Looking for Psychological Barriers in Nine European Stock Market Indices

1. INTRODUCTION

more distant position.The TSE 300,CAC 40,Hang Seng and Nikkei 225 exhibit some significant evidence.They also analysed the distribution of the returns with regard to expected returns and volatility in a modified GARCH model to conclude that upward movements through barriers tended to have a consistently positive impact on the conditional mean return and also that conditional variance tended to be higher in pre-crossing subperiods and lower in post-crossing subperiods.More recently, Bahng (2003) applied the methodology of Donaldson and Kim (1993) to analyse seven major Asian indices including South Korea, Taiwan, Hong Kong, Thailand, Malaysia, Singapore, andIndonesia between 1990 and1999.Their analysis showed that the Taiwanese index did possess price barrier effects and that the price level distributions of the Taiwanese, Indonesian, and Hong Kong indices were explained by quadratic functions.Finally, Dorfleitner and Klein (2009) focused on the DAX 30, the CAC 40, the FTSE-50 and the Euro-zonerelated DJ EURO STOXX 50 for different periods until 2003.They found fragile traces of psychological barriers in all indices at the 1000-level.There were also indications of barriers at the 100-level except in the CAC index.Other studies concerning psychological barriers in stock markets are also related to our analysis.It is the case of those articles that address the presence of barriers in individual stock prices such as Cai et al. (2007) and Dorfleitner and Klein (2009).Cai et al. (2007) assessed the existence of psychological barriers in a total of 1050 A-shares and 100 B-shares from both the Shanghai Stock Exchange and the Shenzhen Stock Exchange during June 2002.A range of measures for price resistance showed the digits 0 and 5 to be significant resistance points in the A-share market.No resistance point was found in the Shanghai B-share market, although digit 0 has had the highest level of resistance compared to others.Dorfleitner and Klein (2009) analysed eight major stocks from the German DAX 30 over the period May 1996-June 2003.The prices were examined with respect to the frequency with which they lied within a certain band around the barrier.In addition, they studied barrier's influence on intraday variances and the daily trading volume.Overall, the authors were not able to identify a systematic and consistent pattern at barriers.Different studies concluded that price barriers or at least significant deviations from uniformity also exist in other asset classes such as exchange rates (De Grauwe and Decupere, 1992), bonds (Burke, 2001), commodities (Aggarwal and Lucey, 2007) and derivatives (Schwartz et al., 2004;Chen and Tai, 2011;Jang, 2013;Dowling et al., 2016).Overall, evidence of price barriers in various asset classes seems to be fairly robust.

Data
The examination window for each of the stock market indices under study is presented in Table 1 below.Starting dates are different since we used the data pertaining each index since its inception.All the data were retrieved from Thomson Reuters Datastream.Summary statistics on the stock prices are presented in Table 2 where it can be seen that the measures of skewness and, especially, kurtosis are in general inconsistent with normality.

Definition of Barriers
Following Brock et al. (1992) and Dorfleitner and Klein (2009), we will use the so-called band technique and barriers will thus be defined as a certain range around the actual barrier.The main reason is that market participants will most certainly become active at a certain level before the index touches a round price level.Considering an index level of 100, for instance, over-excitement is expected to begin for instance at 99 or 101, or even at 95 or 105.Barriers will thus be defined as multiples of the lth power of ten, with intervals with an absolute length of 2% and 5% of the corresponding power of ten as barriers.Formally, we may consider four possible barrier bands: M100: Barrier level l=3 (1000s) 980-20; 950-50 M10: Barrier level l=2 (100s) 98-02; 95-05 (1) M1: Barrier level l=1 (10s) 9.8-0.2;9.5-0.5 M0.1: Barrier level l=0 (1s).0.98-0.02;0.95-0.05

Uniformity Test
Having computed the M-values, the next step consists of examining the uniformity of their distribution.Following Aggarwal and Lucey (2007), this will be done through a Kolmogrov-Smirnov Z-statistic test.Thus we will be testing H0: uniformity of the M-values distribution against H1: non-uniformity of the M-values distribution.
It is important to emphasize that the rejection of uniformity might suggest the existence of significant psychological barriers but it is not in itself sufficient to prove the existence of psychological barriers.Ley and Varian (1994) showed that the last digits of the Dow Jones Industrial Average were in fact not uniformly distributed and even appeared to exhibit certain patterns, but the returns conditional on the digit realization were still significantly random.Additionally, De Ceuster et al. (1998) noted that as a series grows without limit and the intervals between barriers become wider, the theoretical distribution of digits and the respective frequency of occurrence is no longer uniform.

