Spurious seasonality detection: a nonparametric test proposal
Abstract
This paper offers a general and comprehensive definition of the dayoftheweek effect.
Using symbolic dynamics, we develop a unique test based on ordinal patterns in order to detect it.
This test uncovers the fact that the socalled “dayoftheweek” effect
is partly an artifact of the hidden correlation structure of the data. We present simulations based on artificial time series as well.
Whereas time series generated with long memory are prone to exhibit daily seasonality,
pure white noise signals exhibit no pattern preference. Since ours is a non parametric test, it requires no assumptions about the
distribution of returns so that it could be a practical alternative to conventional econometric tests.
We made also an exhaustive application of the here proposed technique to 83 stock indices
around the world. Finally, the paper highlights the relevance of symbolic analysis
in economic time series studies.
Keywords: daily seasonality; ordinal patterns; stock market; symbolic analysis
JEL Classification: G14; C19; C58
1 Introduction
The static capital asset pricing model (CAPM), developed independently by Sharpe (1964), Lintner (1965) and Mossin (1966), has been widely used for a number of financial matters. In its standard form, the CAPM states that the expected risk on a security can be separated into two components: the risk free rate and the risk premium. The latter, in turn, can be explained as the product between the market premium and a modulating coefficient :
(1) 
According to equation 1, return on the security depends only on the risk free rate, market return, and beta. Consequently, return should not be altered by any other circumstances such as the particular day of the week or the time of the year in which the return is measured. According to Fama (1970) a market is informationally efficient if it fully reflects all available information. In fact, as LeRoy (1989) asserts, the Efficient Market Hypothesis (EMH) is just the idea of competitive equilibrium applied to the securities market.
Although early empirical studies (e.g. Blume and Friend (1973); Fama and MacBeth (1973)) support the validity of the CAPM, later research documents departures from this equilibrium model. These departures are called “anomalies” ^{1}^{1}1According to Kuhn (1968), an anomaly is a fact that puts into question an established paradigm.. Among them, there is one specially puzzling feature, the dayoftheweek effect. This anomaly refers to the heterogeneous behavior of returns along the week. Testing the the dayoftheweek effect requires the joint consideration of an equilibrium model, such as equation 1, and of the Efficient Market Hypothesis (EMH).
Empirical research on markets’ daily seasonality can be traced back to Fields (1931, 1934). These papers have the merit of investigating the issue before a market equilibrium model was formally developed. Cross (1973) detects differences in expected S&P 500 on Fridays and Mondays. Gibbons and Hess (1981) finds lower S&P 500 returns on Mondays relative to other days. The effect is subdivided by French (1980) into a Monday effect (abnormal negative return on this day) and a Friday effect (abnormal positive result on this day). Rogalski (1984) analyzes the effect during trading and nontrading hours for the American market. This effect has been widely surveyed and, for brevity, we refer to Keim and Ziemba (2000) and Ziemba (2012) for further discussion on empirical works about this effect. Keloharju et al. (2016) find return seasonalities in commodities and stock indices arround the world.
The standard approach for detecting the dayofthe week effect is based on (or some variations of) the following regression equation:
(2) 
where is the return on day and , are dichotomous dummy variables for each day of the week from Tuesday through Friday. The coefficient represents the mean return on Monday, while , , is the excess return on day , and is an error term. This traditional approach is based on different hypothesis testing on values (for an overview see, for instance, Bariviera and de Andrés Sánchez (2005) and references therein). Working based on Equation 2 forces to make several (sometimes unjustified) assumptions about parameters. For example, Zhang et al. (2017) applies a rolling sample test with a GARCH model in 28 stock indices. Precisely, our original approach, based on ordinal patterns, bypasses this shortcome.
