1 I thank dr. Y.R. Kruse for helpful advice, comments and discussions. *Corresponding author. Email address: s.p.meijer@student.rug.nl
Master Thesis – Msc. Finance Rijksuniversiteit Groningen
Seasonality in time
series of stock
returns
Evidence from the Dutch stock market
Author: Steven Meijer* s2214199 14-1-2015
Supervisor: dr. Y.R. Kruse1
Abstract
This paper investigates seasonality on stock returns of AEX and AMX (historically) listed firms, using simple and multiple regressions. Significant patterns in stock returns exist, namely the first month after a given month and in several annual lags. Portfolio strategies exploit these patterns and report positive decile spreads up to 96bps per month, on average. There appears to be a Halloween effect on the Dutch stock market but it does not explain seasonal patterns in individual stock returns.
JEL Classifications: G12; G14
1
1. Introduction
This paper investigates seasonal patterns in stock returns. Seasonality is caused by seasons, weather, holidays and other factors. It is straightforward that patterns influence economies; at individual scale, i.e. home heating costs during summer and winter, and company scale, i.e. daily amusement-park profits. Putting this into statistical terms, seasonality could be described as a characteristic of time series with predictable patterns during a particular period. Developed
independently by both Fama and Samuelson, the Efficient Market Hypothesis (EMH) states that stock prices embody all relevant information. In such a way, it is impossible to beat the market. Seasonality in share prices could become a violation of the EMH, if it offers an opportunity to profit from
historical returns. This paper tests for seasonal patterns in stock returns, and challenges market efficiency by finding a way to exploit seasonality.
Research investigates seasonal patterns in stock returns extensively. Wachetel (1942) is one of the first to report abnormal stock returns in a particular calendar month. From the seventies Bonin and Moses (1974) pick up on this line and show that stock price patterns inhibit seasonality, supported by findings of Officer (1975). Further research tried to discover patterns between stock returns. For example, Rozeff and Kinney (1976) report a so-called ‘January effect’ on the New York Stock
Exchange, with significant differences in returns between January and other months of the year (see also Agrawal and Tandon (1994), Friday and Hoang (2015)). Tinic and West (1984) extend research on the January effect by controlling returns for risk and still observe significant returns during the first month of the year. Another example of a seasonal pattern is the ‘Halloween effect’, also known by its rhyme ‘Sell in May and go away, but remember to come back in September’, see Bouman and
Jacobsen (2002).Even psychological influencers e.g. the amount of sunlight may play a role in risk and return (Kamstra et al (2003, 2005)).
2
Additionally, Heston and Sadka (2008) discover seasonality in annual intervals lasting up to twenty years. They see this as a continuation of Jegadeesh (1990), who reports positive returns of winners with 12-, 24- and 36 monthly lags. However, there is no effect when measuring time series instead of cross sectional differences.
All in all, these findings are promising. Seasonal patterns offer opportunities to earn profits, if strategies are executed appropriately. Based on previous literature, one could classify three different scopes in this perspect: long-, intermediate- and short-term. Besides the aforementioned Heston and Sadka (2008), long term is investigated by DeBondt and Thaler (1985, 1987) who find evidence of profitability of contrarian strategies over three to five year horizons. Li et al (2010) test long term seasonality in the cross-section of stock returns using Chinese indexes, but do not find significant patterns in historical returns. Heston and Sadka (2010) responded to these contrasts by renewing their investigation at an international level. The authors find supportive evidence in favour of a seasonal impact up to five years for Canada, Japan and twelve European countries.
3
Halloween-effect, seasonal patterns are still observed and winners continue to outperform loser portfolios.
The paper is structured as follows: Section 2 discusses the econometric model used to measure seasonality, and explains investment strategies. Section 3 discusses data selection, data realization, and the structure of the data. Section 4 reports findings, continued by a discussion on validity of results in section 5. Finally, section 6 concludes.
2. Econometric Model
Previous research showed contrasting results. Table 1 provides a brief overview of related literature and investment strategy results. A view at table 1 shows inconsistency among all three different perspectives. This paper tries to solve this puzzle by investigating seasonality and develop successful winner-strategies that outperform market returns. In general, previous literature focusses on multi-month periods to form investment strategies. Moskowitz and Grinblatt (1999) choose winners over January-June and hold them during the following six months.
Table 1
Related literature’s impact of investment strategies
Horizon: Authors: Result:
Short Lehmann (1990) Contrarian
Heston et al (2010) Positive
Jegadeesh (1990) Positive
Intermediate Moskowitz and Grinblatt (1999) Contrarian Jegadeesh and Titman (1993, 2001) Positive
Long DeBondt and Thaler (1985, 1987) Contrarian
Heston and Sadka (2008) Contrarian
Li et al (2010) No Impact
Jegadeesh (1990) Positive
Heston and Sadka (2010) Positive
4
strategies on historical returns using single month periods. Generalizing previous literature, winner loser portfolios generate contrarian returns in the first three months. The 4 – 36 months after the formation period tend to profitable, while investments using a longer horizon appear to be negative. However, Heston and Sadka (2008) find evidence of seasonal influence, up to 20 years. Although most of their investment strategies fail to fully exploit annual patterns, it is include a focus on lags up to 240 months.
The main structure among existing literature measures seasonality on stock returns using a cross-sectional approach
𝑟𝑖,𝑡 = 𝛼𝑘,𝑡+ 𝛾𝑘,𝑡∙ 𝑟𝑖,𝑡−𝑘+ 𝑒𝑖,𝑡 (1)
where i = 1, .. , N ; t = 1, .. , T ; K = 1, .. , 240
Here, 𝑟𝑖,𝑡 is the return of stock i in month t. The small k describes the number of monthly lags. The
econometric interpretation is as follows: It regresses the return of a stock on its lagged variable, lag
k, at a specific point in time. Measuring all stocks together produces a coefficient that explains the
average impact of a historical return with lag 1 at one specific point in time. This regression is repeated for every month in the full time period. The average of all monthly coefficients determines the impact of seasonality on stock returns.
5
𝑟𝑖,𝑡 = 𝛼𝑘 + 𝛾𝑘∙ 𝑟𝑖,𝑡−𝑘+ 𝑒𝑖,𝑡 (2) where i = 1, .. , N ; t = 1, .. , T ; K = 1, .. , 240
Again, 𝑟𝑖,𝑡 is the return of a stock, in which i reflects identity and t time. 𝛼𝑘 is an intercept that
varies for different historical lags, k. Next, 𝛾𝑘 measures the effect a historical returns of lag k has on
the observed return. This historical return 𝑟𝑖,𝑡−𝑘 varies per company. The latter notation is an error
term that is time and company dependent. Lags k range from 1-240 months.
