Bitcoin Returns and the Frequency of Daily Abnormal Returns

This paper investigates the relationship between Bitcoin returns and the frequency of daily abnormal returns over the period from June 2013 to February 2020 using a number of regression techniques and model specifications including standard OLS, weighted least squares (WLS), ARMA and ARMAX models, quantile regressions, Logit and Probit regressions, piecewise linear regressions and non-linear regressions. Both the in-sample and out-of-sample performance of the various models are compared by means of appropriate selection criteria and statistical tests. These suggest that on the whole the piecewise linear models are the best but in terms of forecasting accuracy they are outperformed by a model that combines the top five to produce “consensus” forecasts. The finding that there exist price patterns that can be exploited to predict future price movements and design profitable trading strategies is of interest both to academics (since it represents evidence against the EMH) and to practitioners (who can use this information for their investment decisions).


Introduction
According to the Efficient Markets Hypothesis (EMHsee Fama, 1965), which remains the dominant paradigm in financial economics, asset prices should follow a random walk, and therefore it should not be possible to design trading strategies that exploit predictable patterns to generate abnormal profits. However, there is a large body of empirical evidence indicating that there exist various market anomalies resulting in identifiable price patterns such as contrarian and momentum effects; these include calendar anomalies, price over-and under-reactions, other types of anomalies associated with trading volumes and so on. In the case of the newly emerged cryptocurrency markets various studies have been carried out which have provided mixed evidence on price predictability (Ciaian et al., 2016;Balcilar et al., 2017;Khuntia and Pattanayak, 2018;Al-Yahyaee et al., 2019 and many others).
The current paper contributes to this literature by investigating the relationship between Bitcoin returns and the frequency of daily abnormal returns over the period from June 2013 to February 2020. It extends previous studies by Angelovska (2016) and  by considering a much wider range of econometric models and approaches over a longer sample, assessing the role of an additional regressor (namely the difference between the frequency of positive and negative abnormal returns), and evaluating the in-sample as well as the out-of-sample performance of the rival models. These include standard OLS, weighted least squares (WLS), ARMA and ARMAX models, quantile regressions, Logit and Probit regressions, piecewise linear regressions and non-linear regressions.
The remainder of the paper is organised as follows. Section 2 contains a brief review of the relevant literature. Section 3 describes the methodology. Section 4 discusses the empirical results. Section 5 provides some concluding remarks.

Literature Review
Cryptocurrencies have established themselves in recent years both as an alternative to fiat money and as a tradable asset used for risk-hedging purposes. Various papers have analysed the properties of these newly created markets. For instance, Bartos (2015) and Urquhart (2016) analysed their efficiency; Corbet et al. (2018) focused on price bubbles; other market anomalies were explored by Kurihara and Fukushima (2017) and ; Bariviera et al. (2017) and Caporale et al. (2018) investigated their persistence and long-memory properties; Bouri et al (2019) examined price predictability.
Of particular interest is the issue of whether or not abnormal returns generate stable patterns in price behaviour. This has been a popular topic for investigation since De Bondt and Thaler (1985) developed the overreaction hypothesis. The evidence is mixed: some papers find price reversals after abnormal price changes (Bremer and Sweeny, 1991;Larson and Madura, 2001), whilst others detect momentum effects (Schnusenberg and Madura, 2001;Lasfer et al., 2003). In the specific case of the cryptocurrency markets, Chevapatrakul and Mascia (2019) estimated a quantile autoregressive model and concluded that days with extremely negative returns are likely to be followed by periods characterised by weekly positive returns as Bitcoin prices continue to rise. Caporale and Plastun (2019) used a variety of statistical tests and trading simulation approaches and found that after one-day abnormal returns price changes in the same direction are bigger than after "normal" days (the so-called momentum effect).  provided evidence on the role played by the frequency of overreactions. Qing et al. (2019) applied DFA and MF-DFA methods and found momentum effects in Bitcoin and Ethereum prices after abnormal returns. Momentum effects were also detected by Panagiotis et al. (2019) and Yukun and Tsyvinski (2019). The present study extends the previous one by  as detailed below.

