The Frequency of One-Day Abnormal Returns and Price Fluctuations in the Forex

This paper analyses the explanatory power of the frequency of abnormal returns in the FOREX for the EURUSD, GBRUSD, USDJPY, EURJPY, GBPCHF, AUDUSD and USDCAD exchange rates over the period 1994-2019. Abnormal returns are detected using a dynamic trigger approach; then the following hypotheses are tested: their frequency is a significant driver of price movements (H1); it does not exhibit seasonal patterns (H2); it is stable over time (H3). For our purposes a variety of statistical methods (both parametric and non-parametric) are applied including ADF tests, Granger causality tests, correlation analysis, (multiple) regression analysis, Probit and Logit regression models. No evidence is found of either seasonal patterns or instability. However, there appears to be a strong positive (negative) relationship between returns in the FOREX and the frequency of positive (negative) abnormal returns. On the whole, the results suggest that the latter is an important driver of price dynamics in the FOREX, is informative about crises and can be the basis of profitable trading strategies, which is inconsistent with market efficiency.


Introduction
The FOREX is one of the most liquid (with $6 tn daily turnover) and efficient financial markets (Oh et al., 2006;Serbinenko andRachev, 2009, Kallianiotis, 2017). Nevertheless, several studies have attempted to detect anomalies in the behaviour of exchange rates such as abnormal returns with the associated contrarian or momentum patterns (Parikakis and Syriopoulos, 2008;Caporale et al., 2018), and also investigated whether they can be used as an early warning indicators for financial crises (e.g., the East Asian and the Russian crises of the 1990s, the Dotcom bubble of 1997-2001, and the global financial crisis of 2007-8). The various methods used include price trends and persistence analysis, trade volumes and price volatility analysis, correlation between assets etc. (Granger and Newbold, 1986;Bremer et al, 1997;Eross et al, 2019).
The present paper takes instead a different approach to analyse the explanatory power of the frequency of abnormal returns; this issue has been previously examined in the case of stock markets (Angelovska, 2016;Caporale and Plastun, 2019) and cryptocurrency markets (Caporale et al., 2019), but not in that of the FOREX, which is the focus of this study.
Abnormal returns are detected using a dynamic trigger approach. Then the following hypotheses are tested: (i) their frequency is a significant driver of price movements (H1); (ii) it does not exhibit seasonal patterns (H2); (iii) it is stable over time (H3). For our purposes a variety of statistical methods (both parametric and nonparametric) are applied including ADF tests, Granger causality tests, correlation analysis, (multiple) regression analysis, Probit and Logit regression models.
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 offers some concluding remarks.

Literature Review
There exists an extensive literature investigating one-day abnormal price changes. Various explanations have been suggested for their occurrence. For instance, Govindaraj et al. (2014) and Jin et al. (2012) examined the role of new information, noise or liquidity trades. Bartos (2015) argued that new information is immediately absorbed without significant price effects. The most popular explanations rely on cognitive traps and biases (Barberis, Shleifer and Vishny, 1998), as well as emotions and psychological aspects of trading and investment (Daniel et al., 1998, Griffin andTversky, 1992;Madura and Richie, 2004). Aiyagari and Gertler (1999) and Hong and Stein (1999) see their roots in the presence and activity of "noise" traders. Duran and Caginalp (2007) argued that abnormal returns result from the use of technical and fundamental analysis by investors for decision-making. Other studies have considered the impact of market liquidity (Jegadeesh and Titman, 1993), news (Kocenda and Moravcova, 2018) etc.
Abnormal price changes can generate different price patterns. Atkins and Dyl (1990) and Bremer et al. (1997) found contrarian effects (price reversals) after large price changes. By contrast, Cox and Peterson (1994) did not detect a negative correlation between abnormal returns on the day prices fall and the following three days. Schnusenberg and Madura (2001) and Lasfer et al. (2003) provided evidence of momentum effects. Savor (2012) and Govindaraj et al. (2014) found both effects in the US stock market (momentum effects when analysts issue revisions or price reversals after large daily price shocks).
Various other studies also analyse some of the implications of abnormal returns. For instance, Pritamani and Singhal (2001) showed that information about large price changes can be used to design profitable trading strategies. Govindaraj et al. (2014) also found that a trading strategy based on these effects can generate significant excess returns. Similar conclusions were reached by Caporale et al. (2018), who tested price effects after abnormal price returns in different financial markets; they showed that the reversal effect is exploitable in the stock market, whilst the momentum effect produces profits in the case of the FOREX and commodity markets. By contrast, Cox and Peterson (1994) and Lasfer at al. (2003) argued that trading strategies based on price patterns after one-day abnormal returns can hardly be profitable because of the presence of trading costs and the relatively small size of price reversals. According to Sandoval and Franca (2012), abnormal price changes can also be informative about future price movements and be used as a crisis identifier.
Typically abnormal returns are analysed in the case of stock markets (Atkins and Dyl, 1990;Cox and Peterson, 1994;Bremer et al. 1997;Govindaraj et al., 2014;Sandoval and Franca, 2012; Angelovska, 2016 and many others) or cryptocurrency markets; in particular, Caporale and Plastun (2019) and Caporale et al. (2019) showed that the frequency of abnormal returns can provide useful information in the case of the cryptocurrency markets. Much less evidence is available for the FOREX, which is the focus of the present paper. An exception is the study carried out by Parikakis and Syriopoulos (2008), who investigated patterns following excess oneday fluctuations for various currencies and found that a contrarian strategy is profitable in the FOREX.

