Calendar Anomalies in the Nordic Stock Markets: A quantitative study of the Sell in May effect, January effect & Monthly Anomalies

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Calendar Anomalies in the Nordic Stock Markets

- A quantitative study of the Sell in May effect, January effect & Monthly Anomalies

Bachelor Thesis

Authors: Christopher Edberg & Oliver Kjellander

Supervisor: Magnus Willesson Examiner: Håkan Locking

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Abstract

This study has applied a geographical perspective with the ambition of evaluating the presence of the Sell in May effect, January effect and monthly anomalies in the Nordic stock markets. In extension the study examines the relationship between corporate size and the returns of calendar anomalies. The study has conducted statistical tests based on Newey-West regressions as well as a Generalized Auto-Regressive Conditional Heteroscedasticity model. The findings suggest that the Sell in May and January are present in the Nordic region and partially abide by theory and results of previous research. The findings suggest that the Sell in May and January effect are independent, however, tendencies when the January effect has a considerable influence on the Sell in May effect are also evident. Additionally, the “April Effect” is an unexpected outlier with positive excess returns that was identified through this study.

Key words

Calendar Anomalies, January effect, April effect, Sell in May effect, Monthly Anomalies, Newey-West & Generalized Auto-Regressive

Conditional Heteroscedasticity (GARCH), Nordic Stock markets, NASDAQ OMX Nordic

Acknowledgments

We wish to express our gratitude to our supervisor Magnus Willesson for the support throughout this study. We also wish to thank our opponents for their constructive criticism presented during the thesis seminars, ultimately helping us to further improve this study.

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1 Introduction

1.1 Background

Since the inception of financial markets there has been an unmistakable desire to identify trends in pursuit of abnormal returns. Forecasting financial markets have hence become the subject of immense debate across generations and has tended to “generate strong and conflicting opinions” among economists (Dimson & Marsh 1998 p.1). Financial trends can differ greatly in response to an almost infinite number of factors which also has made it increasingly challenging to identify a definite cause.

Calendar anomalies are market irregularities dependent on changes in time.

The frequency and duration of these irregularities are specific to each individual anomaly. This effect falls under what is known as the calendar time theory, which “states that the market behaves differently on different hours of the day, miscellaneous days of the week, various times of the month and year (Khan et al. 2017 p.56-57). The Sell in May (SIM) effect is one such calendar anomaly which will be a central part of this study.

The Sell in May (SIM) effect states that stock returns are significantly higher during the period November-April then during the period May-October (Bouman & Jacobsen 2002 p.1618). Furthermore, the SIM effect presents a fascinating financial opportunity since seemingly it does not fall victim to its own discovery. These unique characteristics of the SIM effect have intrigued both investors as well as the academic community and have motivated a considerable quantity of research on the topic. However, even with continuous attempts from the academic community any conclusive or absolute results have yet to be identified.

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The existing literature on well-known calendar effects is extensive and some long-standing. The literature forms an important basis for the theoretical discussions and analysis throughout this study. The stipulated delimitations that center this study on the Nordic countries as well as the impact of corporate size on calendar anomalies has not previously been researched and hence allows for a new perspective and potential for new results.

1.2 Problem Discussion

The SIM effect also known as the Halloween effect was first described by Bouman & Jacobsen in 2002 and has the potential of being particularly profitable as a consequence of low transaction costs and its simple application (Haggard & Witte 2010 p.379). In 2002, Bouman and Jacobsen published the first academic research on the topic in which they identified the significance and unmistakable presence of the SIM effect (Andrade et al. 2013 p.94).

The SIM effect is seemingly present at a similar “economic magnitude” as when it was first discovered (Andrade et al. 2013 p.94) which has contributed to the mystery surrounding the SIM effect. In contrast to other market anomalies which are rapidly consumed by the efficient market hypothesis due to arbitrage opportunities the SIM effect has persisted and has proven its ability to withstand arbitrage forces on the financial market (Bouman &

Jacobsen 2002 p.382; Andrade et al. 2013 p.102-103). Research into market anomalies and being able to unveil the cause of the SIM effect would allow the market to approach a higher degree of efficiency.

The current literature on the SIM effect and calendar anomalies are heavily focused on analyzing large macro factors in an attempt to explain the stock anomalies. Research on the SIM effect tends to adopt a very wide perspective based on broad indies from the stock market. Through this perspective it is likely that microeconomic (corporate specific) factors or other causes may be

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overlooked, simply by looking too broadly. Market capitalization is a widely accepted measurement of corporate value (Berk & DeMarzo 2014 p.27), however, it is also commonly used to indirectly measure the corporate size.

Corporate size shall hence be considered a microeconomic factor by definition of Encyclopedia Britannica (2020). In extension, it is possible to use capitalization indices to differentiate between corporate sizes.

Furthermore, the January effect is a calendar anomaly where the academic literature has suggested ties between corporate size and market returns.

Perhaps similar relationships may be applicable in the case of the SIM effect or other undiscovered calendar effects related to delimitations of this study.

Therefore, this study strives to further investigate the role corporate size has on specific regional markets, more specifically the Nordic financial market.

Naturally, this provokes questions regarding how corporate sizes have different influences on stock returns across different regions as well as the underlying causes of such anomalies.

The contribution of this study strives to develop the understanding of calendar anomalies dependency on corporate size in a geographical context. The contributions aim to be applicable in the real world and potentially in extension aid the development of new investment strategies. This study will evaluate the Nordic nations as an entity rather than as autonomous nations, due to their many shared economic similarities (Hilson 2018). This enables a review of calendar anomalies on the basis of geographical location and an analysis of how geographical regions perform in comparison to the benchmark of international stock market indices. Thereby this study also seeks to present a new perspective on calendar anomalies and possibly inspire the idea that investment strategies may be based on geographical regions.

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It is possible that particular geographical areas favor different anomalies, sparking the discussion that perhaps stock investments shall be tailored to the geographical calendar anomalies in which the corporation operates. Prior to 2016, the Sell in May effect was relatively unexplored in Europe context compared to America (Carrazedo et al. 2016 p.491). Therefore, research based on European markets consequently have a greater potential to contribute and advance the current knowledge of market anomalies.

