Should Investors on Equity Markets Be Superstitious ? ( Example of 7 World Stock Indexes Components )

The problem of efficiency of financial markets, especially the weekend effect has always fascinated scholars and practitioners due to its relationship with the financial market efficiency. The issue is significant from the point of view of assessing the portfolio management effectiveness and behavioral finance. This paper tests the hypothesis of the unfortunate dates effect upon 7 equity indexes components (CAC40, DAX, DJIA, FTSE30, FTSE MIBTEL, NIKKEI225 and SENSEX), i.e. 419 companies. For all these equities the following rates of return were analyzed: Close-close, Overnight, Open-open, Open-close. As unfortunate days, the sessions falling on the following dates were selected: 13 th and 4 th day of the month, Friday the 13 th and Tuesday the 13 th . The research proved the presence of all kinds of the “unfortunate dates” effects on analyzed markets. The effects were registered for all analyzed rates of return. The most dominating “unfortunate dates effects” resulted to be Tuesday the 13 th , proceeding the 4 th day of the month effect. This is the first analysis of the presence of the “unfortunate dates effect”, in which other than Close-close returns were examined and fulfils the research gap.


Introduction
Efficient market hypothesis (EMH), introduced by Fama (Fama, 1970) belongs to the most important paradigms of the traditional financial theories.According to this hypothesis, efficient market is defined as a market with a large numbers of rational individuals, maximizing their profit and actively competing with each other undertaking the attempt to predict future market values of specific securities, and where all relevant information is freely available to investors (Latif et al., 2011).The presence of calendar anomalies has been presented extensively for the last three decades in financial markets.The most common ones are the day-of-the-week effect, monthly effect, weekend effect, holiday effects, within-the-month effect, turn-of-the month effect (Agrawal, Tandon, 1994;Boudreaux, 1995;Smirlock & Starks, 1986;Aggarval & Rivoli, 1989;Barone, 1990;Kato et al., 1990;Gu, 2003;Schwert, 2002;Sutheebanjard & Premchaiswadi, 2010).
Another issue related to the financial market efficiency is the behavior of investors during the days considered by them to be unlucky.In Western Europe, every 13 th day of the month, especially the 13 th day of the month when falling on a Friday is to be believed unlucky.In turn, in Spanish-speaking countries (e.g.Spain, Uruguay, Argentina, Chile, Peru, Venezuela and Colombia), it is assumed that the date of bringing bad luck is Tuesday the 13 th , what is expressed in the following Spanish proverb: trece martes ni te cases, ni te embarques (Tuesday the 13 th , don"t get married and don"t travel).On the other hand, in China, an unlucky date is every fourth day of the month.Many Chinese people believe the number 4 is to be unlucky whilst considering the number 8 is a lucky one (Agarwal et al., 2014).In some Chinese dialects, the number 8 is pronounced like the word "prosperity", while the number 4 similar to the word "death".Apparently the Chinese vary in their definition of which numbers are lucky.Shum et al. (2012) defined both 6 and 8 as lucky, while Hirshleifer et al. (2018) considered 6, 8 and 9 to be lucky.
Statistically important difference between daily average rates of return registered on the stock market considered by investors as an unlucky date and daily average rates of return calculated for the others days of the month can be called "the unfortunate dates effect".The number of papers dedicated to "the unfortunate dates effect" in scientific literature is rather low.
The aim of this paper is to examine the prevalence of "the unfortunate dates effect" on the markets of 7 world equity index components.The paper is divided into five parts.The first four parts analyze of "the unfortunate dates effect" that apply to the returns calculated on the basis of the following prices: (1) last session closeprevious session close (close-close), ( 2 (open-close).All calculations will be carried out for the following two populations: (1) the 13 th day of the month rates of return vs rates of return for all other sessions, (2) Friday the 13 th rates of return vs rates of return for all other sessions, (3) Tuesday the 13 th rates of return vs rates of return for all other sessions and (4) the 4 th day of month rates of return vs rates of return for all other sessions.In the fifth part of the paper the one-session rates of return for Friday the 13 th session will be compared with the one-session rates of return for all other Fridays.In turn, in the second part of the fifth part of the paper the similar analysis for rates of return for Tuesday the 13 th and all other Tuesdays will be conducted.Previous researches focused on the calculation of rates of return only for the following scheme: Friday the 13 th close -others Fridays" close.The author is not aware of the papers analyzing the Friday the 13 th effect with the use of rates of return different to the close-close scheme.This article attempts to fill this gap, as well as expand research for Tuesday the 13 th and for the sessions falling on the 4 th day of the month.

