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arXiv:1803.05663v1 [econ.EM] 15 Mar 2018

Are Bitcoin Bubbles Predictable?

Combining a Generalized Metcalfe’s Law and the LPPLS Model

Spencer Wheatley1∗, Didier Sornette1,2∗, Tobias Huber1, Max Reppen3, and Robert N. Gantner

1ETH Zurich, Department of Management, Technology and Economics, Switzerland

2Swiss Finance Institute, c/o University of Geneva, Switzerland

3ETH Zurich, Department of Mathematics

e-mails: swheatley@ethz.ch, dsornette@ethz.ch

∗corresponding authors

March 16, 2018

Abstract

We develop a strong diagnostic for bubbles and crashes in bitcoin, by analyzing the coincidence

(and its absence) of fundamental and technical indicators. Using a generalized Metcalfe’s law based

on network properties, a fundamental value is quantiﬁed and shown to be heavily exceeded, on

at least four occasions, by bubbles that grow and burst. In these bubbles, we detect a universal

super-exponential unsustainable growth. We model this universal pattern with the Log-Periodic

Power Law Singularity (LPPLS) model, which parsimoniously captures diverse positive feedback

phenomena, such as herding and imitation. The LPPLS model is shown to provide an ex-ante

warning of market instabilities, quantifying a high crash hazard and probabilistic bracket of the crash

time consistent with the actual corrections; although, as always, the precise time and trigger (which

straw breaks the camel’s back) being exogenous and unpredictable. Looking forward, our analysis

identiﬁes a substantial but not unprecedented overvaluation in the price of bitcoin, suggesting many

months of volatile sideways bitcoin prices ahead (from the time of writing, March 2018).

1

1 Introduction

In 2008, pseudonymous Satoshi Nakamoto introduced the digital decentralized cryptocurrency,

bitcoin [1], and the innovative blockchain technology that underlies its peer-to-peer global payment

network1. Since its techno-libertarian beginnings, which envisioned bitcoin as an alternative to the

central banking system, bitcoin has experienced super-exponential growth. Fueled by the rise of bit-

coin, a myriad of other cryptocurrencies have erupted into the mainstream with a range of highly

disruptive use-cases foreseen. Cryptocurrencies have become an emerging asset class [3]. At the end

of 2017, the price of bitcoin peaked at almost 20’000 USD, and the combined market capitalization of

cryptocurrencies reached around 800 billion USD.

The explosive growth of bitcoin intensiﬁed debates about the cryptocurrency’s intrinsic or funda-

mental value. While many pundits have claimed that bitcoin is a scam and its value will eventually

fall to zero, others believe that further enormous growth and adoption await, often comparing to the

market capitalization of monetary assets, or stores of value. By comparing bitcoin to gold, an analogy

that is based on the digital scarcity that is built into the bitcoin protocol, some markets analysts pre-

dicted bitcoin prices as a high as 10 million per bitcoin [4]. Nobel laureate and bubble expert, Robert

Shiller, epitomized this ambiguity of bitcoin price predictions when he stated, at the 2018 Davos World

Economic Forum, that “bitcoin could be here for 100 years but it’s more likely to totally collapse” and,

“you just put an upper bound on [bitcoin] with the value of the world’s money supply. But that upper

bound is awfully big. So it can be anywhere between zero and there.” [5].

There is an emerging academic literature on cryptocurrency valuations [6, 7, 8, 9, 10, 11, 12, 13, 14]

and their growth mechanisms [15]. Many of these studies attribute some technical feature of the

bitcoin protocol, such as the “proof-of-work” system on which the bitcoin cryptocurrency is based, as

a source of value2. However, as has been proposed by former Wall Street analyst Tom Lee [4], an early

academic proposal (see Ref.[17]), by now widely discussed within cryptocurrency communities, is that

an alternative valuation of bitcoin can be based on its network of users. In the 1980s, Metcalfe proposed

that the value of a network is proportional to the square of the number of nodes [18]. This may also

be called the network eﬀect, and has been found to hold for many networked systems. If Metcalfe’s

law holds here, fundamental valuation of bitcoin may in fact be far easier than valuation of equities

3—which relies on various multiples, such as price-to-earnings, price-to-book, or price-to-cash-ﬂow

ratios—and will therefore admit an indication of bubbles.

Although it seems relatively obvious that bubbles exist within cryptocurrencies, it is not a straw

1In this network, transactions, which do not rely on an intermediary, are veriﬁed by network nodes and, through

cryptography, immutably recorded in a decentralized publicly distributed ledger [2].

2The question of what constitutes the value of money has preoccupied generations of thinkers. About 2050 years ago,

Aristotle was probably the ﬁrst to argue that money needed a high cost of production in order to make it valuable. In

other words, according to Aristotle, the larger the eﬀort to create new money, the more valuable it is. This was later

elaborated into the labor theory of value, starting with Adam Smith, David Ricardo, and becoming the central thesis

of Marxian economics. Nowadays, this concept is archaic and it is well understood that money is credit (see e.g., [16]).

It is thus puzzling that cryptocurrencies with proof-of-work designs, which aim at revolutionizing money and exchanges

between individuals, use a very old and obsolete concept that has been mostly abandoned in economics.

3See however Cauwels and Sornette [19, 20], who developed an original valuation method for social network ﬁrms

based on the economic value of the demographics of users, and were able to predict ex-ante the performance of companies

such as Facebook, Zynga and Groupon after their IPO’s.