The study strives to present new conclusions through analysis of empirical data that may contribute to clarifying uncertainties regarding this unexplored topic in the Nordic countries. These contributions aim to develop a more comprehensive understanding of the calendar anomalies in the Nordic countries and aid future research and investment strategy development.

1.3 Purpose

This study intends to identify and investigate calendar anomalies in the Nordic financial markets with a primary focus on the Sell in May and January calendar anomalies. The purpose is to illuminate the existence of calendar anomalies (stock return calendar patterns) and how they are affected by corporate size.

More specifically the market returns of the GI (Gross Index) OMX Nordic Large Cap, Mid Cap and Small Cap will function as the dependent variable.

1.3.1 Research Question

To what extent are monthly calendar anomalies, the Sell in May and the January effect evident in the Nordic financial markets and explore how

market returns are affected by corporate size?

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1.4 Delimitations

In the pursuit of evaluating corporate size and its effect on calendar anomalies it may be useful to also limit the regional scope. A limited regional scope aims to minimize external factors such as cultural, political and legislative influences. Consequently, this study will exclusively review the OMX Nordic Large, Mid and Small which will function as a representation of the Nordic region consisting of Denmark, Sweden, Iceland and Finland (Nordic Co- operation) as well as the corporations in this region. Norway is excluded from this study since it is not part of the indices.

A few advantages can be pointed out by looking at a rather homogenous regional area. The OMX Nordic Gross Index allows for an evaluation of a specific region rather than an individual nation. Additionally, the OMX Nordic Gross Index will consequently include corporations over the entire region providing more empirical data which will enhance the accuracy of the statistical test and improve the analysis of the region as a whole. The size of the regional delimitation is essential to enable an analysis that is still representative for a region, whereas a regional scope of a continent would lose its value in this study since the exogenous factors would make an accurate analysis increasingly difficult.

Delimitations regarding the calendar anomalies lengths are restricted to anomalies lasting 1 month or longer. This strives to eliminate shorter return deviations that may be outliers and hence not representable of the period as a whole. Consequently, special recurring calendar events such as: holidays, specific weekdays et cetera will not be reviewed.

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2 Theory

2.1 The Sell in May Effect

Jacobsen and Bouman published in 2002 the first academic paper on the SIM effect where it was identified and its remarkable characteristics were reviewed for the first time (Andrade et al. 2013 p.94). This was the start of many studies to come over the following two decades. Since the initial study by Jacobsen and Bouman countless studies have attempted to prove and explain the existence of the effect.

A study by Jacobsen, Mamun and Visaltanachoti (2005 p.9) showed a significant Sell in May effect in the US stock market from 1970 to 1998, by using indices of the MSCI. The study constructed portfolios based on corporate specific variables, one of which was corporate size. Their conclusion was that most of these portfolios showed both significance in statistics and in economic terms (Jacobsen et al. 2005 p.2). Additionally, the returns of each portfolio were all higher during winter compared to summer which was described as a market wide phenomenon (Jacobsen et al. 2005 p.2).

Furthermore, the study published a few more findings such as: weak evidence for seasonality in dividend payments (Jacobsen et al. 2005 p.13), there’s not a connection between the SIM effect and the Size effect (Jacobsen et al. 2005 p.15) and finally they show that the summer period returns are hard to avoid for most investors (Jacobsen et al. 2005 p.5). At the time of the study the authors found evidence that the SIM effect is solely based on stock market indices (Jacobsen et al. 2005 p.3). This does not mean that it does not exist anywhere else, but the research has been focusing on the stock market. In addition, they conclude that the SIM effect has not been very much written about in the academic literature (Jacobsen et al. 2005 p.3).

A contradictory perspective is given by Lucey and Zao (2008) in their paper

“Halloween or January? Yet another puzzle”. This study evaluates the period

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1926 to 2002, exclusively on the US markets. They argue that the effect may be caused by the January anomaly (Lucey & Zao 2008 p.1055) hence, suggesting that the SIM effect published by Bouman and Jacobsen in 2002 does not exist and that this effect dissipates over the recent period both in statistical significance and in absolute figures (Lucey & Zao 2008 p.1068).

Published in 2010 the study “The Halloween Effect: Trick or Treat?” by Haggard and Witte supports the presence of the SIM effect in the US from 1954 to 2008. The effect seems to be present throughout the entire period even after being corrected for the January effect while it shows robustness towards measurement of outliers and results in statistical significance the authors conclude (Haggard & Witte 2010 p.380). The anomaly is not significant before the period using this data (Haggard & Witte 2010 p.386). Furthermore, a study published by Haggard et al. concluded that the effect has increased since its discovery (Haggard et al. 2015 p.655).

Auer and Degenhardt published a large review on the topic. The authors look at the SIM effect from 1989 to 2016. Their method utilizes regressions with dummy variables and adjustments for accounting for the January effect (Auer

& Degenhardt 2018 p.171). The result in the paper shows that the SIM effect is unstable, in other words, it comes and goes over different time periods when using the DJIA index (Auer & Degenhardt 2018 p.190). The significance is in both statistical terms and in economical terms for the DJIA index even when controlling for the January effect (Auer & Degenhardt 2018 p.200). The authors have also noted that the effect is more present in industrial and basic materials sectors (Auer & Degenhardt 2018 p.200).

Jacobsen and Visaltanachoti (2009) analysed 19 developed markets and their results indicated a SIM effect from 1998 to 2007. Also, Jacobsen and Zhang showed that 87 of 114 countries had higher mean returns for the SIM period

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compared to the remaining period with a statistical significance in 42 of these (Jacobsen & Zhang 2021 p.31), and that the effect seems especially strong in the last 50 years in Western Europe (Jacobsen & Zhang 2014 p.32).

The last finding from the above paragraph is in line with the review from Auer and Degenhardt when they summarize that the SIM effect is strong for the studies analyzed especially in the second half of the 20th century (Auer &

Degenhardt 2018 p.175). Further studies analyzing the SIM effect more often than not indicate that it is stronger in developed countries (Auer & Degenhardt 2018 p.175). In addition, the SIM effect is not clearly present in every market the authors announce (Auer & Degenhardt 2018 p.175).

Zhang and Jacobsen (2013) looked at the UK stock market data for 300 years.

The SIM effect was prevalent over the last three centuries. The study indicated that the returns during summer were very weak overall, sometimes negative.

