The emergence of digital currencies, particularly cryptocurrencies, has brought significant changes to the financial landscape. As these digital assets gain popularity and attention, it becomes crucial to understand the dynamics of their market efficiency, especially in relation to time-varying conditions. Market efficiency, a concept extensively studied in neoclassical finance, determines whether prices fully reflect all available information. However, traditional efficient market hypothesis (EMH) frameworks have been challenged because of their assumption of stability in market efficiency over time. To address this limitation, the adaptive market hypothesis (AMH) was introduced by Lo (2004), proposing that market efficiency evolves and adapts to changing conditions.
The AMH suggests that market efficiency is not a constant, but a dynamic and evolving process. It acknowledges the presence of both efficient and inefficient periods in markets, driven by factors such as human errors, changing investment opportunities, and varying business situations. Unlike the EMH, which assumes a static relationship between risk and reward, the AMH recognizes the instability of this association. Furthermore, the AMH emphasizes the importance of adapting to changing market conditions for survival and success in the financial markets.
While past research has predominantly focused on testing the weak form of efficiency in cryptocurrency markets, the AMH provides a valuable perspective to analyze the time-varying nature of market efficiency in these digital asset markets. By considering the adaptive nature of markets, the AMH offers insights into the dynamics of price movements and the implications for forecasting changes in cryptocurrency prices over time. By employing various statistical tests and methodologies, researchers can shed light on the efficiency periods and inefficiencies in the market, allowing for a more nuanced understanding of cryptocurrency price fluctuations.
However, there remains controversy regarding the efficiency of cryptocurrency markets, with proponents and opponents of the EMH presenting conflicting findings. Some studies support the notion that cryptocurrencies, such as Bitcoin, exhibit a high level of efficiency by incorporating all available information. Conversely, other researchers have identified calendar anomalies and significant deviations from the EMH in the cryptocurrency market. This disparity of results highlights the need for a comprehensive examination of market efficiency that considers the adaptive nature of cryptocurrencies and their evolving efficiency periods.
An analysis of the adaptive market hypothesis
The relationship between market efficiency and the adaptive market hypothesis (AMH) has been explored by previous researchers, although their focus was limited to specific implications of the AMH. Charles, Darné, and Kim (2012) conducted tests using the Martingale difference hypothesis, variance ratio, and automatic portmanteau tests. Their findings indicated that returns in the market are predictable over certain periods, supporting the implications of the AMH. Similarly, Lim, Luo, and Kim (2013) employed wild bootstrap automatic variance ratio and automatic portmanteau Box-Pierce tests, revealing that predictable patterns vary over time, which aligns with the AMH. In the stock market, Urquhart and McGroarty (2016) applied variance ratio tests on a moving window and found varying return predictability, further supporting the AMH. Charfeddine and Khediri (2016) examined market efficiency using the weak-form and a rolling sample test with long-memory factors. They reported different degrees of changing market efficiency in GCC markets, supporting the AMH.
The AMH has also been analyzed in various asset classes, including stocks and currencies. Studies by Urquhart and McGroarty (2015), Kim, Shamsuddin, and Lim (2011), and Urquhart and McGroarty (2016) focused on stocks and found that the AMH serves as an efficient predictor. In the case of currencies, Cialenco and Protopapadakis (2011) and Neely, Weller, and Ulrich (2009) also confirmed the AMH’s predictive capabilities. Cryptocurrencies have garnered significant attention from the media, investors, and regulators due to their unique features and technological innovation. These digital assets have experienced remarkable price growth in short periods, but with extreme volatility (Chu et al., 2017; Katsiampa, Gkillas, & Longin, 2018; Kristjanpoller & Minutolo, 2018; Urquhart, 2017). The market value and trading volume of cryptocurrencies, such as Bitcoin, have surged substantially in recent years.
In this study, we examine the efficiency of the AMH in relation to four significant digital currencies: Bitcoin, Monero, Litecoin, and Stellar. These currencies were selected based on their high tradability, fast processing, and privacy-oriented characteristics identified in previous studies (Chu et al., 2017; Kristjanpoller & Minutolo, 2018; Urquhart, 2017). We analyze the market efficiency variation of these currencies over the period from 2014 to 2018.
Additionally, we conduct tests to assess the non-normality and stationarity of the series using the Jarque-Bera and Augmented Dickey-Fuller (ADF) tests. Furthermore, we employ Ljung-Box statistics and ARCH-LM tests to examine serial correlation and volatility clustering in returns and squared returns of the selected cryptocurrencies. To determine changing linear and non-linear dependence in price movements, we utilize the Generalized Spectral (GS) test by Escanciano and Velasco (2006) and the Dominguez-Lobato (DL) test by Domínguez and Lobato (2003) as part of the martingale difference hypothesis (MDH). We also apply the automatic portmanteau (AP) test to examine autocorrelation and confirm the absence of significant positive or negative correlations.
