Ghent University

FACULTY OF ECONOMICS AND BUSINESS ADMINISTRATION

ACADEMIC YEAR 2013 – 2014

Eugene Fama versus Robert Shiller – Is the Belgian market efficient?

Master’s thesis submitted to obtain the degree of

Master of Science in Business Administration

Siemen Six & Randy Van der Auwera

under supervision of

Prof. Koen Inghelbrecht

PERMISSION Ondergetekenden verklaren dat de inhoud van deze masterproef mag geraadpleegd en/of gereproduceerd worden, mits bronvermelding.

Siemen Six Randy Van der Auwera

I

Table of contents

Abstract ............................................................................................................................ II

List of abbreviations ......................................................................................................... III

I. Introduction ............................................................................................................... 1

II. Literature Review ....................................................................................................... 3

1. The efficient markets hypothesis (EMH) ............................................................................ 3

2. Empirical evidence ............................................................................................................ 6

3. Behavioural finance ........................................................................................................ 10

4. The adaptive markets hypothesis (AMH) ......................................................................... 18

III. Research Design & Methodology .............................................................................. 21

1. Design ............................................................................................................................ 21

2. Summary statistics ......................................................................................................... 24

3. Methodology .................................................................................................................. 26

IV. Results ................................................................................................................. 33

Is the Belgian stock market efficient as a whole? And what about the different size-based

compartments of the stock market? ....................................................................................... 33

1. The Random Walk .......................................................................................................... 33

2. The Variance ratio .......................................................................................................... 40

Do we observe a difference in market efficiency between growth and value stocks? ................ 47

1. The Random Walk .......................................................................................................... 47

2. The Variance ratio .......................................................................................................... 51

Did the financial crisis marked by the 15th of September 2008 as the starting date have an effect on market efficiency? ............................................................................................................. 53

1. The Random Walk .......................................................................................................... 53

2. The Variance ratio .......................................................................................................... 60

V. Conclusion ............................................................................................................... 67

1. Conclusions Random Walk autoregression models .......................................................... 67

2. Conclusions Variance ratios ............................................................................................ 69

3. Differences and matches between the RW autoregression models & Variance ratios ....... 69

4. Fama vs. Shiller, who fits the Belgian stock market? ........................................................ 70

VI. References ........................................................................................................... 71

VII. Appendix .............................................................................................................. 75

II

Abstract

This paper analyses the market efficiency of the Belgian stock market. Therefore, we focus on

data from 1996 till 2014. We evaluate if a random walk is present in the Belgian stock indices

by the use of autoregressive models and variance ratios. We also examine whether differences

in market efficiency occur when we divide the Belgian market into two indices containing

only growth and the other only value stocks. In addition, taken into account the financial

crisis of 2008, we examine if changes occurred in market efficiency. Our results indicate that

the whole Belgian stock market does not follow a convincing random walk over a 1996-2014

period. We find the market compartment including only growth stocks to be more efficient

than the one including just value stocks, although they are both inefficient. And lastly, our

subperiod results show that the financial crisis of 2008 mainly increased market inefficiency

on the Belgian stock market, whereas in the three-year period before the outbreak of the crisis

the market was efficient.

Keywords: Market efficiency, random walk, variance ratio

We would like to thank Prof. Koen Inghelbrecht for his excellent advice, his assistance with

our statistical tests and his positively constructive and deeply valued feedback throughout the

semester.

III

List of abbreviations

ACF Autocorrelation function

AMEX American Stock Exchange

AMH Adaptive markets hypothesis

AR model Autoregressive model

BAS index BEL All-Share index

CAPM Capital asset pricing model

EMH Efficient market hypothesis

NASDAQ National Association of Securities Dealers

Automated Quotations

NYSE New York Stock Exchange

OECD Organization for Economic Co-operation

and Development

RW Random walk

VAR Vector autoregressive regression

VR Variance ratio

1

I. Introduction

In 2013 a Nobel prize was awarded to E. Fama, L.P. Hansen and R. Shiller for their work on

security pricing on capital markets1. This was a remarkable decision as Fama and Shiller have

opposing beliefs about efficient capital markets. Fama is a great advocate of efficient capital

markets, which simply put, means that all available information about a security is included in

the price of that security. He further believes that the markets are still efficient today even

after several anomalies occurred, e.g. the IT-bubble in 2000 or the recent financial crisis of

2008. Fama explains these anomalies by unidentified risk factors and maintains his

convictions about the efficient market hypothesis (Cassidy, 2013).

Shiller, on the other hand, contradicts the efficient market hypothesis and he was one of the

first ones to recognize that economic agents are not always rational but are in fact influenced

by multiple psychological factors. The irrationality of the agents contribute to market

anomalies, as was the case with the IT-bubble and undervaluation or overvaluation of assets

in general as well. For Shiller, the anomalies are the best proof that capital markets are not

(always) efficient (Cassidy, 2013).

It is not the purpose of this paper to firmly choose the side of Fama or Shiller, as we will

explain that the efficient markets domain is not a strict black and white story but has a grey

zone as well. Instead we will explain what efficient markets are, why the efficient market

hypothesis is so important and how the work of Shiller and other likeminded behavioural

finance-economists fit so well in the debate. With our own research, we want to find out

whether the Belgian stock market is efficient or not and therefore we will examine several

market indices based on the capitalization of underlying stocks (i.e. stock size). Four market

indices are included in our research: the BEL All-Share index (BAS), which represents all

stocks noted on the Belgian market, the BEL-20 index comprised out of the 20 most liquid

stocks, the BEL Mid containing medium capitalized stocks and the BEL Small index with

small capitalized stocks as the underlying securities. We will apply weak-form tests by

running autoregression models and calculating variance ratios, which we will compare in

order to determine the market efficiency. Depending upon the results we will have an

indication of whether the prices on the examined (part of the) market reflect the fundamental

value of the underlying security, i.e. in an efficient market or if they do not reflect the true

intrinsic value, as in an inefficient market (Fama, 1965). The results have important

consequences, especially for investment decisions. In an efficient market one of the best

investment choices an investor can make over time is a passive fund mimicking the market

index (Bodie et al., 2013; Inghelbrecht, 2013a). These results can be found in the conclusion

section of this paper.

In the next section, namely the literature review, we start off with the efficient markets

hypothesis in general, followed by important empirical research examining markets in their

efficiency. Thereafter, we pay attention to the behavioural finance view on capital markets

1 Article from The New Yorker by J. Cassidy (Inefficient Markets: A Nobel for Shiller (and Fama))

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also with attention for empirical research. To end the literature section, we refer shortly to the

fairly new adaptive markets hypothesis that combines the two opposing views of market

efficiency and behavioural finance. After the literature review we explain our research design

and methodology approach and then follows a separate section for the research itself. The

next and also last section contains the results of our research in which we will end the paper

making our conclusions on market efficiency and compare with the discussed literature in the

literature review.

