Calendar anomalies on the Dutch stock market

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Maurice Beaumont

10422234

Calendar Anomalies on the Dutch Stock

Market

29-‐06-‐2016

University of Amsterdam

Bachelor of Economics and Business: Economics and Finance

Supervisor: Stephanie Chan

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Tabel of contents

ABSTRACT 4. INTRODUCTION 5. THEORETICAL BACKGROUND 6. LITERATURE REVIEW 9.

EXPECTATIONS AND HYPOTHESES 13.

DATA AND METHODOLOGY 15.

RESULTS 25.

CONCLUSION 30.

LIMITATIONS AND SUGGESTIONS FOR FURTHER RESEARCH 32. REFERENCES 33. APPENDICES XX.

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This document is written by Student Maurice Beaumont who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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ABSTRACT

This thesis examines the presence of seasonality in stock returns on the Dutch stock market by investigating so-‐called ‘calendar effects/anomalies’ between the establishment of the main index AEX on 3 January 1983 until 20 May 2016. Two types of calendar effects are being researched, Day of the Week (DOW) and Month of the Year (MOY) effects. ARCH and GARCH techniques are being used to find the best fitting models and the regression

outcomes are tested on different levels of significance. The tests for DOW effects result in slightly ambiguous results, with big differences between indices for Monday, Tuesday and Wednesday and both Thursday and Friday being significantly positive in all three indices. The tests for MOY effects result show similar level of disunity of the results in the first three months of the year, a strong and significant December effect and also some small negative effects in June, August and September. It is concluded that there are strong indications for seasonality in the Dutch stock market with regard to the Thursday, Friday and December effect. However, other effects showing strong deviations between indices have to be taken into consideration.

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INTRODUCTION

Since the invention of stocks and stock markets, investors have been trying to maximize their gains by using strategies based on the behaviour of stock prices. Some strategies do well, some do not, but the most interesting discussion is whether the performances of these strategies are based on skill and analysis or just simple luck.1_{Some
scientists
state
the
stock} market follows a random pattern, so investors might as well bet on a coin toss. Others believe in analytical skills, which lead to superior strategies to systematically gain profits. This thesis examines whether there are anomalies present in the Dutch stock market that can be exploited by investors using those strategies. The anomalies that are being

investigated are based on different returns in different months or days and therefore called calendar anomalies. Most of the literature about calendar effects focusses on U.S. stock markets, others examine multiple countries at the same time. As far as known, this is the first study about calendar effects on the Dutch stock market (Amsterdam Stock Exchange) only.

In most studies only the main index of a country’s stock market is considered, but to give an extended view of the whole market, the main three Dutch indices are used to calculate the returns on multiple stocks. Logartihmic returns are calculated and ARCH/GARCH methods are used to find the best fitting model per dataset. From the results found, it is concluded that there is a Thursday, Friday and December effect in stock returns on the Amsterdam Stock Exchange.

1_{See,
for
example,
Coval,
Hirshleifer
and
Shumway
(2002),
and
Goetzman
and
Massa
(2002).} 2_{Calcul
des
Chances
et
Philosophie
de
la
Bourse
(1863).}

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THEORETICAL BACKGROUND

Random Walk Hypothesis

Predicting the movements of stocks has been a subject of interest for a long period of time. Jules Regnault, a French stock broker from the 19th century, was the first to describe stock price movements using statistical analysis.2_{He
influenced
many
others
as
the
subject
grew} in popularity. French mathematician Louis Bachelier further developed Regnault’s ideas and came up with the idea of modelling price changes of stock options making use of Brownian motion.3_{Eventually,
these
and
other
scientific
publications
led
to
the
Random
Walk}

Hypothesis (RWH). This hypothesis can best be described by this quote from an article by economists Kendall and Hill4_:

“The series looks like a "wandering" one, almost as if once a week the Demon of Chance drew a random number from a symmetrical population of fixed dispersion and added it to the current price to determine the next week’s price”

Many scientists have worked on RWH and elaborated it. Eugene Fama argued for RWH and mentioned it has two underlying hypotheses: (1) successive price changes are independent, and (2) the price changes conform to some probability distribution.5_{Later
he
and
Burton} Malkiel formulated the Efficient Market Hypothesis (EMH)6_{,
which
has
been
one
of
the} prominent paradigms in financial research ever since.