Besides (2) a second regression is calculated. Equation (3) regresses stock returns on multiple historical lags in order to measure incremental effects of historical returns. Additionally, it provides a robustness check of simple regressions. If historical returns document significant effects in a simple regression, robustness would imply that these effects would be present in a multiple regression too. The specification is as follows:
𝑟𝑖,𝑡 = 𝛼𝑘,𝑡+ ∑12 𝛾𝑘
𝑘=1 ∙ 𝑟𝑖,𝑡−𝑘+ 𝛾24∙ 𝑟𝑖,𝑡−24+ 𝛾36∙ 𝑟𝑖,𝑡−36+ ⋯ + 𝑒𝑖,𝑡 (3)
where i = 1, .. , N ; t = 1, .. , T ; K = 1, .. , 240
Regression (3) thus measures the incremental effect of historical lags 1-12, 24, 36, .. , on stock returns. The outcomes of (2) and (3) form the basis of investment strategies that try to exploit seasonal patterns, using:
𝛾𝑘 = ∑𝑁 𝑤𝑖(𝑡, 𝑘)
𝑖=1 ∙ 𝑟𝑖,𝑡 (4)
When a slope coefficient is statistically significant in both (2) and (3), it suggests that there is a mispricing of stocks. This coefficient 𝛾𝑘 then differs from zero and equation (4) applies. Equation
6
high to low. These are then separated into decile spreads, from top 0-10, 10-20, .. to the losing 90-100. The strategy holds winners of the formation period during the event month t. Every month portfolios are rebalanced by selling losers and buying historical winners. To measure profitability, every strategy starts at a base value of 100 and is every month adjusted for the related monthly portfolio return. This strategy is repeated for every decile spread, making it possible to compare the effect of historical returns among different spreads.
∑𝑁 𝑤𝑖(𝑡, 𝑘)
𝑖=1 = 0. (5)
This implies that the portfolio has no net investments. Specifically, it depends solely on historical returns observed in the formation period, where
∑𝑁𝑖=1𝑤𝑖(𝑡, 𝑘)∙ 𝑟𝑖,𝑡−𝑘= 1.
7
To continue, regression (2) and (3) depend on time and identity. As a result this paper utilizes an unbalanced panel data set, a method that controls for both time and identity. The main advantage of panel data is a larger sample size than cross-section or time series data (Hsiao (1986)). This improves quality of estimates, and minimizes the impact of collinearity in time series, see Baltagi (1988). Furthermore, a panel data set is capable of capturing effects that would not be observable in cross-section or time series analysis alone. Drawbacks are data collection issues, as described by Kasprzyk, Duncan, Kalton and Singh (1989).
In addition, regression (2) and (3) control for fixed effects and random effect. Fixed effects are required when measuring variables that vary over time. This is essential, i.e. industry-effects could affect stock returns. Section 4 describes this in further detail. The following section continues by discussing data.
3. Data
3.1 The Dutch stock market
The analysis investigates stock returns of shares trading on the Dutch stock market. The Dutch market is chosen for three reasons: First, the first stock index in Europe originated in the
Netherlands, hence it provides a considerable time period. Second, data standards are of high quality. And third, as mentioned by Lo and MacKinlay, an unexplored market could provide new insights.
8
index. Tele2 owned a substantial part of the cable network in the Netherlands, and Gucci Group several international clothing brands. Also, mergers between Dutch and foreign companies caused that several companies trading on a Dutch index are partially under control of foreign companies (i.e. Air France-KLM).
The AEX started on 4 March 1983 under the name EOE-index. EOE is the short notation of
‘European Option Exchange’, a phenomenon that existed in the United States and inspired Europe to launch option trading (1978). Two times a day the option index was updated, what motivated the idea to develop an index that updated on a continuous basis. This became the AEX index, the first index in Europe that was computed continuously. The index began with thirteen different shares. Although the EOE-index originated from March, the index was fabricated to start at the first trading day of 1983 at 100 index points.
Over time, supplementary stocks increased the size of the index. On 5 October 1989, the decision was made to include a maximum of 25 companies in the EOE-index, dependent on trading volume per company. The official name ‘Amsterdam Exchange Index’ was adopted in 1994. Nowadays, the index revaluates two times per year, with an extra interim review on the first trading day of September.
The Amsterdam Midkap Index is the second Dutch index, composed of the 26-50 largest firms trading on the exchange. In was introduced one year after the AEX, on 4 October 1995. The AScX followed in 2005, and captures the 50-75 largest stocks.
Despite usefulness of the AScX with 10 years of information, it is excluded from the sample. This index trades smaller funds on the Dutch stock market. While the larger firms in the AScX still feature substantial market capitalization (Refresco Gerber NV, 1258 million euro2), a smaller firm i.e.
MacIntosh Retail Group NV. captures a minimal of 6.4 million euros1. This unsubstantial market capitalization could be prone to speculative investing, moreover affecting returns of smaller stocks. This could create measurement errors when testing the impact of seasonality, as speculative trading
2
9
is hard to observe. The AEX and AMX together form the fundamental of this paper. Every stock that once traded on the AEX and/or on the AMX is included in the sample. No action is undertaken if a particular stock is downgraded to the AScX, assuming that the market capitalization is still substantial of the downgraded stock. Therefore, the sample size occasionally exceeds the 50 individual stocks from the two largest indexes combined.
3.2 Data collection issues
A critical issue in forming a correct sample is dependency on historical compositions.
10
Historical returns are obtained via Datastream. An advantage of Datastream is accessibility to a wide variety of data. It contains data of all companies currently trading on the AEX and AMX, and companies historically trading. An additional benefit is existence of data on Dutch stocks prior to the start of the AEX in 1983. Datastream includes data up to the first trading day of the year 1973, enhancing the sample size with 10 additional years. Consequently, this papers sample incorporates data from 01-01-1993 up to 15-10-2015, with pre-sample data on historical return performance ranging back to 01-01-1973. The reason that October 2015 is included is because it the sample still captures half of all trading days of this particular month (11 out of 22). It would be a loss to neglect these additional days, as seasonality could be observed here. Hence, October does weight as a full month, merely capturing 11 trading days.
The data set consists of monthly returns. Monthly returns are calculated using daily values at the beginning of the month and daily values at the end of the month in combination with a natural logarithm. The main advantage of including a natural logarithm is that returns will be time consistent, or in other words, continuously compounded. This makes it possible to calculate returns over any time period, if the return over a particular time interval is known. Hence, it is possible to calculate annual average stock returns using a monthly average return and time as scale.
Given the time period 1993-2015, a concern is to control for mergers, acquisitions or insolvency. When two companies trading on a Dutch index merge, it is important to cut off both time series on the precise same date. If the newly established company maintains tradable, a new time series starts the next trading day after the event (if possible). An example of merger in the sample is between the two Dutch companies Van Ommeren and Pakhoed, into Vopak. A second issue is acquisitions. Often information is available on the precise date of exit, making it possible to cease time series on a correct date. There are several exceptions on this rule, i.e. Fortis. This Dutch bank acquired a
11
price of Fortis. In despite of this, shares were still trading, aiming at a narrow escape. Ultimately, the Dutch government stepped in and purchased all outstanding shares. The price closed on 12 January 2008 at €1.49. Therefore, December 2007 is the last month included. Insolvency is the third and most complex matter. Companies operating under the threat of bankruptcy reflect this risk in the related share price, directly affecting the profitability of the stock. This observed difference in return is no consequence of seasonality, and should thus not be included in the sample. One could imagine that the correct exit date is difficult to determine, as a decision needs to be made on when risk of insolvency starts to control share prices. This paper combines shares price information and other relevancy regarding insolvency of a company to define an end date. The Dutch company Imtech became insolvent on 13-08-2015 at €0.39 per share. Clearly, the risk of bankruptcy determined the price. In the months preceding the event the stock fluctuated relatively stable between 3 and 4. Therefore, the last trading day of July is the exit date.
