Metcalfe's law and log-period power laws in the cryptocurrencies market

In this paper the authors investigate the statistical properties of some cryptocurrencies by using three layers of analysis: alpha-stable distributions, Metcalfe’s law and the bubble behaviour through the LPPL modelling. The results show, in the medium to long-run, the validity of Metcalfe's law (the value of a network is proportional to the square of the number of connected users of the system) for the evaluation of cryptocurrencies; however, in the short-run, the validity of Metcalfe’s law for Bitcoin is questionable. According to the bidirectional causality between the price and the network size, the expected price increase is a driver for more investors to join the Bitcoin network, which may lead in the end to a super-exponential price growth, possibly due to a herding behaviour of investors. The authors then used LPPL models to capture the behaviour of cryptocurrencies exchange rates during an endogenous bubble and to predict the most probable time of the regime switching. The main conclusion of this paper is that Metcalfe’s law may be valid in the long-run, however in the short-run, on various data regimes, its validity is highly debatable.


Introduction
law for Bitcoin, proving the existence of a log-linear relationship between the market capitalization and a proxy for the number of network users (the number of unique addresses). If Metcalfe's law is valid for cryptocurrencies, then a significant correlation between the number of users and the market price should be present. If the correlation is also a causality (in one way or another), then there may be room for the occurrence of some herding behaviour: if the market is driven by expected future price increases, then more and more players will enter the market, causing the price to develop a bubble which will end eventually in a crash.
The herding behaviour has been studied extensively in relation to the cryptocurrencies market; for example, Bouri et al. (2018) found that there is direct relationship between the uncertainty level of the cryptocurrencies market and the probability of a crash. Cryptocurrencies market can be seen as an ecosystem driven by implicit herding: the smallest cryptocurrencies are herding with the largest ones, this being a signal of market inefficiency (Vidal-Tomás et al., 2018). The Bitcoin inefficiency was also documented by Urquhart (2016), who argues that Bitcoin's market is inefficient, but its dynamic may lead to efficiency. The Bitcoin market has experienced several crashes during its lifetime, the first one being in 2012, due to a Ponzi fraud involving Bitcoin. Another crash occurred in 2014, when Mt. Gox, a Bitcoin exchange handling over 70% of all Bitcoin transactions worldwide, closed its website and exchange service, and filed for bankruptcy protection from creditors; the value of Bitcoin then dropped by 50 percent in just two days. The most recent collapse, at the end of 2017, occurred after the intention of the South Korean regulators to shut down the cryptocurrencies exchange market.
As for the third area, LPPL (Log-Periodic Power Law) models are widely used to describe the behaviour of stock prices during an endogenous bubble and to predict the most probable time of the regime switching (see Johansen et al., 2000), as the aggregated behaviour of the investors is reflected in a log-periodic evolution of the trading price before the crash. As the industry of cryptocurrencies has grown exponentially over the past several years, there are many applications of the LPPL models to the study of this new market. Malhotra and Maloo (2014) investigated the evolution of Bitcoin exchange rates in 2013-2014, showing evidence of a super-exponential growth in Bitcoin exchange rates.
MacDonell (2014) used the LPPL model to forecast the Bitcoin price crash that took place on December 4 th , 2013, showing how the model can be a valuable tool for detecting bubble behaviour in digital currencies. Cheah and Fry (2015) used the LPPL models to test the presence of a bubble in Bitcoin prices before the price crash of December 2013 and they concluded that LPPL models are a valuable tool for understanding the bubble behaviour in digital currencies. More, they proved that the Bitcoins prices contain a considerable speculative component and the fundamental value of Bitcoin is zero. These conclusions are augmented by the findings from Fry and Cheah (2016), who showed that the cryptocurrencies market is susceptible to develop speculative bubbles. Wheatley et al. (2018) have also used a variant of the LPPL model to estimate the most probable time of the crash for the 2017 Bitcoin bubble.
In this study we are solely focusing on applying three major statistical methods for studying the behaviour of cryptocurrencies market.
First, we are using the alpha-stable distributions to emphasize the heavy-tails property of the distribution of cryptocurrencies daily log-returns.
Second, we employ the generalized Metcalfe's law for the most important cryptocurrency, the Bitcoin, in order to understand the relationship between the Bitcoin's price, Bitcoin's market capitalization and the number of network users, deriving from there the potential for herding behaviour.
Third, we use the LPPL model to fit the bubble dynamics for the Bitcoin and for one major cryptocurrencies index, the CRIX index, showing the value of log-periodic power laws in anticipating the regime switching.
In the light of the findings from the literature, our contribution to the research on the statistical analysis of the cryptocurrencies is mostly empirical. By using the alpha-stable distributions to emphasize the heavy-tails property of the distribution of cryptocurrencies daily log-returns, our results extend the findings from Zhang et al. (2018), where the presence of heavy tails for cryptocurrencies is highlighted by using the Hill method to estimate the tail index.
By estimating the generalized Metcalfe's law in order to understand the relationship between the Bitcoin's price, our paper extends the results from Wheatley et al. (2018): precisely, we find reverse causality to Metcalfe's law -with price causing users growth. As a consequence, the potential for herding behaviour is detected, we apply the LPPL models to detect the most probable time of regime switching, in case of the CRIX index and of the Bitcoin.
The most important contribution of this paper is the fact that on the long term Metcalfe's law tends to be valid for Bitcoin; however, on the short term, its validity is questionable. More, we found to be valid another version of Metcalfe's law, where the number of users influence the transaction price; also, there seems to be a bivariate causality between the number of users and the Bitcoins price.
The rest of the paper is organized as follows: Section 2 outlines the methodology; Section 3 presents the dataset and the empirical results and Section 4 concludes.