Efficient Market Hypothesis

The Fama and Malkiel article states that financial markets tend to be “informationally efficient”, which implies that all of the information known to financial agents is displayed in the price of the asset. They enquire three models of market efficiency. The weak form of EMH states that all public information from the past that might influence the price of an asset, is incorporated in the price of the asset. The semi-‐strong form market efficiency

2_{Calcul
des
Chances
et
Philosophie
de
la
Bourse
(1863).}

3_{Théorie
de
la
spéculation
(1900);
Brownian
motion
is
a
random
movement
first
described
by}

physician Robert Brown in A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies (1828).

4_{The
Analysis
of
Economic
Time-‐Series-‐Part
I:
Prices
(1953);
the
quote
refers
to
their
analysis
of}

weekly prices on the Chicago wheat market.

5_{See
the
Behaviour
of
Stock-‐Market
Prices
(1965).}

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contains the theory that all financial agents directly respond to news that might influence the value of assets. Prices directly adjust to the new ‘true’ value of the asset. The strong form EMH states that in addition to public information, privy infomration is also reflected in asset prices. In either form, price adjustments are based on newly revealed information, whether it is public (influences all three forms) or private (only influences strong-‐form EMH). Market participants might make predictions about upcoming events, but they will not be sure of the outcome. If they would, the information is already reflected in the equilibrium price.

Besides, shocks that influence the prices might occur, which also can be predicted, but not known. Due to this uncertainty, no one knows what the next price movement will be. That’s why EMH is consistent with RWH and the relationship between these two is well

documented.7

Opponents RWH and EMH

The acceptance of EMH as the way a stock market operates leads to some serious

implications about expected returns and investor performance, summerarized by this quote from Konstantinidis et al.8_:

“The argument that investors are able top revail over markets by employing

information as a major investment device-‐weapon seems to be rather unsubstantiated. Investors cannot outperform markets and, as a result, they cannot achieve high returns, in view of the fact that information is not exclusive, but available to everybody. Thus, individuals cannot be characterized as investment experts or market specialists as the specific attributes can be equally applied to all investors.”

The article states that until the 1990’s, EMH was the most prominent view of scientists on the stock market in the absence of an alternative paradigm. From the early 1990s,

alternative views emerged as many economists had been trying to find alternative models to predict the movement of stock markets.9_{Some
of
these
publications
focus
on
finding}

patterns or rules in returns or volatility as a rejection of the RWH to be the best description

7

_{See,
for
example,
Mandelbrot
(1971),
Shiller
(1980),
and
Fama
(1995).}

8_{Konstantinidis
et
al.
(2014).}

9_{See,
for
example,
Fromlet
(2001),
;
Andrew
Lo
and
Craig
MacKinlay
dedicated
a
book
to
the
subject}

called A Non-‐Random Walk Down Wall Street (2002) referring to Malkiel’s A Random Walk Down Wall Street (1973).

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of stock market movements. For example, some studies examine the performance of stock per firm size and find that small companies gain an abnormal return above big firms.10

Seasonality

This thesis will not focus on individual firms, but on three main Dutch stock market indices and investigates another type of anomly that is in conflict with EMH and RWH: seasonality. Seasonality in stock markets is a periodically recurring event, such as a high return or low volatility in a certian period. These events are called calendar effects, as they mark a period on the calendar with a characteristic that differs from other periods, indifferent of which stock is enquired. This implies that the whole market moves in a certain direction more often or stronger on certain times. These ‘certain times’ can be days, weeks months, or even years before an election or the first half hour after opening of the bourse. Therefore

seasonality can take place in many forms and shapes. A well documented effect is the

January effect for example, where the outcome is that stocks yield in general a higher return in January than in other monhts.

In this thesis two types of calendar anomalies are examined: Day of the Week (DOW) effects and Month of the Year (MOY) effects. As the names reveal, these calendar effects are about whether the day of the week/month in a year has an influence to the stock return. In other words: has the day of the week/month of the year influence on the way market participants act in their trading choices? The next section will provide a brief overview of the previous research about these two types of calendar effects.

10

_{See,
for
example,
Penman
(1991).}

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LITERATURE REVIEW

Day of the Week effects

One of the very first articles describing a calendar anomaly was published by Frank Cross in 1973.11_{It
is
a
small
paper
containing
just
three
pages.
He
noticed
that
between
1953
and} 1970, the S&P’s Composite Stock Index showed a positive daily return in 62.0% of the trading days on Friday. On Mondays the closing level of the S&P Composite was in 39.5% of the times higher than the opening level. The supposition that there was a systematic

calendar anomaly in stock markets attracted the attention of other scientists. The list of research that has been conducted is too long to mention all individually. Some examples will be given below.