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II. Literature Review

1. The efficient markets hypothesis (EMH)

The term efficient market was introduced by Fama, who laid the foundation for efficient

markets in his own ‘efficient capital markets’-paper in 1970 in which he makes a review of

the then existing literature and empirical research. This paper is seen as Fama’s greatest

contribution to this field of research and still gives a good explanation of what the hypothesis

essentially includes (Fama, 2010). Later on Fama wrote a second review paper on efficient

capital markets in 1991. Now, first of all, we need to acquaint ourselves with the base work to

comprehend efficient markets before we move on to more recent research papers that test

efficiency.

Fama (1970) essentially describes an efficient market as “a market in which security prices

fully reflect all available information at any time” (p. 1). Important elements in this definition

are ‘prices’, ‘all available information’ and ‘at any time’. First, prices must be obtained by a

certain model in order to make everything testable. Second, the information element is crucial

in the hypothesis and therefore the criteria for information to consider, are classically divided

into three information subsets. Third, ‘at any time’ signifies that efficient markets are always

efficient, i.e. in stable economical and financial times, as well as in times of recession or

(excessive/rapid) growth (Fama, 1970). It is favourable to discuss some aspects of these

elements in greater detail.

Prices can be determined by different models in order to make them testable for empirical

research. Fama (1970) includes base models in his overview paper and two special forms of

these base models. As a general rule for these pricing models it is imperative that the

assumptions made by the model are valid. Otherwise when testing market efficiency with a

model that is based on wrong/non-valid assumptions, the efficiency test will be non-valid as

well. Nevertheless, certain assumptions have to be made with each model (Fama, 1970). As

these assumptions are hardly ever consistent with reality and differ from model to model,

according to Fama (1991) this implies that market efficiency cannot be tested in essence.

Fama (1991) describes this problem as the ‘joint-hypothesis’ problem that refers to the need

of an asset-pricing model in order to test for market efficiency. Fama (1991) states that

evidence or indications of inefficient capital markets could also be caused by a faulty or

invalid pricing model used in certain research. Therefore it is unlikely that we will ever be

able to measure market efficiency to its full extent and made assumptions will stay rather

theoretical. Nevertheless research on efficient markets is all but futile. It has already changed

the way investors and academics regard returns on securities etc. (Fama, 1991).

The base pricing models are the expected return or fair game models. The most well-known

expected return model must be the Capital Asset Pricing Model (CAPM or Sharpe-Lintner

model). The type of risk that is taken into account by the CAPM to determine an expected

return is the systematic (market) risk. In some research on efficient markets, different models

can be used that take other (or additional) risks into account to get to the expected returns.

These other models should always be in equilibrium, just as the CAPM. Regardless the risks

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that are included in the model, all expected returns are random and they are a function of the

price of the previous period, the random return of one period and an assumed information set

(Fama, 1970; Bodie et al., 2013). Further, the equilibrium expected return given by the model

implies that there cannot be made any excess returns over the equilibrium return based on the

information set, as the set is already included in the equilibrium expected return. The ‘fair

game’ denomination stems from the fact that the (expected) returns are random and are not

influenced by the previous returns, i.e. are a martingale (Fama, 1970). This is the base model

of which there are two important applications: the submartingale model and the random walk

model.

The submartingale model has a submartingale in a price series. This means that the price of

this period is equal or higher than the price of the previous period. The same goes for

expected returns. If the price is just the same as the one from the previous period, the series of

prices is just called a martingale which is the same as a ‘fair game’ model (Fama, 1970).

The random walk model is used since the beginning of market efficiency research and has two

main conditions that need to be fulfilled in order to speak of a random walk model. First,

Fama (1970) states that successive price changes (i.e. returns) need to be independent, thus

not influence each other. Basically this implies that returns should be uncorrelated. The

changes of these prices or returns are determined by the information of the particular period

regarding the underlying security. Second, these changes in prices or returns need to have an

identical distribution.

Fama (1970) considers this random walk model as an extension of the base expected return or

fair game models, which is logical as it is a very similar. Nevertheless there is an important

difference namely the second condition: identical distribution of returns. These distributions

are assumed to repeat themselves trough time, under influence of changing information and

investor preferences that lead to new price/return equilibriums. In these distributions, the

order of returns is not important, only that they are identically distributed is of importance

(Fama, 1970). Practically this means that there need to be as many overvalued returns as there

are undervalued returns so the market price will float around the intrinsic value (Fama, 1965).

Fama (1970) strongly prefers tests using the random walk model (or variation on the model)

over tests that ascertain the pure independence between returns in a time series, on account of

the fact that the random walk model is based on characteristics of the base (‘fair game’)

model.

Also it is worthwhile to notice the difference between prices and returns. Returns are

stationary which lack long memory, whereas prices are non-stationary and have a trending

behaviour which makes it easier to estimate values for the following period(s). These

characteristics have certain implications for empirical handling that will be considered in our

research design (Inghelbrecht, 2013b; Koop, 2006). There are also implications for the

random walk model. When using non-stationary price time series to make forecasts,

assumptions have to be made for the order of price changes: they need to follow each other

subsequently while being dependent to each other. This obviously means that we cannot

longer speak of a random walk model as the independence condition is not met. Instead it

5

becomes a ‘chartist technique’, which is much less used to predict prices and less reliable

(Fama, 1965). We will revisit this in the Methodology section when discussing ways to test

the random walk.

The theory of random walks assumes capital markets to be efficient. This implies that the

market price will be more or less the same as the intrinsic (‘fundamental’) value of the

underlying security. Because of the market efficiency assumption, past and expected future

information is reflected in the market price and thus in the intrinsic value. Market prices will

float randomly and closely around the intrinsic value (Fama, 1965). This is exactly why we

think it is so important to test the efficiency of the Belgian capital market. If the market is

proven efficient, investors would be able to use the market price as a good estimate for the

intrinsic value of a security in the security selection-decision. This could imply lower

information and search costs for investors as they can just invest in a market index fund that is

passively managed (Bodie et al., 2013; Inghelbrecht, 2013a). Nevertheless, Fama (1965)

stresses that this does not mean that additional fundamental analysis is redundant, especially

not when there is new information that is not yet reflected in the current market price. So

active management can surely pay off, provided that the management in control has new

(superior) information or that they interpret it in a better way (Bodie et al., 2013;

Inghelbrecht, 2013a).

Moving on from the pricing element to the informational element, the three different

information subsets into which Fama (1970) classifies empirical work are weak form tests,

semi-strong form tests and strong form tests. The weak form tests focused originally on

information sets that contained only historical prices and later the set also widened to tests

that aim to forecast returns using past returns or certain ratios and variables (e.g. dividend

yields, earnings to price...). Semi-strong form tests also take into account all publicly made

available information (such as announcements of earnings, new product launches, takeovers,

changes in management...) and mainly research the time that is needed to reach adjusted

market prices. This usually happens by means of event studies. And finally, strong form tests

that take public and private information into account (Fama, 1970, 1991; Bodie et al., 2013).

Private information in this context is actually inside information that only a handful of people

know about. The information requirements for strong form test are very extensive as the

assumption is made that inside information is also reflected in the price (Bodie et al., 2013).

