al. (2013) highlight that this weakness is more prominent in analyzing long
time series or rapidly changing market data, where more than one episode of exuberance
is suspected. To overcome this pitfall, they propose an alternative approach, Generalised
Supremum ADF (GSADF) test, by allowing the starting point of each subsample to change.
Similar to forward recursive ADF test in essence, rolling ADF test is built on the premise
that now both, the start (r1) and endpoints (r2), are not static. They expand in increments
(of one) such that a fixed-sized window of subsamples is rolled forward in each regression.
The GSADF statistic is the largest ADF statistic obtained by conducting the rolling ADF
test across all subsamples and is defined as:
GSADF(r0) = sup
r2∈[r0,1]
r1∈[0,r2−r0][ADFr2
r1
] (3)
3.2.4. Reverse Recursive Augmented Dickey-Fuller Test
According to Astill et al. (2016), the reverse (or ‘backward’) recursive ADF test has a
higher level of explanatory power in detecting an end-of-sample bubble as opposed to either
a forward or a rolling ADF. More importantly, this approach is robust to serial correlation
and conditional heteroskedasticity (Andrews and Kim, 2006).
This test is similar to the forward recursive in having a fixed initial window size (r0) and
repeatedly generating ADF statistical values for the corresponding regressions of every sub-
sample. However, we now fixate the endpoint (r2) of our sample and proceed by recursively
incrementing our start-point (r1) backwards until the entire sample has been included. This
is the Reverse SADF (RSADF) and is defined as:
RSADF(r0) = supr1∈[0,1−r0][ADFr
1
1] (4)
4. Empirical Results
The results from both, the ADF and PP tests, fail to reject the null hypothesis of a unit
root process. Intuitively, this makes sense as standard unit root procedures fail to detect
bubbles in the presence of multiple/collapsing bubbles due to mean-reversion issues (Evans,
1991).
Given the limitations of standard unit root procedures, we resort to conducting the
standard ADF on smaller subsamples through econometric approaches developed by Phillips,
Wu and Yu (2011) and Phillips Shi and Yu (2013). While the forward recursive ADF statistics
fail to reject the null hypothesis for Ethereum, they detect financial exuberance in Bitcoin
prices at the 95% level of significance.
5

6. Table 1: Test statistics for Log-Bitcoin Prices (Lags=8)
Test Test Statistic 1% C.V. 5% C.V. 10% C.V.
ADF 2.354 3.430 2.860 2.570
PP 2.046 3.430 2.860 2.570
Recursive ADF 2.873** 3.430 2.860 2.570
Rolling ADF 4.496*** 3.458 2.879 2.570
R-Recursive ADF 2.799* 3.458 2.879 2.570
***
p < 0.01, **
p < 0.05, *
p < 0.1
Table 2: Test statistics for Log-Ethereum Prices (Lags=1)
Test Test Statistic 1% C.V. 5% C.V. 10% C.V.
ADF 0.446 3.430 2.860 2.570
PP 0.102 3.430 2.860 2.570
Recursive ADF 2.270 3.430 2.860 2.570
Rolling ADF 6.432*** 3.495 2.877 2.577
R-Recursive ADF 3.496*** 3.495 2.877 2.577
***
p < 0.01, **
p < 0.05, *
p < 0.1
2011 2014 2017
0
1
2
3
Year
Recursive-ADFt-stat
(a) Recursive ADF (Ln-BTC)
99% C.I.
95% C.I.
90% C.I.
2011 2014 2017
0
2
4
Year
Rolling-ADFt-stat
(b) Rolling ADF (Ln-BTC)
99% C.I.
95% C.I.
90% C.I.
2011 2014 2017
0
1
2
3
Year
R-RADFt-stat
(c) R-Recursive ADF (Ln-BTC)
99% C.I.
95% C.I.
90% C.I.
Jan2016
Jun2016
Jan2017
Jun2017
Jan2018
0
1
2
Year
Recursive-ADFt-stat
(d) Recursive ADF (Ln-ETH)
99% C.I.
95% C.I.
90% C.I.
Jan2016
Jun2016
Jan2017
Jun2017
Jan2018
0
2
4
6
Year
Rolling-ADFt-stat
(e) Rolling ADF (Ln-ETH)
99% C.I.
95% C.I.
90% C.I.
Jan2016
Jun2016
Jan2017
Jun2017
Jan2018
0
1
2
3
Year
R-RADFt-stat
(f) R-Recursive ADF (Ln-ETH)
99% C.I.
95% C.I.
90% C.I.
Fig. 2. Time-Series Analysis on Bitcoin and Ethereum
6

