Bitcoin fluctuations and the frequency of price overreactions
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Abstract
This paper investigates the role of the frequency of price overreactions in the cryptocurrency market in the case of BitCoin over the period 2013–2018. Specifically, it uses a static approach to detect overreactions and then carries out hypothesis testing by means of a variety of statistical methods (both parametric and nonparametric) including ADF tests, Granger causality tests, correlation analysis, regression analysis with dummy variables, ARIMA and ARMAX models, neural net models, and VAR models. Specifically, the hypotheses tested are whether or not the frequency of overreactions (i) is informative about Bitcoin price movements (H1) and (ii) exhibits no seasonality (H2). On the whole, the results suggest that it can provide useful information to predict price dynamics in the cryptocurrency market and for designing trading strategies (H1 cannot be rejected), whilst there is no evidence of seasonality (H2 cannot be rejected).
Keywords
Cryptocurrency Bitcoin Anomalies Overreactions Abnormal returns Frequency of overreactionsJEL Classification
G12 G17 C631 Introduction
Cryptocurrencies have attracted considerable attention since their recent creation and experienced huge swings. For instance, in 2017 Bitcoin prices rose by more than 20 times, but in early 2018 fell by 70%; similar sharp drops had in fact already occurred 5 times before (June 2011, January 2012, April 2013, November 2013, December 2017). Such significant deviations of asset prices from their average values during certain periods of time are known as overreactions and have been widely analysed in the literature since the seminal paper of De Bondt and Thaler (1985), various studies being carried out for different markets (stocks, FOREX, commodities etc.), countries (developed and emerging), assets (stock prices/indices, currency pairs, oil, gold etc.), and time intervals (daily, weekly, monthly etc.). However, hardly any evidence is available to date on the cryptocurrency market, which is particularly interesting because of its extremely high volatility compared to the FOREX or stock market (see Caporale and Plastun 2018a for details). In the most recent years interest in the cryptocurrency market has increased even further, and price prediction has been investigated in various studies (Ciaian et al. 2016; Balcilar et al. 2017; Khuntia and Pattanayak 2018; AlYahyaee et al. 2019 and many others). However, the evidence is still mixed.
The present paper aims to analyse the role of the frequency of overreactions, specifically whether or not it can help predict price behaviour and/or exhibits seasonality, by using daily prices for BitCoin over the period 2013–2018. Overreactions are detected by plotting the distribution of logreturns. Then, the following null hypotheses are tested: (i) the frequency of overreactions is informative about BitCoin price movements (H1), and (ii) it exhibits no seasonality (H2). For this purpose a variety of statistical methods (parametric and nonparametric) are used such as ADF tests, Granger causality tests, correlation analysis, regression analysis with dummy variables, ARIMA and ARMAX models, neural net models, and VAR models.
The remainder of the paper is organised as follows. Section 2 contains a brief review of the literature on price overreactions in the cryptocurrency market. Section 3 describes the methodology. Section 4 discusses the empirical results. Section 5 provides some concluding remarks.
2 Literature review
According to Hileman and Rauchs (2017) there were more than 300 academic papers devoted to the cryptocurrency market published before the crypto boom; their number has increased further since then. The cryptocurrency market is still relatively young and as a result papers have initially analysed some of its general features (Dwyer 2015a, b; Elbahrawy et al. 2017) or properties such as competitiveness (Halaburda and Gandal 2014). There is only a limited number of studies examining instead its long memory and persistence (Caporale et al. 2018c; Bariviera 2017; Urquhart 2016), efficiency (Urquhart 2016; Bartos 2015), correlations between different cryptocurrencies (Halaburda and Gandal 2014), price predictability (Brown 2014), volatility (Cheung et al. 2015; Carrick 2016).
Bariviera (2017) finds evidence of long memory in the daily dynamics of BitCoin; they also show that persistence in the cryptocurrency market is decreasing. Similar conclusions are reached by Bouri et al. (2016) and Catania and Grassi (2017).
Aggarwal (2019) examines Bitcoin returns and finds strong evidence of market inefficiency (see also Urquhart 2016). Calendar anomalies in the cryptocurrency market are analysed by Kurihara and Fukushima (2017) and Caporale and Plastun (2018c), intraday patterns are explored by Eross et al. (2017), the overreaction hypothesis is tested by Caporale and Plastun (2018a).
Ma and Tanizaki (2019) analyse the dayoftheweek effect for both returns and their volatility in the cryptocurrency market, and find significantly high volatilities on Monday and Thursday. Similar results are reported by Aharon and Qadan (2018). Eross et al. (2019) analyse the intraday dynamics of Bitcoin and find that the trade volume in the cryptocurrency market increases during the day and falls from around 4 pm until midnight.
Comparative analysis of the average daily price amplitude in different financial markets. Source: Caporale and Plastun (2018a)
Asset  Market  2014 (%)  2015 (%)  2016 (%)  2017 (%)  Average (%) 