The aim of this paper is to provide a more general definition of the dayoftheweek effect and to develop an alternative test to assess the existence of seasonal effects in daily returns. This paper contributes to the literature in several ways. First, it generalizes the definition of the dayoftheweek effect. Second, it develops an alternative nonparametric test to detect it. Third, it shows that the results of the tests are not obtained by chance, since timecausality is taken into consideration. Fourth, most of the dayoftheweek findings in the literature are related with the underlying returngenerating process, and not with the causes indicated previously in the literature. Consequently, from a theoretical point of view, this paper introduces a new nonparametric test that is able to detect the intrinsic characteristics of the time series, and uncover spurious seasonality detection causes. We would like to point out that the methodology we use here is unique in that it is nonlinear, ordinal, requires no model and provides statistical results in term of a probability density function. Ours is a statistical methodology that, to the extent of our knowledge, no one using time series analysis has used it before.
The remaining of the paper is organized as follows. Section 2 presents the notion of ordinal patterns. Section 3 redefines the dayof the week effect and proposes a nonparametric test. Section 4 displays results of the test on theoretical simulations of different stochastic processes. Section 5 performs an empirical application to the New York Stock Exchange. Finally, Section some conclusions are drawn in 6.
2 Ordinal pattern analysis
Estimations based on equation 2 require the assumption of an underlying stochastic process for returns. For these processes, symbolic analysis becomes a suitable alternative to study the dynamics of a time series. Bandt and Pompe (2002) develop a method for estimating the probability distribution function (PDF) based on counting ordinal patterns. The comparison of neighboring values of a time series requires no model assumption Bandt and Pompe (1993). The advantage of this method is that can be applied to any time series, and takes into account time causality Bandt and Pompe (1993). If returns fulfill the Efficient Market Hypothesis (EMH), there should be no privileged pattern. If there were, it would be exploited by arbitrageurs and any possibility of abnormal return should be rapidly wiped out. Thus, if the time series is random, patterns’ frequency should be the same, provided
If patterns are not equally present in the sample, three anomalous situations might be the cause:

Forbidden pattern: a pattern that does not appear within the sample.

Rare pattern: a pattern that seldom appears.

Preferred pattern: a pattern that emerges oftener than expected by the uniform distribution.
In any of such cases we are in presence of a time series with daily seasonal behavior. Consequently the dayofthe week effect needs to be redefined.
In this line, Zanin (2008) applies the concept of forbidden patterns in order to assess market efficiency, and shows that different financial instruments could achieve different informational efficiency. According to Amigó et al. (2006), forbidden patterns can be used as a means of distinguishing chaotic and random trajectories and constitute a satisfactory alternative to more conventional techniques.
Ordinal patterns have been previously used by Zunino et al. (2010, 2011, 2012) in order to compute quantifiers like permutation entropy and permutation complexity, which, in turn, allow one to quantify the degree of informational efficiency of different markets. Rosso et al. (2012) demonstrates that forbidden patterns are a deterministic feature of nonlinear systems. Bariviera (2011); Bariviera et al. (2012) show that the correlation structure and informational efficiency are not constant through time and could be affected by several factors such as liquidity or economic shocks.
Given a time series of daily returns ^{2}^{2}2Let us assume that the time series is characterized by a continuous distribution. beginning on Monday and a pattern length , following the Bandt and Pompe (2002) method, partitions of the time series could be generated. Each partition is a 5dimensional vector , which represents a whole trading week. Each return is associated with a day of the week. For simplicity, we have standing for Monday through Friday. The method sets the elements of each vector in increasing order. Doing so, each vector of returns is converted into a symbol. For example, if in a given week , where represents return on day , the pattern is . There are possible permutations. Each permutation produce a different pattern () and the associated frequencies can be easily computed. Each pattern has a frequency of appearance in the time series. Carpi et al. (2010) asserts that in correlated stochastic processes, patternfrequency observations do not depend only on the time series’ length but also on the underlying correlation structure. Amigó et al. (2007, 2008) show that in uncorrelated stochastic processes every ordinal pattern has an equal probability of appearance. Given that the ordinalpattern’s associated PDF is invariant with respect to nonlinear monotonous transformations, Bandt and Pompe (2002) method results suitable for experimental data (see e.g. Saco et al. (2010); Parlitz et al. (2012)). A graphical meaning of the ordinal pattern can be seen in Parlitz et al. (2012).