Methodology
The selected sample includes daily and monthly BitCoin data over the period 06.2013-02.2020. The data source is CoinMarketCap: https://coinmarketcap.com/currencies/bitcoin/. For forecasting purposes two subsamples are created, namely 01.06. 2013-30.12.2018 and 01.01.2019-28.02.2020 at the daily frequency, and June 2013-December 2018 and January 2019-February 2020 at the monthly frequency; various models are estimated over the first subsample, forecasts are then generated in each case for the second subsample using the estimated parameters and their accuracy is evaluated by means of various statistical criteria.
As a first step abnormal returns are computed using the daily series. The dynamic trigger approach is based on relative values, specifically abnormal returns are defined on the basis of the number of standard deviations to be added to average returns (Wong, 1997). By contrast, the static approach requires setting a threshold; for example, Bremer and Sweeney (1991) use a 10% price change as a criterion for abnormal returns.  compared the suitability of these methods in the case of the cryptocurrency markets and concluded that the latter is preferable; therefore this will be applied here.
Returns are defined as: Electronic copy available at: https://ssrn.com/abstract=3614279 where stands for returns, and and −1 are the close prices of the current and previous day. To analyse their frequency distribution histograms are created. Values 10% above or below those of the population are plotted. Thresholds are then obtained for both positive and negative abnormal returns, and periods can be identified when returns were above or equal to the threshold. Such a procedure generates a data set for daily abnormal returns. We then calculate their frequency, namely the cumulative number of positive / negative abnormal returns detected during a month (which is a time-varying parameter changing on a daily basis) and use the end-of-the-month values for the following regression analysis.
Next the data set for the frequency of abnormal returns is divided into 3 subsets including respectively the frequency of negative and positive abnormal returns, and their difference, known as delta. The relationship between the frequency of one-day abnormal returns and Bitcoin returns is investigated by using a number of regression techniques and model specifications including standard OLS, weighted least squares (WLS), ARIMA and ARMAX models, quantile regressions, Logit and Probit regressions, piecewise linear regressions and non-linear regressions.
The specification of the standard OLS regression is the following (2): where -BitCoin log returns in period (month) t; a 0 -BitCoin mean log return; a 1 (a 2 )coefficients on the frequency of positive and negative one-day abnormal price respectively; F t + (F t − )the frequency of positive (negative) one-day abnormal price days during period t; -Random error term at time t.

An OLS regression including the single parameter
The size, sign and statistical significance of the estimated coefficients provide information about the possible effects of the frequency of daily abnormal returns on BitCoin log returns. The weighted least squares regressions are similar, but instead of Electronic copy available at: https://ssrn.com/abstract=3614279 treating all observations equally they are weighted to increase the accuracy of the estimates.
To obtain further evidence an ARMA(p,q) model is also estimated (4): coefficients the lagged log returns and random error terms respectively; ε trandom error term at time t; This is a special case of an ARIMA(p,d,q) specification with d=0, which is appropriate in our case since all series are stationary, as indicated by a variety of unity root tests which imply that differencing is not required (the test results are not reported for reasons of space but are available from the authors upon request).
Next, in order to improve the basic ARMA(p,q) specification exogenous variables are added, namely the frequency of negative and positive one-day abnormal returns in (5) and Delta in (6), to obtain the following ARMAX(p,q,2) and ARMAX(p,q,1) models: A non-parametric method not requiring normality is also used; specifically, quantile regressions are run to estimate the conditional median instead of the conditional mean. More precisely, the quantile regression model for the -th quantile is specified as follows (7-8): wherethe -th quantile and ∈ (0,1); Next, Probit and Logit regression models are estimated. These are specific cases of binary choice models that provide estimates of the probability that the dependent variable will take the value 1. In a Logit regression, it is assumed that -is the logistic function, and the parameter z is obtained from the regression (9-10): where is a binary variable equal to 1 if the return in month t increased compared to day t-1, and 0 otherwise.
To allow for the possibility that the linear relationship between the dependent variable and the independent ones changes between subsamples a piecewise linear regression is then run to obtain estimates of the coefficients of interest before and after a given breakpoint, specifically: where С 1 and С 2 are the breakpoints.
Possible non-linearities are also considered by estimating a non-linear regression model (NLS) such as: Specifically, we run the following regression: where a 0 ,b,c, p, q are the model parameters.
Information criteria, namely AIC (Akaike, 1974) and BIC (Schwarz, 1978), are used to select the best model specification for Bitcoin log returns. To compare the forecasting performance of different models various measures such as the Mean Absolute Error (MAE) and Theil's statistic are computed instead.