Barrier Tests
Barrier tests are used to assess whether observations are less frequent near barriers than it would be expected considering a uniform distribution.The existence of a psychological barrier implies we will observe a significantly lower closing price frequency within an interval around the barrier (Donald and Kim, 1993;Ley and Varian, 1994).Therefore, the objective of the barrier tests is to investigate the influence of round numbers in the non-uniform distribution of M-values.We will use two types of barrier tests: the barrier proximity test and the barrier hump test.

a) Barrier Proximity Test
This test examines the frequency of observations, f(M), near potential barriers and will be performed according to equation (8).
() =  +  +  (8) The dummy variable will take the value of unity when the index is at the supposed barrier and zero elsewhere.As it was mentioned in section 3.2.1, this barrier will not be strictly considered as an exact number but also as a number of different specific intervals, namely with an absolute length of 2% and 5% of the corresponding power of ten as barriers.The null hypothesis of no barriers will thus imply that β equals zero, while β is expected to be negative and significant in the presence of barriers as a result of lower frequency of M-values at these levels.

b) Barrier Hump Test
The second barrier test will examine not just the tails of frequency distribution near the potential barriers, but the entire shape of the distribution.It is thus necessary to define the alternative shape that the distribution should have in the presence of barriers (Donaldson and Kim, 1993;Aggarwal and Lucey, 2007).Bertola and Caballero (1992), who analysed the behaviour of exchange rates in the presence of target zones imposed by forward-looking agents, suggest that a hump-shape is an appropriate alternative for the distribution of observations.The test to examine this possibility will follow equation ( 9), in which the frequency of observation of each M-value is regressed on the M-value itself and on its square.
() =  +  +  2 +  (9) Under the null hypothesis of no barriers ϒ is expected to be zero, whereas the presence of barriers should result in ϒ being negative and significant.

Conditional Effect Tests
The rejection of uniformity on the observations of M-values is not sufficient to prove the existence of psychological barriers (Ley and Varian, 1994).Therefore, it is necessary to analyse the dynamics of the returns series around these barriers, namely regarding mean and variance in order to examine the differential effect on returns due to prices being near a barrier, and whether these barriers were being approached on an upward or on a downward movement (Cyree et al., 1999;Aggarwal and Lucey, 2007).
Accordingly, we will thus define four regimes around barriers: BD for the five days before prices reaching a barrier on a downward movement, AD for the five days after prices crossing a barrier on a downward movement, and BU and AU for the five days respectively before and after prices breaching a barrier on an upward movement.These dummy variables will take the value of unity for the days noted and zero otherwise.In the absence of barriers, we expect the coefficients on the indicator variables in the mean equation to be non-significantly different from zero.
=  1 +  2   +  3   +  4   +  5   +   (10) Following Aggarwal and Lucey (2007), we started with an OLS estimation of Eq. (3.9) but heteroscedasticity and autocorrelation were clearly present across our data base.Therefore, the full analysis of the effects in the proximity of barriers required us to apply the former test also to the variances.Equation ( 11) represents this approach assuming autocorrelation similar to one as in Cyree et al. (1999) and Aggarwal and Lucey (2007).Besides the abovementioned dummy variables it includes a moving average parameter and a GARCH parameter.
The four possible hypothesis to be tested are the following: H1: There is no difference in the conditional mean return before and after a downward crossing of a barrier.H2: There is no difference in the conditional mean return before and after an upward crossing of a barrier.H3: There is no difference in conditional variance before and after a downward crossing of a barrier.H4: There is no difference in the conditional variance before and after a upward crossing of a barrier.