2

man argument that, in ﬁnance and economics, ﬁnancial bubbles are often excluded based on market

eﬃciency rationalization4, which assume an unpredictable market price, for instance following a kind

of geometrical random walk (see e.g., [22]). In sharp contrast, Didier Sornette and co-workers claim

that bubbles exist and are ubiquitous. Moreover, they can be accurately described by a deterministic

nonlinear trend called the Log-Periodic Power Law Singularity (LPPLS) model, potentially with highly

persistent, but ultimately mean-reverting, errors. The LPPLS model combines two well documented

empirical and phenomenological features of bubbles (see [23] for a recent review):

1. the price exhibits a transient faster-than-exponential growth (i.e., where the growth rate itself is

growing)—resulting from positive feedbacks like herding [24]—that is modeled by a hyperbolic

power law with a singularity in ﬁnite time, i.e., endogenously approaching an inﬁnite value and

therefore necessitating a crash or correction before the singularity is reached;

2. it is also decorated with accelerating log-periodic volatility ﬂuctuations, embodying spirals of

competing expectations of higher returns (bullish) and an impending crash (bearish) [25, 26].

Such log-periodic ﬂuctuations are ubiquitous in complex systems with a hierarchical structure

and also appear spontaneously as a result of the interplay between (i) inertia, (ii) nonlinear

positive and (iii) nonlinear negative feedback loops [27].

The model thus characterizes a process in which, as speculative frenzy intensiﬁes, the bubble

matures towards its endogenous critical point, and becomes increasingly unstable, such that any small

disturbance can trigger a crash. This has been further formalized in the so-called JLS model where

the rate of return accelerates towards a singularity, compensated by the growing crash hazard rate

[25, 28], providing a generalized return-risk relationship. We emphasize that one should not focus on

the instantaneous and rather unpredictable trigger itself, but monitor the increasingly unstable state

of the bubbly market, and prepare for a correction.

Here, we combine—as a fundamental measure—a generalized Metcalfe’s law and—as a technical

measure—the LPPLS model, in order to diagnose bubbles in bitcoin. When both measures coincide,

this provides a convincing indication of a bubble and impending correction. If, in hindsight, such

signals are followed by a correction similar to that suggested, they provide compelling evidence that a

bubble and crash did indeed take place.

This paper is organized as follows. In the ﬁrst part, we document a generalized Metcalfe’s law

describing the growth of the population of active bitcoin users. We show that the generalized Metcalfe’s

law provides a support level, and that the ratio of market capitalization to “the Metcalfe value” gives

a relative valuation ratio. On this basis, we identify a current substantial but not unprecedented

overvaluation in the price of bitcoin. In the second part of the paper, we unearth a universal super-

exponential bubble signature in four bitcoin bubbles, which corresponds to the LPPLS model with a

reasonable range of parameters. The LPPLS model is shown to provide advance warning, in particular

with conﬁdence intervals for the critical bursting time based on proﬁle likelihood. An LPPLS ﬁtting

algorithm is presented, allowing for selection of the bubble start time, and oﬀering an interval for the

crash time, in a probabilistically sound way. We conclude the paper with a brief discussion.

4For instance, the Eﬃcient Market Hypothesis (EMH) assumes that prices quasi-instantaneously reﬂects all available

information. Thus, market crashes result from novel very negative information that gets incorporated into prices [21].

3

2 Fundamental value of bitcoin: active users & a generalized Met-

calfe’s law

Metcalfe’s law states that the value, in this case market capitalization (cap), of a network is,

p=eα0uβ0, β0= 2,(1)

where uis called the number of active users, imperfectly quantiﬁed by a proxy, being the number of

active addresses5. It is a single factor model for a fundamental valuation of bitcoin, and plausibly

for other cryptocurrencies. From Figure 1, we indeed see a surprisingly clear log-linear relationship.

Rather than taking Metcalfe’s law as a given, we estimate the relevant parameters by a log-linear

regression model, which we refer to as the (generalized) Metcalfe law,

ln(p) = α+βln(u) + ǫ. (2)

The result of this ﬁt, on 2’782 daily values, from 17-07-2010 to 26-02-2018, is a slope β= 1.69

(standard error 0.0076), intercept α= 1.51 (0.087), and coeﬃcient of determination R2= 0.956.

Forcing the exponent βto be equal to 2 would result in an intercept of −2.01 (0.018), but this

regression is signiﬁcantly worse than the above7. Further, a slope of 2 (or larger) is robustly rejected

on moving windows8. On this basis, it seems that the value 2 proposed by Metcalfe is too large, at

least for the bitcoin ecosystem.9

It should be noted that this regression severely violates the assumption that the errors be inde-

pendent and identically distributed, as there are persistent deviations from the regression line. This

statement deserves to be made in more salient terms: the residuals are in fact the bubbles and crashes!

This is the focus of the second part of this paper. Ignoring this egregious violation of the so-called

Gauss–Markov conditions is well known to give the false impression of precise parameter estimates.