Literature Review
Belief in the ill-fortune that supposedly accompanies the of 13 th as well as the date of Friday the 13 th is widespread across the Western world and has ancient and somewhat uncertain origins (Boyle et al., 2004).Both the number 13 and Friday are characterized by long and separate histories associated with "bad luck".It is believed that these two were combined in order to create an unfortunate date at the beginning of the 20 th Century (Chaundler, 1970).In the literature there are a lot of explanations for these two lines of superstitions: Christ was crucified on Friday, and the number of people seated at the table for the Last Supper was 13.Even in developed countries, people are prone to superstitions such as daily newspapers publishing horoscopes to guide their readers.Nowadays many buildings skip the thirteenth floor, streets lack the number 13 th and hospitals in many countries decline to label their operating theatres with that number (Hira et al., 1998;Reilly & Stevenson, 2000;Boyle et al., 2004;USA, Today, 2007;Kramer & Block, 2008).Of more interest is the fact that admittance to hospitals seems to cluster around unlucky days, as reported by Blacher (1983) and Scanon et al. (1993).Fudenberg and Levine (2006) state that superstitious beliefs can persist if the probability of being exposed as untrue is sufficiently low.If there is always any chance of a bad outcome when following superstition and some chance of a good outcome when not following superstition, any person might not realize that the belief is untrue, and, persists in the superstition (Agarval et al., 2014).Jiang et al. (2009) found that Asians exposed to lucky numbers, give higher estimates of winning a lottery and are more willing to participate in a lottery or a risky promotional game, and express greater willingness to make risky financial investments.Chong and Du (2009) estimated the value of superstition: a lucky (unlucky) number can bring good (bad) luck, and the value of superstitions can be economically significant.Psychology and anthropology researchers suggest that people rely on superstition as a way to cope with misfortune and uncertainty, and to rationalize a complex world (Vyse, 1997;Tsang, 2004;Lepori, 2009;Liu, 2013, Zhang et al., 2014, Robiyanto & Puryandani, 2015, Robiyanto et al., 2015).Scanlon et al. (1993) founded that the number of traffic accident in UK is higher on Friday the 13 th , in spite of the smaller number on cars being on the roads.Kolb and Rodriguez (1987), in one of the first studies linking superstition with the stock market, proved that in the period of 1962-1985, the average Friday 13 th rates of return of CRSP Index are significantly lower than the average rates of return for all other Fridays.Later papers of Dyl and Maberly (1988), Agrawal and Tandon (1994), Coutts (1999), Lucey (2000) and Lucey (2001) conceded the reverse pattern: average returns on Fridays the 13 th were higher than those on regular Fridays.Dyl and Maberly (1988) proved that in the analyzed time horizon of 1940-1987, in five out of the six analyzed periods, Friday the 13 th rates of return turned out to be positive and higher compared to other Fridays and the only period when the Friday the 13 th rates of return were in red compared to other Fridays rates of return, fell during the 1970s.The similar conclusion was reached in the research of Agrawal and Tandon (1994) as well as of Mills and Coutts (1995).Chamberlain et al. (1991) examining behavior of rates of return falling on Friday the 13 th during the period of 1930-1985 found no stronger evidence of lower mean returns for Fridays falling on the 13 th .Fortin et al. (2014) investigated the effect of superstition on the prices of single-family homes in Great Vancouver in Canada.In neighborhoods with relatively more Chinese residents and in repeated transaction, the sales of prices of houses with street address numbers ending in "4" were 2.2% lower, while those ending in "8" were 2.5% higher than other houses.According to Agarwal et al. (2014), on a per square meter basis, units with numbers ending in "4" were discounted by 1.1%, units on floor with numbers ending in "4" were discounted by 0.5%, while units with numbers ending in "8" commanded a 0.9% premium.There are also reports of a link between the superstition beliefs of certain time periods and the demographics of two nations: Japanese (Kaku, 1972;Kaku & Matsumoto, 1975;Kaku, 1975) and Koreans (Kim, 1979).Ng et al. (2010) studying the auction prices between 1997 and 2009 proved that the prices of license numbers including the lucky number 8 were systematically higher while prices of license numbers with the unlucky number "4", were lower.Besides the premium for "8" could also be interpreted as conspicuous spending to signal wealth or status (Feltovith et al., 2012).Boyle et al. (2004), analyzing daily returns of the index NZSE40, the value-weighted capital index of the 40 largest