Spurious seasonality detection: a non-parametric test proposal

Spurious seasonality detection: a non-parametric test proposal

Aurelio F. Bariviera
Department of Business, Universitat Rovira i Virgili, Av. Universitat 1, 43204 Reus (Spain)
aurelio.fernandez@urv.cat
Angelo Plastino
IFLP-CONICET-UNLP, C. C. 727, 1900 La Plata (Argentina)
SThAR - EPFL Innovation Park, Lausanne (Switzerland)
George Judge
Professor of the Graduate College, 207 Giannini Hall, University of California Berkeley (United States)
Abstract

This paper offers a general and comprehensive definition of the day-of-the-week 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 so-called “day-of-the-week” 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” 111According to Kuhn (1968), an anomaly is a fact that puts into question an established paradigm.. Among them, there is one specially puzzling feature, the day-of-the-week effect. This anomaly refers to the heterogeneous behavior of returns along the week. Testing the the day-of-the-week 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 non-trading 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 day-of-the 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 day-of-the-week 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 day-of-the-week effect. Second, it develops an alternative non-parametric test to detect it. Third, it shows that the results of the tests are not obtained by chance, since time-causality is taken into consideration. Fourth, most of the day-of-the-week findings in the literature are related with the underlying return-generating process, and not with the causes indicated previously in the literature. Consequently, from a theoretical point of view, this paper introduces a new non-parametric 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 day-of the week effect and proposes a non-parametric 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 day-of-the 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 222Let 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 5-dimensional 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, pattern-frequency 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 ordinal-pattern’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 Day-of-the-week effect: a redefinition of the problem

As recalled in Section 1, the conventional definition of the day-of-the-week 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 (Al-Loughani and Chappell (2001)). Additionally, markets are not open simultaneously, due to the different time zones (Koh and K.A. (2000)). Consequently, spill-over 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 day-of-the-week 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 day-of-the-week effect is required.

Definition 1

The day-of-the-week 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 day-of-the 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 chi-squared 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 so-called “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 5-day patterns in the sample.

Our definition assumes that the day-of-the-week 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
Table 1: Ordinal patterns. Each number of a pattern represents a day of the week, beginning on Monday. The position of the numbers in a pattern represents the increasing order of returns within a week.

4 Simulation of fractional Brownian motion

In this section we apply the above-outlined 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 data-points 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 day-of-the-week behavior than anti-persistent time series. Additionally, Monday, Wednesday and Friday are the days most affected by the value-change 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 data-sets 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 day-of-the-week. In particular, we claim that, in view of our results, the day-of-the-week 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.
Table 2: Test of Hypothesis 1 on simulated series
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.
Table 3: Test of Hypothesis 2 on simulated series
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
p-value 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.
Table 4: Test of Hypothesis 3 in simulated series
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.
Table 5: Test of Hypothesis 4 in simulated series
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.
Table 6: Test of Hypothesis 5 in simulated series

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 Data-Stream. We split the sample into four non-overlapping 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 week-day.

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 non-trivial 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 day-of-the-week 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
Table 7: Absolute frequency of each pattern. Whole period. Each number of a pattern represents a day of the week, beginning on Monday. The position of the numbers in a pattern represents the increasing order of returns within a week.
Figure 1: Histogram of pattern frequency for the whole period.
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.
Table 8: Absolute frequency of each day in each position. Whole period: 03/01/1966–08/12/2017. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6. Hurst=0.5471.
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.
Table 9: Absolute frequency of each day in each position with shuffled data for the whole period. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6.
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.
Table 10: Absolute frequency of each day in each position. Subperiod 1: 03/01/1966–09/09/1977. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6. Hurst=0.5633
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.
Table 11: Absolute frequency of each day in each position. Subperiod 2: 12/09/1977–19/05/1989.Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6. Hurst=0.5168
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.
Table 12: Absolute frequency of each day in each position. Subperiod 3: 22/05/1989–26/01/2001. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6. Hurst=0.4453
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.
Table 13: Absolute frequency of each day in each position. Subperiod 4: 29/01/2001–05/10/2012. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6. Hurst= 0.4983
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.
Table 14: Absolute frequency of each day in each position. Subperiod 5: 08/10/2012–08/12/2017. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. is the statistic defined in equation 6. Hurst= 0.4914

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 Hurst-levels 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 day-of-the-week 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 so-called day-of-the-week 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 long-range correlation causes of this financial phenomenon.

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