Further, endogeneity is an issue, as the number of active users may determine market cap in the long

term, but large ﬂuctuations in market cap can also plausibly trigger ﬂuctuations in active users on

shorter time scales (see Figure 1). We address this by smoothing active users10, assuming that this will

5The data is collected from bitinfocharts.com. Limitations: It is diﬃcult to know the true number of active users, in

particular because a single user can have multiple addresses that, to an outsider, cannot be distinguished from addresses

belonging to multiple users. Moreover, bitcoin.org’s Developer Guide [29] discourages key reuse, advising that each key

should only be used for two transactions (to receive, then send), and that all change should be sent to a new address,

generated at the time of transaction (belonging to the sender). Depending on to what extent this advice is followed, this

measure is thus an unclear mix between the number of daily users and the number of daily transactions (their activity).

6Such high values are of limited value as one often obtains high coeﬃcients of determination when regressing unrelated

trending/non-stationary series onto each-other (so-called “spurious regression”). In this case, the causal link between

active users and market cap is assumed.

7An ANOVA/F-test comparing the two models gives a p-value of less than 10−16. Further, the calibrated value of the

slope, β= 1.69, with standard error 0.0076, is clearly far from Metcalfe’s value 2.

8On 83% of 1-year windows, the parameter βis less than 2, and on 75% of windows the parameter βis signiﬁcantly

less than 2, at level p= 0.05.

9Note, however, that the measure of uis overestimating the true number of daily users. It is possible that this does

aﬀect the precise value of the exponent β. On the other hand, it could provide an underestimate of the number of active

users if the typical user does not transact daily.

10This is done with the R library loess with 5 equivalent degrees of freedom.

4

Figure 1: Left panel: Scatterplot of the bitcoin market cap versus the number of active users, with

logarithmic scales. The points becomes darker as time progresses, and the three latest crashes are

indicated by colored points, and arrows indicating the size of the correction. The generalized Metcalfe

regression is given in solid red, and with slope forced to be 2 given by the dashed red line. Right panel:

Active users (rough black line), again in a logarithmic scale, as a function of time, with linear scale

inset. A scaled bitcoin market cap is overlaid with the grey line. The red and dashed yellow lines are

the nonlinear regression ﬁts of active users, ﬁtting on diﬀerent time windows.

average out the eﬀects of short term feedback of market cap onto active users. A multiplier eﬀect is

also a plausible consequence of this endogeneity: a jump in user activity causes an increment in market

cap, which triggers a (smaller) jump in user activity, feeding back into market cap, etc. Therefore, we

do not claim to isolate the eﬀect of a single increment in active users on market cap, and do not need

it. Finally, we omit formal tests for causality, given the plausibility of the general mechanism behind

Metcalfe’s law, as well as the very turbulent and only long-term adherence to it11 .

In view of these limitations, the generalized Metcalfe’s law here is still rather impressive, and

will be shown to be highly useful, despite its radical simplicity and uncertain parameter values. Of

course, one may add other variables to the regression, which further characterize the network, such as

degree of centralization, transaction costs, volume, etc. However, the actual volume (value of authentic

transactions) for instance is not only diﬃcult to know, but, in general ﬁnancial markets, is known to

be highly correlated with volatility, of which bubbles and bursts are the most formidable contributors,

and may therefore be too endogenous to soundly indicate a fundamental value. Therefore, the variable

‘active users’ is retained as the focal quantity.

Looking at Figure 1, a clear and important feature is the shrinking growth rate of active users

11The exponent value 2 in the standard Metcalfe’s law embodies the idea that the value of the network is proportional

to the total number of interactions or exchanges, which themselves scales as the total number of possible connections. In

other words, Metcalfe’s law assumes full connectivity between all users. This does not seem realistic. Our ﬁnding of a

smaller exponent β≈1 + 2/3 expresses a more sparsely connected network in which each user is on average linked to

∼N2/3other users in the total network of N users. For instance, for N=1 Million, a typical user is then connected to

“only” 10’000 other users, a more realistic ﬁgure.

5

which we model by a relatively ﬂexible ecological-type nonlinear regression,

ln(u) = a−be−ctd+ǫ, (3)

which saturates at a “carrying capacity”, u→eaas t→ ∞, and where the log transform stabilizes the

noise level. As in the case of the generalized Metcalfe regression, here there is clear structure in the

residuals, as feedback loops develop between the number of active users and price during speculative

bubbles. We opt to ﬁt the curve (3) by OLS (ordinary least squares) and treat it as a rough estimate:

Fitting from 2012-01-01 to 2018-02-2612 , the annual growth rate is expected to decrease over the next

ﬁve years from 35% to 21%, taking the expected level of active users from 0.79 Million currently to 2.60

Million in 2023 with 5% and 8% standard errors, respectively. Comparing with a ﬁt starting earlier, in

2010-10-2413, again a similarly decreasing growth rate is conﬁrmed, but with predictions for 2018 and

2023 respectively being 7% and 28% larger than predictions for the ﬁrst ﬁt. More generally, within the

sample, the ﬁtted curves are similar, but, beyond the sample, diﬀerences explode such that there are

4 orders of magnitude diﬀerence between the predicted carrying capacities. Here, model uncertainty

dominates uncertainty of estimated parameters. There is also likely to be some non-stationarity and

regime-shifts as the bitcoin network evolves and matures, contributing another level of uncertainty

in the long-term extrapolation of our models. Therefore, precise inference based on a single model—

notably omitting any limitation imposed by the physical bitcoin network—is misleading, and long-term

predictions eﬀectively meaningless. However, smoothing of past values is not problematic, and short

term projections may be reasonable.