Methodology
As previously mentioned, the methodology used in this paper has three layers: first, we study the statistical properties of the daily log-returns of the selected cryptocurrencies and we estimate the parameters of alpha-stable distributions, in order to derive their propensity for non-Gaussianity and heavy tails behaviour. Additionally, we investigate the validity of Metcalfe's law for the most popular cryptocurrency, Bitcoin, showing the existence of a potential for herding behaviour.
Subsequently, we apply the Log-Periodic Power Law models (Johansen et al., 2000) to identify the bubble regime in Bitcoin prices and in the evolution of the CRyptocurrency IndeX (CRIX).
The reason for applying these three different techniques lies on the following argumentation: if through the alpha-stable modelling, the cryptocurrencies market turns out to be far from Gaussianity, then the propensity for extreme returns is in line with the speculative behaviour of cryptocurrencies market, leading eventually to a crash. The speculative behaviour, which may also be caused by herding, can be tested using LPPL models. At the same time, the herding behaviour may be derived from a modified version of Metcalfe's law, where the price increase is an incentive for more investors to join the market, driven by high expected returns.

Stable distributions
In order to characterize the tail behaviour of the cryptocurrencies we fit the distribution of daily log-returns through the alpha-stable approach. A random variable X follows an alpha-stable distribution ( , , , ;0) S α β γ δ if its characteristic function has the form (Nolan, 2011): is the stability index, controlling for probability in the tails (for Gaussian distribution is the scale parameter and δ ∈R is the location parameter. The tail behaviour of the stable distributions is driven by the values of stability index α : small values are associated to higher probabilities in the tails of the distribution. In this paper we are using a regression-based method for estimating the parameters of an alpha-stable distribution (Kogon and Williams, 1998). This method is implemented as a SAS macro in Pele (2014) and can be used to obtain estimates for the parameters of stable distributions (see the Appendix A).

Metcalfe's law
In the 1980s, Robert Metcalfe, the co-inventor of Ethernet, stated what was called later Metcalfe's law (Gilder 1993): the value of a network is proportional to the square of the size of the number of connected users. Metcalfe's law was validated in various contexts, by using social network data: Zhang et al. (2015) proved the validity of the law for Facebook and Tencent (Chinese social network). Other researchers (Madureira et al., 2013, Van Hove, 2014, Metcalfe, 2013 have shown the validity of the law, mostly regarding internet networks. Peterson (2018) showed that Metcalfe's law can be used to explain the evolution of the Bitcoin transaction price, by using factors relating to supply (number of Bitcoins) and demand (number of wallets). More, Van Vliet (2018) showed the validity of a modified Metcalfe's law for Bitcoin, incorporating the logistic diffusion of the innovation. In this study we are using the Metcalfe's law version from Wheatley et al. (2018): where: t C is the Bitcoin's market capitalization at time t; Economics: The Open-Access, Open-Assessment E-Journal 13 (2019-29) www.economics-ejournal.org 6 t u is the Bitcoin's number of unique addresses at time t.
If Metcalfe's law is valid for Bitcoin, then the coefficient b 1 =2; we are testing the Equation (3) over the entire sample and by using a rolling window approach.
In addition to the classical form of Metcalfe's law, we are testing the hypothesis that the Bitcoin's price itself is driven by the Bitcoin's network size, showing potential for some herding behaviour. Also, by using cointegration analysis and Granger causality, we infer that the expected price increase is a driver for more investors to join the Bitcoin network, which may lead in the end to a super-exponential price growth, due to the herding behaviour of investors.