Having discussed the most important elements of this rudimentary definition of efficient

market, there are also some additional ‘softer’ conditions that can be added. First of all, there

are many profit-maximizing investors in direct competition, each making analyses of

individual securities and looking for additional information. Secondly, the information is

available to all investors, free of charge or almost freely. Thirdly, new information becomes

available at random, i.e. unpredictable which means that the prices are random and

unpredictable as well. Fourthly, security prices adjust to new information very quickly, almost

at once. This fourth condition entails that investors all judge the new information in a very

similar way so that market prices change correctly in line with the average opinions of all

investors (Fama, 1965, 1970; Bodie et al., 2013; Inghelbrecht, 2013a). Fama (1970) also adds

6

the assumption of no transaction costs on the market. In real markets this is hardly ever the

case, just as some of the four ‘soft’ conditions mentioned here above. Fama (1970) does not

reject the efficient market hypothesis if one of these conditions is not met. He sees the

absence of some of these conditions as merely potential sources of market inefficiency in a

limited extent. The real extent depends on the influence these conditions have on the price

formation of a security.

2. Empirical evidence

We already noticed the division in three information subsets above and this is important for

our own research and the research that we will pay attention to in this section of our paper. As

our own research can be classified into the weak form test division we will concentrate on

weak form testing. We will first discuss some important aspects of weak form tests that Fama

(1970, 1991) reviewed in his two overview papers that we did not mention above due to the

empirical nature and relevance.

As mentioned before, weak form tests also aim to forecast returns using past returns or certain

ratios and variables (e.g. dividend yields, earnings to price...). Fama (1991) arranged weak

form tests based on time periods from short term to long term. He found that in the short term

(daily, weekly and monthly) returns had just a very small part in their variance that could be

forecasted. But on the long term (two to ten years), roughly 40% of the variance in returns

could be explained. Fama (1991) reports disagreements between academics who contribute

this predictable part in the return variance to either irrational bubbles or rational changes in

expected returns instead. Once again this questions the existence of efficient capital markets

and refers to the opposing views of the behavioural finance-economists and irrationality in

capital markets.

Fama (1991) reviews research in which random walk tests are carried out by using an

autoregressive model of the first order (AR(1)-model). These studies yield different results

depending on their time horizon. On the short term, Fama (1991) notices that these models

often have very low statistical explanation power, especially when examining individual

stocks opposed to portfolios. The autoregressive models to predict returns of an individual

stock usually have less than one percent explanation power, which makes these models

useless in real life. Then there is also the significance of the lagged variable itself. Fama

(1991) acknowledges the statistical but therefore not its economical significance. As the

values of autocorrelations usually are around zero, they are not economically significant.

Returns with no or very little autocorrelation are evidence in support of a random walk (Fama,

1991).

Fama (1991) pays special attention to Lo and MacKinlay (1988), who divided stocks on

account of their stock size before testing efficiency using a random walk/autoregressive

model. They found significant positive autocorrelations and stronger autocorrelations for

smaller stocks. Fama (1991) noticed that this could mean there was spurious positive

autocorrelation that is a consequence of the nonsynchronous trading effect of Fisher. The

Fisher nonsynchronous trading effect is well explained by Lo and MacKinlay (1990) as the

7

problem that arises when multiple time series are sampled at the same time when actually they

are not. This problem seems meaningless at first but it can result in biases which is given by

the example of Lo and MacKinlay (1990) of two stock returns of which one stock is traded

less than the other. The return of the most traded stock will most likely be more accurate as it

is traded more and new information regarding the stock value is better adopted. In this

context, Lo and MacKinlay (1988) remark that small caps are less traded than large caps and

thus it takes the small stock longer to soak in new information.

On the long term, Fama (1991) cites his earlier work with K. French (1988a, in Fama, 1991)

in which they find negative autocorrelations for three to five year returns. Fama (1991)

notices the similar findings in research of Shiller but does not agree with Shiller’s

interpretation of the development of irrational bubbles. Instead of developing irrational

bubbles, Fama (1991) contributes the negative autocorrelations to temporary price swings and

as one of the conditions of the random walk hypothesis is that random walks must have

identical distributions, there should be as many prices undervalued as overvalued to neutralize

each other. This is the phenomena of mean-reversion (Fama, 1991). Over the long term this

should be the case and thus no reason to reject the efficient market hypothesis is given. Fama

(1991) did not find any strongly significant evidence to reject the random walk hypothesis

(i.e. absence of autocorrelation or no economical significance) up to then.

Finally, variables and ratios used to forecast returns (such as dividend yields and earnings to

price) do not prove or disprove efficient markets. Information, that also is reflected in the

price is needed to compute these variables or ratios. A lot depends on what kind of

information is used (and if the quality is of sufficient quality) which implies that the results of

these forecasting indicators are not always rational (Fama, 1991). Furthermore, Fama (1991)

warns us about new predictors that may look very promising and trustworthy at first sight but

are actually spurious later on.

Now we can move on to more recent work. Tóth & Kertész (2006) examine market efficiency

on the New York Stock Exchange in two different ways, each with an appropriate dataset.

They compute time-dependent as well as equal time cross-correlations between the most

traded stocks on the NYSE and link these to the Epps effect. Tóth & Kertész (2006) describe

the Epps effect as the decrease of correlations that occurs when the interval length of returns

is decreased. If the Epps effect diminishes over time, which is the main conclusion of the

paper, the NYSE should be more efficient according to Tóth & Kertész (2006). The two

datasets they use are a high frequency set, containing information on all trades for the 190

most traded stocks from 1993 to 2003 and a daily return dataset for 116 large stocks from

1982 to 2000. Before carrying out cross-correlation tests, Tóth & Kertész (2006) had to

classify the stocks into two same-sized groups based on market capitalization.

The first important conclusion they drew out of the time-dependent tests, was that the average

correlation in daily returns between the large caps group and the smaller caps group decreased

very strongly to a level at which correlation is negligible (values around zero). Tóth &

Kertész (2006) deduced out of this result that large stocks do not ‘pull’ the stock prices of

smaller stocks, which implies that price changes of large stocks have no effect on price

8

changes of smaller stocks and misguided price changes of smaller stocks based on irrational

elements do not occur. Secondly, distributions of time-dependent correlation functions were

also studied in the high frequency set, establishing that correlations actually became higher

(shown in higher peaks) but nevertheless kept decreasing over time and the time intervals in

which correlations decreased became shorter. Tóth & Kertész (2006) explain the shorter

intervals by improvements in market processes such as better technical infrastructure (faster

computers and networks) which in turn speeded up the trades on the market and information

flows for traders. Due to the shorter intervals and more uniform adaptations to new

information combined with short time correlation peaks of returns in the high frequency set,

the first indication of a diminishing Epps effect is presented. Thirdly, the equal time tests in

the high frequency set leads to the finding of a rising average correlation over time, except for

the year 2000 due to the market crash of that year. This similar finding also implies a decrease

of the Epps effect. Tóth & Kertész (2006) explain the rising correlation by the increase in

trading which makes the time scale grow and which in turn results in growing correlations.