Inclusion of data on firms that became insolvent is essential. Not only does this improve the explanatory power of single and multiple regressions, it overcomes the influence of survivorship bias. Survivorship bias exists if defaulted companies are excluded from a study. This can result in skewed findings as mere survivors are included.
To conclude, the data set consists of 127 different companies’ share returns in a time period from January 1993 till October 2015, with lagged historical returns ranging back to January 1973. This unbalanced panel data set contains 20710 unique observations in total.
3.3 Data discussion
12
stocks per lag is shown, described by the dotted line. Additionally, figure 1 provides an estimate of the sample distribution as lags increase, in the form of a trend line. All firms in the sample have return histories of one month. The minimum amount of observations is two, the company
Buhrmann. On the other extreme, there are 33 unique firms in the sample that cover the full time period 1993 - 2015. In the full sample, 98% of all stocks is able to provide data with an annual
historical lag (12 months). If the historical lag extends to 120 months, the sample size decreases to 91 cross sections. Even when testing for seasonality on stock returns using 240 monthly lags, still 54% of all companies reports returns with 240 monthly lags. This equals 68 companies.
Overall, it is reasonable to state that the data set is extensive. Even when stock returns depend on 240 monthly lags, the sample size still contains a large amount of different identities.
Figure 1. The sample describes the number of firms available for regression analysis, using (historically) listed firms on the AEX and AMX during the period January 1993 – October 2015.The dotted line represents the real fraction, whereas the thick line is a trend line estimating the fraction.
Figure 1 provides a detailed overview of the full sample and availability of data when historical lags extend. Nevertheless, it is still ambiguous how company-specific data develop over time.
13
Regularly in the sample, a new company enters or an incorporated company leaves the exchange. Therefore, it is important to control for differences in the time period between firms. Figure 2 shows the development of stock return over time, uniquely for every individual firm. It shows that there are different observation periods within the sample, an unbalanced panel. This could become
problematic when regressing at a cross-sectional level. However, a time-series regression partially overlooks this problem of unbalanced time periods by testing within time instead of at a specific point in time. To control for unavailable historical values of newly entered stocks, a restriction is imposed in the regression. The intuition is the following: If a company does not have a matching historical return 𝑟 𝑡−𝑘 for lag k with a realized return at time t, the observation is excluded from the
regression. Additionally, EViews has an automatic function build-in that controls for the unbalanced time series in the regression analysis in a similar way. Ultimately, the number of unique stock in the sample varies over time, and for different values of k. Here, an extensive data set is able to overcome this mismatching of sample periods.
From figure 2 it is obvious to see that the observations cluster around their mean, with some scattering extremes. The clustering around the mean confirms the exclusion of moving averages. A moving average exists if observations cluster around an upward or downward trend. A moving average could be problematic as it biases regression coefficients. This is an effect of trends following observations, hence lagging the real momentum. Despite the fact practically every stock includes a (temporary) moving average, this problem is easy to overcome by controlling for relative returns, a first step in computation of original data. The scattered extremes could indicate some
14
15
In addition to the previous two figures, Table 2 presents descriptive statistics of the complete data set as it gives a numerical overview of the distribution and details of the sample. It describes the distribution of the monthly return and historical returns for lag k = 1, 12, 120 and 240 months. Monthly return is the realised return in the period ranging from January 1993 to October 2015, comprising all companies. The additional historical lags give an overview of data set development as lags increase. The monthly return has a mean return of 0.218%, with a higher median return. This suggests that average returns are affected by negative extremes. The negative extreme is indeed significantly large. For a historical lag of k = 1 there are no considerable differences. The dissimilar number of observations of lag 1 and the monthly return is a consequence of stocks that newly enter the index, and therefore have no historical value in the first month. The same can be said about the other three historical lag variables. Here, the lags are longer, what makes the historical lags even more selective. The mean return decreases among companies with minimally 12 monthly lags, similarly the median return. If k equals 120, surprisingly the mean shows an increase while the standard deviation decreases. This is an interesting insight, as it offers higher returns with less risk. Finally, when a regression or strategy requires 240 historical returns, the sample decreases to 10751 observations. This equals 0.519 as a fraction of monthly returns, with 39 unique companies as aforementioned. While the average return is lower for 240 lags than k = 120, it is still larger than the mean of the other series. Contrasting, median return decreases, suggesting less negative extremes in the sample. Additionally, the standard deviation decreases.
Table 2
Descriptive statistics of monthly returns in percentages
Variable: Mean: Median: Maximum: Minimum:
16
To summarize, the sample consists of firms that currently trade on either the AEX or AMX, or had historically traded on one of the two exchanges. The sample size ranges from January 1993 to October 2015, with pre-sampling data on lagged returns ranging back to January 1973. There are 127 different stocks with 20710 unique return observations during this period. The data set decreases in size as historical lags extend, as not every stock has historical observations for a given lag. Yet, the data set is extensive to overcome potential problems regarding extension of lags. Additionally, the data set is an unbalanced panel data set, with a fluctuating number of unique identities at specific points in time, and over time.
4. Results
To ease the reader, the simple regression (2) is duplicated below:
𝑟𝑖,𝑡= 𝛼𝑘 + 𝛾𝑘∙ 𝑟𝑖,𝑡−𝑘+ 𝑒𝑖,𝑡 (2) where i = 1, .. , 127 ; t = 1, .. , 274 ; k = 1, .. , 240
17
cross-sectional basis, in contrast to the time-series approach here. It is interesting to notice a similar finding.
Panel B shows a smoothed version of the regression estimates. Smoothed returns are calculated over average returns using a moving window. Specifically, the returns used in Panel A are averaged in a [-5, +5] window, and replace the returns in equation (2). This implies that 𝑟𝑖,𝑡 is an average return of
t-5, t-4, .. , 0 , t+1, t+2, .., t+5 for stock i. Next, the returns are regressed on the lagged average return, for every k = 1, 2, .., 240. An advantage of a moving window in this respect is the representation of trends in the return among lags.
Panel B is consistent with findings from Panel A. Estimates computed using a moving window are below zero in general. Regression estimates have positive predictive power on the first 12 months after the event month. This is in line with findings from Jegadeesh and Titman (1993, 2001), who conclude that the stock performance has a positive influence on future performance up to twelve months. In Panel B, signs change after 12 months and become negative. There is a negative effect on returns up to 60 months. DeBondt and Thaler (1985, 1987) report a negative effect during year three to five. Figure 3 is thus consistent with their findings. For higher values of k the effect is
predominantly negative, except around lag 84 and 120. It is interesting to see that, at k = 120, the regression estimate culminates at an annual lag. The same can be said at annual intervals 72, 144, 192, 204 and 216. Jegadeesh was the first to report positive pulses at annual lags (24 and 36). Heston and Sadka recognize this as a periodical pattern that lasts up to 240 months. This paper reproduces multiple positive pulses at annual intervals for the Dutch exchange, while several annual intervals describe no distinct pattern. Overall, figure 3 is consistent with previous literature.