Given the number of active users, and calibrations of the generalized Metcalfe’s law, which maps

to market cap, we can now compare the predicted market cap with the true one, as in Figure 2. Also,

using smoothed active users, the local endogeneities—where price drives active users—are assumed

to be averaged out. The OLS estimated regression, by deﬁnition, ﬁts the conditional mean, as is

apparent in Figure 2. Therefore, if bitcoin has evolved based on fundamental user growth with transient

overvaluations on top, then the OLS estimate will give an estimate in-between and thus above the

fundamental value. For this reason, support lines are also given, and—although their parameters are

chosen visually—they may give a sounder indication of fundamental value. In any case, the predicted

values for the market cap indicate a current over-valuation of at least four times. In particular, the

OLS ﬁt with parameters (1.51,1.69), the support line with (0,1.75), and the Metcalfe support line (-3,2)

suggest current values around 44, 22, and 33 billion USD, respectively, in contrast to the actual current

market cap of 170 billion USD. Further, assuming continued user growth in line with the regression of

active users starting in 2012, the end of 2018 Metcalfe predictions for the market cap are 77, 39, and

12Details of the ﬁt: The interval spanned by the natural log of the number of active users was transformed to (0,1)

by shifting by 9.483 and dividing by 4.46. The time span was also transformed to (0,1). The nonlinear regression was

then ﬁt by OLS, giving parameters and standard deviations a=1.72 (0.14), b= 1.76 (0.15), c=0.79 (0.09), d=0.70 (0.26).

Predicted values (transformed back to original scale) for the ﬁrst day of each year from 2018–2023 in Millions of active

users, and percentage standard error are 0.788 (0.05%), 1.06 (0.06%), 1.39 (0.07%), 1.75 (0.07%), 2.16 (0.074%), and 2.60

(0.08%). Finally, the estimated carrying capacity is 2.76 ×107with standard error of 86%.

13Doing the same as for the previous ﬁt, but starting from 2010-10-24, gives parameters: a= 2.86 (0.59)b=

3.03 (0.61), c = 0.46 (0.11), d = 0.40 (0.02), with predicted values for ﬁrst day of 2018–2023: 0.812, 1.14, 1.54, 2.04,

2.63, and 3.35 (Millions). The predicted growth rates over the next ﬁve years are 40%, 36%, 32%, 29%, and 27%. And a

massive carrying capacity of 9.39 ×1011 is predicted with 180% standard error.

6

106

108

1010

-5

0

5

10

2018

2016

2014

2012

Metcalfe Exponent on Window

Bitcoin Market Cap

Figure 2: Comparing bitcoin market cap (black line) with predicted market cap based on various

generalized Metcalfe regressions of active users. The rough red line is given by plugging the true

number of active users into the generalized Metcalfe regression shown in Figure 1, having OLS estimated

coeﬃcients (α, β) = (1.51,1.69). The remaining lines plug smoothed active users (non-parametric up

to 2018 and the nonlinear regression starting in 2012 to project beyond) into the generalized Metcalfe

formula with diﬀerent parameters: The smooth green line for the estimated coeﬃcients (1.51,1.69); the

orange dashed line is proposed as a “support line”, having coeﬃcients (0,1.75) speciﬁed by eye; the blue

dash-dotted line being a Metcalfe support line with coeﬃcients (-3,2). The grey line, plotted against

the right axis, is the exponent of the generalized Metcalfe regression onto smoothed active users on

a causal 60 day moving window (i.e., window on the previous 60 days). It is truncated to emphasize

ﬂuctuations around the value 2 (solid grey line).

64 billion USD respectively14, which is still less than half of the current market cap. These results are

found to be robust with regards to the chosen ﬁtting window15.

On this basis alone, the current market looks similar to that of early 2014, which was followed by

a year of sideways and downward movement. Some separate fundamental development would need to

exist to justify such high valuation, which we are unaware of.

14With standard errors already above 10% induced by estimated parameters, excluding additional prediction uncertainty

due to persistent ﬂuctuations of active users about the mean.

15Although the parameters vary depending on the ﬁtting window, even allowing for ﬁtting windows starting in 2016,

where one obtains a high exponent β(above 2.5), an overvaluation of about a factor of two is still indicated.

7

3 Bitcoin bubbles: universality of unsustainable growth?

3.1 Identiﬁcation and main properties of the four main bubbles

Using the generalized Metcalfe regression onto smoothed active users as well as its support lines,

one can identify in Figure 2 four main bubbles corresponding to the largest upward deviations of the

market cap from this estimated fundamental value. These four bubbles in market cap are highlighted

in Figure 3, and detailed in Table 1—in some cases exhibiting a 20 fold increase in less than 6 months!

In all cases, the burst of the bubble is attributed to fundamental events, listed below, in particular for

the ﬁrst three bubbles, which corrected rapidly at the time of the clearly relevant news. The fourth

and very recent bubble was much longer, and it is plausible that the main news there was really the

20’000 USD value of bitcoin, i.e., it ﬁnally collapsed under its own weight16. Market participants often

lament that crashes are unforeseeable due to the unpredictability of bad news.