To conclude, the decrease of the Epps effect is attributed by Tóth & Kertész (2006) to

strongly decreased lagged autocorrelations and cross-correlations and increased correlations

brought by the increased trading. Market efficiency should thus have increased over the

examined time span, implying that the extent of the efficiency can change over time. Finally,

Tóth & Kertész (2006) generalize their findings for the NYSE to other markets as they argue

that the causes of the decrease of the Epps effect can be found on those other markets too.

Another important research paper is the one of Lee et al. (2010) in which market efficiency is

examined by using the real stock price indexes in different countries, each within different

states of economical development. This paper uses price indexes while other research mostly

uses return indexes. In this paper the use of price indexes is explained by the adjustments that

were made to the indexes to account for inflation effects, which makes them real price

indexes.

Lee et al. (2010) start with giving an overview of the most important research that is carried

out examining price series for the presence of a unit root. The presence of a unit root in a

price series implies a random walk which is also stated by Fama (1970) and here again by Lee

et al. (2010). The overview shows that univariate tests in search for a unit root all seem in

favour of efficient markets, i.e. a unit root is present in the examined price series. The

research by Chaudhuri & Wu (2003) is the only exception out of the nine univariate research

papers stated in the overview. It examines the price series of 17 emerging stock markets. The

result of Chaudhuri & Wu (2003) is remarkable but cannot simply be generalized. Evidence is

found that 10 out of the 17 emerging markets do not follow a random walk and thus seem to

be inefficient. This is a plausible result according to Bodie et al. (2013) and Inghelbrecht

(2013a), as there is less competition in emerging markets due to a fewer number of investors

seeking to maximize their profit in comparison to developed markets. And also, the partial

unavailability of qualitative information to all of these investors results in slower adjustments

of asset prices. Chaudhuri & Wu (2003) also take so-called structural breaks into account and

the effect they have on price series. In their research, the structural breaks are mostly formed

by market characteristics (e.g. liberalization) and thus are specific for each examined market.

9

Other listed research in the overview of Lee et al. (2010), all examining indexes of developed

markets and also taking structural breaks into account, finds that the developed markets are

efficient.

Next to univariate time series analysis, the overview also discusses unit root analysis on panel

data. Again, a division is made between research accounting for structural breaks and research

that does not. The research, not accounting for structural breaks finds that emerging markets

seem to be inefficient whereas developed markets seem efficient. This is basically the same

result as with the univariate analyses. A different story applies to research that does take

structural breaks into account. Narayan & Smyth (2005) find that the 22 developed OECD

countries they examined had efficient markets, i.e. presence of a unit root. But then more

empirical evidence of the contrary is handed for the G7 countries and a group of Asian

countries. And in this category, the own research of Lee et al. (2010), covering the data period

of 1999 to May 2007, also finds market inefficiency for 32 developed (incl. Belgium) and for

26 developing markets. The results of Lee et al. (2010) are based on a stationarity test for

panel data that accounts for multiple structural breaks opposed to most other tests that take

maximum two structural breaks into consideration. Also important is, that Lee et al. (2010)

find that after a structural breaking point the price level will return to some kind of

equilibrium level, which is in accordance with stationarity in prices and thus market

inefficiency. Under this assumption it should be possible to predict future price movements

based on past prices using technical analysis (e.g. price charting) and in a weak-form efficient

market, technical analysis and forecasts based on past prices in general should not be possible

(Inghelbrecht, 2013a; Lee et al., 2010).

Lee et al. (2010) highlight the importance of structural breaks, as did Chaudhuri & Wu

(2003). They explain that ignoring such breaks can lead to biases that wrongly accept the

presence of a unit root and therefore the random walk hypothesis which implies market

efficiency. Furthermore, structural breaks can occur due to many reasons such as crises,

regulations and other events that influence stock markets globally as well as domestically.

Allowing for structural breaks increases the econometric power of the tests but when

comparing end results of several research papers, the answer whether markets are efficient

remains inconclusive (Lee et al., 2010). Perhaps the most important conclusion that we can

draw is that the random walk and thus market efficiency seems stronger in developed markets.

Moving on to other research, Kim et al. (2011) study predictability of returns using the Dow

Jones Industrial Average index with a vast data set covering the period of 1900 to 2009. They

use several tests based on autocorrelations to examine weak-form efficiency. Although they

perform weak-form efficiency tests, they do not link their outcomes to the efficient market

hypothesis but to the adaptive market hypothesis instead (see section on the adaptive market

hypothesis). Nevertheless, the test results apply to both theories. Kim et al. (2011) use the

automatic variance ratio test, the automatic portmanteau test and a generalized spectral test,

all based on autocorrelations. These test results differ over time and evidence is found that

return predictability on the market is higher before 1980 than after, which implies an increase

in market efficiency after 1980. These results correspond with Tóth & Kertész (2006) who

10

also found rising efficiency in their datasets starting from the eighties and nineties. And just

as Tóth & Kertész (2006), Kim et al. (2011) also explain the rise in efficiency by

infrastructural innovations on the stock markets in the sixties and seventies of which the

effects manifested from the eighties onward. But does this mean that markets were inefficient

before 1980? Kim et al. (2011) do not necessarily think so because there still is a difference

between theoretical and economical gains. Although return predictability seemed possible at

times (from an ex post view), the possibility of attaining financial benefits is questionable as

there are still transaction costs and uncertainties associated with the used forecast models that

need to be considered. However, evidence is presented that market efficiency is not proven

for the whole period and it is shown that the efficiency changes over time.

In addition to examining return predictability (i.e. weak-form efficiency), Kim et al. (2011)

also run a regression model to find out if and if so, what market conditions contribute to

return predictability. On the one hand, dummy variables are used to allow for the effects of

market turmoil (crashes, crises, bubbles...) and on the other hand economic variables (interest

rates, inflation, market price-earnings ratio...) are included. The findings of the regression are

that return predictability is influenced by market conditions and the degree of predictability

differs over time. For example, return predictability lacks when the market crashes but

predictability is fairly high in times of economical or political crises. Furthermore, Kim et al.

(2011) find that risk-free rates, inflation and market volatility are significant economic

variables in the regression results.

Although the part of the research of Kim et al. (2011) involving the effect of market

conditions is mainly to test a part of the adaptive markets hypothesis, it also has important

implications for all research on weak-form efficiency and may help explain the differences

and contradictions in weak-form results found by the many researchers. Kim et al. (2011)

point out that the results of previous empirical research are influenced by the market

conditions that are present at the time of the studied dataset, which is referred to by the

authors as the ‘data-snooping bias’. The need to put an examined dataset in a larger timeframe

is hereby put forward but is not always achievable by restrictions in data collection. The data-

snooping bias is a plausible explanation for the many conflicting results that are found in the

literature and also is of importance for Kim et al. (2011) as their results are also deviating

from most general conclusions of previous work that may be (out)dated and in need of

revision.

3. Behavioural finance

Behavioural finance, that challenges the rationality of stock markets, is the counterpart of the

Efficient Market Hypothesis (EMH). One of its important founders is Robert J. Shiller, one of

the three Nobel Prize winners of 2013, who contradicts the EMH with his empirical research.