EURUSD  FOREX  0.6  1.1  0.8  0.6  0.8 
DowJones Industrial  Stock market  0.8  1.2  1.0  0.5  0.9 
CSI300  1.5  3.0  1.5  0.9  1.8  
Gold  Commodities  1.3  1.4  1.5  0.9  1.3 
Oil  1.8  3.9  3.9  2.1  2.9  
BitCoin  Cryptocurrency  5.0  4.2  2.4  6.3  5.1 
LiteCoin  6.6  6.4  2.9  9.6  7.3  
Dash  22.0  9.0  7.1  11.3  12.1  
Ripple  7.1  4.2  3.2  12.7  7.3 
Further, the log return distribution of prices has unusually fat tails (see Table 18), which suggests their being prone to overreactions, which can be helpful to predict future prices and crises. Catania and Grassi (2017) show that price behaviour in the cryptocurrency market is quite complex, with outliers, asymmetries and nonlinearities that are difficult to model.
AlYahyaee et al. (2019) try to predict Bitcoin prices using information from a Volatility Uncertainty Index (VIX), whilst Mensi et al. (2019) find evidence of comovement between Bitcoin and five major cryptocurrencies (Dash, Ethereum, Litecoin, Monero and Ripple). Balcilar et al. (2017) show that information about trade volumes can be used to predict returns in the cryptocurrency market. Aharon and Qadan (2018) show that normally used variables have limited forecasting power for Bitcoin prices. Khuntia and Pattanayak (2018) explore timevarying linear and nonlinear dependence in Bitcoin returns. Kristoufek (2014) finds that the tradeexchange ratio plays an essential role in driving Bitcoin price fluctuations in the long run. Ciaian et al. (2016) show that the total number of unique Bitcoin transactions per day is an important determinant of Bitcoin price fluctuations.
Another issue investigated in the literature is whether overreactions exhibit seasonality. De Bondt and Thaler (1985) show that they tend to occur mostly in a specific month of the year, whilst Caporale and Plastun (2018b) do not find evidence of seasonal behaviour in the US stock market. Note also that according to Khuntia and Pattanayak (2018) market efficiency in the cryptocurrency market is evolving over time. Caporale and Plastun (2018a) find evidence in favour of the overreaction hypothesis, whilst Bartos (2015) report that the cryptocurrency market immediately reacts to the arrival of new information and absorbs it; as a result prices are not affected by overreactions.
Whilst most studies examine abnormal returns and the subsequent price behaviour (in general, contrarian movement) for a given time interval (day, week, and month), the current paper focuses on the frequency of abnormal price changes. Only a few papers have considered this issue in the case of the FOREX or stock market (see Govindaraj et al. 2014; Angelovska 2016), and none in the case of the cryptocurrency market. We will aim to show that the frequency of abnormal price changes can be a useful tool for price predictions in the cryptocurrency market.
3 Methodology
The first step in the analysis of overreactions is their detection. There are two main methods. One is the dynamic trigger approach, which is based on relative values. Wong (1997) and Caporale and Plastun (2018a) in particular propose to define overreactions on the basis of the number of standard deviations to be added to the average return. The other is the static approach which uses actual price changes as an overreaction criterion. For example, Bremer and Sweeney (1991) use a 10% price change as a criterion. Caporale and Plastun (2018b) compare these two methods in the case of the US stock market and show that the static approach produces more reliable results. Therefore, this will also be used here.
Then, the following hypotheses are tested:
Hypothesis 1 (H1)
The frequency of overreactions is informative about price movements in the cryptocurrency market.
The size, sign and statistical significance of the coefficients provide information about the possible influence of the frequency of overreactions on BitCoin log returns.
To assess the performance of the regression models a multilayer perceptron (MLP) method will be used (Rumelhart and McClelland 1986). This method is based on neural networks modelling. The algorithm is as follows. The data is divided into 3 groups: the learning group (50%), the test group (25%), and the control group (25%). The learning process in the neural network consists of 2 stages: the first stage is based on an inverse distribution method (number of periods − 100, training speed − 0.01) and the second uses a conjugate gradient method (number of periods − 500). This procedure generates an optimal neural net. The results from the neural net are then compared with those from the regression analysis.
Information criteria, specifically AIC (Akaike 1974) and BIC (Schwarz 1978), are used to select the best ARMAX specification for BitCoin log returns.
Hypothesis 2 (H2)
The frequency of overreactions exhibits no seasonality.
We perform a variety of statistical tests, both parametric (ANOVA analysis) and nonparametric (Kruskal–Wallis tests), for seasonality in the monthly frequency of overreactions, which provides information on whether or not overreactions are more likely in some specific months of the year.
4 Empirical results
The data used are BitCoin daily and monthly prices for the period 01.05.201331.05.2018; the data source is CoinMarket (https://coinmarketcap.com/). As a first step, the frequency distribution of log returns is analysed (see Table 18 and Fig. 6). As can be seen, two symmetric fat tails are present in the distribution. The next step is the choice of thresholds for detecting overreactions. To obtain a sufficient number of observations we consider values ± 10% of the average from the population, namely − 0.04 for negative overreactions and 0.05 for positive ones. Detailed results are presented in Appendix 2.
Results of ANOVA and nonparametric Kruskal–Wallis tests for statistical differences in the frequency of overreactions between different years
F  pvalue  F critical  Null hypothesis 

