Efficiency in the Markets of Crypto-Currencies

We show that the level of market-efficiency in the five largest cryptocurrencies is highly time-varying. Specifically, before 2017, cryptocurrency-markets are mostly inefficient. This corroborates recent results on the matter. However, the cryptocurrency-markets become more efficient over time in the period 2017-2019. This contradicts other, more recent, results on the matter. One reason is that we apply a longer sample than previous studies. Another important reason is that we apply a robust measure of efficiency, being directly able to determine if the efficiency is significant or not. On average, Litecoin is the most efficient cryptocurrency, and Ripple being the least efficient cryptocurrency.


Introduction
In this paper we analyze the market efficiency of the five largest cryptocurrencies 1 . We find that the markets for these five currencies are currently mostly efficient, but has been significantly inefficient in the past. The market for cryptocurrencies has received much attention the last three years, both from regulators, the public, and traders. The trading volume in the largest such currencies has grown exponentially, and with this increase in liquidity, the prices increased as well. We analyze how this increased interest and trading volume affects the efficiency of the cryptocurrency markets. We find that the efficiency increases significantly during 2017 and remains efficient midway through 2018. Our results also shows that these markets are sensitive to various events. For example, in June 2016 the DAO hack leads to a separation of Ethereum into Ethereum and Ethereum Classic, which caused increased uncertainty in the market. The level of efficiency dropped significantly following this event. The markets also stayed highly inefficient for several months, before stabilizing at weakly inefficient in late 2016 and early 2017.
Market efficiency has received much attention since Fama (1970) and the follow-up paper by Fama (1991). In the papers, the Efficient Market Hypothesis (EMH) is introduced and and the author sorts the efficiency of the market into three segments: strong efficiency, semi-strong efficiency, and weak efficiency. Furthermore, the author argues that financial markets are, to a large extent, strongly efficient. This implies that all available information is reflected in the price of the security. The challenge was for a long time to quantify market efficiency. Lo and MacKinlay (1989) proposed a method to test if markets are efficient or not. Furthermore, Lo (2004) proposed an alternative to the static view of market efficiency, proposing that the efficiency evolves over time. This is denoted the Adaptive Market Hypothesis, (AMH). The papers by Urquhart and Hudson (2013), Ito et al. (2014), 1 The size is measured by market capitalization as of Feb 28th, 2019. The currencies includes Bitcoin (BTC), Ethereum(ETH), Ripple (XRP), Litecoin (LTC), EOS (EOS). The market capitalization varies substantially over time, so other currencies might be larger in other periods. Noda (2016), Ito et al. (2016), and Urquhart and McGroarty (2016) investigates the market efficiency with methods derived with the AMH in mind. Furthermore, Chu et al. (2019) investigates the AMH for the two largest cryptocurrencies, and find evidence that supports the hypothesis of a time-varying market efficiency. However, some specific measures of market efficiency has potential challenges. For example, the efficiency estimator can be given as a fraction, where the denominator can be zero, or fluctuate between positive, zero, and negative. This can lead to discontinuities in the estimates, yielding unreliable estimates, and implies crucial challenges for applications in testing market efficiency. We apply a novel measure of the level of market efficiency, derived in Tran and Leirvik (2019). This measure allows for any values for both nominator and denominator. The estimator is continuous for all input parameter values.
The cryptocurrency market, and in particular the market for Bitcoin, is found to be largely inefficient, see, for example, Urquhart (2016), Vidal-Tomás and Ibañez (2018) spill-over, volatility co-movement, lead-lag effect, market co-movement. The systematic risks involved in such markets are also thoroughly investigated in Corbet et al. (2019). One reason for risks and inefficiencies can be that the markets has been difficult to trade in, and hence liquidity has been relatively low compared to other markets. The ease of trading one cryptocurrency can be significantly different from the ease of trading another such currency, thus the liquidity in various such currencies varies substantially, see Phillip et al. (2018).
Liquidity and market efficiency is closely related, and different markets show different levels of liquidity, see for example Amihud (2002), Chordia et al. (2008), Leirvik et al. (2017), Wei (2018, Brauneis and Mestel (2018), and de la Horra et al. (2019). In this paper we show that the level of market efficiency is varying over both time and individual currencies. In particular, we show that the Bitcoin market is largely inefficient until early 2017. In contrast with other cited studies, we find that Bitcoin becomes significantly efficient after 2017. The other currencies under investigation shows similar time-varying patterns.