Instead assuming that investors are fully rational and that share prices do reflect all available

information, proponents of the behavioural finance believe that investors are not completely

rational like the EMH assumes. In contrast to the EMH, irrational investor characteristics have

to be incorporated in asset pricing models (Barberis and Thaler, 2003). Therefore, behavioural

11

finance is based on the psychology and the sociology of investors to explain changes in stock

prices.

Stock return predictability

Since the EMH does not cope with irrational behaviour which can be found in stock market

prices, behavioural finance tries to find the answer to the question why market inefficiencies

exist. Behavioural finance first started to challenge the EMH with research on dividends, real

interest rates and changes in the intertemporal rate of substitution. It first started with Shiller

(2003) who mentioned that changes in the stock prices are partly due to psychological

elements and not completely due to the fundamental value that changes when new

information is made publicly. Furthermore, Shiller (1987) examined if the dividend changes,

changes in the real interest rates and changes in the intertemporal marginal rate of substitution

could explain the volatility in share prices. He concluded that the volatility could only be

partly explained by these indicators while the extra volatility remained unexplained.

Concerning these indicators, dividend changes contribute little to the variability of share

prices and the other two indicators are also a small part of the observed variability (Shiller,

1987). Like in Shiller (1987), we present a formula of how the price of a stock is obtained:

Pt= Dt/(1+r) + Dt/(1+r)² + Dt/(1+r)³+ …

The price of a stock, which is shown in the formula above, presents all dividends (i.e. future

and present ones) discounted by r, which is the real rate of return. Regarding to this formula,

the EMH assumes that all future dividends are known by investors. Yet, Shiller criticizes this

assumption with evidence from Marsch and Merton (1986; in Shiller, 1987) who say that

dividends may follow a random walk and that future dividends are more uncertain than

presumed by the efficient markets model. Grossman and Shiller (1981) researched price

swings of stock market indices in order to find evidence of market inefficiency. The goal of

their paper was to find determinants other than new information. They found that price

changes were not only represented by new information but also by changes in the real interest

rates. Their paper shows a positive relation between real interest rates and share prices which

have a stable dividend in real terms. They stress that very high real interest rates cause serious

increases in stock prices, even dramatically. In their paper, it is also assumed that when

present consumption is abnormally low in comparison with the future consumption, real

interest rates will be large. People will try to hold their current consumption level and for that

reason, stock prices have to be lower than future stock prices in order to prevent dissavings

(i.e. people will tend to sell their stocks).

Because dividends, real interest rates and the intertemporal rate of substitution could not fully

explain excess volatility, research started to investigate serial correlation. In the opinion of

Fama, extra volatility that consists of overreactions and underreactions on new events in the

stock markets is equally distributed. However, Shiller (2003) contradicts this criticism

because there is no psychological principle that explains why people always react too strongly

on new circumstances. The fact that market anomalies disappear after a certain time, which

would provide evidence of market efficiency, can be contradicted by Shiller. He argues that

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anomalies also disappear in inefficient markets and therefore is not an evidence of market

efficiency. Clearly, the random walk model has to be innovated.

As mentioned in the EMH section, the random walk hypothesis implies that price changes are

independent (Fama, 1970). Put differently, the changes of the stock prices are uncorrelated

with each other which indicates that the probability of a negative return is the same as for a

positive return. This implies that returns are not predictable as well. In contrast to the efficient

market hypothesis and its literature, autocorrelations are found by several researchers who

contradict the EMH. One of these contradicting papers is from Barberis et al (1998), who

observed autocorrelations. Barberis et al. (1998) found positive and negative autocorrelations

which caused overreactions and underreactions of share prices. After good news, they

analyzed positive autocorrelations during three to five years. They had two remarks about

autocorrelations. First, their study shows an underreaction to sole announcements from the

company which results in negative autocorrelation in stock returns. Second, an overreaction

happens when these sole announcements are followed by the same information. Actually,

investors underreact because there is some uncertainty about the effect on the long term;

therefore, investors will wait for more confirming news. Investors will believe, when the

announcement is confirmed by extra news, that the effects will be persistent on the long term.

Lo and McKinley (1988) also find, positive autocorrelations on the long term but do not find

negative autocorrelations. Their findings are derived from weekly and monthly returns. They

tested if the random walk model could be rejected when weekly trading data was observed.

Results confirmed that share prices do not follow the random walk hypothesis, and that

weekly returns do not follow the stochastic behaviour as assumed in the random walk

hypothesis. Furthermore, the rejection is even stronger when the model is tested on small

capitalization stocks. But as seen before when discussing the EMH, Fama (1991) disregards

these results due to a possible nonsynchronous trading effect. Some papers argue that small

stocks are different due to infrequent trading (i.e. less trading implies that new information is

being observed slower). In fact, the intermittently trading of shares is a possible source of

bias (Dimson, 1979). As a result, he finds each estimated beta of his asset-pricing model, for

instance a CAPM model, to be underestimated. However, the use of weekly trading data

minimizes the different biases that come from infrequent trading (Lo and McKinley, 1988).

After all, their study concludes that infrequent trading is not the whole reason why they are

able to reject the random walk hypothesis. The literature provides sufficient arguments against

the criticism of biases due to the infrequent trading of shares.

Serial correlation

Because serial correlation is a valid explanation for excess volatility, several researchers tried

to clarify why serial correlation exists and why arbitrage is not a realistic solution. If under

and overreactions occur, this implies that returns are not independent, which is not in line

with the first assumption of the random walk. We will further discuss the second assumption

that goes about the rationality of investors and the effectiveness of arbitrage. Depending on

the level of rationality by investors, we can distinct two kinds of stock market traders. One

type of stock market traders are irrational investors, which are also called noise traders. The

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other type of investors, are rational stock market traders. Barberis and Thaler (2003) disprove

the assumption of the EMH, stating that any mispricing will quickly disappear because of

riskless arbitrage opportunities. Irrational investors who drive prices away of its fundamental

value will attract rational investors. Therefore, rational investors will bring the price back to

its fundamental value, eventually leading to an equilibrium. However, Barberis and Thales

(2003) argue that arbitrage opportunities can be risky and costly. Consequently, this

investment strategy becomes unattractive which leads to prices that are not in accordance with

all publicly available information. They sum up several reasons why arbitrage can be

unattractive. First, investing in stock market produces costs such as bid-ask spreads,

commissions and fees for shorting stocks and options. Second, irrational markets are not

always able to move the price to its fundamental value which makes it risky for an

arbitrageur. The infamous risk is noise trader risk which is the risk that irrational traders pose,

on the short term, as the capability of pushing the price away from the fundamental value.