By conducting these analyses, we aim to shed light on the efficiency of the AMH in the context of the selected cryptocurrencies. This study will provide insights into the time-varying nature of market efficiency and its implications for predicting changes in cryptocurrency prices. Additionally, we examine the normality, stationarity, serial correlation, and volatility clustering of the cryptocurrency returns. The results will contribute to a deeper understanding of the dynamics of cryptocurrency markets and emphasize the importance of considering time-varying market conditions for efficient forecasting.
Empirical analysis
In this empirical analysis, we utilize the daily prices of four cryptocurrencies: Bitcoin, Monero, Litecoin, and Stellar. The data spans from January 2014 to December 2018, and Figure 1 illustrates the price fluctuations of each cryptocurrency examined in this study.
Figure 1. Price Fluctuations of Cryptocurrencies
The data is sourced from https://www.coindesk.com/price. It should be noted that the starting dates differ for each cryptocurrency, resulting in varying numbers of observations, as indicated in Table 1. To calculate the logarithmic percentage returns, we transform the raw data using the formula:
Percentage Return = ln(Price_t / Price_(t-1)) * 100
Table 1 provides descriptive statistics based on the daily percentage returns of the four cryptocurrencies. The statistics reveal that the distributions of all return series are positively skewed, except for Litecoin’s returns, which exhibit a negative skew. Additionally, the excess kurtosis values indicate that the returns distributions have heavier tails.
Table 1. Summary Statistics of Cryptocurrencies Returns
The results of the Jarque-Bera test indicate the non-normality of the return series for all cryptocurrencies. On the other hand, the Augmented Dickey-Fuller (ADF) test confirms the stationarity of the series. Furthermore, the Ljung-Box statistics and ARCH-LM test results demonstrate the presence of both serial correlation and volatility clustering in the returns and squared returns of all the cryptocurrencies.
Data and methodology
Data Collection
The daily price data for Bitcoin is obtained from Quandl.com, while data for other assets (gold, USD/EUR exchange rate, S&P 500 index) are sourced from the World Gold Council and the Federal Reserve Bank of St. Louis, respectively. The data collection period spans from September 1, 2010, to March 31, 2019, resulting in a total of 2,134 observations of daily data. To match the trading dates of Bitcoin and other assets, data was available only on trading days. The logarithmic percentage returns are calculated by taking the first difference of the log prices and annualizing the returns.
Descriptive statistics
Table 1 presents the summary statistics of the annualized log returns for Bitcoin and other assets. The mean and standard deviation of Bitcoin’s log returns are comparatively larger than the other assets, indicating higher volatility. Bitcoin’s returns exhibit positive skewness, suggesting a risk-loving behavior among investors. The excess kurtosis values for Bitcoin’s returns are extremely high, indicating heavy-tailed distributions that deviate significantly from normality.
Variance Ratio (VR) Test
The VR test is employed to examine whether the log price series of Bitcoin follows a random walk. The test compares the variance of increments in a random walk with the variance of the first difference of the log prices. The results in Table 2 reveal mixed evidence regarding the random walk hypothesis. While Bitcoin’s log price series follows the efficient market hypothesis (EMH) in the long run (VR(q) ≈ 1), it exhibits mean-reverting behavior in the short run (VR(q) < 1).
QHO (Quasi-Harmonic Oscillator)
The QHO is used to model the evolution of the log return distribution of Bitcoin. It is a stochastic differential equation that describes various random behaviors in financial markets. The Fokker-Planck equation, derived from the QHO, provides insights into the probability density function (PDF) of the log returns. The PDF estimated using the QHO fits the data better compared to a random walk model (RWM), as indicated by goodness-of-fit tests and likelihood ratio tests (Tables 4 and 5).
Results and discussion
The VR test suggests that the efficient market hypothesis holds for Bitcoin in the long run but not in the short run. However, the QHO analysis consistently supports market efficiency in Bitcoin, indicating that its log return distribution follows a random walk with a probability of 90% or more. The QHO provides a better description of the log return distribution, considering its non-Gaussian features and capturing both Gaussian and non-Gaussian characteristics. The analysis reveals the presence of heavy-tailed distributions and deviations from normality in Bitcoin’s returns.
By employing the QHO model, this study contributes to understanding the efficiency of the Bitcoin market and the dynamics of its log return distribution. The findings suggest the Bitcoin market exhibits market efficiency, implying that future price movements cannot be predicted based on past price patterns.
Empirical findings
Efficiency of Bitcoin
The analysis of Bitcoin’s price movements using the Dominguez-Lobato (DL) and Generalized Spectral (GS) tests reveals periods of efficiency and inefficiency. The DL test indicates that Bitcoin remained efficient in the market for longer periods, specifically from November 2014 to January 2017 and from April to December 2018. In contrast, inefficiency periods were observed from May to October 2014 and from February 2017 to March 2018. These findings support previous studies that highlight Bitcoin’s efficiency and widespread use as a cryptocurrency and universal payment method.
The GS test shows similar efficiency periods for Bitcoin, spanning from January 2015 to December 2017 and from February 2018 onwards. The only inefficiency period observed was in December 2017 to January 2018. While there are slight differences between the DL and GS test results, both tests confirm Bitcoin’s long periods of market efficiency.