1010

108

106

Bitcion Market Cap

2014 2016 2018

1

2

4

18-08-12

t0=0

Normalized Log Mcap

tc=1

3

4

1

3

11-04-13

23-11-13

18-12-17

2

Figure 3: Upper triangle: market cap of bitcoin with four major bubbles indicated by bold colored

lines, numbered, and with bursting date given. Lower triangle: The four bubbles scaled to have the

same log-height and length, with the same color coding as the upper, and with pure hyperbolic power

law and LPPLS models ﬁtted to the average of the four scaled bubbles, given in dashed and solid black,

respectively.

However, focusing on the news that may have triggered the crash is akin to waiting for “the

ﬁnal straw”, rather than monitoring the developing unsustainable load on the poor camel’s back. Of

16This large valuation is likely to have attracted “whales” to cash a part of their bitcoin portfolios, ei-

ther to realize their proﬁt or due to operational constraints. For instance, it was revealed on March

2, 2018 that Nobuaki Kobayashi, bankruptcy trustee for Mt. Gox, once the largest bitcoin exchange

in the world, has sold oﬀ about $400 million in bitcoin and bitcoin cash since late September 2017

(https://www.zerohedge.com/news/2018-03-07/bitcoins-tokyo-whale- sells-400m-bitcoin-bitcoin-cash).

8

Bubble Start End Days M-Cap0M-Cap1Growth Mean Return

1 2012-05-25 2012-08-18 84 4.65 ×1071.45 ×1083.1 0.013

22013-01-03 2013-04-11 98 1.39 ×1082.84 ×10920.4 0.031

32013-10-07 2013-11-23 47 1.45 ×1099.8×1096.8 0.042

42015-06-08 2017-12-18 924 3.17 ×1093.27 ×1011 103 0.005

52017-03-31 2017-12-18 155 1.69 ×1010 3.27 ×1011 21 0.02

Table 1: Bubble statistics. Columns: Start, end (time of peak value, prior to correction), duration in

days, starting and peak market cap, growth factor (peak divided by start value: M-Cap1/ M-Cap0),

and average daily return. The bubbles correspond to the numbering in Figure 3. Bubble 5 corresponds

to approximately the last six months of the fourth bubble, and will be used in the next section. The

price data for bitcoin is from Bitstamp, in USD, hourly from 2012-01-01 to 2018-01-08; the bitcoin

circulating supply comes from blockchain.info.

particular interest here is that, although the height and length of the bubbles vary considerably, when

scaled to have the same log-height and length, a near-universal super-exponential growth is evident, as

diagnosed by the overall upward curvature in this linear-logarithmic plot (lower Figure 3). And in this

sense, like a sandpile, once the scaled bubble becomes steep enough (angle of repose), it will avalanche,

while the precise triggering nudge is essentially irrelevant.

Below, events thought to trigger crashes/corrections, corresponding to bubbles 1–4 in Table 1 are

mentioned17:

0. 2011-06-1918: Mt. Gox was hacked, causing the bitcoin price to fall 88% over the next 3 months.

1. 2012-08-28: Ponzi fraud of perhaps hundreds of thousands of bitcoin under the name bitcoins

Savings & Trust; charges ﬁled by Securities and Exchange Commission.

2. 2013-04-10: Major bitcoin exchange, Mt Gox, breaks under high trading volume; price falls more

than 50% over next 2 days.

3. 2013-12-5: Following strong adoption growth in China, the People’s Bank of China bans ﬁnancial

institutions from using bitcoin; bitcoin market cap drops 50% over the next two weeks. 07-02-

2014: operational issues at major exchanges due to distributed denial of service attacks, and two

weeks later Mt Gox closes.

4. 2017-12-28: South Korean regulator threatens to shut down crypto currency exchanges.

3.2 Log-periodic ﬁnite time singularity model

Following Sornette and colleagues [25, 28, 30], as mentioned in the introduction, we consider bubbles

to be the result of unsustainable (faster than exponential) growth, achieving an inﬁnite return in ﬁnite

time (a ﬁnite time singularity), forcing a correction / change of regime in the real world. We adopt

the LPPLS model, as parameterized in [31], for the log market cap, piat time ti,

yi:= ln(pi) = a+ (tc−ti)mb+ccos wln(tc−ti)+dsin wln(tc−ti)+ǫi, ti(4)

17Events taken from https://99bitcoins.com/price-chart-history/

18This trigger is for the “zeroth” bubble, being before our data window.

9

where 0 < m < 1, ln(pc) = a, and T1≤ti< tc.T1is the starting time, and tcthe stopping or so-called

critical time by which the bubble must burst. This model combines two well documented empirical and

phenomenological features of bubbles: (1) a transient “faster-than-exponential” growth with singularity

at tc, modeled by a pure hyperbolic power law (the above equation with c=d= 0), resulting from

positive feedbacks, which is (2) decorated with accelerating periodic volatility ﬂuctuations, embodying

spirals of fear and crash expectations.