securities by market capitalization on the New Zealand Stock Exchange in the period 01.01.1967-30.11.2001 proved that the average rates of return for the Fridays the 13 th were not statistically different form the rates of return for regular Fridays.The name of "the Friday the Thirteenth effect", introduced by Kolb and Rodriguez (Kolb & Rodriguez, 1987) has been regularly used by different researchers (Chamberlain et al., 1991;Coutts, 1999;Patel 2009;Botha, 2013;Auer & Rottmann, 2013).Coutts (1999) examining the Friday the 13 th effect in the UK with the use of FTSE index in the period of 59 years, proved that in most cases the rates of return registered for Friday the 13 th were positive and higher compared to other Fridays rates of return, but statistical significance was not observed.Patel (2009), analyzing the period of 58 years for NASDAQ and S&P 500 index, discovered that in four out of the seven periods rates of return for Friday the 13 th were positive and higher than the rates of return calculated for other Fridays.Brown et al. (2002) and Brown and Mitchell (2008) discovered that the daily opening and closing prices tend to cluster at the number "8" in Asian Pacific and Chinese Stock markets.Hirshleifer et al. (2018) found that the superstition affected the pricing of initial public offerings in China in the period of 1991-2005.On Shanghai and Shenzhen stock exchanges, listed companies are identified by a numerical code, which is the equivalent of the US ticker.Consistent with superstition, newly listed equities with lucky listing codes (that included at least one lucky digit and no unlucky digit) initially traded at a premium dissipated within three years.Botha (2013) analyzed the Friday the 13 th effect for samples from stock exchanges in Kenya, Morocco, Nigeria, South Africa and Tunisia.Auer and Rottmann (2013) investigating the presence of Friday the 13 th effect for seven emerging markets in Asia (India, Indonesia, Malaysia, Philippines, South Korea, Taiwan and Thailand) during the period of July 1996-August 2013, proved that the effect was registered on the Stock Exchange in Phillipines.They also found that the Friday the 13 th effect had a significant influence on the stock market volatility in Indonesia and the Philippines.Chung and Darrat (2014) examined the potential effect of superstitious beliefs on stock trading in four Asian-Pacific countries with deep Chinese cultural heritage (China, Hong Kong, Singapore, and Taiwan).The regression results from daily data over 2 January 1991 to 30 December 2011 suggest that unlucky days (particularly day 4 and Friday the 13th) generally exhibit higher stock returns.Kalayaan (2016) found out that the mean returns for Friday the 13 th were inferior than that of other Fridays or other days and that the Friday the 13 th effect was evident during the period of June 1992 to May 2015 for the PSEI index.Pinto (2015) by analyzing the rates of return (in the period of  noticed them falling on the fourth day of the month on the Tokyo Stock Exchange (TSE) and proved that the effect of bad luck numbers started to lose its power in the middle of 1980s.This can be explained by the increasing internationalization of equity investors in Japan.More foreigners, less prone to be influenced by Japanese folk beliefs, trading the TSE , diluted the strength of the Fourth Day effect.Haggard (2015) examining the stock returns impact of days with lucky numbers on Chinese equity market, demonstrated a lucky number date trading strategy for the Shenzhen market.Suganda et al. (2018) studying the influence of the scared days between daily cycles in Georgian calendar and Javanese calendar on the basis of rates of return of Jakarta Composite Index in the period of January 2009 -June 2016, found that investment decision were sill influenced by superstition, leading to behavior biases.Bhattacharya et al. (2017) proved on the example of Taiwan Futures Exchange that the individual investors, but not institutional investors, submitted disproportionately more limit orders at"8" than at "4".This imbalance, defined as superstitions index for each investors seed to be positively correlated with trading losses.Superstitious investors lose more money because of their bad market timing and stale orders.
Taking into consideration the fact that some traders try to avoid making investments during unlucky days, it seems reasonable to study the returns during trading days before and after Friday the 13 th (Peltomaki & Peni, 2010;Peltomaki & Vahamaa, 2014).Stefanescu and Dumitriu (2018) on the basis of the daily rates of return for three American stock indexes: S&P 500, FJIA and NASDAQ, found no evidence for the traditional form of the Friday the 13 th effect, but thy concluded that the returns during two trading days before Friday the 13 th tended to be higher than the average returns, while the returns during one or two trading days after, resulted to be lower than the average.