ANOVA test  
7.24  0.000  2.81  Rejected 
Adjusted H  pvalue  Critical value  Null hypothesis 

Kruskal–Wallis test  
14.98  0.001  9.49  Rejected 
Correlation analysis between the frequency of overreactions and different BitCoin series indicators
Parameter  BitCoin close prices  BitCoin returns  BitCoin logreturns 

Over_negative  0.50  − 0.21  − 0.34 
Over_positive  0.41  0.62  0.53 
All_over  0.53  0.25  0.13 
Over_mult  0.15  − 0.40  − 0.60 
There appears to be a positive (rather than negative, as one would expect) correlation between BitCoin prices and negative overreactions. By contrast, there is a negative correlation in the case of returns and log returns. The overreaction multiplier exhibits a rather strong negative correlation with BitCoin log returns. Finally, the overall number of overreactions has a rather weak correlation with prices.
Augmented Dickey–Fuller test: BitCoin log returns and overreactions frequency data
Parameter  Logreturns  Over_all  Over_negative  Over_positive 

Augmented Dickey–Fuller test (intercept)  
Augmented Dickey–Fuller test statistic  − 7.55  − 2.87  − 5.48  − 3.39 
Probability  0.0000  0.0549  0.0000  0.0152 
Test critical values (5% level)  − 2.89  − 2.89  − 2.89  − 2.89 
Null hypothesis  Rejected  Not rejected  Rejected  Rejected 
Augmented Dickey–Fuller test (trend and intercept)  
Augmented Dickey–Fuller test statistic  − 7.47  − 2.91  − 5.59  − 3.37 
Probability  0.0000  0.1677  0.0001  0.0650 
Test critical values (5% level)  − 3.41  − 3.41  − 3.41  − 3.41 
Null hypothesis  Rejected  Not rejected  Rejected  Not rejected 
Augmented Dickey–Fuller test (intercept, 1st difference)  
Augmented Dickey–Fuller test statistic  − 6.86  − 12.21  − 13.95  − 11.65 
Probability  0.0000  0.0001  0.0000  0.0000 
Test critical values (5% level)  − 3.41  − 3.41  − 3.41  − 3.41 
Null hypothesis  Rejected  Rejected  Rejected  Rejected 
Regression analysis results: BitCoin closes
Parameter  All overreactions  Negative and positive overreactions as separate variables  Regression with dummy variables 

a _{0}  − 100.64 (0.85)  − 158.22 (0.77)  368.88x (0.32) 
Slope for the overreactions (all overreactions)  350.77 (0.00)  –  – 
Slope for the overreactions (negative overreactions)  –  475.44 (0.00)  551.28 (0.00) 
Slope for the overreactions (positive overreactions)  –  237.43 (0.10)  514.33 (0.00) 
Ftest  22.55 (0.00)  11.69 (0.00)  16.32 (0.00) 