Data
The markets for cryptocurrencies is relatively new, and our sample covers the period April 29th, 2013 through February 28th, 2019. Bitcoin, Litecoin and Ripple enter the sample from the very beginning (2013). Some of the currencies has been developed after April 29th, 2013 (Ethereum from 2015; EOS from 2017), but has shown to rely on a solid technology compared to other currencies, and has quickly become some of the largest currencies by market capitalization. We apply freely available data from Coinmarketcap.com, and import all available data via a statistical package named "crypto" in the software R. The data is at daily frequency which contains open, high, low, close prices, volume, and market capitalization 2 . Table (1) shows the descriptive statistics for the simple returns of the five currencies we analyze. We use simple returns because log-returns might give unreliable estimates for assets with extremely high volatility. In fact, for our sample the minimum daily log-return is -130.2% 3 . This is clearly not economically sound. To eliminate the chance of using uneconomic reasonable estimates for returns, we exclusively use simple returns as inputs to our calculations. 3 Reader can find the summary statistics using log return in the appendix. Not surprisingly, there is a significant variation between the various cryptocurrencies.
The differences seems to be very heterogeneous. Figure  media. We will analyze whether the prices can be considered efficient, and whether the level of efficiency varies over time.

Methodology
To estimate the level of efficiency, we apply a recently derived method to quantify the level of market efficiency, see Tran and Leirvik (2019). In this paper, the authors derive a measure for the level of Adjusted Market Inefficiency Magnitude (AMIM). In short, to compute the AMIM, we start with representing the returns of a currency as If markets are efficient then the coefficients (β 1 , β 2 ,. . . , β q ) should be zero, or at least insignificantly different to zero. If not, the β coefficients are (significantly) non-zero. Lo (2004) used the first auto-regressive coefficient to characterize the inefficiency level. One can argue that the Efficient Market Hypothesis (EMH) is based on a random walk or martingale dynamics of the price or the log-price. This has a direct implication in which the future price differences and log-price differences (log-returns) cannot be predicted. In this study, we mainly use the simple return, not log-returns. That can potentially be problematic.
However, it turns out that if price follows a RW process then the future simple return cannot be predicted either 4 . Hence a regression of simple return on its lag should yield a non-significant coefficient if the market is efficient.
We estimate Equation (1) (2016) and Ito et al. (2016). Specifically, the M IM is given by: As Equation (2) sums up the standardized auto-regression coefficients of Equation (1), it should be statistically equal to zero in a strongly efficient market. To reduce the impact of 4 Consider a RW process: y t+1 = y t + ε t+1 . Where ε t+1 is a shock at time t + 1 in the future which cannot be predicted, hence E[ε t+1 ] = E t [ε t+1 ] = 0, and ε t+1 is independent with y t . The simple return at time t+1 is : r t+1 = εt+1 yt . We will show that E[r t+1 ] = E t [r t+1 ] = 0, which means both the conditional and unconditional expectation of simple returns cannot be predicted. First, it is clear that in addition ε t+1 is independent with y t hence we can believe that cov (E t [ε t+1 ] , 1/y t ) = 0. Therefore E [r t+1 ] = 0. insignificant parameter estimates, we subtract the range of the confidence interval under the null hypothesis of efficient market from the M IM and divide by one minus the range of the confidence interval under the null hypothesis of efficient market. We call this the AM IM .
The measure is thus robust against insignificant autocorrelation. In short, we estimate the AM IM for any financial asset price by the estimator The R CI is the range of the confidence interval for the M IM under the null hypothesis of efficient market. For further explanations and derivations, the reader is encouraged to read the article by Tran and Leirvik (2019).
Because the M IM is constrained between zero and one, the R CI < 1. Thus, Equation (2, and 3) makes sure that both M IM and AM IM are continuous functions. Accounting for equation (3), the AM IM t cannot be larger than one. It can, however, be zero, or negative.
A positive value, i.e. AM IM t > 0, indicates an inefficient market. If AM IM t is less than zero, i.e. AM IM t ≤ 0, then the market is efficient. Hence, the measure is simple to compute, and very easy to interpret. In addition, it is also very easy to use AM IM to compare the level of efficiency for different assets in different points in time. Another quality of AM IM that we want to exploit is that it reflects very well economic events influencing the assets.