(Jacobsen & Zhang 2013 p.1)

Exploiting the SIM effect in the stock market with a trading strategy which is based on buying and holding a stock market portfolio during the winter months according to the SIM effect and during the non-SIM period holding treasury bonds, the authors concluded that this strategy had higher returns than the market over 10 year horizons in 90% of the cases. When the horizon was 5 years the strategy was better 80% of the time. (Jacobsen & Zhang 2013 p.20)

In addition to the previous studies a review on the SIM effect revealed that using indices that reinvest dividends contribute for a larger effect and the authors found that 34 of 65 of these dividend included indices show statistical significance. (Auer & Degenhardt 2018 p.175)

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A more recent and extensive study on the SIM effect on all stock market indices was published by Jacobsen and Zhang in 2021. Their conclusion of the study was:

”In sum what we can say for certain is that the Halloween effect is alive and kicking. The claims that the effect is not present in earlier studies using limited data are likely caused by the specific samples used” - Jacobsen &

Zhang, 2021 p.32

Moreover, they write that the effect is strong and has been increasing in the recent period prior to the study (Jacobsen & Zhang 2021 p.32). The quote of the referee of the paper perspective on the topic is as follows:

‘‘While the empirics are super strong, there is a lack of seriousness from the academic community of treating this topic with respect. Perhaps this

overview will help.” - Jacobsen & Zhang, 2021 p.1-2

The 62 962 observations that form the basis of the study results in about 4%

higher return for the winter months (November to April) than the summer period (Jacobsen & Zhang 2021 p.1). The stock indices used, are according to the authors all available indices and therefore a first study that takes all stock indices into account (Jacobsen & Zhang 2021 p.2). In the summer period 45 of 65 markets have a negative risk premium on average (Jacobsen & Zhang 2021 p.31). When taking the total return the results are more pronounced than just looking at price returns. Statistical significance was found in 36 countries and point estimates were positive for 63 of 65 countries (Jacobsen & Zhang 2021 p.32).

The study of Jacobsen and Zhang also states that the SIM effect is more pronounced in developed countries and emerging countries compared to

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“frontier and rarely studied markets.” Consequently, the effect is more common to identify in Europe, North America and Asia. In addition, the authors specifically mention that the effect is more pronounced in western parts of Europe throughout the past seven decades. (Jacobsen & Zhang 2021 p.32)

2.1.1 Theoretical explanations of the SIM effect

Even though numerous attempts have tried to explain the SIM effect with some research being more fruitful than others the SIM effect is somewhat of a mystery and is not easily explained. The initial study on the SIM effect by Bouman and Jacobsen published 2002 found that factors such as risk, the January effect (not on the US data) or correlation between markets do not explain the SIM effect and can be ruled out (Bouman & Jacobsen 2002 p.1618). The SIM effect has increased its presence since it was first identified over many countries worldwide (Haggard et al. 2015 p.655). Citing Jacobsen et al. (2005) regarding the factors that drive the SIM effect, they contribute with the perspective: “An important consequence of our analysis is that the search for the potential cause of this Halloween effect should explain the phenomenon market wide.” (p.15) and with the quote “The Halloween effect might be a result of an effect that affects all investors or a returning macroeconomic phenomenon that has so far not been discovered but which causes stock returns to fluctuate so consistently in a predictable way.” (p.15).

Other authors explored questions, for example: on how risk aversion relates to temperature changes (Cao & Wei 2004) and how SAD (or seasonal affective disorder) affects investor behavior (Kamstra et al. 2003).

One possible explanation that has been researched is how vacations may affect the SIM effect. Bouman and Jacobsen gives two possible hypothetical explanations what links the vacations and the SIM effect, it could be explained

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by change in liquidity premium with investors or the risk aversion is affected (Jacobsen & Bouman 2002 p.1629). Airline travel and consumption of ice also have some similar pattern as the effect and is statistically significant as do several other variables that have a similar seasonal characteristic (Auer &

Degenhardt 2018 p.181). The cause may be the growing significance and the view on vacations in more developed countries (Jacobsen & Zhang 2021 p.32).

Bouman and Jacobsen (2002 p.1630) found that the SIM effect is significant in relation to vacation length and when vacations occur. In addition to the findings, trading activity seems to be affected by vacation (Bouman &

Jacobsen 2002 p.1630). This reasoning makes sense for why the SIM effect is more pronounced in Europe the authors propose. Further Hong and Yu (2009 p.672) found that during summer months, which is linked to holidays, the trading activity is lower, supporting the claim by Bouman and Jacobsen (2002) above. Another study by Jacobsen and Visaltanachoti (2019 p.439) concluded from their empirics that vacation seems to not affect liquidity.

Another possible explanation proposed by Cao and Wei is that temperature affects the mood which in turn affects the trading decisions. They found an inverse correlation between the temperature and the stock market returns when looking at several stock markets around the world. The argument is that lethargy during summer makes investors more risk averse and cold temperatures make investors more aggressive. (Ciao & Wei 2005 p.1572)

The optimism cycle hypothesis by Doeswijk explains the calendar returns by how optimism changes throughout the year following a cycle. The explanation is that investors are more optimistic at the end (the last quarter) of the year because they are looking forward towards the next year. The optimism reverses itself a few months into the next year, thereby lowering the stock returns. (Doeswijk 2008 p.195)

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Another possible explanation for the SIM effect in the article “The “Sell in May” effect: A review and new empirical evidence” is presented by Kamstra et al. (2003) is the seasonal affective disorder (SAD) hypothesis. SAD is prevalent and affecting many people during months of less daylight. The links put forward by the authors explains a possible causation between SAD and risk aversion, because SAD is connected with depression, and in turn depression is connected to risk aversion. So, SAD and its correlation with seasonal daylight fluctuations therefore affects equity returns. Based on the result from the study the authors propose the support of the SAD hypothesis.

(Kamstra et al. 2003 p.324)

Contradictory evidence for the SAD hypothesis is that the SIM effect has been growing the last five decades which could be problematic for the SAD hypothesis, because the sunlight cycle is constant (Jacobsen & Zhang 2021 p.5).