3 Dayoftheweek effect: a redefinition of the problem
As recalled in Section 1, the conventional definition of the dayoftheweek effect refers to the abnormal negative or abnormal positive returns on Monday and Friday, respectively. Since not all the markets are open on the same days, comparisons among countries could be difficult. For example, the Israeli market is open from Sunday through Thursday (Lauterbach and Ungar (1992)) and the first day of the week in the Kuwait Stock Exchange is Saturday (AlLoughani and Chappell (2001)). Additionally, markets are not open simultaneously, due to the different time zones (Koh and K.A. (2000)). Consequently, spillover effects can influence returns and could distort results if such influence is not incorporated into the model.
In order to overcome these difficulties, we develop here a more general definition of the dayoftheweek effect that exploits the potential of the symbolic analysis of time series. Instead of estimate the return on each day by means of equation 2, we will look at the relative position of the return on each day within its week. If there is no seasonal effect, the order in which the days appear in each position (from the worst until the best return of the week), should be random. Otherwise, a seasonal pattern would be detected.
First, we need to give an specific definition of our seasonal effect. It must be emphasized that, according to our proposal, we are not interested in detecting abnormal negative or positive returns on a given day. Instead, we are looking for the features of the return on a given day within its week, from the worst return of the week to the best return of it, independently of its sign. Thus, a new definition of the dayoftheweek effect is required.
Definition 1
The dayoftheweek effect occurs whenever a pattern appears much more or much less frequently than expected by the uniform distribution.
From this definition a natural null hypotheses arises:
Hypothesis 1
(3) 
where , , stands for “absolute frequency of pattern ”.
Since we are interested in studying the dayofthe week effect, testing this hypothesis is insufficient for our purposes.
We should count the number of times in which a given day exhibits the worst return of the week, the next to worst return, and so on, until the best return of the week is detected. In other words, we should count the number of times a given day occupies the first, second, third, fourth or fifth position in a pattern and place the absolute frequencies in a matrix as follows:
Definition 2
Let be a 5x5 matrix. Element is the absolute frequency of return on day at the position .
Displaying results in this way, we count how many times a given day is in position 0 (the worst return of the week), position 1, position 2, position 3, and position 4 (the best return of the week). As a consequence, we advance two additional hypothesis:
Hypothesis 2
(4) 
This hypothesis says that a given day could occupy any position, from the worst to the best return within a week.
Hypothesis 3
(5) 
This hypothesis says that a given position in the week could be occupied by any day of the week.
All these null hypotheses could be tested using Pearson’s chisquared test. This test is useful to verify if there is a significant difference between an expected frequency distribution and an observed frequency distribution. Following Fernández Loureiro (2011) the test statistic is:
(6) 
where is the observed frequency of day at position and is the expected frequency . is distributed asymptotically as a with 4 degrees of freedom.
We advance two additional hypotheses focused on the socalled “Monday effect”.
Hypothesis 4
(7) 
This hypothesis tests whether patterns with Monday having the largest return are preferred patterns or not. Pattern numbers correspond to those displayed in Table 1.
Hypothesis 5
(8) 
This hypothesis tests whether the patterns with Monday exhibiting the lowest weekly return and Friday the largest are preferred patterns or not.
These hypotheses are tested using the binomial test, which for large samples can be approximated by the normal distribution. The test statistic is:
(9) 
where is the observed frequency, , is the expected frequency, and is the number of “weeks”, i.e. the number of 5day patterns in the sample.
Our definition assumes that the dayoftheweek effect could be produced by the dependence among days of the same week. However, by splitting time series into weeks, we implicitly assume the independence among weeks. Even though this later assumption may be questionable, we do this way in order to emphasize the order of the returns within the week. We could relax this assumption, by moving data daily, instead of weekly. In this way, we could compare if, e.g. Friday in week t influences Monday in week t+1 However, it could result in a more confuse analysis, and we let it for further research.