The model needs to be ﬁt with data ((y1, t1),...,(yn, tn)), on a window (T1, T2), where T1≤t1<

··· < tn≤T2< tc. The window (T1, T2) needs to be speciﬁed, with selection of the start of the bubble

T1often being less obvious. As is typical in time series regression [32], the errors ǫiare correlated and

may have changing variance (hetero-skedasticity), which if ignored leads to sub-optimal estimates, and

conﬁdence intervals that are too small (over-optimistic). In this case, generalized least squares (GLS)

provides a conventional solution, which has been used with LPPLS [33, 34, 35] and, if well-speciﬁed,

has optimal properties. Here, we opt for a simple speciﬁcation of the error model, being auto-regressive

of order 119,

ǫi=φǫi−1+ηi,|φ|<1,(5)

to model the rather persistent deviations from the overall trend. We then estimate the LPPLS model

by proﬁling over non-linear parameters (m, w, tc, φ), which allows the conditionally linear parameters

(a, b, c, d) to be estimated analytically, by GLS, or by OLS if φ= 0. Assuming white noise normal errors

ηi, this is maximum likelihood, and allows for proﬁle likelihood conﬁdence intervals of all parameters.

Here, we focus on tc, the critical time at which the bubble is most likely to burst. Before taking the

Metcalfe fundamental value into account, and to provide a curve to compare with the data in Figure 3,

we ﬁt the pure hyperbolic power law (obtained by putting c=d= 0 in (4)) and the LPPLS model to

the average of the four scaled bubbles20, with results summarized in Table 2. The hyperbolic power

law and LPPLS ﬁts provide a similar trend, and the forward-looking predicted critical/bursting time

hugs the lower bound of 1.01 (the true peak being by construction at 1).

Perhaps curiously—despite ﬁtting on an average of unsynchronized disparate bubbles with similar

overall trajectories—the LPPLS ﬁt is signiﬁcantly better, based on log-likelihood (p < 10−5) as it

captures some of the persistent ﬂuctuations, and allows for a signiﬁcantly smaller φ, i.e. a reduction

of the memory time ∼1/(1 −φ) of the residuals by a factor 1321 .

19Higher order ARMA models can also be considered, and are seen to leave residuals with little auto-correlation. Given

the regression based de-trending, truly long memory in the errors is not expected, and the auto-correlation of residuals is

seen to decay clearly faster than a power law. Further, Dickey-Fuller tests tend to reject that the residuals are unit-root,

strongly when signiﬁcant log periodic oscillations are ﬁt.

20These ﬁts contain future information, in the sense that the end time of each ﬁtted bubble is the time at which the

price peaked, which can only be determined after the crash occurred. These ﬁts are thus not for prediction purpose but

for assessing the quality of the hyperbolic power law versus LPPLS models.

21This suggests the existence of an intrinsic phase of the log-periodic oscillations with respect to the ﬁnite-time rounding

of the mathematical singularity at the market peak before the crash [36, 37].

10

a b c d ω m tcφ

2.00 -1.97 -0.020 0.013 10.79 0.23 1.03(1.01,1.06) 0.87

1.54 -1.52 =0 =0 NA 0.31 1.02(1.01,1.05) 0.99

Table 2: LPPLS (second row) and pure hyperbolic power law (c=d= 0) (third row) ﬁts on the

average of the four scaled bubbles shown in Figure 3. The sample is taken at 200 equidistant points.

The 95% proﬁle likelihood conﬁdence interval is given for tc.

3.3 Bubbles in the Market-to-Metcalfe Ratio

Given our proposed fundamental value of bitcoin based on the generalized Metcalfe regressions

presented above, we deﬁne the Market-to-Metcalfe value (MMV) ratio,

MMVi=pi

e−3u2

i

,(6)

as the actual market cap (piat time ti) divided by the market cap predicted by the Metcalfe support

level, with parameters (α0=−3, β0= 2) in (1), with smoothed active users (ui) plugged in22. We

sample the value every three hours over the time periods corresponding to bubbles 1–3 and 5 in Table

1.

As shown in Figure 4, bubbles are persistent deviations of the Market-to-Metcalfe value above

support level 1, which are well modelled by the LPPLS model. In particular, the parameters of the

hyperbolic power law and LPPLS models ﬁtted on the Market-to-Metcalfe ratio data, for the full bubble

lengths, are given in Table 4. For the diﬀerent bubbles, the key nonlinear parameters fall within similar

ranges, and calibration of tcis accurate. Again, the LPPLS ﬁts dominate the pure hyperbolic power

laws, according to likelihood ratios. Further, based on our methodology (see appendix), none of these

ﬁts can be rejected on the basis of their residuals.

22Note that whether the value β= 2 or β= 1.75 are used, the results for this analysis will be eﬀectively identical.

11

2012 2014 2016 2018

0 1

0 1 2

Normalized

et to Met atio

10

1

Figure 4: Left panel: Market-to-Metcalfe value ratio (MMV) over time. The apparent bubbles, which

radically depart from the fundamental level 1, are colored and given in Table 1 as bubbles 1–3 and

5. Right panel: for the same four bubbles, the MMV ratios are shown in log-scale as a function of

linear rescaled time, with 0.25 vertical oﬀset for visibility. The hyperbolic power law and LPPLS ﬁts

on the 4 full bubbles are shown. Values of the MMV ratio after the bubble peak are shown on the grey

background, where the colored vertical lines indicate the upper limit for tcof the 95% proﬁle likelihood

conﬁdence interval for each of the four bubbles. The three thin vertical black lines gives the rightmost

edge of the 95, 97.5, and 100% data windows on which ﬁts were done, with parameters summarized in

Table 3 and Appendix Table 5.