Macroeconomic news may be, or at least partly, another possible cause for the SIM effect. Through the analysis of the result the author proposes that macroeconomic news announcements cause calendar, but also weather anomalies. The reasoning is that investors are affected by news. Consequently, the reaction from traders is not related to factors of psychology. (Gerlach 2007 p.295)

Importantly, the finding from previous research, even though several attempts and hypotheses have been attempted to try to explain the phenomenon, is that none of the different explanations can be fully satisfactory for understanding the SIM effect fully (Auer & Degenhardt 2018 p.199).

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2.1.2 Previous literature on the relationship between corporate size and the Sell in May effect

Only a few studies have analyzed how the size of corporations are affected by the SIM effect. Two of them are based on the US stock market, another article could not be accessed and therefore the decision not to include it in this study was made. To the best of the author's knowledge none cover the European theatre and the Nordic region. The method for categorizing corporations often tends to be based on decile portfolios with market capitalization as measurement.

One such study is Lucey and Zao where they constructed securities based on market capitalization into equal portfolios, from which indices were created.

The securities with highest market capitalization adhered to one portfolio and the smallest ones to the opposite end of all the portfolios. The involved securities were stocks from the New York Stock Exchange, American Stock Exchange and Nasdaq national market, where American Depository Receipts were not added. (Lucey & Zao 2008 p.1059)

The study wraps up with the conclusion that the SIM effect is weak (Lucey &

Zao 2008 p.1068). The proposed argument is that the SIM effect may be a disguise of the January effect across the deciles analyzed (Lucey & Zao 2008 p.1068).

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Table 1: Table 3 from their study (Lucey & Zao 2008 p.1061). Here deciles are based on corporate size. The smallest; decile 1 through to the largest decile 10.

In addition, supporting evidence regarding the irrelevant influence of size on the Halloween effect can also be found in the US study by Jacobsen et al.

(2005). Portfolios constructed on different metrics and one of those, size, was explored in the study. Similarly stocks from New York Stock Exchange, American Stock Exchange and companies from NASDAQ. The time period used is from July 1926 to December 2004. (Jacobsen et al. 2005 p.10)

The authors described that the SIM effect is unrelated to the size effect (Jacobsen et al. 2005 p.12). Further, the SIM effect was noted to be of a broader scale across the whole market (Jacobsen et al. 2005 p.15). A similar explanation that a size factor seems to be unassociated with the SIM effect is also used by Auer & Degenhardt (2018 p.199).

2.2 The January effect

In 1942, The Journal of Business of the University of Chicago published the article “Certain Observations on Seasonal Movements in Stock Prices”.

Written by Sidney Wachtel this study is regarded as the first definite

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identification of the January effect. The January effect is a market anomaly that suggests that stock returns are higher during January than in other months of the year (Urquhart & McGroarty 2014 p.156). Previous research on the January effect demonstrates a pronounced difference in stock returns, in which Kinney and Rozeff (1976) investigated the New York Stock Exchange (NYSE) between the years of 1904 and 1974. This study provided empirical evidence that concluded that the average January returns (3.48%) far exceeded the average return of the other months (0.42%) of the year (Haug & Hirschey 2006 p.76).

The January effect has historically been proven to correlate with firm size.

This dependency was proven significant between the years of 1963-1979.

Figure 1 is a graphical illustration where firms have been compartmentalized into deciles based on corporate size and plotted against abnormal returns. It is clear that smaller firm sizes (decile 1-4) enjoy abnormal returns throughout the month of January (Keim 1983 p.21).

Figure 1: “Decile of market value” from Keim (1983 p.21)

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Keim (1983 p.17) concludes that average annual returns based on January returns would exceed 20% for decile 1 (smallest firms) and hence create abnormal returns in relation to its decile risk coefficient (beta) whereas returns for decile 10 (largest firms) would only equal 9.6% and not be a viable reflection of the beta value.

2.2.1 Theoretical Explanations of the January effect

Tax Loss Selling Hypothesis

The theoretical discussion suggests that the January effect may be a consequence of the “tax loss selling” hypothesis. This hypothesis states that investors tend to sell securities at losses when approaching the “year-end” to reduce net capital gains and hence reduce taxation, causing a rebound of securities prices throughout January (Starks et al. 2006 p.3050). Furthermore, Keim (1983 p.29) hypothesized that since firm size is a measured as a function of total value of equity there is bias towards smaller firms that have experienced a price decline to be used for tax loss selling purposes.

However, the hypothesis has also encountered well-grounded critique. Haug and Hirschey (2006 p.86-87) question the reliability by suggesting that the presence of January effects after the Tax Reform Act of 1986, devalues the hypothesis.

“Since passage of the Tax Reform Act of 1986, any seasonal tendencies related to tax-motivated selling by institutional investors should not occur at

calendar year-end.” - Haug & Hirschey (2006 p.86-87)

As a consequence of the Tax Reform Act of 1986 mutual funds are required to distribute 98% of income and capital gains to avoid a 4% excise tax on the undistributed proportion (Chen et al. 2011 p.109) for the past 12-month period

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which ends on the 31 of October (Haug & Hirschey 2006 p.79). Due to this reform, the months of November and December apply to the following fiscal year and hence tax planning such as Tax Loss Selling shall be observable during the end of October, rather than the beginning of January.

Window Dressing Effect Hypothesis

The Window Dressing Effect Hypothesis argues that as investors approach the year-end they sell securities that have performed poorly. The purpose of window dressing is to minimize necessary disclosures of unsuccessful investments (Poterba & Weisbenner 2001 p.354).

“Nobody wants to be caught showing last quarter’s disasters. … You throw out the duds because you don’t want to have to apologize for and defend a stock’s presence to clients even though your investment judgment may be to

hold.” - quote by money manager from Jansson (1983)

The above quote clearly reflects a reality where money managers are willing to forgo potential returns in favor of an improved short term appearance (Agarwal et al. 2018 p.1968). As a consequence, it is suggested by Mai et al.

(2020 p.10) that the window dressing effect causes a downward pressure on poorly performing securities and a positive pressure on profitable securities.

However, this pressure dissipates once the year has ended, subsequently causing the unfruitful securities to recover throughout the month of January, which may be an underlying cause of the January effect.

Poterba and Weisbenner (2001 p.354) derived a statistical test with the purpose of differentiating between the effects caused by window dressing and tax-loss selling. They argue that untaxed institutions are unaffected by “changes in capital gains taxation” whereas individual investors are not. Furthermore, Poterba and Weisbenner (2001 p.354) state that if window dressing caused

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year-end selling pressure and thereby the January effect, the relationship between stock returns throughout the year and at the “turn-of-year” would remain unaffected. However, their hypothesis was rejected since the empirical evidence suggested that tax rates significantly influenced the relationship between past and turn-of-year returns.