Pattern  Pattern  Pattern  Pattern  

1  01234  25  10234  49  20134  73  30124  97  40123 
2  01243  26  10243  50  20143  74  30142  98  40132 
3  01324  27  10324  51  20314  75  30214  99  40213 
4  01342  28  10342  52  20341  76  30241  100  40231 
5  01423  29  10423  53  20413  77  30412  101  40312 
6  01432  30  10432  54  20431  78  30421  102  40321 
7  02134  31  12034  55  21034  79  31024  103  41023 
8  02143  32  12043  56  21043  80  31042  104  41032 
9  02314  33  12304  57  21304  81  31204  105  41203 
10  02341  34  12340  58  21340  82  31240  106  41230 
11  02413  35  12403  59  21403  83  31402  107  41302 
12  02431  36  12430  60  21430  84  31420  108  41320 
13  03124  37  13024  61  23014  85  32014  109  42013 
14  03142  38  13042  62  23041  86  32041  110  42031 
15  03214  39  13204  63  23104  87  32104  111  42103 
16  03241  40  13240  64  23140  88  32140  112  42130 
17  03412  41  13402  65  23401  89  32401  113  42301 
18  03421  42  13420  66  23410  90  32410  114  42310 
19  04123  43  14023  67  24013  91  34012  115  43012 
20  04132  44  14032  68  24031  92  34021  116  43021 
21  04213  45  14203  69  24103  93  34102  117  43102 
22  04231  46  14230  70  24130  94  34120  118  43120 
23  04312  47  14302  71  24301  95  34201  119  43201 
24  04321  48  14320  72  24310  96  34210  120  43210 
4 Simulation of fractional Brownian motion
In this section we apply the aboveoutlined technique to simulated time series. We used MATLAB wfbm function, in order to simulate fractional Brownian motion for , where is the Hurst exponent. Then, we take first differences in the time series in order to obtain the corresponding fractional Gaussian noise (fGn). The Hurst exponent characterizes the scaling behavior of the range of cumulative departures of a time series from its mean. The study of long range dependence can be traced back to seminal paper by Hurst (1951), whose original methodology was applied to detect long memory in hydrologic time series. This method was also explored by Mandelbrot and Wallis (1968) and later introduced in the study of economic time series by Mandelbrot (1972). If the series of first differences is a white noise, then its . Alternatively, Hurst exponents greater than 0.5 reflect persistent processes and less than 0.5 define antipersistent processes.
We perform 1000 simulations consisting of 10000 datapoints for each value of . Accordingly, we obtain 2000 “weeks”, which will be classified into one of the possible patterns. If the underlying stochastic process is purely random and uncorrelated, the frequency of patterns should be uniform. On the contrary, if some correlation is present, some patterns could be preferred over others. All the tests are performed at a significance level.
In Table 2 we test the equality of patterns (Hypothesis 1). When (ordinary Gaussian noise), we cannot reject, on average, the null hypothesis of equal appearance of patterns. Out of the 1000 simulations, only in 50 cases is the null hypothesis rejected. When we move away form , in both directions, rejection increases almost symmetrically. This clearly shows that some kind of correlation affects the distribution of ordinal patterns.
As commented in the previous section, this analysis is not sufficient. Therefore, we proceed to test Hypothesis 2 and present the pertinent results in Table 3. We test the hypothesis (for every value) for each of the samples and for the average of the samples. We find that for we cannot reject the null hypothesis. In fact, rejection occurs in only 51 to 59 times out of 1000 samples. In other words, when the generating stochastic process is a white noise, any day is equally prone to occupy any of the positions in the pattern. This is the same to say that any day could exhibit the best or the worst return of the week, or any intermediate value among them. When we move away from the value 0.5, rejections increase. However, the effect is stronger for than for . This could mean that a positive long range correlation (i.e. a persistent time series) is more likely to exhibit a dayoftheweek behavior than antipersistent time series. Additionally, Monday, Wednesday and Friday are the days most affected by the valuechange of .
Regarding Hypothesis 3, results are displayed in Table 4. In the case of the uncorrelated process (), one encounters that how good or bad is the return within a trading week is independent of the day of the week. This hypothesis is only rejected only 59 times out of the 1000 simulations. When we move away, and then correlations become stronger, patterns exhibits some degree of preference, increasing the number of rejections. As in the case of Hypothesis 3, rejections are more frequent in case of persistent time series.