7. Rolling ADF statistics, on the other hand, provide an even stronger evidence for bubble
activity in both, Bitcoin and Ethereum, at the 99% level of significance.
Interestingly, despite rejecting the null hypothesis, the rolling ADF test fails to capture
bubble formation towards the latter end of the sample. Contextually, this result is particu-
larly counter-intuitive given the meteoric rise in cryptocurrency prices and volatility in 2017.
Reverse recursive ADF statistics provide statistical evidence for the explosive behaviour in
Bitcoin and Ethereum but more importantly, this test captures end-of-sample instability in
both cryptocurrencies.
5. Analysis and Discussion
Historically3
, there have been 5 instances of major price shocks in Bitcoin (Roberts,
2018) dating back to the Chinese government’s decision to ban financial institutions’ usage
of Bitcoin in March 2013. Bitcoin’s price soared from $30 to $230 over the span of a few
months but collapsed by 71% in a single day during April 2013. Following this collapse,
Bitcoin prices recovered and rallied, witnessing a ten-fold rise to reach $1150 before tumbling
to less than $500 in December 2013. The closure of Mt. Gox (Bitcoin’s biggest exchange at
the time) in February 2014, on the account of hacking, caused another major sell-off wherein
Bitcoin prices dropped by 49% and this doldrums period lasted till 2016 (Roberts, 2018).
As graphically illustrated in Fig. 2(b), rolling ADF statistics exceed their respective critical
values to successfully detect these three bubbles.
Reverse recursive ADF statistics (Fig. 2(c)) capture not only Bitcoin’s crash by 68% in
June 2011 but also Bitcoin’s historical price run of 2017 – theoretically, linking it to the end-
of-sample price explosiveness. The reverse recursive ADF test successfully detects the year-
long persistent exuberance in Bitcoin which temporarily culminated in September 2017. This
was due to the Chinese government’s crackdown on ‘Initial Coin Offerings’ causing Bitcoin’s
price to plummet by 37%. Contextually, this crash was crucial in highlighting China’s de
facto monopoly in cryptocurrency mining (Roberts, 2018). Despite this relatively ‘minor’
blip, Fig. 2(c) effectively captures a steeper upward trend (after November 2017) followed
by a steep crash towards the end of 2017. This can be explained by Bitcoin prices flash
spiking from $2,000 to $20,000 on 7th December4
before nose-diving in January 2018 due to
a major sell-off discussed later in this section5
.
However, our findings fail to detect relatively smaller bubbles in Bitcoin prices such as the
3
Until December 2017
4
Prices rose by more than 900% in a 12-month period.
5
Roubini, who predicted the 2008 financial crisis, labels this event as “The Mother of all Bubbles and
Biggest Bubble in Human History”
7