Hischleifer (2001) also argues that arbitrage by irrational investors can arbitrage efficient

prices away. An arbitrageur bears not only noise trader risk, but also fundamental risk. For

instance, an arbitrageur accepts fundamental risk when he shorts an overpriced stock, of

which the dividend news caused an overreaction. What if the news about the future dividend

suddenly seems better than expected? Consequently, the fundamental value increases and the

arbitrageur’s short position will become a losing position. Third, investors who apply the

wrong models to determine the fundamental value (i.e. model risk), bear the risk of

arbitraging a stock which price equals the fundamental value. Last, research like Shiller

(2003), Long et al. (1990) and Barberis & Thaler (2003) indicate the existence of positive

feedback. This phenomenon happens when investors buy when prices rise and sell when

prices go down. As a result, when irrational traders purchase shares and prices go up, rational

traders follow suit. Eventually, prices quickly increase and an overreaction happens.

Admittedly, stop loss orders can influence stocks which are sold after price declines (Long et

al., 1990). The fact that rational traders act irrational in order to benefit from the momentum

and that irrational traders do not leave the market after a price correction is explained by Long

et al. (1990). Long et al. (1990) provide three reasons why noise traders do not leave the

market after participating in it. First, the capital market’s circumstances change over time, and

so learning from past mistakes is rather limited. Second, when positive feedback traders leave

the market, they can reconsider to turn back later. Last, if noise traders hold riskier positions

than rational investors, higher returns can be earned. Shiller (2003) found that these

anomalies, like panics and crashes, disappear over time but that overreactions and

underreactions are common phenomena that lead to excess volatility. Cutler, Poterba and

Summer (1991) plead for a model where rational and irrational investors interact, thus,

forming market equilibria. Equilibria which are formed by the demand of rational investors,

who base their decisions on expected returns, and irrational investors (i.e. feedback traders),

who base their decisions on realized returns. Eventually, excess volatility can take place when

behavioural effects arise. In the next paragraph, excess volatility will be further discussed and

linked with empirical results.

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Empirical studies

Before we discuss the behavioural characteristics of stock market traders and their effects,

which cause serial correlation and excess volatility, we investigate the literature thoroughly on

excess volatility in stock markets globally. First, we begin with the paper of Cuthbertson and

Hyde (2002). They investigated the efficiency of German and French stock markets. The

Campbell-Shiller VAR methodology is used and applied on monthly data from January 1973

till June 1996 (Cuthbertson and Hyde, 2002). Results conclude that excess volatility exists in

both stock markets, but only when they assume that excess volatility remains constant.

Nevertheless, if the model assumes that the risk premium changes over time then no evidence

of excess volatility can be found. Cutler, Poterba and Summer (1991) explain that the existing

models in the field of finance are not capable to justify changes in risk premia. They also

assert that autocorrelations of the stock returns cannot be produced by changes in the risk

factors. Since 1926, excess returns are found in the US stock market by the study of Mehra

and Prescott (1985; in Cutler, Poterba and Summer (1991)) and they state that the found

deviations are not consistent with the empirical risk of the stock returns.

DeLong and Becht (1992) analyzed the German market from 1876 till 1990 and come to

different conclusions. Before the First World War, the market has no excessive volatility

which supports the EMH. In contrast to the previous period, later periods are marked with

excess volatility. DeLong and Becht (1992) owe this to the structure of the capital markets.

Before the First World War, the capital market was monopolized by six major banks. This

institutional structure made it able for these banks to have almost perfect information since

they were stockholders, investment banks and receivers of deposits at the same time.

Furthermore, these banks had a long-term investment horizon which made the capital market

very stable. On the contrary, the capital markets became well-developed after 1914. As a

result, more investors could trade on the stock market which became less informed and in

which these investors were not sufficiently able to price the stock to its correct fundamental

value. Now, we will discuss the effects caused by behavioural characteristics.

Positive-feedback effects Lakonishok et al. (1992) researched institutional traders in order to evaluate positive-feedback

effects and herding. Their study made use of data coming from the SEI, which is an

investments company with more than five hundred funds. Results showed that, for small

stocks, positive-feedback effects could be found but herding did almost not occur. While for

large stocks, there was weak evidence of herding and positive-feedback effects. As a result,

larger stocks seem to be less inefficiently priced than smaller stocks. Like Barberis and Thaler

(2003) argue, some groups of shares have better average returns than others, in the literature

also known as the cross-section of average returns. These differences are not justified by the

CAPM, therefore, seen as anomalies in capital markets. They find a higher average return for

the small stock decile than for the big stock decile. Despite the use of the same data like Fama

and French (1992; in Barberis and Thaler, 2003), they conclude that the higher return is more

than a compensation for the additional risk.

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Excessive volatility cannot be explained by the efficient markets hypothesis. In other words, if

markets are efficient, how can bubbles arise? Kindleberger and Aliber (2011) studied different

bubbles throughout the history and found that during times of manias (i.e. when prices deviate

significantly from their fundamental value) a general irrationality is present. Waves of

excessive optimism lead to bubbles which eventually implode by excessive pessimism. Long

et al. (1990) justify this process of up- and downswings, which may lead to speculation, with

evidence of positive return correlations at short time periods till the price is restored to the

fundamental value. Kindleberger and Aliber (2011) conclude that stock prices are influenced

by insiders who follow the trend and by outsiders who act irrationally. The investment

strategy where high-priced securities are sold and low-priced securities are bought is normally

executed by the insiders. However, the outsiders adopt a reversed investment strategy, which

leads to destabilizing speculation. If all future dividends are known, markets would be

rational. This will also imply, in theory, that all investors have sufficient knowledge of the

discount rate (Shiller, 1987). In reality, future dividends are uncertain and are the cause for

market irrationality among rational and irrational investors.

The part of excessive volatility was an unexplained mystery in the beginning of behavioural

finance. Nowadays, many researchers succeeded in finding explanations for this excess

volatility but it still remains a complex process. Many studies appear to find different effects,

which are very divergent, forming a base to explain anomalies. After a brief description of

behavioural critiques on conventional financial theory, we will take a closer look at one of the

most important biases in the literature that explain excess volatility and other anomalies in the

stock markets.

When individuals participate in stock markets, they must make difficult and complex choices

that cause irrationalities (Bodie et al., 2013, p. 266); these irrationalities can be distinct and

divided into two major categories. The first category contains anomalies caused by investors

which are not always able to interpret information correctly, thus, estimate faulty probability

distributions of future returns. The second category contains investors which are aware of the

probability distributions, but still make suboptimal decisions. Another point of criticism,

besides the complex choices made by investors, is the limitation of the realization of arbitrage

opportunities (Bodie et al., 2013, p. 266). We start with the discussion of biases due to

incorrect information processing and we will end with biases due to investors’ behavioural

characteristics. To sum up, we start with the representativeness bias, overconfidence, stock

splits, conservatism and end with regret avoidance, mental accounting and prospect theory.

Representativeness bias Investors who make prognoses do not know the true probabilities of their forecasted events,

which is why they base their forecasts on experience and recent events. These past events are

largely representative but do not completely represent the properties of the population

(Tversky and Kahneman, 1973). Consequently, investors are subject to the representativeness

bias. Tversky and Kahneman (1972, 1973; in Bodie et al., 2013, p. 267) indicate that forecasts

can be too extreme, thus, earning expectations will be too high which would eventually cause

a stronger surge in share prices than normally (i.e. overreaction). De Bondt and Thaler (1990;

in Bodie et al., 2013, p. 267) researched the link between P/E effects and the

16

representativeness bias and found that companies with current exceptional high earnings tend

to have a high P/E due to an optimistic trend in the share price. Eventually, investors will

recognize their error and the price will fall back which makes firms with a high P/E into bad

investments.