Table 3 reports the regression estimates of simple and multiple regressions and t-statistics
18
Panel A. Simple regression estimates as a function of historical lag
Panel B. Smoothed graph of regression estimates as a function of average returns, using a moving window [-5, +5]
Figure 3. Time series regression of returns. Simple regressions are calculated for each lag k, over a period January 1993 – October 2015. The regression takes the form ri,t= αk + γk∙ ri,t−k+ ei,t. Here, ri,t is the return of stock i in month t. αk is a constant, dependent on k. The regression estimate is γk, a coefficient calculating the impact of lagged return (ri,t−k) of stock i at time t-k. It has a maximum of 274 monthly observations and historical lags range from 1-240. Panel A shows the regression estimates γk for every lag k. Panel B plots a smoothed version of Panel A. Here the regression estimate is calculated using average returns using a moving window [-5, +5].
19
Simple regression of the form (2) produces estimates that are generally positive for lag k. Negative returns are small in size compared to positive estimates. Nevertheless, the estimates are mainly insignificant. Estimates that are significantly different from zero at a five percent level are lag 1, 96, 120 and 204. At a ten percent level, lag 108 and 168 join. In the multiple regression lag 8 and 10´s incremental effect is significant at a 10 percent level. The question arises if seasonality is actively influencing the Dutch exchange, as several simple regressions fail to detect significant effects in 23 out of 31 times. The answer to this question is yes. There is still significance in 8 different historical lags. Additionally, significance is discovered in the time series of stock returns. Several related studies failed to detect seasonality using time series (ie. Heston and Sadka, 2008). They are only able to detect seasonality on a cross-sectional level. Hence, there is an rare degree of seasonality on the Dutch index.
Panel B produces multiple regressions of the form (3), where regression estimates are calculated using multiple historical lags. An advantage is the possibility to measure incremental effects and test robustness of simple regressions. Robustness can be tested by regressing additional equations to determine if effects stay significant among other regressions. Lag 168 is an example, in which its incremental effect in the multiple regression is indifferent from zero. Multiple regression estimates are consistent with simple regressions, as they show a positive incremental effect for historical lags. Since this measures incremental effects, estimates decrease in size, similarly for white controlled t-statistics. Lags that are robust in the simple and three multiple regressions are 1, 96, 108, 120 and 204. The fact that, of these five lags three complementary annual lags are significant, is remarkable. Testing a multiple regression on lags 96, 108 and 120 together still detects a positive effect.
Unreported statistics show a significant positive effect on returns for 121 lags, robust in simple and multiple regressions. A similar pattern is unobserved for other lags after an annual interval.
20
Table 3
Simple and multiple regression of returns
Simple regressions are calculated for each lag k, over a period January 1993 – October 2015 (274 months in total). The regression takes the form ri,t= αk + γk∙ ri,t−k+ ei,t. Here, ri,t is the return of stock i in month t. αk is a constant,
dependent on k. The regression estimate is γk, a coefficient calculating the impact of lagged return (ri,t−k) of stock i at time
t-k. The historical lags range from 1-240. Panel A reports γk and the relating t-statistic. Panel B shows estimates of a multiple regression of the form ri,t= αk,t+ ∑12k=1γk∙ ri,t−k+ γ24∙ ri,t−24+ γ36∙ ri,t−36+ ⋯ + ei,t. Multiple coefficients are calculated in a time series regression. Similarly, 𝛾𝑘 is the regression estimate. Three multiple regressions are analysed. The first consists of lag 1 to 12, 24 and 36. The second adds each 12th lag up to 120. The final regression adds every 12th lag up to 240. The reported t-statistics are corrected for cross equation correlation and heteroskedasticity (using White cross-section standard errors). The analysis uses AEX- and AMX- (historically) listed stocks.
Lag:
Panel A. Simple regressions Panel B. Multiple regressions
Estimate: t-statistic: Estimate: t-statistic: Estimate: t-statistic: Estimate: t-statistic:
21
Table 4
Specification of simple regressions in table 3
Simple regressions are calculated for each lag k, over a period January 1993 – October 2015. The regression takes the form ri,t= αk + γk∙ ri,t−k+ ei,t. For every regression the joint significance of cross section fixed effects estimates is calculated, represented in terms of probabilities in the second column. The third column shows the probability of the Hausman test. NA stands for not available. Model specification shows the specification of the regression type used for simple regressions in table 3, based on the probabilities of the two tests.
Lag: Fixed effects probability: Random effects probability: Model specification:
1 0.000 0.087 Fixed 2 0.000 0.000 Fixed 3 0.000 0.004 Fixed 4 0.000 0.019 Fixed 5 0.000 0.000 Fixed 6 0.000 NA Fixed 7 0.000 NA Fixed 8 0.000 NA Fixed 9 0.000 NA Fixed 10 0.000 NA Fixed 11 0.000 NA Fixed 12 0.000 NA Fixed 24 0.000 0.086 Fixed 36 0.000 0.465 Random 48 0.000 0.834 Random 60 0.000 NA Fixed 72 0.008 0.463 Random 84 0.034 NA Fixed 96 0.015 0.753 Random 108 0.003 NA Fixed 120 0.000 NA Fixed 132 0.001 NA Fixed 144 0.000 0.932 Random 156 0.000 0.739 Random 168 0.000 0.723 Random 180 0.000 0.891 Random 192 0.000 0.747 Random 204 0.000 0.849 Random 216 0.000 0.500 Random 228 0.000 0.779 Random 240 0.000 0.401 Random
22
random effects (RE). In order to determine the correct approach, table 4 presents probabilities that estimate the correct estimation form of simple regressions, specified per lag. The column fixed effects probability rejects the null hypothesis of no evidence of cross section FE at a five percent level for every lag. Hence, every simple regression in table 3 prefers FE over pooled OLS. RE are a special form of FE, assuming that independent variables are uncorrelated with individual specific effects. If this assumption holds, RE estimates are preferred over FE estimates because they are more efficient. However, if the RE assumption is violated, the model produces inconsistent estimates. Therefore it is critical to be correct when accepting RE estimation. A Hausman test determines preference of RE over FE. Specifically, under the null hypothesis both estimators are consistent, but RE estimators are efficient due to a lower asymptotic variance. A rejection of the null hypothesis implies that FE are preferred over RE, as the latter produces inconsistent estimates. Because it is extremely important to avoid inconsistency, the null hypothesis is rejected at a 25 percent level. The specification of simple regressions in table 3 is presented in the latest column of table 4. In a special case, the Hausman test statistic is not positive definite. Hence it is not possible to compute probability statistics, represented by NA. In this case the FE model is preferred, since there is no sound proof for RE preference. The same holds for multiple regressions. All three multiple regressions accept FE, as pooled OLS is rejected at a ten percent level (probabilities of 0.000, 0.066 and 0.007, respectively).