Empirical Results
As a first step, thresholds are calculated by analysing the frequency distribution of log returns to detect abnormal returns (see Table A.1 and Figure A.1). As can be seen, two symmetric fat tails are present in the distribution for log returns: -0.04 for negative returns and 0.05 for positive ones; these are then used as the thresholds to detect negative and positive abnormal returns respectively.
Next we carry out correlation analysis for negative and positive abnormal returns and Bitcoin log returns as in . Specifically, we compute the correlation between Delta and Bitcoin log returns, which is equal to 0.87, and to make sure that there is no need to shift the data we calculate the cross-correlations at the time intervals t and t+i , where I = {-10, . . . , 10}. Figure 1 shows them over the whole sample period for different leads and lags. The highest coefficient corresponds to lag length zero, which means that there is no need to shift the data.  Electronic copy available at: https://ssrn.com/abstract=3614279 The OLS and WLS regression results are reported in Table 1. Models 1 and 2 are the standard OLS regressions given by (2) and (3), whilst models 1.1 and 2.1 are the WLS ones, where the weights are the inverse of the standard error for each observation used. As can be seen the two sets of estimates are very similar. The selected specification, on the basis of the R-squared for the whole model, the p-values for the individual estimated coefficients as well as AIC and BIC criteria, is the following: Bitcoin log return i = 0.0650 + 0.0993 × F i which implies a significant positive (negative) relationship between Bitcoin log returns and the frequency of positive (negative) abnormal returns. Any difference between the actual and estimated values suggests that Bitcoin is over-or undervalued, and therefore that it should be sold or bought till the observed difference disappears, at which stage positions should be closed.
The estimates from the selected ARMA(p,q) models on the basis of the AIC and BIC information criteria, namely ARMA(2,2) and ARMA(3,3), are presented in Table 2. As can be seen, although most coefficients are significant, the explanatory power of these models is rather low.  To establish whether it can be improved by taking into account information about the frequency of abnormal returns, ARMAX models (4) are estimated. First  t F (the frequency of positive abnormal returns) and  t F (the frequency of negative abnormal returns) are added as regressors. The estimated parameters are reported in Table 3. Model 6 and 7 correspond respectively to Model 3 and 4 with the frequency of negative and positive abnormal returns as additional regressors. They outperform Model 5, namely the best ARMAX specification with p=1. Table 4 reports instead the estimates from the ARMAX models with Delta as a regressor. As can be seen all coefficients in Tables 3 and 4 are statistically significant.
The best model on the basis of the AIC and BIC criteria is the one with Delta as a regressor. The R 2 indicates that the ARMAX (3,3,1) is the most adequate model (Model 10). Tables 5, 6, and 7 report the estimates from the quantile regression models with quantiles equal to 0.4, 0.5 and 0.6 respectively, where the 0.5 quantile corresponds to the regression using the median.   In Models 11, 13 and 15 the regressors are the frequency of negative and positive daily abnormal returns, whilst in Models 12, 14, 16 Delta is the independent variable. In the case of the quantile regression with Q=0.5 Model 13 is the most adequate according to AIC.
The Logit and Probit regression results are presented in Table 8. As a selection criterion the percentage of correctly predicted cases is used; this suggests that the best specification is Model 19 which includes the frequency of negative and positive daily abnormal returns.  Table 9 shows the piecewise linear regression results. Model 2 includes the frequency of negative and positive daily abnormal returns and С 1 =0 is used as a breakpoint: for С 1 >0 BitCoin returns are positive, otherwise (С 1 <0) they are negative. Model 22 includes instead the Delta parameter with С 2 =0 as the breakpoint. Both R 2 and AIC imply that Model 21 should be preferred.   Non-linear models of two types are estimated next: non-linear in the regressors (but linear in the parameters) and in the parameters respectively. In the first case the model can be transformed into a linear one by replacing the variables, and then the parameters can be estimated using OLS. In the second case iterative procedures have to be used instead.
The first type can be formulated as follows (13) random error.
The modified variables (selected after some experimentation) are the following: Table 10 reports the corresponding parameter estimates. As can be seen both models 23 and 24 have statistically significant coefficients, but according to R 2 and AIC Model 24 should be preferred. The second type of non-linear model incorporates a new variable, namely x 6 = x 1 x 5 , and is specified as follows: The corresponding estimates are shown in Table 11. All coefficients are statistically significant. Model 27 is the most data congruent:  Table 12 reports the ranking of the top five models (of the 29 considered) according to the AIC criterion. As can be seen the non-linear and piecewise linear regressions appear to be the most data congruent.  Table 13 ranks the rival models in terms of their forecasting performance using the Mean Absolute Error (MAE) and Theil's U criteria. It can be seen that Models 21 and 22 (piecewise linear regressions) are still in in the top five specifications, and therefore the overall evidence based on both in-sample and out-of-sample performance suggests that they are the best models for Bitcoin returns.
Finally, we evaluate the accuracy of the "consensus" forecasts produced by a model that combines the top five selected above and therefore is specified as follows: F=4.1356 (0.0374) where the weights have been estimated by running a standard multiple linear regression. As can be seen from the forecasting accuracy measures reported in Table  C.1, this model outperforms all the individual ones.

Conclusions
This paper carries out a comprehensive examination of the role played by the frequency of daily abnormal returns in driving Bitcoin returns over the period from June 2013 to February 2020. It extends the work of  by considering a much wider range of models over a longer sample period, exploring the role of the difference between the frequency of positive and negative abnormal returns as well, and assessing the forecasting accuracy of the rival models in addition to their in-sample performance. The results indicate that, if one takes into account both in-sample and out-of-sample performance, piecewise linear models are the best for Bitcoin returns. However, in terms of forecasting accuracy they are outperformed by a model that combines the top five to produce "consensus" forecasts.
The finding that there exist price patterns that can be exploited to predict future price movements and design profitable trading strategies is of interest both to academics (since it represents evidence against the EMH) and to practitioners (who can use this information for their investment decisions).