2.3 Relationship between the Sell in May and January Effect

Multiple studies on calendar anomalies have recognized the possibility that the SIM and January effect in fact are the same anomaly. Jacobsen et al. addresses this possibility and mentions that rather than being an independent anomaly the SIM effect may be propelled by the January effect (Jacobsen et al. 2005 p.9) and the potential of the SIM effect being the January effect in disguise or the other way around (Jacobsen et al. 2005 pp.10-11).

A US study from 2008 suggested that the SIM effect may not exist when controlling for the January effect for the period 1926-2002, because the SIM effect variable had no statistical significance (Lucey & Zao 2008 p.1059). The same study did show a statistically significant January effect variable (Lucey

& Zao 2008 p.1062), thereby reaching the conclusion that the SIM effect was weak in the data analyzed (Lucey & Zao 2008 p.1068).

On the other hand, articles prior to 2018 indicate that the SIM effect is not explained by any other calendar anomalies (Auer & Degenhardt 2018 p.199).

Moreover, the authors write that the empirics for the SIM effect is solid in multi-continent studies (Auer & Degenhardt 2018 p.175). The conclusion from Jacobsen et al. is that the SIM and January effects are two different calendar effects (Jacobsen et al. 2005 p.4).

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2.4 The April Effect

The study “The Halloween effect in European sectors” mentions the April effect more closely when looking at indices in Europe, with the Nordic indices included which are of relevance for this study.

The findings from the study when analyzing monthly average returns shows that all months except April and September are somewhat similar in average returns (Carrazedo et al. 2016 p.497). The month of April had the highest average return of all the months from the indices used in the study (Carrazedo et al. 2016 p.498). The statistical significance of the April effect is prevalent in many of the indices, and when including an April dummy, the statistical significance of the SIM dummy decreases (Carrazedo et al. 2016 p.498). As a consequence the authors propose the argument that the Sell in May effect, called the Halloween effect in their study which is essentially the same, is driven to some extent by the April effect, or the high returns during April (Carrazedo et al. 2016 p.498).

The literature on the April effect has been found to be very limited, at least in the calendar anomalies research area. Therefore it is uncertain to draw any conclusions from previous literature to base a more thorough analysis on in this study. The absence also makes it worthwhile to further investigate this calendar anomaly.

2.5 The Efficient Market Hypothesis

The efficient market hypothesis (EMH) is probably one of the most applicable economic theories to date as it both functions as a cornerstone in both academic research as well as practical application. According to Fama (1965 p.56) an efficient market is a market driven by market forces with a pursuit of abnormal profits. Fama further states that the players on this competitive (efficient) market are rational and that the information is “almost” freely

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accessible to most investors. The EMH also exists under the assumption that information is absorbed into the market without delay (Lo 2004).

The EMH is in simplicity a theoretic explanation of how information seamlessly is integrated and exploited in financial markets (Ying et al. 2019 p.3). From an EMH perspective it is indisputable the theory poses a challenge for the livelihood Sell in May and January effect in the long run (Lucey & Zao 2008 p.1056). Other researchers argue that the persistence of such calendar anomalies instead challenge the EMH itself (Auer & Degenhardt 2018 p.170).

The EMH consists of three degrees of perceived market efficiency:

• Weak Form Efficiency states that all current market prices are reflections of past prices. Additionally, this market belief proposes that no technical analysis can be applied successfully to generate abnormal profits. Nonetheless, this theory believes that fundamental analysis may be used to evaluate stocks.

• Semi-strong Form Efficiency suggests that market prices are a perfect reflection of all available public information. Therefore, semi-strong form efficiency states that neither technical or fundamental analysis can be used to generate returns that exceed the market.

• Strong Form Efficiency is a belief that market prices reflect all available information, both public and non-public. Consequently, it is not possible for investors to access any information that will be advantageous from an investment perspective (Ying et al. 2019 p.4).

All the degrees of market efficiency have different theoretical beliefs of market arbitrage opportunities, however modern technology has come to affect the very perception of efficient markets. Due to technological advances stock

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analysis can be based on practically limitless quantities of data and has extended to include factors such as behavioral and psychological aspects where certain patterns have been observable under short periods of time (Ying et al. 2019 p.5-6).

Even though the EMH functions well and has obvious values it has come to be critiqued for some of its weaknesses. One such critique is an old economics joke…

“…about an economist strolling down the street with a companion. They come upon a $100 bill lying on the ground, and as the companion reaches down to pick it up, the economist says” “Don't bother if it were a genuine

$100 bill, someone would have already picked it up” - (Lo 2004 p.2)

Though simplistic and exaggerated this joke carries an underlying critique that reflects a valid issue of the EMH, that still remains unresolved. This has led to the development of the Adaptive Market Hypothesis.

2.6 Adaptive Market Hypothesis

The Adaptive Market Hypothesis (AMH) is an evolution of EMH that seeks to explain the shortcomings of the efficient market hypothesis. AMH

originates in a belief that investors partially learn from trial and error, but even more importantly the theory acknowledges that investors are

susceptible to committing mistakes but adapt in response to results

(McGroarty & Urquhart 2014 p.154). The adaptive market hypothesis is a theory that strives to incorporate elements of behavioral finance and hence the psychological aspects of investors. Lo (2004) states that investors are driven by fear and greed, consequently Lo argues that the EMH is

incomplete.

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In contrast to the efficient market hypothesis, AMH states that arbitrage opportunities do arise (McGroarty & Urquhart 2014 p.154), which

potentially are a strong contributing factor as to why investors are willing to purchase information or allocate a lot of time in developing it. These theories also differ as the AMH suggests that the relationship between risk and

returns varies over time and that investors therefore would benefit from applying different strategies in response to a change in the market (Lo 2004 p.23).

The adaptive market hypothesis is supported by Almail and Almudhaf (2017 p.51) who proved that return predictability in the UK stock market shifts between periods of dependency and independency on market efficiency. This finding supports AMH and indicates that the return predictability changes in response to differing degrees of market efficiency. McGroarty & Urquhart (2014 p.161) also identified fluctuations in the calendar anomalies that they tested for which further supports the AMH.