Methodology
To analyse the frequency of abnormal returns and their role as drivers of price dynamics we use daily and monthly data for the main exchange rates, specifically for EURUSD, GBRUSD, USDJPY, EURJPY, GBPCHF, AUDUSD and USDCAD over the period 03.01.1994-28.05.2019; the data source is Yahoo! Finance (https://finance.yahoo.com).
There are two main approaches to detecting abnormal returns, namely a static one (which uses a specific threshold as an abnormal price criterion, as in Bremer and Sweeney, 1991) and a dynamic one (which is based on relative valuesnormally abnormal returns are defined on the basis of the number of standard deviations to be added to the average return as in Caporale and Plastun, 2018). Since they can perform rather differently depending on the dataset (Caporale et al., 2018) the first step is to choose the most appropriate method for the data in hand.
Let returns be defined as: where stands for returns, and and −1 are the close prices of the current and previous day. The static approach introduced by Sandoval and Franca (2012) and developed by Caporale and Plastun (2019) is based on creating histograms with values 10% above or below those of the population; 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.
In the dynamic trigger approach (Wong, 1997;Caporale et al., 2018) abnormal price changes are defined by the following inequality: (2) and negative abnormal price change are defined as: ( 3) where k is the number of standard deviations used to identify them (specifically, k=1), is the average size of daily returns for period n and is the standard deviation of daily returns for period n Both procedures (static and dynamic) generate a data set for the frequency of abnormal returns (at a monthly frequency), which is then divided into 4 subsets including respectively the frequency of negative and positive abnormal returns, the ) ( Electronic copy available at: https://ssrn.com/abstract=3558607 difference between them and the overall frequency of abnormal returns (positive as well as negative).
Then the following hypotheses are tested: (i) the frequency of abnormal returns is a significant driver of price movements (H1), (ii) it does not exhibit seasonal patterns (H2), (iii) it is stable over time (H3).
To test H1, we regress monthly returns (and any observed momentum or contrarian effects) against the frequency of abnormal returns over a 1-month period; specifically we estimate the following regressions: wherereturns on day t; a 0 -mean return; 1 ( 2 )coefficients on the frequency of positive and negative abnormal returns respectively; F t + (F t − )the number of positive (negative) abnormal returns days during a period t; ε t -Random error term at time t.
wherereturns on day t; a 0 -mean return; 1coefficient on the delta frequency; F t deltathe difference between the number of positive (negative) abnormal returns days during a period t; ε t -Random error term at time t.
As an alternative, Logit and Probit regressions are run. These are binary choice models producing estimates of the probability that the dependent variable will take the value 1 depending on the values of the regressors. In a Logit regression, it is assumed that the probability of event y being equal to 1 is given by , Electronic copy available at: https://ssrn.com/abstract=3558607 where -is the logistic function, and the parameter z is determined on the basis of regression (6).
where is a binary value equal to 1 if the return on day t increased compared to day t-1; otherwise, this value is 0.  If the probability predicted by the model , then the dependent variable is equal to 1, whilst -implies that it is equal to 0. The Probit regression is based on the assumption that the variable under investigation is normally distributed.
The size, sign and statistical significance of the coefficients provide information about the possible effects of the frequency of abnormal returns on returns in the FOREX. A number of diagnostic tests are also carried out; these include Lilliefors's test, Durbin-Watson's test, White's test, Ramsey's Regression Equation Specification Error Test (RESET) and Chow's test. Table 1 specifies the null hypothesis in each case. To test H2 and H3 we perform both parametric (ANOVA analysis) and nonparametric (Kruskal-Wallis) tests.