Kenneth French was testing two models for predicting stock returns in five periods of five years each between 1953 and 1977 on the S&P Composite, when his results showed that

“

Surpnsmgly, although the average return for the other four days of the week was positive, the average return for Monday was slgndicantly negatrue durmg each of five five-‐year

subperiods.”12_{Besides,
he
discovered
that
the
Friday
returns
was
significantly
higher
than} the Tuesday, Wednesday and Thursday returns. He suggested that investors delay their purchases they would have made on Thursdays or Fridays to Mondays and their sales scheduled on Mondays to preceding Fridays.

Gibbons and Hess examined the Monday effect with S&P 500 data and indices composed by the Center of Research in Security Prices.13_{They
conclude
that
there
are
‘persistent’}

negative returns on Monday, using a simple Ordinary Least Squares (OLS) regression. They give as possible explanation (1) assymetry in settlement period, i.e. the days it takes to complete a transaction, (2) measurement errors in observed prices. However, they admit that none of these two problems adequately describes their data.

Richard Rogalski argued that the Monday anomaly might appear because of the two non-‐ trading days between Friday and Monday.14_{He
used
data
from
the
Dow
Jones
Industrial} Average index between 1974 and 1984. He decomposed the returns on Monday into a non-‐

11

_{The
Behaviour
of
Stock
Prices
on
Fridays
and
Mondays
(1973).}

12_{French
(1980)}

13_{Day
of
the
Week
Effects
and
Asset
Returns
(1981).}

14

_{New
Findings
Regarding
Day-‐of-‐the-‐Week
Returns
over
Trading
and
Non-‐Trading
Periods:
A
Note}

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trading component from Friday’s close to Monday’s open and a trading component during the trading on Monday. He finds that the negative returns occur during the non-‐trading period.

Although most papers investigate the Monday and Friday effects, anomalies in other days of the week have been enquired as well. Agarwal and Rivoli find beside a negative Monday effect also a negative Tuesday effect in all four emerging Asian markets they examined.15 They dedicate this Tuesday anomaly to spillover effects from the Monday effect in American stock markets, due to the different time zones.16_{Later
on,
Wong,
Agarwal
and
Wong
state} that these anomalies have disappeared over time on the Singaporean stock market.17

Month of the Year effects

A pioneer study about MOY effects has been conducted by Bonin and Moses.18_They examined 30 Dow Jones industrial stocks individually between and 1962 and 1971 and conclude that 7 stocks show significant seasonal patterns in general.

A more comprehensive study on MOY effects on the New York Stock Exchange (NYSE) was later performed by Rozeff and Kinney.19_{They
employed
a
linear
model
and
used
different} stock compositions between January 1904 and December 1974. They find a particular high return in January, with also above average returns in July, November and December and below average returns in February and June. Three hypotheses to explain the January effect are given by Rozeff and Kinney: 1. tax loss selling: it can be fiscally favorable to sell stocks that performed badly just before the end of the year20_{;
2.
accounting
information:}

information about past year and forecasts for the next year are published in January; 3. Seasonality in cash demand, as suggested by Chen, Kim and Kon.21

The pronounced January effect had later been subject to various studies. For example, Marc Reinganum researched the January effect in small market value firms that have been traded

15_{Agarwal
and
Rivoli
(1989).}

16_{See
also
Jaffe
and
Westerfield
(1985).} 17_{Wong
et
al.
(2006).}

18_{Bonin
and
Moses
(1974).} 19_{Rozeff
and
Kinney
(1976).}

20_{Dyll
(1977)
presents
evidence
of
high
trading
volume
of
poorly
performing
stocks
at
the
end
of
the}

year.

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on the NYSE and the American Stock Exchange since July 1962.22_{The
results
of
his
research} were remarkable: small firms earned larger returns in January (and especially the first few days) compared to large firms.23_{This
size-‐related
anomaly
cannot
be
dedicated
to
tax-‐loss} selling only according to Reinganum.

Donald Keim also investigated the relation between returns and size of firms in January. He found out that 55% of the premium in returns that small firms have over large firms is realised in the month January.24_{Keim
made
use
of
a
Weighted
Least
Squares
model,
a} variant of OLS, to estimate the results per month. In a more specific examination of the month January he found out that most of this premium is gained in the first five trading days of the month. He copied two of the suggested explanations for the January effect from Rozeff and Kinney: tax-‐loss selling and accounting information. Besides, he took into consideration that there might be some errors in the database causing this effect.