Multiple R  0.53  0.54  0.46 
Regression analysis results: BitCoin returns
Parameter  All overreactions  Negative and positive overreactions as separate variables  Regression with dummy variables 

a _{0}  − 0.0442 (0.72)  0.0395 (0.55)  0.0119 (0.88) 
Slope for the overreactions (all overreactions)  0.0328 (0.00)  –  – 
Slope for the overreactions (negative overreactions)  –  − 0.1597 (0.00)  0.0023 (0.00) 
Slope for the overreactions (positive overreactions)  –  0.2076 (0.00)  0.0922 (0.00) 
Ftest  3.93 (0.05)  77.64 (0.00)  8.71 (0.00) 
Multiple R  0.25  0.86  0.36 
Regression analysis results: BitCoin log returns
Parameter  All overreactions  Negative and positive overreactions as separate variables  Regression with dummy variables 

a _{0}  − 0.0200 (0.72)  0.0645 (0.04)  0.0368 (0.35) 
Slope for the overreactions (all overreactions)  0.0084 (0.33)  –  – 
Slope for the overreactions (negative overreactions)  –  − 0.0939 (0.00)  − 0.0122 (0.32) 
Slope for the overreactions (positive overreactions)  –  0.1013 (0.00)  0.0355 (0.00) 
Ftest  0.98 (0.33)  96.48 (0.00)  6.85 (0.00) 
Multiple R  0.13  0.88  0.32 
Comparative characteristics of neural networks
Architecture  Performance  Errors  

Learning  Control  Test  Learning  Control  Test  
MLP 2231:1  0.4484  0.4547  0.5657  0.0811  0.0392  0.0630 
L 221:1  0.3809  0.6265  0.8314  0.0664  0.0801  0.0836 
Quality comparison of neural networks
Parameters  Type of neural net  

MLP 2231:1  L 221:1  
Average  0.0677  0.0677 
Standard deviation  0.3103  0.3103 
Mean error  0.0067  − 0.0158 
Standard deviation error  0.1450  0.1576 
Mean absolute error  0.1106  0.1244 
Standard deviation error and data ratio  0.4673  0.5078 
Correlation  0.8844  0.8719 
As can be seen, the neural net based on the multilayer perceptron structure provides better results than the linear neural net: the control error is lower (0.0392 (MLP) vs 0.0801(L)); the standard deviation error and the data ratio are also lower (0.4673 vs 0.5078); the correlation is higher (0.8844 vs 0.8719).
As can be seen the estimates (from the regression model and the neural network, respectively) are very similar and very close to the actual values, which suggests that the regression model (Eq. 8) captures very well the behaviour of BitCoin prices.
Parameter estimates for the best ARIMA models
Parameter  Model 1: ARIMA(2, 0, 2)  Model 2: ARIMA(1, 0, 0)  Model 3: ARIMA(0, 0, 1) 