The ex-ante predictive aspect is important as, in addition to verifying the LPPLS bubble in hind-

sight, one would like to have a sound advance warning of the bubble’s existence and a reasonable

conﬁdence interval for its bursting time. Here, we provide a simple indication of this potential with

two additional sets of ﬁts: ﬁtting with bubble data up to 95% and 97.5% of the bubble length. The

overall parameter estimates (see appendix Table 6) are similar to the 100% window, in Table 4, with

key nonlinear parameters typically in ranges 0.1< m < 0.5, and 7 < w < 11. Focusing on the criti-

cal bursting time, in Table 3, the estimated tcand 95% conﬁdence intervals are given, showing quite

stable advance-warning. That is, point estimates and conﬁdence intervals are consistent with the true

bursting time, noting that tcis in theory both the most probable and latest time for the burst of the

bubble [25, 28, 30], as the market is increasingly susceptible as it approaches tc, and can therefore be

toppled by bad news.

12

Fit a b c d w m tcφ

1 2.74 -2.72 -0.051 -0.044 8.37 0.10 1.02 (1.01,1.09) 0.92

23.74 -3.73 -0.005 0.012 10.80 0.10 1.05 (1.03,1.06) 0.90

34.56 -4.53 -0.031 -0.013 8.97 0.10 1.09 (1.05,1.12) 0.92

51.09 -0.96 -0.071 0.053 12.00 0.38 1.01 (1.01,1.07) 0.98

1a 2.71 -2.68 =0 =0 NA 0.10 1.02 (1.01,1.04) 0.98

2a 2.13 -2.13 =0 =0 NA 0.18 1.04 (1.02,1.04) 0.99

3a 4.61 -4.59 =0 =0 NA 0.10 1.09 (1.05,1.23) 0.97

5a 1.046 -0.94 =0 =0 NA 0.43 1.00 (1.01,1.20) 0.99

Table 3: Estimated parameters of the LPPLS and hyperbolic power law models on the Market-to-

Metcalfe value ratios for the four bubbles, indicated by the ﬁt number. The suﬃx ‘a’ corresponds to

the hyperbolic power law ﬁts of the Market-to-Metcalfe value ratios for these four bubbles. The 95%

proﬁle likelihood conﬁdence interval for tcis given. The likelihood ratio test of the LPPLS versus the

hyperbolic power law (null) gives p-values of 0.01, 10−5, 0.02, and 0.07, for these four bubbles. A lower

bound for m of 0.1 was enforced.

Fit 0.95 0.975 1

1 0.99 (0.98,1.08) 1.01 (0.99,1.08) 1.02 (1.01,1.09)

20.99 (0.98,1.0) 1.07 (1.05,1.07) 1.05 (1.03,1.06)

31.02 (1.01,1.02) 1.07 (1.04,1.08) 1.09 (1.05,1.12)

50.97 (0.97,0.98) 0.98 (1.01,1.06) 1.00 (1.01,1.07)

1a 0.99 (0.97,1.4) 1.01 (0.98,1.32) 1.02 (1.01,1.04)

2a 1.00 (0.99,1.04) 1.06 (1.04,1.11) 1.04 (1.02,1.04)

3a 1.08 (1.0,1.4) 1.08 (1.01,1.25) 1.09 (1.05,1.23)

5a 0.95 (0.95,1.4) 0.98 (0.98,1.4) 1.00 (1.01,1.20)

Table 4: Estimated critical time and 95% conﬁdence interval, for LPPLS and hyperbolic power law

ﬁts of the Market-to-Metcalfe value ratios of the four bubbles, indicated by the ﬁt number and suﬃxed

with a, as deﬁned in Table 3. The three columns are for ﬁts on data up to T2, being 95, 97.5, and

100% of the bubble length, as indicated by bubbles 1–3 and 5 in Table 1.

4 Discussion

In this paper, we have combined a generalized Metcalfe’s law, providing a fundamental value based

on network characteristics, with the Log-periodic Power law Singularity (LPPLS) model, to develop

a rich diagnostic of bubbles and their crashes that have punctuated the cryptocurrency’s history. In

doing so, we were able to diagnose four distinct bubbles, being periods of high overvaluation and

LPPLS-like trajectories, which were followed by crashes or strong corrections. Although the height

and length of the bubbles vary substantially, we showed that, when scaled to the same log-height and

length, a near-universal super-exponential growth is documented. This is in radical contrast to the

view that crypto-markets follow a random walk and are essentially unpredictable.

Further, in addition to being able to identify bubbles in hindsight, given the consistent LPPLS

bubble characteristics and demonstrated advance warning potential, the LPPLS can be used to provide

ex-ante predictions. For instance, a reasonable conﬁdence interval for the endogeneous critical time

13

indicates a high hazard for correction in that neighborhood, as any minor event could topple the

unstable market. Of course, massive exogeneous shocks, although rare, could occur at any time, and

the LPPLS model can provide no warning there.

Focusing on the outlook for bitcoin, the active user data indicates a shrinking growth rate, which a

range of parameterizations of our generalized Metcalfe’s law translates into slowing growth in market

capitalization. Further, our Metcalfe-based analysis indicates current support levels for the bitcoin

market in the range of 22–44 billion USD, at least four times less than the current level. On this basis

alone, the current market resembles that of early 2014, which was followed by a year of sideways and

downward movement. Given the high correlation of cryptocurrencies, the short-term movements of

other cryptocurrencies are likely to be aﬀected by corrections in bitcoin (and vice-versa), regardless of

their own relative valuations.