Uniformity Test
Table 3 provides the results of a uniformity test concerning the distribution of digits for the nine stock market indices under scrutiny.Overall, there is robust evidence that the M-values do not follow a uniform distribution.Uniformity is clearly rejected for all data series.Considering a statistical significance level of 5%, uniformity is never rejected.These findings are in line with the ones obtained by other authors (e.g., Cyree et al, 1999;Dorfleitner and Klein, 2003) although their results were more heterogeneous than ours.3 shows the results of a Kolmogorov-Smirnov test for uniformity.Each test was performed for the daily closing prices of each stock index.D stands for the value of the test statistic while P-value gives the marginal significance of this statistic.H0: uniformity in the distribution of digits, H1: non uniformity in the distribution of digits.***: significant at the 1 percent level; **: significant at the 5 percent level.

Barrier Tests 4.2.1. Barrier Proximity Test
Results for the barrier proximity tests are shown in Tables 4 to 6 for the intervals mentioned in sections 3.2.1 and 3.2.4.As referred above, in the presence of a barrier we would expect β to be negative and significant, implying a lower frequency of M-values at these points.Considering a barrier in the exact zero modulo point, evidence in Table 4 shows that all the data series seem to reject the no barrier hypothesis at a statistical significance of 10%.All the significant results were detected on the two highest levels, i.e., the 100-and the 1000-barrier levels.
When we widen the barrier interval, evidence of psychological barriers appear to be weaker.In fact, if we assume a barrier to be in the interval 98-02, only Finland, Germany and the Netherlands seem to reject the no barrier hypothesis at a statistical significance of 1% (see Table 5).Considering the 95-05 interval, Table 6 shows that the no barrier hypothesis is again rejected for the same three countries and also for Belgium.All the other series are either not significant or β is not negative.Overall, evidence suggests that psychological barriers are a relevant phenomenon for the all indices of the sample but only at 100-and 1000-barrier levels.R-squares are significantly low, which is in line with previous studies.Table 5 shows the results of a regression f(M)=α+βD+ε, where f(M) stands for the frequency of appearance of the M-values, D is a dummy variable that takes the value of unity when M=value is in the 98-02 interval and 0 otherwise.Refer to section 3.2.4 for details.*** significant at the 1% level; **: significant at the 5 percent level.6 shows the results of a regression f(M)=α+βD+ε, where f(M) stands for the frequency of appearance of the M-values, D is a dummy variable that takes the value of unity when M=value is in the 95-05 interval and 0 otherwise.Refer to section 3.2.4 for details.***: significant at the 1 % level; **: significant at the 5% level.

Barrier Hump Test
Table 7 shows the results for the barrier hump test, which is meant to test the entire shape of the distribution of M-values.Assuming it should follow a hump-shape distribution, we thus expected ϒ to be negative and significant in the presence of barriers.The results of the barrier hump test partially confirm the evidence presented previously with the barrier proximity tests.The stock market indices of Germany and Finland stand out again as they are the only ones that exhibited a persistent barrier, namely at the 1000-level barrier, at a statistically significant level of 1%.All the other series are either not significant or ϒ is not negative.