a _{0}  0.0717 (0.1019)  0.0676^{*} (0.0931)  0.0676^{*} (0.0929) 
\( \psi_{t  1} \)  0.2622 (0.1568)  0.0048 (0.9702)  – 
\( \psi_{t  2} \)  − 0.6935^{***} (0.0000)  –  – 
\( \theta_{t  1} \)  − 0.2938^{***} (0.0052)  –  0.0044 (0.9714) 
\( \theta_{t  2} \)  1.0000^{***} (0.0000)  –  – 
AIC  35.7555  35.8773  35.8788 
BIC  48.3215  42.1617  42.1618 
As can be seen, Model 1 captures best the behaviour of BitCoin log returns: all regressors are significant at the 1% level, except \( \psi_{t  i} \), and AIC has the smallest value.
Estimated parameters for the ARMAX models
Parameter  Model 4  Model 5 

a _{0}  0.0669 (0.3870)  0.0821 (0.3027) 
\( \psi_{t  1} \)  − 0.1316 (0.3155)  0.7101^{***} (0.0003) 
\( \psi_{t  2} \)  0.8245^{***} (0.0000)  0.8895^{***} (0.0000) 
\( \psi_{t  3} \)  –  − 0.7811^{***} (0.0000) 
\( \theta_{t  1} \)  − 0.3383 (0.1969)  − 1.1925^{***} (0.0000) 
\( \theta_{t  2} \)  − 0.1307 (0.4683)  – 
\( \theta_{t  3} \)  –  0.5468^{***} (0.0000) 
\( a_{t  1} \)  − 0.0590^{***} (0.0008)  − 0.0513^{***} (0.0026) 
\( a_{t  2} \)  0.0663^{***} (0.0000)  0.0585^{***} (0.0000) 
\( a_{t  3} \)  − 0.0493^{***} (0.0006)  − 0.0476^{***} (0.0005) 
\( a_{t  4} \)  0.0333^{**} (0.0107)  0.0345^{***} (0.0068) 
\( b_{t  1} \)  0.0467^{***} (0.0025)  0.0410^{**} (0.0212) 
\( b_{t  2} \)  − 0.0498^{***} (0.0000)  − 0.0506^{***} (0.0002) 
\( b_{t  6} \)  0.0103 (0.3459)  0.0069 (0.4879) 
AIC  22.2441  19.3993 
BIC  48.3395  47.5019 
Granger Causality Tests between BitCoin log returns and both negative (OF^{−}) and positive overreactions (OF +)
Null hypothesis: no causality  

Excluded  Y  \( OF^{  } \)  \( OF^{ + } \)  
Chisq  Probability  Chisq  Probability  Chisq  Probability  
Y  –  –  3.6428  0.0563^{*}  8.6296  0.0033^{***} 
\( OF^{  } \)  1.2724  0.2593  –  –  7.8424  0.0051^{***} 
\( OF^{ + } \)  1.4541  0.2279  0.0011^{***}  0.9725  –  – 
All  1.4902  0.4747  14.4342  0.0007^{***}  9.0730  0.0107^{**} 
Null hypothesis  Not rejected  Rejected  Rejected 
VAR lag length selection criteria
Lags  AIC  BIC 

1  7.4380  7.8969 
2  7.5663  8.3694 
3  7.7687  8.9160 
4  8.0171  9.5085 
5  7.9496  9.7852 
6  8.0571  10.2368 
7  8.2555  10.7793 
8  8.0228  10.8908 
9  7.6051  10.8173 
10  7.6916  11.2480 
VAR(1) parameter estimates
Parameter  Y  \( OF^{  } \)  \( OF^{ + } \) 