Log-periodic power laws (LPPL)
According to the field theory (Goldenfeld, 1992), an imitative process can be described through and c t being the critical time, so the price dynamics prior to the crash should be As the crash probability should be compensated by larger price changes, prior to the stock market crash (Blanchard, 1979), the hazard rate could be expressed via the Ising model:  . Thus, the trading price before the crash follows a log-periodic power law (Johansen et al., 2000): where p(t) is the price at moment t, c t is the critical time (the most probable moment of the crash), and 0 1 , , , , b B B ω ϕ are the parameters of the model which give its log-periodic feature. In order to have a proper specification of the model, there are several constrains applied to the parameters: A>0, B<0, . We are using the LPPL models to test the propensity for herding behaviour in the case of Bitcoin and the CRIX Index, following the methodology used in Fantazzini et al. (2016), who applied the LPPL modelling to Bitcoin exchange rates, finding evidence of explosive behaviour in the Bitcoin-USD exchange rates during August -October 2012 and November, 2013 -February, 2014.

3
Empirical results

Dataset and alpha-stable modelling of log-returns
The dataset presented in the Table 1 consists of daily cryptocurrency data (transaction count, on-chain transaction volume, value of created coins, price, market capitalization and exchange volume). 3 One market index was also used for the analysis: Cryptocurrency Index 4 as a reference for the cryptocurrencies market (Trimborn and Härdle, 2018).
The dataset deals only with cryptocurrencies for which at least 2 years of daily transaction data (at least 500 daily observations) were available at the moment of the data collection (October 2 nd , 2018). For the purpose of data analysis, the statistical software SAS 9.3 was used.  (Pele, 2014) was applied and the results are presented in the Table 2.
As depicted in Table 2 and Figure 1, in most of the cases, all the analysed cryptocurrencies exhibit large departures from normality, the values of the stability index α being significantly lower than 2, the value that corresponds to the Gaussian distribution. Figure 2 shows the correlation between the scale parameters γ (the equivalent of the volatility in the classical approach) and the stability index α, controlling for the tail probability. www.economics-ejournal.org 9 Figure 1: Distribution of the stability index α for log-returns distribution of the selected assets Based on this correspondence, we are able to cluster the selected cryptocurrencies based on their propensity to heavy-tails and the likelihood of high volatility.\ For example, the cryptocurrency Theter (USDT) has the lowest stability index α (large departure from normality), but the scale parameter is low, so USDT is placed in the orange zone. The closest to the normal distribution is Aragon (ANT), yet his scale parameter is around the sample average, so it is placed in the yellow zone. As the cryptocurrencies market turns out to be far from Gaussianity, this type of assets exhibits a high propensity for extreme returns.

Metcalfe's law for Bitcoin
In order to evaluate the applicability of Metcalfe's law for cryptocurrencies, we limit the research to the most known and traded cryptocurrency, the Bitcoin, also due to the availability of transaction and network data. 5 As stated in the original formulation of Metcalfe's law, the value of the network should be proportional to the squared number of network users; however, in the case of cryptocurrencies, the actual number of users is unknown and we need to use a proxy for it, i.e. the number of unique addresses. Unique addresses in the Bitcoin ecosystem are the payment addresses that have a non-zero balance; this metric can be used as a proxy for the number of network users, although we cannot assume that the number of users is equal to the number of unique addresses. The number of unique addresses is not constant over time: when fees are high, investors leave their cryptocurrencies in multiple addresses, because a consolidation into a single address will require a high cost. When fees are low, investors can consolidate their funds into a single address. As the Bitcoin network grows, the number of unique addresses will also grow over time, but when the market is going down, less unique addresses are in use as the number of transactions decreases, as seen in Figure 3.

Metcalfe's law for the entire data sample
In this section, we are estimating the generalized Metcalfe's law, which is a log-linearization of Equation (2): where: t C is the Bitcoin's market capitalization at time t; t u is the number of unique Bitcoin addresses used at time t.
The estimation results for Equation (4) are shown in the Table 3, using daily data for the period 2010/08/24 -2018/10/05.   Although the slope of the Equation (4) is b 1 =1.696, below the theoretical value of 2, the model has a high explanatory power ( 2 0.924 adj R = ), supporting the validity of Metcalfe's law for Bitcoin.
The log-linear relationship between the Bitcoin's market capitalization and the number of unique Bitcoin addresses used is illustrated in Figure 4.
From the validity of Metcalfe's law for Bitcoin one can infer the existence of a possible herding effect: as an increase of the number of users is reflected in an increase of the market capitalization, this may be explained by the fact that there is a mimetic effect among users, making the price to have an ascending trend.
One insight into this direction can be found by estimating the generalized Metcalfe's law for the Bitcoin's price: The estimation results of Equation (5) are shown in the Table 4.  The results of the estimation show that there is strong log-linear relationship between the Bitcoin's market price and the number of unique addresses, as a proxy for the number of Bitcoin's network users (see also Figure 5); moreover, the price increase may be a direct effect of the increasing network size, through a possible mimetic behaviour.