Hypothesis 4 tests whether the presence of patterns with Monday as the largest return is in agreement with the uniform distribution. There are 24 patterns with Monday as the last element. Consequently, its expected frequency is 0.2. Table 5 displays the results of the simulations. In the case of a pure Gaussian noise, we cannot reject the null hypothesis, as in only 117 out of the 1000 trials we reject it. However, increasing the Hurst exponent produces an increment in the number of rejections. Additionally, we observe that larger Hurst values are associated with greater observed frequency of patterns with Monday as largest return. However, we cannot reject the null hypothesis until or higher.
Hypothesis 5 tests the presence of a weekly seasonality with Monday as the smallest return of the week and Friday as the largest. There are 6 patterns with this structure. In analogy with the preceding finding, for an uncorrelated noise, this pattern is neither a preferred nor a rare one. Nevertheless, the increase of the Hurst exponent produces a quick increase in the number of rejections: 309 out of 1000 when . More impressive is how preferred this pattern is in most of the simulations. For , in 698 simulations, the observed frequency of these 6 patters was above expectation, and for , in 873 simulations.
Symbolic analysis is powerful to detect nontrivial hidden correlations in data. As shown by Rosso et al. (2012); Carpi et al. (2010) a correlated structure as produced by fractional Gaussian noise processes generates an uneven presence of patterns. Provided a sufficiently long time series, no pattern is forbidden. However, strongly correlated structure produces the emergence of preferred and rare patterns.
Our artificial time series are larger than the usual datasets used in economics. Consequently, the presence of preferred patterns as the ones evaluated in Hypothesis 5 casts doubts on the validity of previous findings of dayoftheweek. In particular, we claim that, in view of our results, the dayoftheweek effect is mainly produced by a complex correlation structure of the pertinent data.
0.1  224.82856  0.6  20.42699  
#rejections  1000  #rejections  348  
0.2  140.50932  0.7  86.57005  
#rejections  1000  #rejections  996  
0.3  67.89548  0.8  204.51209  
#rejections  970  #rejections  1000  
0.4  18.10740  0.9  386.14031  
#rejections  317  #rejections  1000  
0.5  0.13315  
#rejections  50  
: significant at 10%, 5% and 1% level. 
M  T  X  T  F  
0.1  5.47646  1.94578  5.40877  2.06613  5.80607  
#rejections  461  176  437  176  464  
0.2  3.83407  1.34267  3.30037  1.33138  3.83153  
#rejections  308  124  275  139  318  
0.3  1.95798  0.62585  1.79598  0.59399  2.03637  
#rejections  186  87  146  86  162  
0.4  0.62419  0.17731  0.47961  0.22011  0.60417  
#rejections  82  54  78  57  71  
0.5  0.00666  0.00304  0.00078  0.00500  0.00159  
#rejections  51  53  54  57  59  
0.6  0.76829  0.25922  0.67148  0.25509  0.87231  
#rejections  95  68  79  66  93  
0.7  3.94460  1.16245  3.40845  1.22150  3.72159  
#rejections  318  123  293  114  320  
0.8  9.90281  3.44099  8.96147  3.61403  9.96604  
#rejections  706  270  657  295  719  
0.9  20.42595  7.95607  20.65903  7.86588  20.71557  
#rejections  960  572  966  596  966  
: significant at 10%, 5% and 1% level. 
worst return  best return  
0.1  5.66761  2.40628  4.47306  2.35024  5.80600  
#rejections  461  193  357  188  471  
0.2  3.69734  1.58367  2.83837  1.46318  4.05746  
#rejections  299  133  220  144  335  
0.3  2.08654  0.68270  1.44706  0.80686  1.98700  
#rejections  187  95  117  82  181  
0.4  0.48517  0.21129  0.42396  0.22292  0.76205  
#rejections  73  60  68  51  89  
0.5  0.00390  0.00301  0.00335  0.00009  0.00673  
#rejections  58  57  59  49  37  
0.6  0.78880  0.34282  0.58457  0.24124  0.86898  
#rejections  95  66  85  56  91  
0.7  3.72085  1.27291  3.09096  1.35108  4.02279  
pvalue  0.44510  0.86595  0.54272  0.85265  0.40293  
#rejections  294  115  264  127  315  
0.8  10.08089  3.67766  8.59256  3.66076  9.87346  
#rejections  700  306  631  296  700  
0.9  20.63329  8.33357  20.41387  7.67672  20.56505  
#rejections  965  617  963  570  962  
: significant at 10%, 5% and 1% level. 