Empirical Results
Table (2) shows some descriptive statistics of the estimation of Equation (3). In our study we have applied daily observations of cryptocurrency prices. We largely follow Tran and Leirvik (2019) by estimating the AM IM daily using over-lapping window data of 1 year. Tran and Leirvik (2019) shows that the non-overlapping and overlapping window approaches give the same AMIM results. Ripple are all inefficient (AM IM > 0). This is statistically significant, as seen by the small size of the standard error. The median, however, is zero for BTC, ETH, and LTC. This indicates that the efficiency of these currencies experiences substantial periods of efficiency. AM IM is very high but does not last long because Ethereum community responded very quickly to eliminate doubt and uncertainty. Indeed, to save Ethereum from the hack, the Ethereum community decided to "hard-fork", which are upgrades to the programming code that add new rules to the Ethereum software that are incompatible with earlier versions.
Basically, the Ethereum community rewrote the ledger, which is the common transaction book, to eliminate all the hacked transactions 7 .
All in all, many of the spikes and drops in market inefficiency, as shown in figure (3), can be related to idiosyncratic events. This is particularly true for the time early in the sample period of each cryptocurrency. In the later stages of our sample period, all currencies show a significant improvement in efficiency, as shown by a negative estimate of the AM IM . This means that the currencies are significantly efficient. However, it seems that the prices are turning less efficient in the last half of Q1-2018, and the first half of Q2-2018 but return  -bitcoin-remains-500-amid-china-uncertainty/ 7 This process violates the core motivation of using blockchain technology which prevents any modification of past information. This process requires a lot of computing power and is a collective work. Therefore debates were going at that time. This leads to a separation of Ethereum into Ethereum and Ethereum Classic. The Ethereum Classic does not accept the hard fork, thus accept all the hacked transactions, and being a separate community from Ethereum. See Leising (2017) for more details.
to the efficient level at the end of 2018. These results corroborates the main idea of the Adaptive Market Hypothesis of Lo (2004), where market efficiency is changing over time, and reacting to events in the market. We also redo the above exercises with AMIM using log-return. The results are in the appendix. The AMIM result from log-return is qualitatively the same with the AMIM result with simple return. However, we should also aware that the log-return series will be mechanically smoother than the simple return series in the positive return region. For example, a simple return of 5% will only be log (1.05) = 4.88% in log return, and a simple return of 10% will only be log (1.1)= 9.53% in log return. In addition, large deviations in the negative return region will blow up the log return. For example, a price moving from 50 to 10, yielding a simple return of -80%, will give a log-return of about -160%, which is not economically sound. Therefore, we need to be more cautious on the interpretation of any result from log-returns with highly volatile financial assets.

Conclusion
In this paper we investigate the inefficiency of the prices of five different cryptocurrencies.
Using a rolling window, we find that the prices has been significantly inefficient during our sample. However, there are signs that the efficiency of all cryptocurrencies are improving, with all having significant drop in AM IM in the last 6 quarters. These results are consistent with recent research on the topic. The markets for cryptocurrencies are improving at an exceptional pace, with volume improving and becoming less volatile. This invites more research in the near future, both on the topic of efficiency of these markets, but also other aspects, such as for example price-return volatility, liquidity, and the relationship to other assets. Table 3: Summary Statistics of daily log returns for the 5 crypto-currencies: Bitcoin (BTC), Ethereum (ETH), Ripple (XRP), Litecoin (LTC), EOS (EOS). The sample is from 29 April 2013 to 28th February 2019. n is the number of observations, mean is the sample average of the returns, sd is the sample standard deviation of the returns, min and max are the minimum and maximum daily returns, respectively. skew is the skewness of returns, kurtosis measures the thickness of the tails of the return distributions, and se is the standard error of the mean.  Table 4: Summary Statistics of AMIM for 5 crypto-currencies: Bitcoin (BTC), Ethereum (ETH), Ripple (XRP), Litecoin (LTC), EOS (EOS). The AMIM is computed daily with one year overlapping data using log return. The sample is from 29 April 2013 to 28th February 2019. n is the number of observations, AMIM is the sample average of AMIM, sd is the sample standard deviation of AMIM, min and max are the minimum and maximum AMIM, respectively. skew is the skewness of AMIM, kurtosis measures the thickness of the tails of the AMIM distributions, and se is the standard error of the mean.