8. summer sell-off of 2017. This can be rationalised through our selection of 280 observations as
the window size. While this is too large a window to capture smaller bubble activity, it elimi-
nates the possibility of falsely identifying explosive behaviour through seasonal idiosyncratic
price fluctuations.
Conversely, the rolling ADF test empirically detects wild swings in Ethereum prices
throughout 2016. The interpretation of this empirical data is consistent with the economic
narrative that Ethereum was trading at $21.50, a 2000% increase YTD6
in June 2016, when
attacks on ‘Distributed Autonomous Organisation7
’ caused the price to face heavy head-
winds. A relatively smaller window size of 144 observations allows us to detect more in-
stances of explosiveness in Ethereum prices. The biggest spike in the rolling ADF statistics
(see Fig. 2(e)) relates to the Thanksgiving hard forking – Ethereum network splitting into
two transaction histories – following which, the price hit a 9-month low in December 2016
(Bovaird, 2016).
Ethereum, according to Goldman Sachs, has dwarfed Bitcoin and Dutch tulipmania to
become the biggest bubble in history (Hargreaves, 2018). Ethereum’s stratospheric price
rise by over 2,300% in 2017 was halted by a flash-crash when it’s price plunged by 99.9%
within seconds on 22nd June 2017. The flash-crash was triggered by a multi-million dollar
sell order on GDAX (a leading Ethereum exchange) which executed 800 stop-loss orders,
thereby creating a domino effect (Williams-Grut, 2017). Both, rolling ADF and reverse
recursive ADF statistics (Fig. 2(e) and 2(f)), successfully detect the occurrence of this crash
and the precursive price explosiveness at the 99% significance level.
Ethereum prices dramatically escalated to $1000 in January 2018 after a Russian bank
(under Vladimir Putin’s tutelage) embraced blockchain technology through striking a deal8
with Ethereum CEO. This dramatic price rise, September 2017 onwards, was a classic exam-
ple of highly speculative Ethereum trading. As illustrated in Fig. 2(e), rolling ADF statistics
successfully detect this bubble behaviour across the aforementioned timespan (at the 99%
critical level).
After reaching all-time highs, Bitcoin and Ethereum prices experienced a series of troughs
in January 2018 following Facebook and Google’s decision to ban cryptocurrency advertise-
ments (Williams-Grut, 2018). While it is common for financial assets to show explosive
behaviour following major announcements, cryptocurrencies (due to an absence of funda-
mental value) are more erratic in their behaviour. Momentum and signalling effects play
a pivotal role in determining the demand (and in turn, prices) for cryptocurrencies. For
6
year-to-date
7
The DAO is aimed to develop an Ethereum-based vehicle through which other projects in the ecosystem
could be crowdfunded (Bovaird, 2016).
8
The deal led to the creation of ‘Ethereum Russia’ through Ethereum forking
8

9. instance, the rallying prices of Bitcoin and Ethereum, in late 2017, were driven by positive
expectations originating from favourable developments in South Korea/Japan and Russia
respectively. Rolling (for Bitcoin) and reverse recursive (for Ethereum) ADF statistics trace
a time-period which roughly coincides with the crash in both cryptocurrencies.
This raises a bigger question – do Bitcoin and Ethereum persistently exhibit correlated
trends? Our empirical results for the last bubble ostensibly suggest a correlation between
Bitcoin and Ethereum. Rather, this implies a spurious relationship and it would be a fallacy
to deduce a strong correlation between the two. Antonopoulos (2017) and Toren (2017) noted
that Bitcoin’s relationship with Ethereum, unlike other cryptocurrencies, is underlined by
the fundamental technological differences between the two. Antonopoulos further stated that
while Ethereum and Bitcoin compete indirectly in the short-run, they operate in segmented
markets and do not compete in the long-run. Independent and unassociated global events
might steer simultaneous price movements in Bitcoin and Ethereum temporarily. However,
it is crucial to highlight the existence of low complementarity between the two.
6. Conclusion
It is beyond an iota of doubt that Bitcoin, along with other cryptocurrencies, is a series
of giant bubbles which is destined to end in grief (Krugman, 2018). We followed the right-
tail ADF approaches of recursive, rolling and reverse recursive testing – all of which have
been extensively developed in literature to detect explosiveness. Through rolling and reverse
recursive ADF tests, we successfully detected and compared multiple collapsing and end-
of-sample bubbles respectively. This enabled us to rationalise meteoric runs of Bitcoin and
Ethereum in 2017 which were halted by major announcements. Contextually, our results
ascertained ‘irrational exuberance’ in both cryptocurrencies. This reiterates their highly
speculative behaviour wherein the aggressive price swings are even more pronounced due to
the absence of a fundamental value.
Interestingly, a reverse rolling ADF test could pose as a possible extension to empirically
investigating cryptocurrency bubbles. This could potentially serve as a single foolproof test
that could deliver robust results in detecting periodically collapsing as well as end-of-sample
bubbles.
Lastly, we emphasize the existence of low complementarity between Bitcoin and Ethereum
which can often be eclipsed in the wake of sporadic global shocks that affect all cryptocur-
rencies alike, albeit transiently.
9

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