Hurst  # of rejections  #  
0.10  0.20  0.18850  0.81150  0.97729  306  161 
0.20  0.20  0.19034  0.80966  0.81787  232  222 
0.30  0.20  0.19397  0.80603  0.50693  172  275 
0.40  0.20  0.19742  0.80258  0.21572  124  359 
0.50  0.20  0.20243  0.79757  0.20089  117  507 
0.60  0.20  0.20842  0.79158  0.68858  107  670 
0.70  0.20  0.21406  0.78594  1.13915  225  839 
0.80  0.20  0.22125  0.77875  1.70120  380  924 
0.90  0.20  0.22867  0.77133  2.26805  627  983 
: significant at 10%, 5% and 1% level. 
Hurst  # of rejections  #  
0.1  0.05  0.04013  0.95987  1.67154  562  45 
0.2  0.05  0.04263  0.95737  1.21287  427  91 
0.3  0.05  0.04531  0.95469  0.74944  302  155 
0.4  0.05  0.04824  0.95176  0.27363  174  290 
0.5  0.05  0.05197  0.94803  0.29438  78  507 
0.6  0.05  0.05588  0.94412  0.85028  119  698 
0.7  0.05  0.06074  0.93926  1.49392  309  873 
0.8  0.05  0.06681  0.93319  2.23719  589  983 
0.9  0.05  0.07376  0.92624  3.01987  856  998 
: significant at 10%, 5% and 1% level. 
5 Empirical application
We use daily data of NYSE Composite Price Index from 03/01/1966 to 08/12/2017, for a total of 13,550 observations. All data used in this paper was retrieved from DataStream. We split the sample into four nonoverlapping periods of equal length (3,050 data points), and a last period of 1350 datapoints, in order to verify the temporal evolution of the seasonal effect. We compute daily log returns in order to apply our test.
Regarding Hypothesis 1 (see Table 7), we find, in the whole sample, no forbidden patterns. Under these circumstances, we should discard chaotic behavior in the time series (Rosso et al. (2012)) The least frequent pattern, with an absolute frequency equal to 7, is 42013 (i.e. ). The most frequent patterns, with an absolute frequency equal to 34, are 03421 and 04312 (i.e. and , respectively). As stated in Section 2, if data were generated at random, i.e., if no seasonal effect exists, patterns should uniformly appear, configurating the histogram of a uniform distribution. However, as seen in Figure 1, ours is a far from uniform distribution.
Table 8 exhibits frequencies and tests for hypothesis 2 and hypothesis 3. Following the horizontal lines of the table, we test whether a given day indifferently occupies any position in the returns of the week. Along the vertical sense of the table, we test whether a given position within a week is indifferently occupied by any day.
Regarding the whole period we observe that we cannot accept the null hypothesis of equal distribution of returns across the week. In fact, if we observe Table 8, Monday acquires the lowest return of the week more frequently than any other weekday.
In order to justify the fact that intrinsic temporal correlations play a significant role in the ordinal patterns, we have also estimated the frequency of the patterns for the shuffled return data. “Shuffled” realizations of the original data are obtained by permuting them in random order, and eliminating, consequently, all nontrivial temporal correlations. From Table 9, we observe that patterns are distributed in a more or less uniform fashion and, consequently, we cannot reject the null hypotheses. Therefore, the results of our test are not due to chance.