Tan and Tat examined four calendar anomalies on the the SES All-‐Singapore Index from January 1975 to December 1994. They documented a higher average return in January compared to other months, but also that the mean daily return in January has fallen by almost 65% between the two subperiods (1975-‐1984 and 1985-‐1994), so a decline in the calendar anomaly over time. Their results also brought to light that mean returns on Monday and Tuesday are lower than Wednesday, Thursday and Friday. By conducting an F-‐ test, they conclude that there is a DOW effect in the Singaporean stock market and it also declined from the first subperiod to the second. They dedicated this decrease in calendar effects to awareness of investors of these effects as the information added to the market makes in more efficient.

Autoregressive conditional heteroskedasticity

In 1982, Robert Engle published a regression technique that would have a major impact on analyses of stock market data.25_{Most
regressions
were
performed
by
the
OLS
technique,} which relies on a couple of assumptions.26_{On
of
those
assumptions
is
that
OLS
assumes} homoskedasticity in the data examined, but Engle provided evidence that this is a mostly unlikely assumption when applied to stock market returns. Therefore he introduces a new

22_{Reinganum
(1981,
1982).}

23_{Small
and
large
firms
were
defined
by
market
capitalization.} 24_{Keim
(1983).}

25_{Engle
(1982).}

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error term distribution with a dependent variance and a new variance equation (see the Methodology part of the Data & Methodology section). This new technique, called

Autoregressive Conditional Heteroskedasticity (ARCH), has been adopted and extended by many economists ever since.

For example, Taufiq Choudhry conducted a study making use of a GARCH (extention to ARCH) in seven emerging stock markets in Asia.27_{In
six
out
of
seven
markets
the
negative} Monday effect was witnessed. He put forward that within emerging markets might calendar anomalies might be present as in most cases there is a lot of regulation which counteracts market efficiency.

Alagidede and Panagiotidis employ GARCH, TGARCH and EGARCH (extentions to the ARCH) to examine day of the week and month of the year effects on returns and volatility in the Ghanaian stock market.28_{Their
findings
are
that
overall
there
is
an
April
effect
with
a
high} mean of about 8% in returns. When using only recent information this effect seems to disapear.

An increasing amount of studies confirm a decrease of the calendar effects over time. Marquering, Nisser and Valla have performed a big literature study on calendar anomalies across countries and periods of time.29_{They
conclude
that
MOY
effects
as
well
as
DOW} effects have diminished over time in established and developed stock markets after the large number of publications about the subject. The results are consistent with Tan and Tat (1998) and Alagidede and Panagiotidis (2009).

27_{Choudhry
(2000).}

28_{Alagidede
and
Panagiotidis
(2009).} 29_{Marquering
et
al.
(2006)}

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EXPECTATIONS AND HYPOTHESES

Expectations

The Literature Review shows that in general a negative Monday effect and a postive Friday effect are witnessed in stock markets, caused by the dynamics of the weekend.30_{Moreover,
a} number of studies find a positive January effect due to tax-‐loss selling, or other causes.31 Other monthly effects aren’t described well enough to expect a strong calendar anomaly. However, an increasing amount of studies report the decline and eventual disappearance of several calendar anomalies.32_{Most
of
these
papers
dedicate
this
development
to
an
increase} in stock market efficiency. This would mainly apply to developed stock markets like the S&P Composite in the United States, as some developing markets still exhibit these effects more profoundly.33_{The
Amsterdam
Stock
Exchange
can
be
regarded
as
a
developed
market,
so
it} is not likely that highly significant calendar effects are present. However, this thesis being the first study examining the Amsterdam Stock Exchange solely, there might be some information that is not fully disclosed to all participants. Besides, it is a small market and therefore might be less efficient than bigger ones.

Therefore, with regard to the DOW effects, a small negative coefficient is expected on Mondays, and a small positive coefficient is expected on Fridays. In the rest of the week, no effect is expected. Within the MOY effects, a small positive coefficient is expected in January. For the rest of the year expectations are ambiguous, so no effect is expected. The following hypotheses are formulated:

Hypotheses

DOW effects

• Hypothesis Monday: negative coefficients at the significance level of 10% or lower in all three indices and at the significance level of 5% or lower in at least one index. • Hypotheses Tuesday, Wednesday, Thursday: no negative or positive coefficients at

the significance level of 5% or lower in either index.