Metcalfe's law on rolling windows
The validity of Metcalfe's law for Bitcoin's market capitalization is questionable: as shown below, due to the different regimes, one will obtain very different parameter estimates by fitting the model on different sub-windows of the data.
For w the length of a rolling window, we estimated the following model: where and T is the number of observations in the sample. 6 The results are presented in the succession of graphs from Figure 6, which depicts the values of the Adjusted R-squared for various rolling windows; there are periods when the explanatory power of the model is close to, or higher than 90%, but there also situations when the number of network users has no explanatory power on the market capitalization of Bitcoin. Figure 7 shows the estimated values of the slope coefficient from the Equation (6), which, according to the classical formulation of Metcalfe's law, should be equal to 2. However, there is a huge volatility in the evolution of this coefficient, its average values being significantly lower than 1, for the rolling windows of 60, 90 and 250 days and significantly lower than 2 for the 500 days rolling window. _________________________ 6 The code used for estimation can be found here: https://github.com/danpele/Stat_fin_markets/tree/master/SFM_Metcalfe . It can be noticed that the R-squared of 500-days rolling window is very close to zero around 2015 and the slope of the Equation (6) for 250-days and 500-days rolling window is lower than -1 around 2015. During 2014 and 2015, the relationship between the two variables from the Equation (6) was not linear, as seen in Figure 8.
Overall, this analysis shows that there is clear pattern of inconsistency over time, questioning the validity of Metcalfe's law for Bitcoin, when considering different sub-windows of the data.

Granger causality and cointegration between the Bitcoin's price and the network size
Granger causality and cointegration analysis has been previously applied to analyse the correlation of the Bitcoin price to macroeconomic indicators (see, for example, Zhu et al., 2017). Shen et al. (2019) are using Granger causality to argue that Twitter activity is a significant driver of the next day trading volume and realized volatility of Bitcoin. Going deeper with the analysis, we have also performed a Granger causality test in order to detect the existence of the causal links between the Bitcoin's price and the number of unique addresses. We consider the two-dimensional vector ' (ln , ln ) where P t is the Bitcoin's price and u t is the number of unique addresses.
As shown in Table 5, these time series are nonstationary (according to the Augmented Dickey Fuller -ADF test) and integrated of order one. Table 6 presents the result of the Johansen test; as we can see, there are two cointegration equations at the significance level of 0.05. Thus, we can draw a conclusion that there exists a long-term dynamic equilibrium between the Bitcoin price and the Bitcoin's network size.  As these time series are not stationary and both of them are integrated of order one, in order to test for Granger causality, the Toda-Yamamoto (1995) procedure was applied, by using the steps below: a. Test the two time-series to determine their order of integration. b. Let the m=1 the maximum order of integration for the group of the two time-series. c. Estimate a VAR model in level. d. Determine the appropriate maximum lag length (p) for the variables in the VAR, using the AIC, criterion. e. Check and correct for serial correlation in the residuals. f. Test for cointegration of the two time-series. g. Estimate the VAR(p+m) model and test the Granger causality using the Block Exogeneity Wald Test.
Based on the Granger causality tests (see Table 7), one can deduce the existence of a unidirectional causal relationship from the Bitcoin's prices to the size of the network, expressed as the number of unique addresses. The temporal dependency can be captured via a Vector Autoregressive (VAR (p)) model, of the following form: One can note from the Table 8 that the past realizations of the Bitcoin's price can be used to forecast the future realizations of the network size. For example, if at time t-1 the Bitcoin's price increases by 1%, then at time t one can expect a 0.212% increase of the number of unique addresses. Moreover, the behaviour of the impulse response function offers an indication that a shock from the Bitcoin's price have a positive effect on the network size, and the effect is permanent and significantly different from zero (see Figure 9).
One can infer from this analysis that the expected price increase is a driver for more investors to join the Bitcoin network, which may lead in the end to a super-exponential price growth, due to a herding behaviour of investors. Note: the optimum number of lags (15) was chosen based on the lag length criteria from VAR specification.