Methodology
The research is divided into five parts.The calculation were proceeded concerning constituents of the following world stock indexes (in brackets the number of the analyzed companies): CAC40 (39), DAX (30), DJIA (30), FTSE30 (30), FTSE MIBTEL (37), NIKKEI225 (223) and SENSEX (30), e.g. for 419 equities.In case of the indexes CAC40 and NIKKEI225 indexes, one and two of their components, respectively were removed due to the short listing period.The list of analyzed companies and the first dated included in the analysis are presented in the Table A1 and Table A2 (Appendix).The last session considered in the process of calculating rates of return was 30.06.2018.
In case of two populations, the null hypothesis H 0 and alternative hypothesis H 1 regarding equality of rates of return in two populations, can be formulated as follows: (1) where: ̅ -average rate of return in the first population,  ̅ -average rate of return in the second population.
On the basis of two independent populations of rate of returns, which sizes are equal n 1 and n 2 , respectively, the hypotheses H 0 and H 1 should be tested with the use of statistics z (Defusco et al., 2001, p. 335 where: -variance of rates of return in the first population, -variance of rates of return in the second population, n 1 -number of observations in the first population, n 2 -number of observations in the second population. In case when the population variances are unknown and cannot be assumed that they are equal, the number of degrees of freedom will be expressed according to the following formula (Defusco et al., 2001, p. 335): (3) In the following part of the analysis, parametric tests of Kruskal-Wallis will be implemented.The Kruskal-Wallis test statistics is given by (Vargha & Delaney, 1998): where: Ntotal number of observations across all groups, average rank of all observations in group i,  number of observation in group i,  the rank (among all observations) of observation j from group i, In all analyzed cases, the p-values will be calculated.If the p-value is less or equal to 0.05, then the hypothesis H 0 is rejected in favor of the hypothesis H 1 .Otherwise, there is no reason to reject hypothesis H 0 .i {\displaystyle i} For each of the analyzed indexes the following rates of return will be calculated: 1) Close -Close: where: The daily rates of return were calculated for all companies included in the analyzed indices.Then the tests for equality of two average rates of return in two populations were exemplified in the following cases: 1) The first population: 13 th day of the month, the second population: all remaining sessions, 2) The first population: Friday the 13 th day of the month, the second population: all remaining sessions, 3) The first population: Tuesday the 13 th day of the month, the second population: all remaining sessions, 4) The first population: 4 th day of the month, the second population: all remaining sessions, 5) The first population: Friday the 13 th day of the month, the second population: all remaining Fridays.In the second part of the fifth part, the test for equality of two average rates of return were computed under the assumption that the first group of data consists of rates of return for sessions falling on Tuesday the 13 th and the second group is composed of rates of return for all remaining Tuesdays.In this part only close-close rates of return were taken into consideration.