2.7 NASDAQ OMX Nordic indices

The included stock markets in the Nordic index family are the following four:

NASDAQ OMX Stockholm, NASDAQ OMX Helsinki, NASDAQ OMX Copenhagen and NASDAQ OMX Iceland (NASDAQ 2013 p.9). The different indices in the Nordic family are denoted in one of the following currencies:

EUR, SEK, DKK and ISK (NASDAQ 2013 p.6). Gross Indices (GI) include dividends and other cash distributions to the stock owners in the index, this calculation is specific for each corporation in the index (NASDAQ 2013 p.15).

The corporations listed on the NASDAQ OMX are also included in one of three segment indices (Large Cap, Mid Cap, Small Cap) based on market capitalization. The boundaries for eligibility are: minimum 1 billion EUR for

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the Large Cap, between 1 billion and 150 million EUR for the Mid Cap, below 150 million EUR for the Small Cap. The three lists are rebalanced according to the market Cap every first trading day for the new year. The calculation of the market Cap for determining the placement in which segment, November average prices are used. (NASDAQ 2013 p.9)

3

Methodology 3.1 Overview

The statistical tests are based on the Gross Index of the Nasdaq OMX Nordic Large-, Mid- and Small Market Capitalization, accessed through Eikon (Datastream) from Thomson Reuters. The index data for all three market capitalizations have been collected with a daily frequency spanning from 2006-10-02 to 2021-04-27. This time interval represents all available data, accumulating in a total of 3702 observations.

The decision to utilize daily index data was based on the advantageous effect that daily data allows for improved accuracy in the statistical test as a consequence of a significantly larger sample size (Dell et al. 2002 p.208).

However, to enable statistical tests of specific time periods of one month or larger (delimitation), all daily data observations were grouped into months, from which dummy variables were constructed, depending on the purpose of the test. This was executed in Microsoft Excel. Daily returns were used as the dependent variable in all regressions and were calculated for each individual index by:

Percentage change = (x2 - x1)/ x1

The specifications are based on linear regressions with the OLS (ordinary least squares) method for all the tests performed. This basic specification is

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common in the literature on the topic of seasonal anomalies, which also is the reason for the selection.

Qualitative independent variables can be incorporated into a linear regression model, which are known as dummy variables (Anderson et al. 2017 p.429).

These dummy variables function as a representation of qualitative data which cannot otherwise be integrated into the model since they do not possess a quantitative value. Instead these variables accept the value of 1 or 0 dependent on their presence or absence respectively (Encyclopedia Britannica 2020).

Throughout this study multiple dummy variables were utilized comprehensively to represent specific months and time periods. The constructed linear regression containing a single dummy variable takes the following expression:

(x1): Y= 𝛽0 + 𝛽1x1 + ε.

When constructing dummy variables the amount of k levels the specification has decides the amount of dummy variables needed (Anderson et al. 2017 p.431), in order to avoid the dummy variable trap. The dummy variable trap exists when the same amount of dummy variables for quantitative categories are the same number as all the existing quantitative categories, which gives good prerequisites for collinearity and multicollinearity (except for when the intercept is dropped)(Gujarati & Porter 2009 p.281). Importantly this issue may be addressed by excluding the regression intercept and thus avoiding the dummy variable trap (Gujarati & Porter 2009 p.282).

3.2 Heteroscedasticity and Autocorrelation

For the dummy variable regression to be valid it must not have heteroscedasticity in different subperiods (Gujarati & Porter 2009 p.299).

Absence of autocorrelation is also an assumption for a classical linear

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regression model, which means that the error term is not correlated with each other (Gujarati & Porter 2009 p.299).

To combat the issue of heteroscedasticity and autocorrelation in the standard errors all regressions applied Newey-West standard errors. The Newey-West standard errors are heteroskedasticity and autocorrelation consistent standard errors (HAC) (Gujarati & Porter 2009 p.447) and is an evolution of White’s heteroscedasticity consistent standard errors (Gujarati & Porter 2009 p.447).

The drawback of White’s heteroscedasticity-consistent standard errors is that it doesn't correct for autocorrelation. Therefore, it is clearly advantageous to apply HAC standard errors since it handles both of these potential problems.

This produces a more reliable output in the statistical tests. The Newey-West standard errors are valid in large samples (Gujarati & Porter 2009 p.448) which makes it highly suitable in this study with over 3 700 observations.

3.3 Statistical testing 3.3.1 Overview

The statistical testing throughout this study is based on linear regressions with Newey-West standard errors as well as related F and T-tests. In total three F- tests and 38 T-tests were performed. The lag-order of these regressions have been determined by the “lag-order selection statistic” (varsoc) test in Stata/IC 15. The Hannan and Quinn information criterion (HQIC) proposed a lag-order of 3 observations which has hence been applied.

The significance level, ⍺, throughout this study considers significance at the 1%, 5% and 10% level. However, significance levels above 10% may be interpreted and analyzed when found appropriate and relevant.

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3.3.2 First Regression with F-test

Firstly, to evaluate the presence of any monthly anomalies F-tests were conducted to establish a foundational understanding and basis for further and more specified testing. F-tests were completed for all three indices including the 12 calendar months and their respective dummy variables.

This study replicates the method of Wuthisatian (2021) and Compton et al.

(2013). The following specification is well used in the literature Wuthisatian (2021) which will be the basis for the F-tests:¹

rt = 𝛽1DJan,t + 𝛽2DJan,t + … + 𝛽12DDec,t + εt

The above regression model estimates the average daily return, for all the months from January to December. The dependent variable is the mean daily return, beta is the average mean return for that specific month, the dummy variable for each month will assume 1 if the observation is within that specific month or 0 otherwise, lastly an error term is added. (Wuthisatian 2021)

The regression could be used to test that all the including months have the same mean daily return. The null hypothesis therefore assumes that all months have the same mean returns. (Compton et al. 2013 p.1145-1146; Wuthisatian 2021)

Consequently a significant F-statistic would reject the null hypothesis and indicate a calendar anomaly in one of the months in application of this study.

The F-critical value was computed using Stata/IC 15 F-distribution right-tail critical value function: invFtail (df numerator, df denominator, probability level). The probability level was set at a 5% significance level.