LPPL models
According to the bidirectional causality between the price and the network size, the expected price increase is a driver for more investors to join the Bitcoin network, which may lead in the end to a super-exponential price growth, possibly due to a herding behaviour of investors. This super-exponential price growth can be modelled using the LPPL approach. In order to capture the bubble regime and to estimate the most probable time of the crash, the algorithm from Pele (2012), using price gyrations and peak detection was applied.

Numerical results for Bitcoin
In case of Bitcoin, the regime switching was recorded in December 2017, the exchange rate hitting a local maximum on December 19 th , 2017. The initial sample for fitting LPPL model in the case of Bitcoin for predicting the phase transition from December 2017 was 1 Jan 2016 -30 Nov 2017 (700 daily observations). Starting from the last observation in the initial sample, we extended the sample using an expanding window, so we estimated at every step the LPPL model for t ∈[1, T+k], k=1…17: .00 .02 .04 .06 .08 .10 . In Equation (7), T corresponds to 11/30/2017, the model being estimated k=17 times, by increasing the sample by one day, at every iteration. The best results of the LPPL models, 7 in terms of minimizing the RMSE, are given in the Table 9.
As a result of the estimation, three models were kept, with the best Root Minimum Squared Error (RMSE). The model with the minimum RMSE anticipated on December 11 th 2017 an imminent crash for the next day, as seen in Figure 10. The other two selected models offer close predictions, for December 2 nd 2017 and December 7 th 2017.

Numerical results for the CRIX Index
The local maximum for the CRIX index was recorded on January 7 th 2018, this being the moment of the regime switching. The initial sample for fitting LPPL model in the case of the CRIX index for predicting the phase transition from January 2018 was 1 Jan 2016 -15 Dec 2017 (716 daily observations). Starting from the last observation in the initial sample, we extended the sample by using an expanding window, so we estimated at every step the LPPL model for t ∈[1, T+k], k=1…20: ; ; In Equation (8), T corresponds to 12/15/2017, the model being estimated k=20 times, by increasing the sample by one day, at every iteration. 8 The best results of the LPPL models, in terms of minimizing the RMSE, are given in the  Table 10. The best fit for the CRIX index was given by the model estimated for the period January 1 st 2016 -December 30 th 2017, for which the estimated critical time was exactly the date of local maximum, January 7 th 2018 (see Figure 11).
In Figure 10 and Figure 11, it seems like that LPPL mode do not reproduce the characteristic oscillations observed in the actual data of BITCOIN price and CRIX index. This may be explained by the fact that the parameter C (Equation (4)), controlling the magnitude of oscillations around the exponential trend, is very close to 0 in our case.

Conclusions
Our paper deals with a new class of assets, cryptocurrencies, from the point of view of their statistical properties and behaviour. One of the main findings is that the daily cryptocurrencies log-returns exhibits large departures from normality, leaving room for high uncertainty levels, as shown the estimated stability indexes of stable distributions.
Moreover, by analysing Bitcoin related data, we prove, in the medium to long-run, the validity of Metcalfe's law for the evaluation of cryptocurrencies; however, in the short-run, the validity of Metcalfe's law for Bitcoin is questionable. This analysis shows that there is clear pattern of inconsistency over time, questioning the validity of the Metcalfe's law for Bitcoin, when considering different sub-windows of the data.
As there is a strong correlation between the size of the network and the market price of cryptocurrencies, this may be a sign for a mimetic behaviour of investors, who enter the market driven by high expected currency rates, which may lead the market into a super-exponential bubble regime. LPPL models could be useful in estimating the most probable time of the regime switching for an endogenous cryptocurrency bubble. By analysing the behaviour of the Bitcoin's price and the CRIX index, we have proven that LPPL models can be a useful tool in recognizing and mapping out the behaviour of a developing bubble. This is a validation of the predictive power of LPPL models in detecting the imitative behaviour of investors in the cryptocurrencies market, our results being useful both from a theoretical point of view and from a business perspective. At the same time, this validation of the LPPL models is another argument questioning the universal validity of Metcalfe's law for Bitcoin, when modelling the relationship between the price and the number of network users.
The econometric method to determine causality is based on cointegration, which requires that the residuals between the linear regression of the log-price onto the log number of users should be stationary (integrated of order 0) -i.e., controlling for the long run Metcalfe law equilibrium, the relative movement of the two series cointegrated series will be stationary. This stationarity assumption contradicts/excludes the existence of LPPL bubbles as well as different market regimes observed within the price series, studied in part 3 -which are radically nonstationary.
The main conclusion of the paper is that Metcalfe's law may be valid in the long-run, however in the short-run, on various data regimes, its validity is debatable.

A.1. Estimating the parameters of an alpha-stable distribution using McCulloch method
McCulloch method (1986) involves the following steps for estimating the parameters of a