CAC40 Index
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:
The results of calculating p values for returns of the CAC40 components with the use of the Z statistic test are presented as an example in the Table A3 (Appendix).For the Kruskal-Wallis test, as well as for other index components, the calculation were proceeded in the same way.

Kruskal-Wallis Test
The null hypothesis regarding equality of two average rates of return was rejected for the following equities (p-value shown in parenthesis): In all other cases, there was no reason to reject the null hypothesis in favor of the alternative hypothesis.

Confirmation of the Results Obtained with Z-Statistics by the Kruskal-Wallis Test
The null hypothesis was rejected using of two tests (the Z statistics and Kruskal-Wallis) for the following companies: For many analyzed companies the results obtained with the Z statistics were not confirmed by the Kruskal-Wallis test.Thus, in case of the French stock index, the effect of Tuesday the 13 th was the strongest and was mainly observed for the Close-close and Open-close rates of return.This is a result that deserves attention, especially since the perception of the Tuesday the 13 th as a unfortunate date is a characteristic for Spain and Hispanic countries.On the French market one could rather expect the dominance of the effect of the Friday the 13 th than Tuesday the 13 th .

DAX Index
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:

Z-Statistics
The null hypothesis regarding equality of two average rates of return was rejected for the following equities (p-value shown in parenthesis):

Kruskal-Wallis Test
The null hypothesis regarding equality of two average rates of return was rejected for the following equities In all other cases, there was no reason to reject the null hypothesis in favor of the alternative hypothesis.The strongest effect on the German stock exchange was Tuesday the 13 th , which preceded the effect of the 4 th day of the month.The first effect was registered mainly for the rates of return: Close-close.On the German market, as in case of France, more expected was the dominance of the Friday the 13 th effect, which was not recorded.The effect of the 4 th day of the month is expected mainly in Asian markets.

DJIA Index
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:

Z-Statistics
The null hypothesis regarding equality of two average rates of return was rejected for the following equities On the American market, the two dominant effects were observed: Tuesday the 13 th as well as the 4 th day of the month.The first of them is associated mainly with Spanish and Latin culture, and the second with Asian.There was no Friday the 13 th effect, characteristic mainly for the European cultural circle.Tuesday the 13 th effect was registered mainly for Overnight and Close-close rates of return and the 4 th day of the month effect for Open-open rates of return.

FTSE30
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:

Z-Statistics
The null hypothesis regarding equality of two average rates of return was rejected for the following equities (p-value shown in parenthesis): In all other cases, there was no reason to reject the null hypothesis in favor of the alternative hypothesis.On the British market, just like on the American market, the following effects dominated: Tuesday the 13 th as well as the 4 th day of the month.The first one was observed most frequently for Clos-close and Open-open rates of return and the second for Open-open returns.The Friday the 13 th effect occurred but sporadically.

FTSE MIBTEL
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:

Z-Statistics
The null hypothesis regarding equality of two average rates of return was rejected for the following equities (p-value shown in parenthesis): In all other cases, there was no reason to reject the null hypothesis in favor of the alternative hypothesis.On the Italian market the two most dominant effects were: 4 th day of the month as well as Tuesday the 13 th .The first one was registered mainly for Open-close rates of return, and the second for Close-close returns.Friday the 13 th effect occurred but sporadically.

NIKKEI
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:
On the Japanese stock market the appearance of all types of effects was observed, both those related to the number 13 and the number 4. Most often, the effects occurred for the following returns: Close-close (13 th day of the month, Friday the 13 th and Tuesday the 13 th ), Open-open (Tuesday the 13 th and 4 th day of the month) and Open-close (13 th day of the month).

SENSEX
The results of testing a zero hypothesis with the use of average rates of returns for two different populations permit to draw the following conclusions:

Z-Statistics
The null hypothesis regarding equality of two average rates of return was rejected for the following equities (p-value shown in parenthesis): On the Indian stock market, just like on Japanese market, were registered all types of effects, related to the number 13 (Friday the 13 th and Tuesday the 13 th ) and the number 4. The only exception is the effect of the 13 th day of the month that was not present.The observed effects most often occurred for the following returns: Close-close (Friday the 13 th and Tuesday the 13 th ), Open-open and Open-close (in both cases: 4 th day of the month).