Const  0.0568 (0.3984)  1.2280^{**} (0.0157)  1.0447^{*} (0.0553) 
Y (− 1)  0.3004 (0.2818)  3.8884^{*} (0.0615)  6.4797^{***} (0.0048) 
\( OF^{  } \) (− 1)  0.0354 (0.2642)  0.4658^{**} (0.0489)  0.7012^{***} (0.0070) 
\( OF^{ + } \) (− 1)  − 0.0391 (0.2330)  0.0082 (0.9726)  − 0.2904 (0.2666) 
Rsquared  0.0264  0.2874  0.2831 
Adj. Rsquared  − 0.0267  0.2485  0.2440 
Fstatistic  0.4971  7.3953  7.2412 
pvalue  0.6857  0.0003^{***}  0.0004^{***} 
Akaike AIC  0.6009  4.5958  4.7547 
BIC criterion  0.7418  4.7366  4.8956 
Durbin–Watson stat  2.0280  2.2936  2.2109 
Akaike AIC  8.3870  
BIC criterion  8.8095 
Variance Decomposition
Variable  Lag  Percentage of the variance accounted for by a variable  

Y  \( OF^{  } \)  \( OF^{ + } \)  
Y  1  100.00  0.00  0.00 
2  97.42  0.19  2.39  
3 >  97.42  0.19  2.39  
\( OF^{  } \)  1  17.04  82.96  0.00 
2  22.02  77.98  0.00  
3 >  22.65  76.74  0.61  
\( OF^{ + } \)  1  36.13  38.65  25.22 
2  37.58  41.79  20.63  
3 >  36.86  43.04  20.10 

The behaviour of Y is mostly explained by its previous dynamics (97.4%); \( {{O}}F^{  } \) accounts for only 0.2% of its variance, and \( {{O}}F^{ + } \) for only 2.4%.

The behaviour of \( {{O}}F^{  } \) is also mainly determined by its previous dynamics (76.7%), with Y explaining only 22.7% of its variance and \( OF^{ + } \) only 0.6%.

The behaviour of \( OF^{ + } \) is mostly accounted for by the \( OF^{  } \) dynamics (43%), with Y explaining 36.9% of its variance and \( OF^{ + } \) 20.1%.
Parametric ANOVA of monthly seasonality in the overreaction frequency
Parameter  Frequency of negative overreactions  Frequency of positive overreactions  Frequency of overreactions (overall) 

F  0.90  0.77  0.72 
pvalue  0.5461  0.6596  0.7055 
F critical  1.99  1.99  1.99 
Null hypothesis  Not rejected  Not rejected  Not rejected 
Nonparametric Kruskal–Wallis of monthly seasonality in the overreaction frequency
Parameter  Frequency of negative overreactions  Frequency of positive overreactions  Frequency of overreactions (overall) 

Adjusted H  7.21  7.16  8.32 
d.f.  11  11  11 
pvalue  0.78  0.79  0.68 
Critical value  19.675  19.675  19.675 
Null hypothesis  Not rejected  Not rejected  Not rejected 
5 Conclusion
This paper investigates the role of the frequency of price overreactions in the cryptocurrency market in the case of BitCoin over the period 20132018. Specifically, it uses a static approach to detect overreactions and then carries out hypothesis testing by means of a variety of statistical methods (both parametric and nonparametric) including ADF tests, Granger causality tests, correlation analysis, regression analysis with dummy variables, ARIMA and ARMAX models, neural net models, and VAR models. Specifically, the hypotheses tested are whether or not the frequency of overreactions (i) is informative about Bitcoin price movements (H1) and (ii) exhibits no seasonality (H2).
On the whole, the results suggest that the frequency of price overreactions can provide useful information to predict price dynamics in the cryptocurrency market and for designing trading strategies (H1 cannot be rejected) in the specific case of BitCoin. However, these findings are somewhat mixed: stronger evidence of a predictive role for the frequency of price overreactions is found when estimating neural net and ARMAX models as opposed to VAR models. As for the possible presence of seasonality, the evidence is very clear: no seasonal patterns are detected for the frequency of price overreactions (H2 cannot be rejected).
Notes
Acknowledgements
Alex Plastun gratefully acknowledges financial support from the Ministry of Education and Science of Ukraine (0117U003936). The authors are grateful to the editor and an anonymous referee for their useful comments and suggestions.
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