• Hypothesis Friday: positive coefficients at the significance level of 10% or lower in all three indices and at the significance level of 5% or lower in at least one index.

30_{Cross
(1973),
French
(1980),
Gibbons
and
Hess
(1981),
Rogalski
(1984),
Agarwal
and
Rivoli}

(1989), Tan and Tat (1998) and Choudhry (2000).

31_{Rozeff
and
Kinney
(1976),
Reinganum
(1981),
Keim
(1983),
Tan
and
Tat
(1998).} 32_{Tan
and
Tat
(1998),
Marquering
et
al.
(2006)
and
Alagidede
and
Panagiotidis
(2009).} 33_{Choudhry
(2000)}

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MOY effects

• Hypotheses January: positive coefficients at the significance level of 10% or lower in all three indices and at a significance level of 5% or lower in at least one index. • Hypotheses Other Months: no negative or positive coefficients at the 5% level or

lower in either index.

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DATA AND METHODOLOGY

Data

For comprehensive analysis, data from the three main stock indices are used. The leading index is the Amsterdam Exchange Index (AEX). It was established at January the 3rd 1983 and contains the 25 companies with the highest share turnover. The second index is the AMX index, which stands for Amsterdam Midkap Index and consists of the next 25

companies measured by share turnover. AMX data goes back to 1983 as well. The last index is the Amsterdam Small Cap Index (AScX), which was established on June 30th 2000. The AScX is composed of the 25 companies that rank number 51-‐75 in share turnover.34_{Each
of} the indices is examined separately to examin if there are differences in results between large, mid and small cap stocks. Data from all three indices is used until May 20th 2016. To calculate the index returns, daily and monthly closing levels are requested from Reuters Datastream. The returns are calculated by taking the logarithm of the index level at a certain moment, divided by the index level of one period before. The data is filtered for non-‐trading days, in such way that weekend and holidays are not included in the analysis.

Methodology

As mentioned in the Literature Review section, Engle introduced the ARCH technique to adress the homoskedasticity problem. This thesis will make use of ARCH and Generalized ARCH (or GARCH), which is an extended form of ARCH introduced by Bollerslev.35

Using ARCH requires the error term to have a zero mean just like OLS, but the variance is given by the value of a dependent variable instead of the constant. This variable is

dependent on one or more lags of the squared error term from the model. When there is just one lag, the model looks like this:

-‐ Model equation:

𝑦! = 𝛼!+ 𝛼!𝑥! + 𝑢! (1)

-‐ Error term distribution:

𝑢! ∼ 𝑁(0, 𝜎!) (2)

34_{https://www.euronext.com/nl/products/indices/NL0000000107-‐XAMS/market-‐information} 35_{Bollerslev
(1984)}

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-‐ Variance equation:

𝜎_! = 𝛽_!+ 𝛽_!𝑢_!!!! ₍₃₎

where 𝑦_! is the dependent variable, 𝛼_! is the constant term, 𝛼_! is the coefficient for 𝑥_!, which is the independent variable, 𝑢! is the error term, 𝜎! is the variance variable of the

distribution of 𝑢_!, 𝛽_! is the constant in the variance equation, 𝛽_! is the coefficient for the lagged error term, which is 𝑢_!!!.

One can easily see that a large residual will lead to a broad distribution of the next residual, which will increase the probability of the next residual to be large as well. The GARCH process is based on the same principle as ARCH, but with GARCH an extra lag of the variance variable itself is added. Where the ARCH lag just increases the probability of the next residual to be large after a large residual is witnessed, the GARCH term directly causes the next residual distribution to increase after a large residual is witnessed. Of course, this works in both cases the other way around as well. In that way, ARCH and GARCH models take into account for the volatility clustering in stock returns. Adding the GARCH lag results in a change in the variance equation:

𝜎_! = 𝛽_!+ 𝛽_!𝑢_!!!!_{+ 𝛽}

!𝜎!!! (4)

where all of the above applies, supplemented with 𝛽_! as the coefficient of 𝜎_!!!, which is the lagged variance variable. Just as with ARCH, multple lags can be added.

The first step of the analysis, which has been done in the statistical program Stata, was finding out whether volatility clustering was present in the dataset. Graph 1 to 6 show the squared residuals of each regression per dataset. Naturally, monthly data show figures similar to daily data, but only with less data points in the graph. In both AEX and AMX data, peaks in volatility are witnessed in 1987 (around Black Monday) and at the start of the finanical crisis in 2008-‐2009.36_{Also,
a
period
of
relatively
high
volatility
ocurred
in
the
first} years of the 2000s. Since the AScX has been established in 2000, the 1987 peak is missing, but the graphs show similar high levels in the early 2000s and around 2008-‐2009. The graphs give an indication of volatility clustering in the datasets.