23
investment strategies are only formed if seasonality in a historical lag is robust over all regressions. Robustness implies a significance among all regressions at a ten percent level. Strategies thus focus on seasonality observed on the Dutch exchange only, instead of following discovered patterns mentioned in related literature. For example, DeBondt and Thalers 2-5 year perspective is ignored as seasonality is unobserved from this data set. Ultimately, investment strategies are formed for historical lags 1, 96, 108, 120 and 204.
Besides comparing between lags, it is interesting to compare stock returns within lags. Therefore, decile portfolios are formed. Decile portfolios rank stock performance from high to low for every stock trading in the formation month. After the formation period, the stocks are held as a portfolio in the specific decile during the event month, based on the historical lag. The event month measures the return of the portfolio per decile. For example, for lag 204, the winner portfolio in December 2009 consists of the 10% best performing stocks in December 1992.
Table 5
Average monthly returns of winner loser strategies based on historical performance
Decile spreads are created using historical returns sorted from high to low. Every stock in a decile has an equal weight, and is being held during the event month. For example, for lag 204, the winner portfolio in December 2009 consists of the 10% best performing stocks in December 1992. Every month portfolios are rebalanced. The average monthly return of every strategy is reported inpercentages in the table below, with standard deviations in brackets. The average return is calculated over the period January 1993 – October 2015. The analysis uses AEX- and AMX- (historically) listed stocks.
24
Table 5 reports average monthly returns of the equally weighted decile portfolios at time t based on stock performance in the month t – k. Standard deviations of the portfolios are shown in brackets. These are reported because t-statistics are negligible. A reason for negligible t-statistics is this
presence of several extreme returns in the strategy. These do not heavily influence mean returns as the sample size is large, but do influence volatility of portfolios. The first row present results for a historical lag of one month. Historical winners produce an average return of 0.072%. Other deciles outperform the winners, except of losing stocks. The worst performing stocks of the previous month perform poorly in the subsequent month, resulting in an average monthly loss of -0.733%. Therefore, the incremental return between winners and losers is large, as shown in the latest column. These findings do not match with conclusions from Lehmann (1990). He finds evidence of contrarian stock performance. Winner portfolios using a historical lag of 96 months outperform other deciles. The table reports a decrease in average returns as deciles decrease. Surprisingly, the loser decile
performs relatively good, with an average return of 0.365% per month. Again the difference between the winner and loser portfolio is positive.
25
Similar winner loser portfolios by Heston and Sadka report a higher profitability of loser deciles compared to winner deciles, all statistically significant. Contrasting, in this paper winner portfolios generally predict a higher return than loser portfolios. This suggests that on the Dutch exchange market efficiency fails to eliminate all seasonal differences in expected returns. Clearly, the best performing stocks based on 204 lags produce an enormous difference from loser portfolios in the period. Similarly, the decreasing average return for lags 96 and 204 is interesting. The question arises whether the returns are truly driven by a form of market inefficiency, or that other factors explain the difference between average strategy return.
A main driver of the return is the exposure to the level of risk of a particular stock. Investors are risk averse by nature, and demand a premium for risk taking. A method that relates risk to return is the Sharpe-ratio. A Sharpe ratio estimates average excessive return in respect to portfolio risk. The risk free rate in the Sharpe equation is the yield on a three-month German treasury bill at 15 October 2015, because the German government is a stable government using the same currency. Longer spreads would require an illiquidity premium and therefore not reflect a risk-free rate. 15 October 2015 is exit date of portfolio strategies. Table 6 shows Sharpe ratios for every decile and every lag. Return controlled for risk produces similar results. For all lags the winner deciles produce a higher ratio than losers, except of lag 108. Additionally, the decreasing patterns in lag 96 and 204 are still present among Sharpe ratios. The pattern is distinct in the latest historical lag, comparing the top 40% of winners with the 30% of losers.
26
difference between winner- and loser portfolios. The main driver of stock returns is risk. However, controlling portfolio returns for risk fails to explain differences between average returns. This can be seen as a form of market inefficiency, as investors fail to incorporate this information in reflected stock prices. The discussion continues.
Table 6
Sharpe ratio per trading strategy
Sharpe ratios (SR) take the form SRk,s= E(R)σk,s−RF
k,s . The function calculates the ratio using respective average returns
and standard deviations per portfolio strategy reported in table 5. k represents the lag, and s the strategy. The risk-free rate is the yield on a 3month German treasury bill at 15 October 2015 (-0.226%).
Strategy: 1 (Winners) 2 3 4 5 6 7 8 9 10 Lag 1 0.039 0.082 0.129 0.107 0.090 0.081 0.105 0.082 0.052 -0.048 Lag 96 0.119 0.107 0.086 0.070 0.083 0.116 0.080 0.046 -0.006 0.086 Lag 108 -0.013 0.128 0.171 0.099 0.089 0.031 0.116 0.060 0.068 0.006 Lag 120 0.059 0.045 0.128 0.113 0.108 0.061 0.067 0.017 0.057 0.055 Lag 204 0.191 0.128 0.106 0.078 0.025 0.036 -0.003 0.062 0.056 0.054
5. Discussion
The previous section shows that seasonality in stock returns can be exploited using simple winner loser strategies based on past performance and hold these during event months. To ensure
robustness of this finding, the first part in this section tests and discusses related factors in respect to periodic patterns in stock returns. For example, there is evidence of a Halloween effect on the Dutch stock market, but controlling returns for a Halloween effect still produces seasonal patterns in stock returns.
5.1 Robustness
A first determinant of seasonality could be seasonal patterns itself. Therefore, this subsection
27
strategies in January. Returns are positive in all other months of the year. Heston and Sadka (2008) continue, and find evidence of positive effects in all months, particularly high in January. For longer horizon strategies, they find both positive and insignificant results. Again, returns are particularly high in January. Table 7 reports average return of winner-loser investment strategies, separated for every calendar month. For example, the average return of May, given lag k, is the average return of all trading portfolios (ten deciles) in the month May, during January 1993 through October 2015. Because earlier research mentions the significance of January, a separated column presents the mean portfolio return over all months except January.
Table 7 offers an overview of the distribution of portfolio returns specified per calendar month. Clearly, portfolios are very profitable during January. Average returns vary between 1.202 and 2.007 percent. Also February and March appear to be fruitful investment months, with all mean returns being positive. During April returns are again positive for every lag, with the comment that t-statistics are high. Namely, three of five are significant at a five percent level, lag 108 joins at ten percent. During May, stock returns are positive but marginal compared to previous months.
28
Table 7
Seasonal returns of investment strategies based on past performance
Every month stocks are distributed over ten trading portfolios based on past stock performance. The stocks are given an equal weight. For example, the trading portfolio that is formed based on past stock performance 96 months ago, classify stocks according to their returns during the historical lag 96. Portfolios are rebalanced every month without additional investments. The average returns of trading strategies are reported in the table below, separated for calendar months during the period January 1993 to October 2015 (in percentages). t-statistics are reported in the corresponding rows, in parentheses. Additionally, average returns is calculated using all strategy returns excl. January. These findings are reported in a separated column. The analysis uses AEX- and AMX- (historically) listed stocks.