Biased self-attribution Bodie et al. (2013, p. 267) states that people tend to be overconfident, also dubbed as biased

self-attribution, regarding their own abilities and forecasts. When they look at the domination

of active management, this dominance is in line with the tendency to overvalue competences.

In this respect, Acker and Duck (2008) set up an experiment consisting of Asian and British

students who had to play a stock-market game. Tests were set up to verify if investors were

overconfident and/or were subject to biased self-attribution. Important conclusions were made

and Acker and Duck (2008) proved that investors are not always acting rational, like the EHM

assumes, and that they were influenced by characteristics as stated above. They found a

stable level of overconfidence for most participants. Only a small part had a varying level of

confidence throughout the study. Additionally, Asian students were more overconfident than

the British students. Consequently, Acker and Duck (2008) assume that this discrepancy is

due to over-optimism or an unawareness of downside risk and as a result, make Asian capital

markets more volatile. Besides, results showed differences between gender for the British

sample, but not for the Asian sample. Bodie et al. (2013, p. 267) also mention the conclusion

of Acker and Duck that men are more overconfident than women.

Stock split effect A highly priced stock can be divided into several new stocks which will make the stock less

expensive. This is what Desai and Jain (1997) call a stock split, the opposite of a split is called

a reverse split. In fact, a stock split is an efficient instrument to stimulate the trading of the

share. Desai and Jain (1997) conclude out of their study, based on data from NYSE, AMEX

and NASDAQ firms, that a negative stock split has a negative effect on the stock and vice

versa. Furthermore, the stock split effect is not directly picked up by the market. In other

words, the market underrates the consequences of a split. Even though Fama (1969) (in Desai

and Jain, 1997) concluded abnormal returns after the announcement of stock splits to be equal

to zero, Desai and Jain (1997) find positive stock splits that generate long-term positive

abnormal returns, which can be analyzed periods after the announcement. In addition, a

simultaneous announcement of a split and a dividend increase makes the positive drift even

stronger. They also stress that this excess return does not originate from risk changes.

Conservatism Ritter (2003) made a study about conservatism in financial markets. With conservatism,

changes in the environment are picked up very slowly by investors (Ritter, 2003). He argues

that this type of bias is the source of an underreaction on a new event and when there is a long

pattern of the same events, the market will overreact. These delayed reactions could be an

explanation for autocorrelations, which are found in empirical papers like Barberis et al.

(1998) and Lo and McKinley (1988). Furthermore, underreactions to new information could

explain excess volatility. A conservatism bias generates momentum in stock markets (Bodie

et al., 2013).

17

Momentum effect Jegadeesh and Titman (2001) explain the momentum effect, which means that investors buy

shares with a high return in the previous period and sell shares that had a low return in that

previous period. They state that momentum profits are due to delayed overreactions in the

capital markets and anticipating on momentum effects yield extra profits. Their analysis of the

US stock market shows that shares with good returns during the previous three to twelve

months tend to maintain this pattern and vice versa. It is noteworthy that Jegadeesh and

Titman (2001) find that, over a period of eight years, their momentum strategy continues to be

beneficial. Although, results show that return reversals do not exist before the fourth year

after the formation date. The excess return (i.e. the alphas) was derived from the CAPM and

the Fama and French three-factor model where the alpha of the Fama and French Three-factor

model is significantly larger than the alpha of the CAPM. Additionally, losers are more

sensitive to the Three-factor model because losing stocks have a smaller market capitalization,

thereby, are more sensitive to the size-risk factor of the Three-factor model.

Disposition effect Shefrin and Statman (1985) studied why investors sell winning stocks too soon and why they

maintain holding losing stocks. The research of Grinblatt and Han (2005) give reasons that

cause the disposition effect like mental accounting, regret aversion and the prospect theory.

All elements which are not considered to be part of the decision process of a rational investor.

Concerning the disposition effect, they find evidence of risk seeking when investors bear

losses and risk averting when investors gain profits. On the one hand, investors pursue pride

which is achieved when an investment has a positive trade-off. On the other hand, investors

try to avoid the regret of selling a share when its price increases after selling the stock. Shefrin

and Statman (1985) general conclusion is that a disposition effect is not only present in

psychological experiments but also in financial markets. The prospect theory represents an S-

shaped value function where investors with gains are risk averse and where investors with

losses are risk seeking. Generally, when profits are made the utility function becomes concave

and when losses are made the function becomes convex (Grinblatt and Han, 2005). The

starting point of the investor’s utility does not depend on the level of wealth but depends on

gains and losses (Bodie et al., 2013). Mental accounting is the idea that people segregate their

investments decisions or their gambles into different accounts (Bodie et al., 2013; Grinblatt

and Han, 2005). Bodie et al. (2013) clarifies mental accounting with an example, that is, an

investor with two different investment accounts. He has one account for his pension and one

account for other purposes. During his investment process, he will tend to be more risk averse

when it concerns his pension, whereas he will be less risk averse for the other account.

Rationally, an investor has to pretend these two accounts as unity. Bodie et al. (2013) find

evidence of mental accounting in the paper of Statman (1997; in Bodie et al. 2013) where

consistency is found with irrational preferences for high dividend shares. Investors prefer

stocks with high dividends since investors try to avoid dipping into capital, even when both

stocks have the same rate of return.

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4. The adaptive markets hypothesis (AMH)

The adaptive market hypothesis is introduced by Lo (2004) who attempts to create a new

theoretical framework in which elements of the efficient markets hypothesis and of

behavioural finance are combined. Lo (2012) sees his AMH as a reworked efficient market

hypothesis and does not label the EMH as incorrect but rather incomplete instead. He further

argues that in normal financial times with economical stability the EMH seems valid but as

soon as the financial and economical conditions change, behavioural finance gets increasingly

popular to explain anomalies. Lo (2004) builds his framework on an evolutionary point of

view and socio-biology with inherent principles such as natural selection, trial-and-error, etc.

that help explain human interactions and emotions. That way, the framework can combine the

EMH with the refuting arguments of behavioural finance. Essential in this framework is the

‘bounded rationality’ of investors as described by Simon (1955, in Lo, 2004). This means that

instead of economic agents being fully rational investors, which is presumed by the EMH and

criticized by behavioural finance, investors actually settle for ending up with a satisfying

result based on their personal possibilities (i.e. intellect and wealth) instead of the most

optimal investment decision. Lo (2004) elaborates on this ‘bounded rationality’ concept and

states that economic agents invest in a way characterized by trial-and-error and natural

selection traits instead of theoretically sound investment models for which many investors do

not possess the knowledge and sometimes costly input information. So these agents will keep

on investing, using their own developed methods that have been successful and satisfying for

them in the past. However when market conditions change over time, the developed methods

of investors can become invalid, resulting in negative return outcomes. These negative

outcomes will bring ‘behavioural biases’, as Lo (2004) names them. These behavioural biases

refer to actions of investors that are unsuited to the situation and leaves investors with a bad

end result. Lo (2004) does not speak of irrational behaviour but of ‘maladaptive’ behaviour

instead, suited with the AMH viewing point and name. As the AMH allows for behavioural

biases, there are many examples such as the ‘fight-or-flight’ response during the financial

crisis which is better known under the ‘flight-to-safety’ denomination when many investors

left the stock markets and invested in “safer” bonds (Lo, 2012). Over time, natural selection

filters the markets of unsuccessful investors and so it brings survival of the fittest or, as Lo

(2004) calls it, ‘the survival of the richest’.