36_{At
these
moments,
markets
all
over
the
world
suffered
great
losses
and
high
uncertainty}

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Graph 1 Graph 2

Volatility in AEX daily returns. Volatility in AEX monthly returns.

Graph 3 Graph 4

Volatility in AMX daily returns. Volatility in AMX monthly returns.

Graph 5 Graph 6

Volatility in AScX daily returns Volatility in AScX monthly returns.

To prove volatility clustering, a Lagrange Multiplier test is conducted with each dataset. This test examines whether ARCH effects are present in the data. The null hypothesis therefore is: no ARCH effects in the data. Each test has been carried out with 1 to 15 lags. The results are given in Table 1 to 6.

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Table 1

Output Lagrange Multiplier test on AEX Daily data.

Table 2

Output Lagrange Multiplier test on AMX Daily data.

lags(p) chi2 df Prob > chi2

1 458.752 1 0.0000 2 1002.525 2 0.0000 3 1095.151 3 0.0000 4 1106.504 4 0.0000 5 1381.404 5 0.0000 6 1374.924 6 0.0000 7 1398.212 7 0.0000 8 1380.792 8 0.0000 9 1373.140 9 0.0000 10 1361.477 10 0.0000 11 1361.519 11 0.0000 12 1359.490 12 0.0000 13 1354.190 13 0.0000 14 1334.458 14 0.0000 15 1335.791 15 0.0000

1 551.235 1 0.0000 2 1196.199 2 0.0000 3 1448.556 3 0.0000 4 1440.514 4 0.0000 5 1668.941 5 0.0000 6 1691.356 6 0.0000 7 1672.640 7 0.0000 8 1653.999 8 0.0000 9 1659.980 9 0.0000 10 1700.381 10 0.0000 11 1686.014 11 0.0000 12 1695.765 12 0.0000 13 1737.596 13 0.0000 14 1715.035 14 0.0000 15 1701.406 15 0.0000

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Table 3

Output Lagrange Multiplier test on AScX Daily data.

1 307.330 1 0.0000 2 597.627 2 0.0000 3 686.373 3 0.0000 4 701.899 4 0.0000 5 877.358 5 0.0000 6 870.666 6 0.0000 7 872.878 7 0.0000 8 872.018 8 0.0000 9 870.954 9 0.0000 10 869.769 10 0.0000 11 865.682 11 0.0000 12 860.688 12 0.0000 13 866.829 13 0.0000 14 881.741 14 0.0000 15 875.172 15 0.0000 Table 3

Output Lagrange Multiplier test on AEX Monthly data.

1 33.834 1 0.0000 2 34.244 2 0.0000 3 37.097 3 0.0000 4 37.663 4 0.0000 5 37.622 5 0.0000 6 38.028 6 0.0000 7 38.404 7 0.0000 8 38.668 8 0.0000 9 38.722 9 0.0000 10 39.142 10 0.0000 11 39.919 11 0.0000 12 40.628 12 0.0001 13 39.434 13 0.0002 14 40.294 14 0.0002 15 40.381 15 0.0004

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Table 4

Output Lagrange Multiplier test on AMX Monthly data.

1 19.972 1 0.0000 2 21.759 2 0.0000 3 21.799 3 0.0001 4 24.509 4 0.0001 5 24.494 5 0.0002 6 24.717 6 0.0004 7 24.664 7 0.0009 8 24.695 8 0.0018 9 25.769 9 0.0022 10 28.582 10 0.0015 11 28.983 11 0.0023 12 28.615 12 0.0045 13 28.307 13 0.0082 14 29.353 14 0.0094 15 29.367 15 0.0144 Table 6

Output Lagrange Multiplier test on AScX Monthly data.

1 24.569 1 0.0000 2 27.107 2 0.0000 3 29.643 3 0.0000 4 29.897 4 0.0000 5 29.811 5 0.0000 6 30.540 6 0.0000 7 30.396 7 0.0001 8 30.530 8 0.0002 9 32.430 9 0.0002 10 33.941 10 0.0002 11 34.168 11 0.0003 12 34.075 12 0.0007 13 35.353 13 0.0007 14 35.627 14 0.0012 15 36.572 15 0.0015

In these tables, lags(p) represent the number of lags for which the test is conducted, chi2 is the chi-‐squared score as the Lagrange Multiplier test is a chi-‐squared distribution, df are the degrees of freedom and Prob > chi2 is the p-‐value of the outcome. It can be noticed that for each number of lags in each dataset the null hypothesis is rejected at the 5% signifcance level as the p-‐values (Prob > chi2) are all lower than 0.05.