1 All regressions in the Wuthisation study applied HAC standard errors.

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The following hypothesis were constructed to evaluate the f-tests:

H0: 𝛽2= 𝛽3= 𝛽k

Suggesting no difference in population variances hence indicating that calendar anomalies are not present.

H1: 𝛽2 ≠ 𝛽3≠ 𝛽k

Suggesting difference in population variances hence indicating the existence of statically significant anomalies.

3.3.3 Second regression with T-tests

Secondly, two-tailed t-tests were used to examine each individual month separately to conclude if the tested month was significantly different from the rest of the year (Anderson et al. 2017 p.229). Furthermore, t-tests were also used to specifically evaluate if the Sell in May period was statistically significant in comparison to the May-October period.

This approach has been adopted because of the fact that previous literature on the topic tends to utilize this method, more specifically the regression specifications and significance testing of the dummy variable coefficients.

Studies with this methodology include: Haggard et al. 2015, Jacobsen &

Zhang 2021 and Whuthisatian 2021.

The two-tailed test is preferable in this study, because the samples may be less or more (Anderson et al. 2017 p.229). Stata software calculates a t-value according to the double-tailed t-test. On the basis of the t-value Stata

consequently calculates a p-value. The p-value calculated by Stata can be used directly as the hypothesis rejection rule when compared to the ⍺ (significance level), if the p-value is lower than the ⍺ the hypothesis H0 can be rejected (Anderson et al. 2017 p.229). In this study a p-value lower or equal than the significance level would indicate a seasonal anomaly, at the specific ⍺.

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The following hypothesis were constructed to evaluate the t-tests:

H0: Populations means are equal. Suggesting that calendar anomalies are not present.

H1: Populations means are not equal. Suggesting that calendar anomalies are present.

3.4 Regressions

3.4.1 The Sell in May effect including and excluding the January

To differentiate the Sell in May effect from potential influence caused by the January effect linear regressions were conducted both including and excluding the month of January. This aims to enable an analysis with both comparing and contrasting perspectives.

Firstly, a linear regression specified for the SIM effect including the month of January was constructed in the following format:

Yt=𝛽0 + 𝛽1 (Dummy SIM) + εt

The dummy variable assumes the numerical value of 1 if the date of the observation falls within the period November 1 - April 30, otherwise the variable assumes the value of 0. Additionally, the error term, εt, is added.

Thereafter, another linear regression was constructed excluding the month of January where the months November, December, February, March and April assumed the value of 1, whereas the remaining months assumed a value of 0.

This regression adopted the same format and was expressed as follows:

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Yt=𝛽0 + 𝛽1 (Dummy SIM excluding January) + εt

3.4.2 Test of all separate months

Firstly F-test were performed on the three indices. The specified regression:

Yt=𝛽1 (DJan. )+𝛽2 (DFeb. )+𝛽3 (DMar. )+𝛽4 (DApr. )+𝛽5 (DMay. )+𝛽6 (DJune. )+𝛽7

(DJuly. )+𝛽8 (DAug. )+𝛽9 (DSept. )+𝛽10 (DOct. )+𝛽11 (DNov. ) +𝛽12 (DDec. )+ εt.

Importantly, the above regression excluded the intercept for the reasons discussed in 3.1 Overview. Also, this format of the regression where all months are included as dummy variables enables the F-test to test all twelve months for significance. This procedure is also used by Wuthisatian in the study “An examination of calendar anomalies: evidence from the Thai stock market”

published in 2021. The author alleges that this model specification has been used to a great extent in determining returns on the stock market (Wuthisatian 2021 p.6 (counted, in the section 3.2 The January effect)).

Similarly, like the testing of the SIM effect, the same regressions are applied to all months of the year for the t-tests, the only difference is the dummy variables which are attributed to each and every month instead of a whole period with several months included. Therefore the functions is consequently expressed:

Yt=𝛽0+𝛽1(Dummy [month])+εt.

3.4.3 Generalized Auto-Regressive Conditional Heteroscedasticity A Generalized Auto-Regressive Conditional Heteroscedasticity (GARCH) model is a time-modelling procedure that “ensures robust and unbiased results” (Wuthisatian 2021). The GARCH model may be used to improve the reliability of the OLS Newey-West regressions by offering an alternative

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method for generating regression coefficients and evaluating the data.

Additionally, it functions well in predicting the conditional variance. The GARCH model functions as a procedure to evaluate the volatility of e.g. stocks and a way to capture volatility clustering (Gujarati & Porter 2009 p.773).

This study uses a lagged variance of 1 observation and 1 lagged error term GARCH (1,1). The model also omits the month of December, which enables the month to be a reference month when evaluating the other months. The decision of omitting December was justified by the reasoning that December is not a specific monthly anomaly itself on the basis of the research of this paper, as well as the fact that Stata/IC 15 suggested the omittance of this month.

The application of the GARCH (1,1) model assumes that it is specified correctly, nonetheless, applying the same model specifications to all regressions drastically improves the ability to compare results. Furthermore, the GARCH (1,1) model is frequently applied in studies of calendar anomalies and studies with similar data and testing procedures (Jacobsen & Dannenburg 2003 p.482). More specifically the GARCH (1,1) specification has been applied in multiple other Sell in May studies such as Dichl & Drobetz 2015 and Jacobsen & Zhang (2014). The GARCH (1,1) model is also known to function well in observing volatility clustering and has been proven through previous studies to function well in modelling financial time series. The GARCH model was the following structure:

𝑟_! = 𝜇 + 𝛽_"#$𝑆𝐼𝑀_!+ ℰ_! ℰ_!|𝜙_!%&~𝑁(0, 𝜎_!^') 𝜎_!^' = 𝛼₍+ 𝛼_&𝜀_!%&^' + 𝛼_'𝜀_!%&^'

Jacobsen & Zhang, 2021 p.31

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3.4.4 Analysis of the Empirical Tests

The analysis of the empirical tests will be focused on identifying, analyzing and comparing different time periods, the periods of different mean returns and significance. Based on this foundational parameters analysis tied to the literature will be analyzed. Hence the importance of the theory section and explains why it is substantial in the study. The purpose is to obtain an understanding of calendar effects and how those are or may be affected by corporate size. Analysis will provide a confirmation of already existing calendar anomalies and if they exist for the data used in the study, but also new other unexplored calendar anomalies may be found for specific months.