Conclusions
The aim of this study was to determine the prevalence of the calendar effect in case of "the unfortunate dates effect", on the example of 7 world equity indexes components.Analysis of the effects of seasonality included an examination of the rates of return calculated for four approaches: In the fifth part the statistical equality of one-session rates of return for the population of Friday the 13 th and the population of other Fridays were compared.The following part of the fifth part of the paper consists of the analysis of equality of rates of return for the sessions falling on Tuesday the 13 th vs rates of return calculated for all remaining Tuesdays.This is the first study known to the author that takes into account other rates of return than Close-close.
Taking into account results of both tests, i.e.Kruskal-Wallis and Z statistics, the calendar effect regarding rates of return of the 13 th day of the month was observed only on the Japanese market (for all calculated types of rates of return).Cultural differences between the analyzed markets would suggest the occurrence of the Tuesday the 13 th effect, possibly on European markets, on which the influence of Spanish investors can be noticed.Meanwhile, as a result of the conducted research, it was proved that this effect occurs on all analyzed markets, including Asian ones.The same applies to the effect of the 4 th day of the month, which should mainly be present in Asian markets.Meanwhile, it was registered in all analyzed markets.This fact entitles to the thesis about capital mobility in contemporary financial markets.
The calendar effects of returns calculated for Friday the 13 th in relation to other Fridays, were observed on all exchanges except for German and American, while the calendar effects of Tuesday the 13 th in relation to the other Tuesdays were registered for all analyzed equity exchanges.
Summing up the values in the individual rows of Table 3, another ranking can be created, e.g.ranking of unlucky number anomalies for all analyzed stock exchanges: DAX (26.67%),CAC40 (33.33%),DJIA (33.33%),FTSE MIBTEL (37.84%),NIKKEI 225 (56.05%),SENSEX (60.00%) and FTSE30 (63.33%).Contrary to the expectations, the unlucky day effects were not the most commonly observed on the Asian Stock Exchanges but on the British Stock Exchange."The unfortunate dates effect" was the most frequently observed for Tuesday and 13 th and then for the 4 th day of the month.Taking into consideration all types of analyzed returns (Close-close, Open-open, Open-close and Overnight), the most frequent effects were registered for the following returns: Close-close and for Open-close, with the exception of the 4 th day of the month effect, in which the order was changed.Results obtained in the paper regarding the Friday the 13 th effect are consistent with those of Kolb and Rodriguez (1987).Notably the results do not support the outcomes reported by Agrawal and Tandon (1994), Coutts (1999) and Lucey (2000).Further research on the occurrence of "the unfortunate dates effect" in the financial markets should cover the currency and commodity market.The conducted studies proved the occurrence of "the unlucky day effect" not only in case of Close-close returns, but also in the remaining three that is Open-close, Open-open and Overnight.
The main limitation of this research is the range of data gained from the Reuters as well as the unequal intervals of observations for different equity indexes.The outcome may be regarded as a part of the ongoing discussions on the hypothesis of financial markets efficiency, which was introduced by Fama (1970).The results of the analysis entitles to the thesis about capital mobility in contemporary financial markets.
a) EDF: Friday the 13 th vs Fridays, Close-close, b) Orange: Tuesday the 13 th , Close-close and Open-close, Tuesday the 13 th vs Tuesdays, Close-close, c) Peugeot: Tuesday the 13 th , Open-close and Tuesday the 13 th vs Tuesdays, Close-close, d) Renault: Tuesday the 13 th , Close-close and Open-close, Tuesday the 13 th , Close-close, e) Sanofi: Tuesday the 13 th , Close-close, f) Schneider Electric: Tuesday the 13 th vs Tuesdays, Close-close, g) Veoila Environement: 4 th Close-close and Open-close.
4.2.3Confirmation of the Results Obtained with Z-Statistics by the Kruskal-Wallis TestThe null hypothesis was rejected with the use of two tests (the Z statistics and Kruskal-Wallis) for the following companies: a) Continental: 4 th , Open-open, b) Dimler: Tuesday the 13 th , Close-close, c) Fresenius: 4 th , Close-close and Open-close, d) Heilderbergcement: Tuesday the 13 th , Close-close and Open-close, 13 th Tuesday vs Tuesdays, Close-close, e) Muench Rueckvers: Tuesday the 13 th , Close-close.