After finding out that each dataset significantly contains ARCH effects for lags, the best fitting model has to be selected for each dataset. Just as with variables in a regular

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Each extra variable (lag) added to the model will increase the goodness of fit, but it also increases complexity. To Aikake Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are two ways to calculate the best trade-‐off between improving the goodness of fit and reducing complexity. The results of the AIC and BIC tests are given in tables 7 to 12 below.

Table 7

Output AIC and BIC test on AEX daily data.

lags(p) Obs ll (null) ll (model) df AIC BIC arch(1) 8267 . 25547.43 7 -‐51078.9 -‐51022.7 arch(2) 8267 . 25353.22 7 -‐50690.3 -‐50634.3 arch(1) garch(1) 8267 . 26652.50 8 -‐52041.0 -‐51951.2 arch(1) garch(1) arch(2) 8267 . 24090.74 8 -‐52041.6 -‐50835.0

Table 8

Output AIC and BIC test on AMX daily data.

Table 9

Output AIC and BIC test on AScX daily data.

lags(p) Obs ll (null) ll (model) df AIC BIC arch(1) 4051 . 13113.01 7 -‐26212.02 -‐26167.88 arch(2) 4051 . 13128.57 7 -‐26243.15 -‐26199 arch(1) garch(1) 4051 . 13543.36 8 -‐27070.71 -‐27020.26 arch(1) garch(1) arch(2) 4051 . 12950.65 8 -‐25885.29 -‐25834.84

Table 10

Output AIC and BIC test on AEX monthly data.

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Table 11

Output AIC and BIC test on AMX monthly data.

Table 12

Output AIC and BIC test on AScX monthly data.

In these tables, the lags(p) column represents which lags are selected. ‘arch(1)’ represents the lagged error term variable, like 𝑢_!!!!_{in
equation
(3)
and
(4),
‘arch(2)’
represents
the} same lagged error term variable, only one period before, which would look like 𝑢!!!! . ‘garch(1)’ represents the lagged variance variable, like 𝜎!!! in equation (4). Obs represent the amount of observed data, ll (null) is the log likelihood of the constant-‐only model, which doesn’t apply to these tests, ll (model) is the log likelihood of the model tested, df are the degrees of freedom, AIC is the value of the Aikake Information Criterion and BIC is the value of the Bayesian Information Criterion. It appeared that for all the datasets, one ARCH and one GARCH lag resulted in the lowest values of AIC and SIC, except for the monthly AScX data, where just one ARCH lag resulted in the lowest AIC and SIC values.

After selecting the right number of lags for the variance equationo, the correct models are formulated. The constant term is dropped. This is done, so the coefficients from the output are a direct representation of the average return of that particular day/month. The model for daily returns looks like:

𝑦! = 𝛼!𝑀𝑂𝑁! + 𝛼!𝑇𝑈𝐸!+ 𝛼!𝑊𝐸𝐷!+ 𝛼!𝑇𝐻𝑈!+ 𝛼!𝐹𝑅𝐼!+ 𝑢! 𝑢! ∼ 𝑁(0, 𝜎!) 𝜎! = 𝛽!+ 𝛽!𝑢!!!!+ 𝛽!𝜎!!!

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where 𝑦_! is the daily return, 𝛼_! until 𝛼_! represent the coefficients per day, which are represented by 𝑀𝑂𝑁_! until 𝐹𝑅𝐼_!, 𝑢_! is the error term, 𝜎_! is the variance variable of the distribution of 𝑢!, 𝛽! is the constant in the variance equation, 𝛽! is the coefficient for the lagged error term, which is 𝑢_!!! and 𝛽_! is the coefficient of 𝜎_!!!, which is the lagged variance variable.