Empirical Results
As a first step, one needs to choose between the static and dynamic approaches to calculate abnormal returns. For this purpose the EURUSD exchange rate is used. Table 2 reports the correlation coefficients between the two sets of results. As can be seen, in the case of the frequency delta parameter the correlation is rather high; however, the other correlation coefficients imply a sizeable difference between the static and dynamic results. To choose between the two, we focus on the correlation between the frequency of abnormal returns and both close prices and returns. The results are reported in Table 3. As can be seen the frequency of abnormal returns is correlated only with monthly returns, and consequently only these will be used to test the hypotheses of interest; further, the dynamic approach produces higher correlations for the frequency of negative and positive abnormal returns, and therefore will be used in the remainder of the analysis to detect abnormal returns. Finally, since the overall frequency of abnormal returns does not appear to be informative about price dynamics, only the frequency of negative and positive abnormal returns, and the frequency delta, will be used.
ADF tests (Dickey and Fuller, 1979) carried out on the series of interest (see Appendix C, Tables C.1-C.7) imply a rejection of the unit root null in all cases (i.e., stationarity). Table 4 reports the correlation coefficients for the number of negative and positive abnormal returns, as well as the frequency delta between the number of positive and negative abnormal returns and monthly returns. As can be seen, there is negative (positive) correlation between the frequency of negative (positive) abnormal returns and price dynamics in the FOREX, and the frequency delta has the highest (positive) correlation coefficient, which implies that this variable is the most informative about price movements.
As a further check, we carry out cross-correlation analysis also at the time intervals t and t+i, where I ∈ {-10, . . . , 10}. Figures D.1-D.7 reports the crosscorrelation between returns and the frequency of (both positive and negative) abnormal returns for 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.
Additional evidence is provided by Granger causality tests (Granger, 1969) between returns in the FOREX and the frequency of abnormal returns (both positive and negative, and also for their delta). The results are presented in Appendix G, Table G.1. As can be seen, the null hypothesis of no causality cannot be rejected in any case (the single exception is USDJPY). The next step is to test H1 by running a number of simple linear regressions for returns against the frequency of negative and positive abnormal returns and the delta frequency, as well as regressions with dummy variables (see Section 3 for details). The results are presented in Appendix E, Tables E.1-E.7. As can be seen, all the regressors are statistically significant. Both actual and estimated values are plotted in Figures H.1-H.7. The latter appear to capture well the behaviour of the former. Various diagnostic tests for the models from Tables E.1-E.7 are reported in Table 5, and suggest that the estimated models have the appropriate functional form and their residuals are not autocorrelated. The model for the EURUSD exchange rate passes all tests, but there is evidence of non-normality of the residuals in the case of EURJPY, USDJPY, GBPCHF, and both heteroscedasticity of residuals and unstable parameters are present in the models for GBRUSD, AUDUSD and USDCAD. The best specifications for the linear regression models with the frequency of positive and negative abnormal returns as regressors (as indicated by the R-square for the whole model and the p-values for the estimated coefficients) are presented in Table 6.  The Logit and Probit regression results for price closes are presented in Appendix F, Tables F.1-F.7. We find that the explanatory power of these models ranges between 73.9% and 76.3%. On the whole, the evidence supports H1. Concerning H2, namely the possible presence of seasonal patterns in in the frequency of abnormal returns, at first we do some visual inspection of the data. Figure 1 displays positive and negative abnormal returns and the delta frequency by month for EURUSD and provides no prima facie evidence of seasonality for the former two, while the latter appears to be negative in January and May and positive in December. Further evidence of seasonal behaviour for the delta frequency is provided by Figure 2, which shows it for all the exchange rates considered. To see whether these seasonal differences are statistically significant we carry out ANOVA analysis and Kruskal-Wallis tests. The results at the 5% confidence level are reported in Table 7 and suggest that in most cases there are no significant seasonal patterns, which implies a rejection of H2. As for H3 (parameter stability), first we compute the average number of abnormal returns per year (positive+negative) based on all exchange rates considered; this is displayed in Figure 3. As can be seen, it was lower in the 1990s, and peaked in 2004 and 2008, the latter date coinciding with the global financial crisis.