14

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17

5 Appendix

97.5%

Fit a b c d w m tcφ

1 2.72 -2.72 -0.035 -0.062 7.76 0.10 1.01 (0.99,1.08) 0.92

23.88 -3.87 -0.004 0.013 11.39 0.10 1.07 (1.05,1.07) 0.90

34.32 -4.30 -0.035 -0.012 8.37 0.10 1.07 (1.04,1.08) 0.92

50.91 -0.79 -0.062 0.074 11.34 0.48 0.98 (1.01,1.06) 0.98

1a 2.62 -2.60 =0 =0 NA 0.10 1.01 (0.98,1.32) 0.96

2a 3.88 -3.87 =0 =0 NA 0.10 1.06 (1.04,1.11) 0.98

3a 4.58 -4.56 =0 =0 NA 0.10 1.08 (1.01,1.25) 0.97

5a 0.86 -0.77 =0 =0 NA 0.58 0.98 (0.98,1.4) 0.99

95%

Fit a b c d w m tcφ

1 2.53 -2.52 -0.059 -0.040 7.76 0.10 0.99 (0.98,1.08) 0.92

21.42 -1.44 0.013 0.007 10.79 0.26 0.99 (0.98,1.0) 0.90

32.96 -2.95 -0.004 0.043 10.18 0.13 1.02 (1.01,1.02) 0.93

50.81 -0.71 -0.054 0.085 11.40 0.57 0.97 (0.97,0.98) 0.96

1a 2.47 -2.45 =0 =0 NA 0.10 0.99 (0.97,1.4) 0.96

2a 1.54 -1.55 =0 =0 NA 0.25 1.00 (0.99,1.04) 0.98

3a 4.60 -4.58 =0 =0 NA 0.10 1.08 (1.0,1.4) 0.96

5a 0.75 -0.68 =0 =0 NA 0.70 0.95 (0.95,1.4) 0.99

Table 5: The same as Table 3, but ﬁts up to 95%, and 97.5% of the bubble length, rather than 100%.

The likelihood ratio test p-values of bubbles 1–3 and 5, with the pure hyperbolic power law ﬁt as the

null, for the ﬁrst sub-table are 0.05, 0. 0007, 0.01, and 0.02; and for the second sub-table, 0.05, <10−6,

0.0004, and 0.003.

5.1 LPPLS algorithm

A rough algorithm for ﬁtting LPPL is given, and illustrated with a data example in Figure 5.

Assumed are existence of a smooth trend in a window before the ﬁnite time singularity at tc, and that

a stationary time series model exists for the—often persistent—errors around that trend. It allows for

selection of a best window, giving the bubble starting time, T1, by a hypothesis test, and conﬁdence

intervals for the critical time tc, which are more realistic than if assuming iid errors.

Steps

1. Identify initial error model: Take broad window (T0, T2) thought to contain the bubble, T0<

T1< T2< tc. Fit the log market cap with a ﬂexible non-parametric curve to obtain an estimate

of the trend. We use the R loess library and Akaike Information Criterion (AIC) to select

degrees of freedom. Then ﬁt the error-model, here an AR(1) time series, onto the de-trended

data, giving an initial estimate for the GLS LPPLS estimation.

18

Figure 5: 2015–2018 bitcoin market cap bubble to serve as an illustration for the algorithm. Plotted

are four accepted pure power law regressions, with upper limit of the ﬁtting window T2placed 2 months

prior to the turning point, and with four diﬀerent values of the bubble starting time, T1. The orange

mode is the average of the four proﬁle likelihoods for tcfor the four ﬁts shown up to the 95% level,

bounded by the light grey bar, giving the 95% interval.

2. Characterize error variance: Bootstrap the residuals from step 1 and feed them through the ﬁtted

AR(1) to simulate errors, allowing for the distribution of the residual standard error on diﬀerent

window sizes to be approximated by Monte Carlo. Due to the autocorrelated errors, a chi-square

distribution will not be valid.

3. Fit LPPLS function by proﬁle-likelihood with GLS: Given a ﬁtting window (T1, T2), take a ﬁne

grid of nonlinear parameters (m, w, tc), and for each point do a GLS ﬁt with, in this case AR(1)

errors, initialized from step 1. A maximum likelihood implementation of this is given in R:gls,

and detailed in Ch. 5 of [38], which internally proﬁles over the AR(1) parameter. An iterative

re-weighting to estimate the AR(1) parameter is also an option. Then, take the ﬁt with the

highest log-likelihood of all ﬁts. One may use whatever numerical optimization algorithm, but

the grid search easily allows for proﬁle likelihoods to be computed.

4. Perform the ﬁt on many windows and choose the best: Here, varying bubble start T1, where T0<

T1< T2, repeat step 3. For each ﬁt, having sample size n, take the residual error, RSS/(n−p),

where p is the degrees of freedom of the LPPLS (take p= 7 as an upper bound), and RSS is

the residual sum of squares. Then compare this value with the distribution of residual errors

generated from step 2, possibly bootstrapping only from the ﬁtted window (T1, T2) rather than

the overall window (T0, T2) which may having unbalanced variance. Then for a single ﬁt, take

19

the ﬁt on the largest window that is not rejected. For robustness, one may also wish to consider

multiple non-rejected ﬁts. The same approach can be used to select T2, which although often

visually obvious, can then be identiﬁed in an objective automated way.

20