29
A reason for excessive returns in January and December could be turn-of-year effect, describing higher trading volume and stock prices during the beginning and ending weeks of the months, respectively. Another explanation could be tax-driven trading. But clearly, negative/small returns during May through November and positive returns during December – April perfectly fit in the ‘Halloween-effect’. This describes a seasonal difference in returns between November to April and May to October, where the latter produces systematically negative or low returns. Furthermore, average returns of May, July and October are marginal in contrast to December – April returns. A way to test for Halloween-effects in the Dutch stock market is to compare portfolios exploiting winner-loser strategies during November – April, and compare average returns with the same strategy during May – October. Here, the trading strategy during the six month period is similar as in table 5. Only, after April (October) all stocks are sold, and cash is being held. At the start of November (May) new shares are bought and the strategy continues. The results are shown in table 8.
Table 8
Average return of winner loser strategies controlled for a ‘Halloween-effect’
30
Table 8 shows average returns of similar winner-loser portfolios as in table 5. For every lag, the first row represents average returns during November – April and the second for May – October. Returns are calculated over November 1993 – October 2015, in order to have an equal amount of observations between the two periods. Clearly, the Halloween-effect exists in the particular
investment strategies. Independent of historical lags or deciles, average returns in November – April outperform returns in May – October.
Comparing return differences between winner loser portfolios (1-10) between this table and returns differences from table 5 shows that a Halloween-effect partially influences patterns in stock returns. Under monthly lags, both May-Oct and Nov-Apr report positive returns between winners and losers. Based on 96 lags losers outperform winners during May through October, but marginally different from zero. Similarly, no large differences between the six month periods are observed for lags 108 and 204. Comparing winner- with loser strategies based on 108 lags reports contrasting results. Therefore, the Halloween effect is further studied.
To analyse if a Halloween effect explains periodic patterns in stock returns, simple regressions of returns are estimated on lagged returns and a dummy variable
𝑟𝑖,𝑡= 𝛼𝑘 + 𝛾𝑘∙ 𝜕𝑡∙ 𝑟𝑖,𝑡−𝑘+ 𝑒𝑖,𝑡. (6)
This equation differs from the simple regression (2) by a dummy variable 𝜕𝑡 that controls for the
Halloween effect. It takes a value of 0 for every month between May – October and 1 otherwise. For every historical lag 𝛾𝑘 is estimated during the period November 1993 – October 2015 and compared
31
Panel A. Simple regressions for May through October
Panel B. Simple regressions for November through April
Figure 4. Time series regression of returns. Simple regressions are calculated for each lag k, over a period November 1993 – October 2015. The regression takes the form ri,t= αk + γk∙ 𝜕𝑡∙ ri,t−k+ ei,t. Here, ri,t is the return of stock i in month t. αk is a constant, dependent on k. The regression estimate is γk, a coefficient calculating the impact of lagged return (ri,t−k) of stock i at time t-k, dependent on a dummy variable ∙ 𝜕𝑡. The dummy variable is 0 for every month May – October and 1 otherwise. The regression has a maximum of 132 monthly observations. Panel A shows the regression estimates for May through October. Panel B shows the regression estimates for November through April.
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0 12 24 36 48 Regressio n est im ate Lag
Return regressed on return May - October (/2)
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0 12 24 36 48 Regressio n est im ate Lag
32
Return estimates based on the full sample are added in the graph. Clearly, estimates follow a similar pattern as estimates based on the full sample. Panel B continues and shows estimates during November through April. In a similar way, regression estimates during November – April follow the pattern of regression estimates based on the full sample. Unreported statistics for both six-month periods show positive regressions estimates at annual lags 96, 120 and 204. Only for lag 108 the estimate during November – April is asymptotically zero, and reporting a t-statistic far different from a ten percent significance level. Three other regressions are significant, and four slightly differ from ten percent significance boundaries. It is very probable that insignificant t-statistic are driven by smaller sample sizes. Therefore, only large deviations from the full sample are considered as proof, which is not the case. Thus, controlling for a Halloween-effect does not remarkably change the impact of historical stock performance on future returns.
A different factor related to periodic patterns in stock returns is the time frame. Barroso and Santa-Clara (2015) report high variable risk of momentum over time. Controlling for risk increases Sharpe-ratios substantially. Therefore, table 9 reports average returns of investment strategies for three different time periods. Time periods are of equal length and cover 21 years of the full time period. Average returns during 1993 through 1999 vary between 0.89% and 1.87% per month. The second period reports both positive and negative returns, dependent on the historical lag. Average returns tend are closer to zero than during the previous period. For 2007 through 2013 average returns are generally negative. They range from -1.10% to 0.44%. Yet, all these returns are
33
Table 9
Average return of winner loser strategies for different time frames
Decile spreads are created using historical returns sorted from high to low. Every stock in a decile has an equal weight, and is being held during the event month (rebalanced monthly). For example, for lag 204, the winner portfolio in December 2009 consists of the 10% best performing stocks in December 1992. Returns are specified for different time frames: 1993 through 1999, 2000 through 2006 and 2007 through 2013 (in percentages). t-statistics are reported in parentheses. The analysis used AEX- and AMX- (historically) listed stocks.
Strategy: ’93-‘99 ’00-‘06 ’07-‘13 Lag 1 1.873 -0.856 -0.956 (1.048) (-0.805) (-0.620) Lag 96 1.515 0.232 0.437 (0.968) (0.182) (0.444) Lag 108 0.979 -0.138 -1.093 (0.546) (-0.089) (-0.806) Lag 120 0.893 0.628 -1.104 (0.861) (0.308) (-0.402) Lag 204 1.346 0.989 -0.029 (1.685) (0.971) (-0.020)
Next is trading volume of a stock. Some stocks tend to trade more in particular months of the year, and less in other periods of the year. This could create liquidity premiums in the months of excessively trading the stock, similarly discounts in months of relative trading tranquillity. Pastor and Stambaugh (2003) argue that the correlation between illiquidity and realized returns is indeed negative. An increase in liquidity often arises from disappointing performance, so that prices fall to rebalance increased supply. Contrasting, Acharya and Pedersen (2005), and more recently Chiang and Zheng (2015), report a positive relation between excessive share returns and illiquidity risk premiums. In order to test for a possible influence of trading volume and illiquidity in stock returns, trading volumes are required. Yet, these are unobserved for all stocks in the sample. Heston and Sadka (2008) do test trading volume and find a strikingly similar pattern between seasonality measured using historical lags (12-60) and abnormal trading volume regressed on lagged
34
similar seasonal patterns. They conclude that trading volume does not explain patterns of returns. This suggests that, if for this paper a similar pattern within trading volume is found, it is ambiguous if seasonality could be explained by this. Further research is suggested
Closely related to calendar events are dividends and earnings announcements. Dividends are seasonal events, often quarterly or annually distributed. Dividends provide an additional profit to shareholders, stimulating demand before dividend-distribution periods. Theoretically, in the post-dividend period stock prices move back to the initial value and return decreases. There could be thus be a dividend-driven seasonal deviation in the return of stocks. Heston and Sadka (2008) test for dividend controlled return and observe that dividends do not explain seasonality in stock returns using NYSE- and AMEX-listed stocks.