Conditional Effects Test
Assuming the existence of psychological barriers, we expected the dynamics of return series to be different around these points.In fact, results in Table 8 provide some interesting evidence of mean effects around barriers as it is observed, on one hand, that stock market returns in all nine markets tend to be significantly higher when a barrier is to be crossed in an upward movement.On the other hand, the coefficients of BD and AD are negative and significant for all indices which means that stock market return tend to be significantly lower in the proximity of a barrier when that barrier is to be crossed on a downward movement.This pattern of conditional effects is similar to the one obtained by Cyree et al. (1999).
Table 8 shows the results of the mean equation of a GARCH estimation of the form Rt=β1+ β2BD+ β3AD+ β4BU+ β5AU+εt; εt ~N(0,Vt); Vt= α1+ α2BD+ α3AD+ α4BU+ α5AU+α6Vt-1+α7ε 2 t-1+ηt.BD, AD, BU and AU are dummy variables.BD takes the value 1 in the 5 days before crossing a barrier on a downward movement and zero otherwise, whereas AD is for the 5 days after the same event.BU is for the 5 days before crossing a barrier from below, while AU is 1 in the 5 days after the same upward crossing.Vt-1 refers to the moving average parameter and ε 2 t-1 stands for the GARCH parameter.***: significant at the 1 % level; **: significant at the 5% level.All coefficients of the lagged squared residuals are positive and significant at the 1% level pointing out to an increase in conditional variance coincident with higher residuals from the previous period.The GARCH term in the conditional variance is positive and significant, suggesting significant GARCH effects around barriers.The GARCH term corresponding to the Finnish market is closer to one which indicates a higher level of volatility persistence.The variance effects are particularly evident before an upward movement through a barrier: the coefficient of BU in the variance equation is negative and statistically significant in most the markets under study.This indicates that the markets tend to calm before having risen through a barrier.This is in sharp contradiction with the results obtained by Cyree et al. (1999) according to which, in most cases, markets tend to be more volatile before crossing a barrier in an upward movement.In the pre-crossing period but in the case of a downward movement, the results are heterogeneous: Germany and the Netherlands have statistically significant results whereas the coefficient corresponding to the European market as a whole is negative.
The results in the post-crossing period are also somewhat heterogeneous.It is not possible to discern a clear trend in the volatility level after crossing a barrier in an upward movement.The volatility has increased after the crossing of a barrier in a downward movement for four of the indices but the markets of Germany and the Netherlands present important exceptions.Table 9 shows the results of the variance equation of a GARCH estimation of the form Rt=β1+ β2BD+ β3AD+ β4BU+ β5AU+εt; εt ~N(0,Vt); Vt= α1+ α2BD+ α3AD+ α4BU+ α5AU+α6Vt-1+α7ε 2 t-1+ηt.BD, AD, BU and AU are dummy variables.BD takes the value 1 in the 5 days before crossing a barrier on a downward movement and zero otherwise, whereas AD is for the 5 days after the same event.BU is for the 5 days before crossing a barrier from below, while AU is 1 in the 5 days after the same upward crossing.Vt-1 refers to the moving average parameter and ε 2 t-1 stands for the GARCH parameter.*, **, *** indicates significance at the 10%, 5% and 1% level, respectively.
Table 10 shows the test results of the four barrier hypothesis mentioned in section 3.2.5.If some kind of barrier indeed existed, we would expect that the restraints in terms of mean and variance would be relaxed after the price crossed that barrier.With this test we are now able to examine the differences in returns and volatility before and after crossing a potential psychological barrier and thus we are also able to assess the relationship between these two parameters.In the time horizon of five days, the stock market indices of Austria and France did not show a statistically significant different behavior before and after crossing a barrier.In the markets of Luxembourg and the Netherlands, the differences in the variance were matched by a corresponding changes in the returns, over the same circumstances.However, in other markets important changes were observed in only one of the parameters (return or variance).In the case of the market of Belgium, for example, there was a statistically significant difference in the market return in the case of an upward crossing of a barrier with no statistically significant change in the variance.In the cases of Finland, Germany and of the European index, it was the volatility that has changed with no significant variation in the observed return.In Ireland, the two parameters showed significant changes but in different situations.The fragility in the relationship between risk and return, both in cross-sectional and in temporal frameworks, has been highlighted by several authors over the last decades.For example, Fama andFrench (1998, 2004) have shown that, after controlling the data for factors such as the book-to-market and the stock capitalization, the relationship between the observed returns and the beta risk parameter becomes statistically non-significant, if not negative.And more recently, Savor and Wilson (2014) have shown that beta is positively related to average stock returns only on days when macroeconomics news regarding employment, inflation, and interest rate are scheduled to be announced.On the remaining days, beta is unrelated or even negatively related to average returns.The results of our study suggest an additional circumstance where the relationship between risk and return tends to be weaker: in the proximity of psychological barriers (in our case, round numbers).Psychological barriers continue to represent a fertile field for future research.It would be interesting to investigate why the incidence of psychological barriers, like other market anomalies (see, e.g., Stambaugh et al., 2012), seem to vary both cross-sectionally and over time.

Table 1 -
Data used in the study

Table 3 -
Kolmogorov-Smirnov test for uniformity of digits

Table 4 -
Barrier proximity test: strict barrier

Table 8 -
Table 9 contains results for the conditional variance equation.The constant is positive and significant for all indices.

Table 10 -
Barrier hypothesis tests This has led us to conclude that the relationship between risk and return became weaker around psychological barriers.