4. 4 . 3
Confirmation of the Results Obtained with Z-Statistics by the Kruskal-Wallis Test The null hypothesis was rejected with the use of two tests (the Z statistics and Kruskal-Wallis) for the following companies: a) 3I: Tuesday the 13 th , Close-close, b) Associated British Food: 4 th , Open-open, c) BAE System: Tuesday the 13 th , Open-close and Tuesday the 13 th vs. Tuesdays, Close-close d) British American Tobacco: Friday the 13 th vs Fridays, Close-close, e) Experian: Tuesday the 13 th , Open-close and Tuesday the 13 th vs Tuesdays, Close-close, f) Glaxo Smith Kline: Tuesday the 13 th , Close-close and Open-close, g) Land Sec.: Tuesday the 13 th , Close-close, h) Man Group: 4 th , Overnight, i) Marks and Spencer: Tuesday the 13 th , Open-open, j) Prudential: Tuesday the 13 th , Close-close, k) Royal bank of Scotland: Tuesday the 13 th vs Tuesdays, Close-close, l) Tate and Lyle: Tuesday the 13 th , Close-close, 4 th , Open-open, m) Wolseley: Tuesday the 13 th , Close-close, Open-close and Tuesday the 13 th vs Tuesdays, Close-close.
4.5.3Confirmation of the Results Obtained with Z-Statistics by the Kruskal-Wallis Test The null hypothesis was rejected with the use of two tests (the Z statistics and Kruskal-Wallis) for the following companies: a) Brembo: 4 th , Open-close, b) Buzzi Unicem: Tuesday the 13 th , Close-close, Overnight, c) CHN Industrial: 4 th , Open-open, d) FIAT: Friday the 13 th , Open-close, Tuesday the 13 th , Close-close, Friday the 13 th vs Fridays, Close-close, e) Luxottica: 4 th , Open-close, f) Recordati: Tuesday the 13 th , Open-close, Tuesday the 13 th vs Tuesdays, Close-close, g) SNAM: 4 th , Open-close, h) Telecom Italia, Friday the 13 th vs Fridays, Close-close, i)Tenaris: Tuesday the 13 th , Overnight, j) Terna Rete: 4 th , Open-close.
cases the statistical equality of one-session rates of return for two population were calculated for: a) Sessions falling on the 13 th day of the month vs all other sessions (first part), b) Sessions falling on Friday the 13 th vs all other sessions (second part), c) Sessions falling on Tuesday the 13 th vs all other sessions (third part), d) Sessions falling on the 4 th day of month vs all other sessions (fourth part), a) SENSEX -23% (Friday the 13 th vs Fridays, C-C) b) FTSE30 -20% (Tuesday the 13 th , C-C), c) FTSE30 -13.3 % (Tuesday the 13 th vs Tuesdays, C-C), d) FTSE30 -13.3 % (Tuesday the 13 th , O-C) e) NIKKEI225 -11.21% (Friday the 13 th vs Fridays, C-C) f) FTSE MIBTEL -10.81% (4 th , O-C) g) DJIA -10% (4 th , O-O).

Figure 1 .
Figure 1.Frequency of cases when p values were lower than 0.05 at the same time for two kind of tests Source: own calculation.

Table 1 .
Number of "the unfortunate day effects", calculated for the analyzed equity indexes components with the use of two statistical tests: Z statistics test and Kruskal-Wallis test (in brackets)

Table 2 .
Number of cases when p values were lower than 0.05 at the same time for two tests: Z statistics and Kruskal-Wallis

Table 3 .
Percentage of cases for each of analyzed indexes when p values were lower than 0.05 at the same time for two kind of tests: Z statistics and Kruskal-Wallis (The results are derived from the Table2by dividing the number form the Table2by the umber of the analyzed components of each index)

Table A1 .
Component of the following indexes included in the analysis: CAC40, DAX, DJIA, FTSE 30, FTSE MIBTEL and SENSEX

Table A3 .
Example of p values calculation for returns of CAC40 components with the use of the Z statistics.Shaded cells represent cases when p value was lower than 0.05