The model for monthly returns of the AEX and AMX data looks like: 𝑦! = 𝛼!𝐽𝐴𝑁!+ 𝛼!𝐹𝐸𝐵!+ 𝛼!𝑀𝐴𝑅!+ 𝛼!𝐴𝑃𝑅!+ 𝛼!𝑀𝐴𝑌!+ 𝛼!𝐽𝑈𝑁!+ 𝛼!𝐽𝑈𝐿!+ 𝛼!𝐴𝑈𝐺! + 𝛼_!𝑆𝐸𝑃_!+ 𝛼_!"𝑂𝐶𝑇_!+ 𝛼_!!𝑁𝑂𝑉_!+ 𝛼_!"𝐷𝐸𝐶_!+ 𝑢_! 𝑢! ∼ 𝑁(0, 𝜎!) 𝜎! = 𝛽!+ 𝛽!𝑢!!!!+ 𝛽!𝜎!!!

where 𝑦_! is the daily return, 𝛼_! until 𝛼_!" represent the coefficients per month, which are represented by 𝐽𝐴𝑁! until 𝐷𝐸𝐶!, 𝑢! is the error term, 𝜎! is the variance variable of the distribution of 𝑢!, 𝛽! is the constant in the variance equation, 𝛽! is the coefficient for the lagged error term, which is 𝑢_!!! and 𝛽_! is the coefficient of 𝜎_!!!, which is the lagged variance variable.

The model for monthly returns of the AScX data looks like: 𝑦! = 𝛼!𝐽𝐴𝑁!+ 𝛼!𝐹𝐸𝐵!+ 𝛼!𝑀𝐴𝑅!+ 𝛼!𝐴𝑃𝑅!+ 𝛼!𝑀𝐴𝑌!+ 𝛼!𝐽𝑈𝑁!+ 𝛼!𝐽𝑈𝐿!+ 𝛼!𝐴𝑈𝐺! + 𝛼_!𝑆𝐸𝑃_!+ 𝛼_!"𝑂𝐶𝑇_!+ 𝛼_!!𝑁𝑂𝑉_!+ 𝛼_!"𝐷𝐸𝐶_!+ 𝑢_! 𝑢! ∼ 𝑁(0, 𝜎!) 𝜎! = 𝛽!+ 𝛽!𝑢!!!!

where 𝑦_! is the daily return, 𝛼_! until 𝛼_!" represent the coefficients per month, which are represented by 𝐽𝐴𝑁! until 𝐷𝐸𝐶!, 𝑢! is the error term, 𝜎! is the variance variable of the distribution of 𝑢_!, 𝛽_! is the constant in the variance equation and 𝛽_! is the coefficient for the lagged error term, which is 𝑢_!!!.

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The coefficients retrieved from the final analyses are compared to several critical values at the 20%, 10%, 5%, 2%, 1% and 0.5% levels of significance. By doing so, a more specific image of the significance can be provided. For the results, see Appendix A. The Conclusion section will interpret these results and judge whether an anomaly is the case.

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RESULTS

This section will go through the results retrieved from the tests described under the previous section. First the hypotheses will be repeated, whereafter the results will be

mentioned and the hypothesis will be rejected or not. The results are compared with several critical values, which is displayed by tables 13 to 18.

Day of the week effect

• Hypothesis Monday: negative coefficients at a significance level of 10% or lower in all

three indices and at the significance level of 5% or lower in at least one index. The

output shows that there are no negative coefficients found on Mondays in either of the three indices. Moreover, a positive coefficient at the significance level of 0.5% is found in the AScX data. The hypothesis is rejected.

• Hypothesis Tuesday: no negative or positive coefficients at the 5% level or lower in

either index. The output shows that there is a positive coefficient found on Tuesdays

in the AScX data at the significance level of 5%. Moreover, a postitive coefficient at the significance level of 2% is found in the AEX data. The hypothesis is rejected. • Hypothesis Wednesday: no negative or positive coefficients at the 5% level or lower in

either index. The output shows that there is a positive coefficient found on

Wednesdays in the AEX data at the significance level of 0.5%. The hypothesis is rejected.

• Hypothesis Thursday: no negative or positive coefficients at the 5% level or lower in

either index. The output shows that there is a positive coefficient found on Thursdays

in the AScX data at the significance level of 5%. Moreover, a postitive coefficient at the significance level of 1% is found in the AMX data and even a postitive coefficient at the significance level of 0.5% is found in the AEX data. The hypothesis is rejected. • Hypothesis Friday: positive coefficients at the significance level of 10% or lower in all

three indices and at the significance level of 5% or lower in at least one index. The

output shows that there is a positive coefficient found on Fridays in the AEX and the AMX data at the significance level of 1% and a postitive coefficient at the significance level of 0.5% is found in the AScX data. The hypothesis is accepted.

Calendar anomalies on the Dutch stock market

Maurice Beaumont

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