If we analyze the evolution of the daily seasonal behavior through time, it is clear that the dayoftheweek effect disappears in daily returns of the NYSE Composite index. Results are reflected in Tables 10, 11, 12, 13 and 14. Considering the last subperiod, only Tuesday effect remains. Tuesday is the most frequent day in the worst position and Friday tends to occupy the best return within each week. However, we cannot reject that the worst return of the week can be occupied by any other week day. According to this analysis we observe, in agreement with the literature, a disappearing weekly effect in daily returns in the US market. This disappearing effect is related to the hidden underlying dynamics of data, rather than with markets participants behavior, as it was classically envisaged in the literature. We would like to emphasize that our test unveils the hidden correlation structure of daily returns. As in the case of the artificial generated series, the pattern behavior in real time series is strongly affected by the long memory of data.
An important difference between real and simulated data is that, whereas in the controlled experiment the Hurst exponent is, by definition, constant across all the time series, in the case of real data, the Hurst exponent tends to vary across time (see e.g. Cajueiro and Tabak (2004a, b); Bariviera et al. (2012)). This situation makes difficult the direct comparison between both results. Moreover, we can observe that the power of the test is more sensitive for in detecting the Monday effect (see Table 6 ). In fact, for , the test rejects 856 out of the 1000 simulated series. Another factor that influences results is the time series length. As recalled by Rosso et al. (2012), short time series could result in the incorrect detection of forbidden patterns.
Pattern  Abs. Freq.  Pattern  Abs. Freq.  Pattern  Abs. Freq.  Pattern  Abs. Freq. 

42013  7  14320  16  42031  20  21403  24 
23041  10  20314  16  42301  20  32140  24 
24013  10  23140  16  43012  20  42310  24 
02413  11  31204  16  43210  20  02431  25 
13240  11  04213  17  10324  21  04123  25 
13402  11  20413  17  12043  21  01423  26 
23104  11  40312  17  13042  21  03412  26 
30124  11  41230  17  14032  21  20134  26 
40123  11  20431  18  34201  21  34012  26 
40132  11  23410  18  02134  22  34210  26 
41203  11  40231  18  03124  22  01342  27 
43102  11  02143  19  10432  22  12403  27 
20143  12  02341  19  21340  22  31420  27 
24310  12  03142  19  21430  22  34021  27 
42103  12  10342  19  24301  22  43201  27 
20341  13  12340  19  32014  22  02314  28 
23014  13  23401  19  34102  22  10423  28 
13024  14  24031  19  41302  22  21304  28 
14203  14  30421  19  01243  23  01234  29 
24103  14  32041  19  03241  23  34120  29 
24130  14  41032  19  10234  23  10243  30 
31024  14  42130  19  12034  23  14023  30 
41320  14  43120  19  14302  23  01432  31 
30142  15  01324  20  21034  23  04132  31 
30214  15  12430  20  32104  23  04231  31 
30412  15  31042  20  03214  24  30241  31 
40213  15  31402  20  04321  24  31240  31 
41023  15  32401  20  13204  24  43021  31 
12304  16  32410  20  14230  24  03421  34 
13420  16  40321  20  21043  24  04312  34 
worst return  best return  
Mo  637  503  537  501  532  22.78967 
Tu  557  572  475  509  597  17.94834 
We  479  540  564  564  563  9.92989 
Th  570  505  564  581  490  12.66052 
Fr  467  590  570  555  528  16.74908 
Total  2710  2710  2710  2710  2710  
36.21402  11.25092  11.56089  9.12177  11.92989  
: significant at 10%, 5% and 1% level. 
worst return  best return  
Mo  506  469  501  484  480  1.91393 
Tu  488  508  472  498  474  1.95082 
We  513  475  491  471  490  2.24590 
Th  479  491  509  482  479  1.32787 
Fr  454  497  467  505  517  5.75410 
Total  2440  2440  2440  2440  2440  
4.47951  2.09016  2.69672  1.49590  2.43033  
: significant at 10%, 5% and 1% level. 