Figure 3: Average frequency of abnormal returns (positive + negative) per year
More detailed evidence is presented in the case of EURUSD in Figure 4, which suggests the presence of time variation. The results of the ANOVA analysis and Kruskal-Wallis tests are reported in Table 8 and imply parameter stability, i.e. H3 cannot be rejected.

Conclusions
This paper investigates the explanatory power of the frequency of one-day abnormal returns in the FOREX for the cases of EURUSD, GBRUSD, USDJPY, EURJPY, GBPCHF, AUDUSD and USDCAD over the period 1994-2019. Using a dynamic trigger approach 4 series are created, specifically the frequency of negative and positive abnormal returns, the difference between the two and the overall frequency of abnormal returns. Then the following hypotheses are tested using a variety of parametric and non-parametric methods: the frequency of abnormal returns is a Positive Delta significant driver of price movements (H1); it does not exhibit seasonal patterns (H2); it is stable over time (H3).
The main findings can be summarised as follows. The frequency of abnormal returns in FOREX has significant explanatory power for returns, is informative about crises (since it increases sharply at the time of a crisis), is not seasonal, and is stable over time. On the whole, our findings suggest that profitable FOREX trading strategies can be designed based on the frequency of abnormal returns, which is evidence of market inefficiency. The difference between actual and estimated returns can be seen as an indication of whether currencies are over-or under-valued and therefore a price increase or decrease should be expected. Obviously currencies should be bought in the case of undervaluation and sold in the case of overvaluation till the divergence between actual and estimated values disappears, at which stage positions should be closed.    2008  54  57  49  49  46  42  52  50  2009  34  41  37  39  44  38  34  38  2010  44  48  36  41  41  41  42  42  2011  40  45  32  45  44  42  33  40  2012  40  41  47  37  39  44  40  41  2013  32  43  47  43  47  44  46  43  2014  45  39  38  45  40  39  41  41  2015  45  44  35  40  45  39  32  40  2016  42  41  39  40  43  41  41  41  2017  36  36  37  37  35  36  43  37  2018  51  42  34  46  44  39  42  43   This table presents         These tables present the results of the Augmented Dickey-Fuller test. The first specifies the parameter of the Augmented Dickey-Fuller test being considered, the second column shows the results for returns ("returns"); the third column for delta frequency data ("delta"); the fourth column shows parameter estimates for negative abnormal returns ("Negative") and the fifth column for positive abnormal returns ("Positive"). The Lag Length was chosen on the basis of the Akaike information criterion. The results are significant at the 5% level.        These tables present the coefficient estimates and p-values (in parentheses) from the regression models. The second column reports the parameter estimates for delta frequency, the third for the frequency of both positive and negative abnormal returns as separate variables.  These tables present results for monthly price closes regressed against frequency of negative and positive abnormal returns as well as delta frequency. Coefficient estimates and p-values (in parentheses) from regression models are provided in these tables. The first column reports the model parameters, the second and third the estimates from the Logit models, and the fourth and fifth those from the Probit models.