The validity of the study is an important consideration. The validity has been tried to be handled by using a sufficient and large enough sample size, with index data back to 2006. If possible data even further back in time would be appreciated for an even more extensive analysis. There is a possibility of spurious regressions and randomness in the performance of the index. Perhaps a period has had higher return by random chance the last decade. Therefore, the authors of this study find it important to also implement a theoretical discussion of why calendar anomalies exist, and if it is applicable to the delimitation of the Nordic region that was used as a data set in the study.

The reliability is obviously high because of the full replicability of the statistical tests, as a result of the mathematical basis and quantitative characteristics of this study. The sample size with more than a decade of index data will possibly help to distinguish if and how calendar anomalies may differ based on corporate size.

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4

Empirics

4.1 Newey-West Regression - F-test

F-test: Newey-West Regression of OMX Nordic Gross Index Returns - Lag 3 (No Constant)

Large Cap** Mid Cap*** Small Cap***

P-Value ^0.0211 ^0.0001 ^0.0000

F-statistic 1.99 3.37 4.38

F-critical value 5% 1.7547921

*** = significant at 1% level

** = significant at 5% level

* = significant at 10% level

Table 2: Results for F-tests on the Newey-West regressions with lag 3 (no constant included) for all Nordic stock indices.

The Small and Mid Cap shows significance at the 1% level and the Large Cap displays significance at the 5% level. A significant value, that is a p-value lower than the alpha, rejects the null hypothesis that all months have the same mean daily returns. This result indicates that a monthly anomaly exists in one or more months in the specific index. This is also supported by the fact that the F-statistic value exceeds the critical value for all corporate sizes and hence confirms that the H0 hypothesis shall be rejected. Therefore it’s worthwhile to test each month within all three indices for significance and for the mean for each separate month.

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4.2 Newey-West Regression: Individual t-test of Monthly Dummy Variables

t-test: Newey-West Regression of OMX Nordic Large Cap Returns - Lag 3

January February March April*** May June*

Constant Coefficient

0.0004492

(0.0002023) 0.0004104

(0.0002003) 0.0004327

(0.0001972) 0.0002878

(0.0002016) 0.0004578

(0.0002025) 0.0005525 (0.0001998)

Monthly Coefficients

-0.0000308

(0.0006353) 0.0004406

(0.0007099) 0.0001569

(0.0007945) 0.0019845

(0.0006164) -0.0001414

(0.0006009) -0.0013171 (0.0006863)

P-Value 0.961 0.535 0.843 0.001 0.814 0.055

July August September October November December

0.0004017

(0.0002031) 0.0005032

(0.0001996) 0.0004831

(0.0002012) 0.0004913

(0.0001945) 0.0004529

(0.0001996) 0.0004358 (0.0002031)

0.0005313 (0.0005852)

-0.0006792 (0.0007291)

-0.0004537 (0.0006638)

-0.0004977 (0.0008178)

-0.0000736 (0.0007129)

0.0001383 (0.000595)

P-Value 0.364 0.352 0.494 0.543 0.918 0.816

Note: Robust heteroskedasticity and autocorrelation consistent (HAC) standard errors displayed below coefficient in parenthesis.

Table 3: Shows results of each individual t-test for each month in the Nordic Large Cap index, based on Newey-West regression with lag 3.

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Monthly testing of the Large Cap index returns suggests that only two months indicate any significance. April is significant at a 1% level with a 𝛽 coefficient of 0.0019845. This indicates that the month of April experiences 0.198% in excess daily returns throughout the month. Whereas the remaining days of the year only experiences average daily returns of approximately 0.029%. The Newey-West standard error is 0.0002016 for the April beta and 0.0002016 for the constant coefficient. June is significant at a 10% level. The mean daily excess return is negative at -0.132% compared with the remaining period of 0.055%. The Newey-West standard error is 0.0006863 for the June beta and 0.0001998 for the constant coefficient.

Overall negative average returns for the monthly dummy occur in January, May, June, August, September, October and November.

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Table 4: Shows results of each individual t-test for each month in the Nordic Mid Cap index, based on Newey-West regression with lag 3.

The Mid Cap returns show significant returns for the months of April, June and December. April is significant at the 1% level with a 𝛽 coefficient of 0.0021078. The daily excess return of April has a N.W. standard error of 0.0005486. The N.W. standard error may hence cause daily excess returns in April to vary with approximately 26%. June is significant at the 5% level with

t-test: Newey-West Regression of OMX Nordic Mid Cap Returns - Lag 3

January February March April*** May June**

0.0004616 (0.0001848)

0.0004578 (0.0001843)

0.0005429 (0.000177)

0.0003161 (0.0001845)

0.0004623 (0.0001851)

0.0005991 (0.0001841)

0.0002658

(0.0005701) 0.0003281

(0.0006036) -0.0006628

(0.0008048) 0.0021078

(0.0005486) 0.0002798

(0.0005603) -0.0014212 (0.0005685)

P-Value 0.6411 0.5867 0.4102 0.0001 0.6175 0.0125

July August September October November December*

0.0004691

(0.0001867) 0.0005247

(0.0001818) 0.0005268

(0.0001851) 0.000554

(0.000175) 0.000497

(0.000184) 0.000417 (0.000186)

0.0001848

(0.0004932) -0.0004793

(0.0006898) -0.0005208

(0.0005696) -0.00066

(0.000809) -0.000014

(0.000603) 0.000878 (0.000496)

P-Value ^0.7078 ^0.4871 ^0.3606 ^0.4152 ^0.8197 ^0.0765

Note: Robust heteroskedasticity and autocorrelation consistent (HAC) standard errors displayed below coefficient in parenthesis.

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a 𝛽 coefficient of -0.0014212 with a related standard error of 0.0005685.

Consequently, the standard errors of June suggest that the daily returns throughout the month may vary with approximately 40%. December shows significance at the 10% level with a 𝛽 coefficient of 0.000878 and has a standard error of 0.000496. The N.W. standard errors of December are the largest of the significant months in relation to the forecasted returns, the standard errors may vary with approximately 56%.

March, August, September, October and November all have negative monthly 𝛽 coefficients.

Calendar Anomalies in the Nordic Stock Markets: A quantitative study of the Sell in May effect, January effect & Monthly Anomalies