worst return  best return  
Mo  182  121  105  97  105  39.37705 
Tu  108  145  107  124  126  7.95082 
We  108  99  120  142  141  12.21311 
Th  116  119  138  132  105  5.65574 
Fr  96  126  140  115  133  9.72131 
Total  610  610  610  610  610  
38.55738  8.88525  9.00000  9.65574  8.81967  
: significant at 10%, 5% and 1% level. 
worst return  best return  
Mo  165  111  110  107  117  19.37705 
Tu  138  117  111  104  140  8.60656 
We  100  125  129  131  125  5.18033 
Th  112  119  129  148  102  10.11475 
Fr  95  138  131  120  126  8.90164 
Total  610  610  610  610  610  
28.01639  3.44262  3.63934  10.73770  6.34426  
: significant at 10%, 5% and 1% level. 
worst return  best return  
Mo  109  105  133  126  137  6.72131 
Tu  136  124  103  119  128  4.96721 
We  89  141  147  125  108  18.68852 
Th  154  117  108  123  108  11.81967 
Fr  122  123  119  117  129  0.68852 
Total  610  610  610  610  610  
20.31148  5.57377  10.75410  0.49180  5.75410  
: significant at 10%, 5% and 1% level. 
worst return  best return  
Mo  134  106  121  128  121  3.59016 
Tu  112  139  117  106  136  7.09836 
We  126  115  125  115  129  1.40984 
Th  131  105  121  125  128  3.40984 
Fr  107  145  126  136  96  13.45902 
Total  610  610  610  610  610  
4.63934  11.57377  0.42623  4.47541  7.85246  
: significant at 10%, 5% and 1% level. 
worst return  best return  
Mo  47  60  68  43  52  7.51852 
Tu.  63  47  37  56  67  10.96296 
We  56  60  43  51  60  3.81481 
Th  57  45  68  53  47  6.22222 
Fr  47  58  54  67  44  6.18519 
Total  270  270  270  270  270  
3.55556  4.03704  14.85185  5.62963  6.62963  
: significant at 10%, 5% and 1% level. 
In the Supplementary Material file we present the simulation and test of hypotheses for shorter time series. Additionally, we perform an exhaustive analysis of 83 stock indices with different Hurst levels. We can observe that greater Hurstlevels are associated with more significant presence of preferred patterns.
It is clear that theoretical and empirical analyses exhibit some differences. We have to acknowledge that real stock markets dynamics do no follow a pure fGn. In fact, there is not only long range dependence in financial time series, but also in volatility, as shown recently by Bariviera (2017). Precisely, more advanced models such as the fractional normal tempered stable process presented by Kim (2012, 2015), allow for long range dependence in both volatility and noise, and asymmetric dependence structure for the joint distribution. There are many economic variables that influence behaviors known as “stylized facts” of financial time series: volatility clustering, fat tails, asymmetric dependence, etc. For example, Kim (2016) finds that long range dependence increases more in volatile markets, during the Lehman Brothers collapse.
We try to emphasize in this paper that, even using a simple model such as a fGn, some part of the seasonal effect is simply due to the correlation structure of data, and not only by economic reasons. This finding could be used as a starting point in further research, in order to apply prewhitening to time series previous to its analysis, in order to obtain more reliable results.
6 Conclusions
We propose a more general definition of the dayoftheweek effect. We use symbolic time series analysis in order to develop a test to detect it. According to Definition 1, this effect takes place when a pattern is much more or much less frequent than expected from the uniform distribution. The nature of the seasonal effect is reflected in a frequency matrix (Definition 2), and a test is performed. The new definition allows for a more general and comprehensive study on return seasonality. We would like to highlight that the methodology we use here is unique in that it is nonlinear, ordinal, requires no a priori model. Additionally, it provides statistical results in term of a probability density function. To the extent of our knowledge, no one using time series analysis has used a similar methodology before.
Both, theoretical and empirical applications show that this method could be useful to discriminate between rare and preferred patterns of a time series. We show that the socalled dayoftheweek effect is influenced not only by traders’ behavior or economic variables. It could be also be induced by the stochastic generating process of data. The findings in this paper could be taken into account in future research, aiming at the separation between the economics causes and the longrange correlation causes of this financial phenomenon.
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