The automatic portmanteau (AP) test assesses correlations in Bitcoin’s price movements and provides insights into market stability. The AP test statistics fluctuate over time, indicating periods of market instability. The market was stable in 2014, but from November 2014 to March 2018, the market became unstable, representing the peak inefficiency period for Bitcoin. From March 2015 to December 2018, the market conditions stabilized, and Bitcoin remained efficient. These results align with the Adaptive Market Hypothesis (AMH), which suggests that market efficiency is highly dependent on market conditions.
Efficiency of Monero, Litecoin, and Stellar
The efficiency and inefficiency periods for Monero, Litecoin, and Stellar are also analyzed using the DL and GS tests, as well as the AP test.
For Monero, the DL test reveals efficiency periods from May 2015 to March 2016, May 2016 to March 2017, and from August 2017 to January 2018. Inefficiency periods are observed from March 2015 to April 2015, April 2016, May to July 2017, February to August 2018, and from October to December 2018. The efficiency periods of Monero exhibit high volatility in relation to changing market conditions, supporting the AMH.
Similarly, the GS test for Monero indicates efficiency periods spanning almost the entire sample period from July 2015 to December 2018, with only a brief inefficiency period at the beginning of June 2015.
The AP test for Monero reflects market efficiency periods that were highly unstable, with several spikes observed from 2015 to 2018. The peak inefficiency periods are depicted in 2016 (April to June) and from April 2018 to December 2018.
For Litecoin, the DL test reveals efficiency periods from May 2014 to November 2014, February 2015 to May 2015, and from September 2015 to December 2018. The inefficiency periods for Litecoin are from December 2014 to January 2015 and from June to September 2015.
The GS test for Litecoin shows that it mostly remained in the efficient zone since May 2014, with the only inefficiency period observed from January 2015 to June 2015.
The AP test for Litecoin depicts highly unstable market efficiency periods, with fluctuations observed from 2014 to 2018. The peak inefficiency periods are from December 2014 to September 2015 and in 2017 (July to September).
For Stellar, the DL test indicates efficiency periods from September 2015 to November 2015 and from January 2017 to May 2017. The inefficiency periods are observed from December 2015 to December 2016 and from April 2017 to December 2018. The volatility of Stellar’s market efficiency periods is evident in the figure, indicating the strong influence of market conditions on its returns.
The GS test for Stellar shows efficiency periods from September 2015 to November 2015, March 2017 to September 2018, and from November 2018 to December 2018. The only inefficiency period is in October 2018.
The AP test for Stellar reflects highly fluctuating market efficiency periods, with spikes observed from 2015 to 2018. The peak inefficiency periods are from March 2016 to September 2016 and from April 2018 to December 2018.
The findings reveal varying efficiency and inefficiency periods for different cryptocurrencies. The analysis supports the Adaptive Market Hypothesis, indicating that the prediction of cryptocurrency prices varies over time and is influenced by changing market conditions.
Conclusion
This study focused on examining the price fluctuations and market efficiency of four digital currencies (Bitcoin, Monero, Litecoin, and Stellar) using the Adaptive Market Hypothesis (AMH). The analysis covered the period from 2014 to 2018 and utilized various tests, including the Dominguez-Lobato (DL), Generalized Spectral (GS), and Automatic Portmanteau (AP) tests.
The findings of this study support the notion that the cryptocurrency market does not strictly follow the Efficient Market Hypothesis (EMH). Instead, the results indicate that the efficiency of digital currencies evolves over time and is influenced by changing market conditions. The DL and GS tests revealed both efficiency and inefficiency periods for each cryptocurrency.
Bitcoin demonstrated longer efficiency periods, spanning from November 2014 to January 2017 and from April to December 2018, as indicated by the DL test. The GS test confirmed similar efficiency periods for Bitcoin, suggesting its strong market value and widespread use as a cryptocurrency. However, it is important to note that inefficiency periods were also observed for Bitcoin, emphasizing the dynamic nature of market efficiency.
Monero, Litecoin, and Stellar exhibited varying efficiency and inefficiency periods. Monero demonstrated efficiency periods that covered almost the entire sample period, while Litecoin showed efficiency periods from May 2014 to November 2014, February 2015 to May 2015, and from September 2015 to December 2018. Stellar experienced efficiency periods from September 2015 to November 2015, and from January 2017 to May 2017. These results highlight the different market dynamics and volatility of each digital currency.
The AP test further supported the AMH by showing that market efficiency is highly dependent on market conditions. Fluctuations and spikes in the AP test statistics indicated periods of market instability, which influenced the efficiency and inefficiency of the digital currencies analyzed.
This study contributes to the understanding of market efficiency in the high-frequency cryptocurrency market. It provides evidence that the price movements of digital currencies are influenced by market conditions and do not strictly adhere to the EMH. The findings suggest that investors and portfolio managers should consider the evolving efficiency and dynamic nature of the cryptocurrency market when making investment decisions.
Further research is warranted to explore the market efficiency of other digital currencies and to investigate the factors that contribute to the evolving efficiency in the cryptocurrency market. Additionally, studies analyzing the impact of regulatory developments and market interventions on market efficiency would provide valuable insights for market participants and regulators.