Moreover, earnings announcements could drive seasonality on the stock market. Earnings announcements are seasonal and last up to 12 historical lags. If markets under- or overreact to seasonal earnings information in a particular year, returns increase in the next period of earnings announcements. If in the next period again earnings announcements are not completely captured in the share price, a seasonal pattern emerges. This could explain profitability of annual portfolio strategies. However, earnings announcement receive lots of attention. It is improbable that this factor drives patterns in stock returns.
Risk plays a role in stock returns and is discussed and tested in section 4. There, risk is based on deviations from average portfolio strategies. It is possible that risk in share returns is also related with size or industry. Gandhi and Lustig (2015) indeed show that size anomalies play a role in risk-adjusted returns in the banking sector. Larger banks tend to require a lower risk risk-adjusted return than intermediate – or small firms, despite higher leverage ratios. The same could be true for the stocks trading on the AEX and AMX. Governments forms a hedge for large-sized firms, protecting
35
Furthermore, Lewellen (2002) shows that industries influence monthly stock returns and trading portfolios. Intra-industry differences lead to return differences of 40bps per month, depending on the industry. It is not possible to control for industry-specific effects in this sample, because the number of stocks trading on the Dutch exchange is limited. Such an investigation would require a large database per industry-group to produce accurate estimates.
5.2 Model specification
This section discusses statistics related to regressions and analyses in the paper. Also, some remarks on portfolio strategies are addressed.
Data sets on stock returns are seldom normally distributed, since multiple extremes influence distributions of that particular sample. From a theoretical point of view, this is logical. Investors are particularly sensitive to negative shocks and overreact to this. Consequently, distributions often incorporate high values of skewness and kurtosis. A normal distribution, also known as a non-Gaussian distribution, does not restrict the Gauss-Markov theorems that are assumed under OLS. Such an approach is applicable if the sample size is extensive, because the distribution becomes asymptotically normal. This is supported by arguments of Campbell et al (1997). They use the Central Limit Theorem (CLT) and state: ‘’Since all moments are finite, CLT applies and long-horizon returns will tend to be closer to the normal distribution’’ (p.19). Additionally, Fama (1976) argues that the distribution of monthly returns of individual stocks are approximately normally distributed. The sample size of this paper consists of 127 unique firms and 274 months, making the distribution of monthly returns asymptotically normal.
36
Rademacher variables, as Davidson (2008) shows superiority of this distribution. This approach is beyond the scope of this paper and suggested for further research.
A more strict assumption of OLS relates to errors terms in the regression. By definition the error terms follow a normal distribution, with a constant variance and no correlation between errors. A constant variance is violated when error-terms incorporate sub-populations with differing deviations, called heteroskedasticity. Since time series include periods of regression and growth, error-terms are not constant over time. Hence the regression is controlled for heteroskedasticity by using White cross-section standard errors and the problem of heteroskedasticity disappears. The second part of the assumption, no correlation between error-terms (also known as autocorrelation) is challenging to test. Normally, several tests are available e.g. Durbin-Watson (DW) or a Breusch-Godfrey test. The latter is not possible in EViews for panel data sets. Simple DW statistics are available, but it is
debatable whether these apply to models that include a lagged dependent variable in the regression. Durbin and Watson (1950) themselves advocate that a typical DW statistic does not apply under these conditions and propose a Durbin h statistic, controlled for a lagged dependent variable. On the other hand, Inder (1984) shows proof in favour of a regular DW statistic. A Durbin h statistic is not possible to calculate if the sample size multiplied by the estimated coefficient of the lagged dependent variable is smaller than one. This particular constraint is violated in this paper since the data set is large. Regular DW statistics report values close to two, with extremes around 1.80. These are higher than critical low bound DW values. In addition to Inder, Bhargava et al p.4 (1982) conclude that for large panel sets it is not required to calculate h statistics. Based on these findings is
concluded that error terms do not inhibit autocorrelation.
37
such a specification is stationarity of observations over time, because non-stationarity would produce biased estimates. This is a consequence of trends, cycles or random walks that produce different means, standard deviations and covariances over time. By calculating logarithmic returns, time series become stationary, as shown in figure 2 in section 3.
Another assumption of OLS relates to correlation between errors and independent variables. OLS assumes no correlation between the two variables. However, Nickell discovered that estimators become correlated with regression errors under fixed effects, as i → ∞ for any fixed t. In this
particular case i is indeed large (68 cross sectional observations at minimum). Hence, it could possibly be that a required assumptions under OLS Fixed effects is neglected. This causes a downward bias, called the Nickell Bias, in estimators, and thus incorrect results. Nevertheless, for larges values of t, the bias decreases in size. Moreover, if t moves to infinity, the Nickell bias eventually approaches zero. This paper regresses with 274 different periods. If it would be the case that i approaches infinity, this would hold for t too. Therefore, the Nickell bias eventually approaches zero and do not create a bias in the estimates.
A second issue in regard to the Nickell bias is proposed by Alvarez and Arellano (2003). They show that t-statistics of FE estimators increase in absolute value when both i and t move to infinity. Higher
t-statistics produce significant findings when there actually would be no significant effect observed
using a correct t-statistic. In the case that t is larger than i, this issue disappears. In this sample the amount of periods is indeed larger than the number of cross sections included. Thus, t-statistics are not influenced by Alvarez and Arellano’s bias.
5.3 Further remarks
38
differences between present and past. It is true that this holds for the trading strategies of this paper. Yet, these trading strategies function as a back test attempting to exploit seasonal patterns in the Dutch stock market, and test market efficiency. Furthermore, the trading strategies are relatively simple winner-loser strategies, based on past performance. These do not require specific knowledge about seasonal patterns, and if a similar lag would be chosen, trading strategies in this paper could become feasible portfolios.
In addition, transaction costs are ignored in the process of measuring portfolio returns. Evidently, transaction costs play a role in terms of profitability. After every month portfolios are rebalanced, selling losers and buying winners of subsequent lagged months. Returns should not be considered as real world examples, as particular strategies might not be profitable due to incurred transaction costs. Nonetheless, results are strong overall and could be considered for portfolio strategies. Overall, it is easy to postpone the purchase or sale of winning- or losing stocks.
6. Conclusion
39
The results are tested for robustness. The paper finds evidence of a Halloween effect on the Dutch market, leading to large differences in average returns between May through October and November through April. Controlling for a Halloween effect produces similar regression estimates as under all months. These are statistically significant in several cases. Hence it appears that the Halloween effect is not the main driver of seasonality on the Dutch stock market. Additionally, the paper tests for a periodical impact of seasonality, but reports positive average returns over three separated periods. Other possible drivers of seasonality are discussed, but further research on this is required.
40
Appendix
This table reports all company names used in the paper:
Aalberts CMG Gucci Group OCI Ten Cate
ABN Amro Corbion Hagemeyer Ordina TKH
Aegon Corio HBG Pakhoed Tnt Express
Ahold Corp. Express Heijmans Pharming Tom Tom
Ahrend Corus Heineken Philips Unibail-Rodamco
Air France-KLM Crucell Hoogovens Pink Roccade Unilever
AkzoNobel DAF Hunter Douglas Polygram Unit4
Altice Delta Lloyd IHC Caland PostNL Univar
AMG Douwe Egberts IMCD Prologis USG People
Aperam Draka Imtech Randstad Van Der Moolen
Arcadis DSM ING RELX Van Leer
ArcelorMittal Endemol KLM Robeco Van Ommeren
ASM Int Eurocommercial KPN Rodamco Asia Vastned Offices
ASML Fagron KPN Qwest Rodamco Europe Vastned Retail
ASR Flow Traders Landis SBM Offshore Vedior
Baan Fokker Libertel Shell Vendex
Bam Fortis Logica Sligro VNU
BE Semiconductor Frans Maas Macintosh Smit Vopak
Begemann Fugro Mediq Sns Reaal VWS
Bijenkorf Galapagos NBM Amstelland Sphinx Wavin
Binckbank Gamma Nedlloyd Stad Rotterdam Wegener
Borsumy Wehry Gemalto NN Group Stork Wereldhave
Boskalis Getronics NSI Super de Boer Wessanen
Buhrmann Gist Brocadis Numico Tele Atlas Wolters Kluwer
Cap Gemini Grandvision OCE Tele2 Ziggo
Cap Gemini SA Grolsch
References
Acharya, V., Pedersen, L.H., 2005. Asset pricing with liquidity risk. Journal of Financial Economics 77, 375-410.
Agrawal, A., Tandon, K., 1994. Anomalies or illusions? Evidence from stock markets in eighteen countries. Journal of International Money and Finance 13, 83-106.
41
Baltagi, B.H., 1988. Panel data methods. Handbook of Applied Economic Statistics, A.. Ullah, D., Giles (eds). Marcel Dekker, New York.
Barroso, P., Santa-Clara, P., 2015. Momentum has its moments. Journal of Financial Economics 116, 111-120.
Bhargava, A., Franzini, L., Narendranathan, W., 1982. Serial correlation and the fixed effects model. Review of Economic Studies 49, 533-549.
Bonin, J., Moses, E., 1974. Seasonal variations in prices of individual Dow Jones industrial stocks. Journal of Financial and Quantitative Analysis 9, 963-991.
Bouman, S., Jacobsen, B,. 2002. The Halloween indicator, ‘sell in May and go away’: another puzzle. American Economic Review 92, 1618-1635
Campbell, J.Y., Lo, A.W., MacKinlay, C.A., 1997. The Econometrics of Financial Markets. Princeton University Press, Princeton.
Davidson, R., Flachaire, E., 2008. The wild bootstrap method, tamed at last. Journal of Econometrics 146, 162-169.
DeBondt, W.F.M., Thaler, R.H., 1985. Does the market overreact? Journal of Finance 40, 793-805. DeBondt, W.F.M., Thaler, R.H. 1987. Further evidence on investor overreaction and stock market
seasonality. Journal of Finance 42, 557-581.
Durbin, J., Watson, S., 1950. Testing for serial correlation in least squares regression I. Biometrica 37, 409-428.
Euronext, AEX-Index history evolution of the composition from 1983. Online published document. Euronext, AMX-Index history evolution of the composition from 1995. Online published document. Fama, E.F., 1976. Foundations of Finance. Basic Books, New York.
Friday, S.H., Hoang, N., 2015. Seasonality in the Vietnam stock index. International Journal of Business and Finance Research 9, 103-112.
42
Garrett, I., Kamstra, M., Kramer, L., 2005. Winter blues and time variation in the price of risk. Journal of Empirical Finance 12, 291-316.
Gaibulloev, K., Sandler, T., Sul, D., 2014. Dynamic panel analysis under cross-sectional dependence. Political Analysis 22, 258-273.
Hsiao, C., 1986. Analysis of Panel Data. Cambridge University Press, Cambridge.
Heston, S.L., Korajczyk, R., Sadka, R., 2010. Intraday patterns in the cross-section of stock returns. Journal of Finance 65, 1369-1407.
Heston, S.L., Sadka, R., 2008. Seasonality in the cross-section of stock returns. Journal of Financial Economics 87, 418-445.
Heston, S.L., Sadka, R., 2010. Seasonality in the cross-section of stock returns: the international evidence. Journal of Financial and Quantitative Analysis 45, 1133-1160.
Inder, B.A., 1984. Finite-sample power of tests for autocorrelation in models containing lagged dependent variables. Economics Letters 14, 179-185.
Jegadeesh, N., 1990. Evidence of predictable behavior of security returns. Journal of Finance 45, 881- 898.
Jegadeesh, N., Titman, S., 1993. Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance 48, 65-91.
Jegadeesh, N., Titman, S., 2001. Profitability of momentum strategies: an evaluation of alternative explanations. Journal of Finance 56, 699-720.
Jegadeesh, N., Titman, S., 2002. Cross-sectional and time-series determinants of momentum returns. Review of Financial Studies 15, 143-157.
Kamstra, M.J., Kramer, L.A., Levi, M.D., 2003. Winter blues: seasonal affective disorder (SAD) stock market returns. American Economic Review 93, 324-343.
Kasprzyk, D., Duncan, G., Kalton, G., Singh, M.P., 1989. Panel Surveys. Wiley, New York.
43
Lehmann, B., 1990. Fads, martingales and market efficiency. Quarterly Journal of Economics 105, 1- 28.
Lewellen, J., 2002. Momentum and autocorrelation in stock returns. Review of Financial Studies 15, 533-564.
Li, B., Qiu, J., Wu, Y., 2010. Momentum and seasonality in Chinese stock markets. Journal of Money, Investment and Banking 17, 24-36.
Liu, R.Y., 1988. Bootstrap procedures under some non-LLD. models. Annals of Statistics 16, 1696-1708.
Lo, A.W., MacKinlay, A.C., 1990. When are contrarian profits due to stock market overreaction? Review of Financial Studies 3, 175-208.
Mammen, E., 1993. Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255-285.
Moskowitz, T., Grinblatt, M., 1999. Do industries explain momentum? Journal of Finance 54, 1249- 1290.
Officer, R.R., 1975. Seasonality in Australian capital markets: market efficiency and empirical issues. Journal of Financial Economics 2, 29-51.
Pastor, L., Stambaugh, R.F., 2003. Liquidity risk and expected stock returns. Journal of Political Economy 111, 642-685.
Reinganum M.R., 1983. The anomalous stock market behavior of small firms in January. Journal of Financial Economics 12, 89-104.
Rozeff, M.S., Kinney Jr., W.R., 1976. Capital market seasonality: the case of stock returns. Journal of Financial Economics 3, 379-402.
Tinic, S., West, R.R., 1984. Risk and Return: January vs. the rest of the year. Journal of Financial Economics 13, 561-574.