So far on how the AMH originated, the hypothesis also brings two important implications that

can be tested empirically. These were also mentioned by Kim et al. (2011), whom we

discussed earlier in this review in the empirical evidence section of the EMH. The first

implication is that the market efficiency extent can differ over time as a result of adapting

behaviour of investors to changing market conditions. These market conditions form the

subject of the second implication as they influence market efficiency (Kim et al., 2011; Lo,

2004, 2012). Lo (2004) examines the first implication by calculating first-order

autocorrelations from the S&P Composite index from 1871 to April 2003, basically testing

the random walk. When plotting these coefficients, Lo (2004) finds that the extent of market

efficiency shows a cyclical walk instead of a random walk, confirming varying efficiency

over time. Although this does not mean that market efficiency can be rejected, due to the

19

identical distribution property of the random walk, it can also confirm the adaptive markets

theory. And as far as market conditions go, we already mentioned that Kim et al. (2011)

proved their significance with their regression model.

It is not a certainty that this model is completely justified, as Lo (2004) acknowledges the

lacking of empirical proof and states that with the passing of time we will know for certain.

Nevertheless, it gives an appealing take on the debate between respectively EMH and

behavioural finance proponents. Finally, the adoptive markets hypothesis has very important

implications for investors. In an efficient market a passive index fund is basically the optimal

choice and arbitrage opportunities should not occur but this changes under the AMH. Lo

(2004) explains that when accepting the AMH, different time-dependent investment strategies

are called for due to amongst others: changing risk-return trade- offs, the existence of

occasional arbitrage opportunities due to more assumed market complexity under the AMH

and due to changing market conditions. This definitely implies that there is an important role

for active management to fulfil.

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21

III. Research Design & Methodology

1. Design

In this section we will pay attention to the main question we want to answer, which data we

need in order to do so and also which tests we will carry out. As our thesis title indicates, we

want to find out if the Belgian market is efficient. Efficient refers to market efficiency as

intended by Fama (1970) and as is explained in the Literature Review as a price series that

follows a random walk. More specifically, we will research and focus on weak-form tests just

like papers that are discussed in the EMH-section of the Literature Review. Furthermore, the

denomination ‘Belgian market’ is very broad, so we will specify what we exactly mean to

avoid confusion. The market as we intend can be viewed as the stock market with all shares

noted on the Brussels Stock Exchange (BAS) but more interesting is to divide the market in

compartments based on size with a corresponding index for which enough data is available. In

that way we will be able to not only make an efficiency statement concerning the whole

market but also for certain parts of the total market, each with different characteristics. For

example size, the larger shares included in the BEL-20 are being traded a lot more than those

included in the BEL Small which could lead us to believe that the BEL-20 index containing

large caps will/should be more efficient compared to the BEL Small index. Perhaps this could

also imply that a more efficient compartment, e.g. of more frequent traded large caps (with

more influence on the total market) such as the BEL-20 could bias the overall efficiency of

the entire market index due to the value-weighted proportion of the BEL-20 in the BAS. By

examining indices of different compartments we account for this potential bias. Another

characteristic could be the distinction between growth and value stocks. Ex ante, we could

expect value stocks to be more stable as they mostly are from well-established firms and thus

could logically be believed to be more efficient. Our research will show if this is actually the

case.

Finally our use of indices and the decision to only examine indices also has a reason. Why not

examine only BEL-20 shares on an individual basis instead and simply generalize our

findings to fit the whole market as the most important shares are represented? Our answer is

simple, when examining individual stocks we should also account for firm-specific

information which is not always available and clear and could taint our research and limit the

potential to generalize our results. In our opinion, this alternative design would better suit

researchers conducting semi-strong form tests and strong form tests without the intent to

generalize their findings for the stock market as a whole.

We retrieved the data suited to our research requirements from Datastream & MSCI. We

collected the following indices, which are originally all price time series from Datastream:

BEL All Shares (BAS), BEL-20, BEL Mid, BEL Small. The BEL Mid and the BEL Small

both contain the 36 most representative companies, based on mid and small market

capitalization which display the state of the Belgian economy. Euronext requires a minimum

market capitalization of the level of the BEL-20 multiplied with €55000 for the BEL Mid

shares and multiplied with €5500 for the BEL Small shares (Euronext, 2014). From MSCI we

22

have the total Belgian index as well as large, mid and small stocks and besides the Belgian,

large, mid and small growth and value stocks indices. Moreover we have both original price

and return MSCI indices whereas the indices of Datastream are price series that can be

converted into return indices by calculating logarithmic differences. We went as far back in

time as the availability of the data permitted us. For each index, we can analyze three

frequencies of time: daily, weekly and quarterly data. Nevertheless, as our quarterly dataset

does not contain a lot of data entries, we could question the results this data brings forth and

therefore we will not always pay as much attention to quarterly data unless it raises interesting

outcomes. Our Datastream dataset starts from 19 February 1996 and ends on 31 January 2014,

i.e. from the furthest starting point back in time for which we have data on all chosen indices.

The dates from MSCI all vary and will be mentioned by the tests that were run on the data.

Lastly, each index is continuously traded and value-weighted (Euronext, 2014).

In order to analyze the market efficiency on the Belgian stock market, we make use of two

time dimensions. The first one is a comparison of time series with different frequencies, for

instance, daily data compared with monthly data. There are several reasons why we do not

only investigate daily but also weekly and monthly data. First, daily data is extensively used

in the literature and therefore it is better to enrich the existing literature with weekly data.

Second, daily data is subject to biases which are not common with weekly and monthly data.

For example, Lo and McKinley (1988) found that daily data was not appropriate due to biases

like infrequent trading and bid-ask spreads. On the one hand, daily data provides large

datasets which are more accurate. On the other hand, the dataset for daily trading data will be

biased by infrequent trading. The infrequent trading problem is closely related to the size of

the shares as we alluded on already. By taking indices of several capitalizations and time

frequencies into account we try to avoid these potential biases. Weekly data, which could be

the proper answer on these problems, will thus minimize the influence of biases caused by

limited trading.

The second time dimension comes forth out of a comparison between several sub periods.

Analyzing several sub periods gives us the possibility to examine the changes in market

efficiency over time. We decided to make an analysis of the three years before the outburst of

the financial crisis and an analysis of the three years after the beginning of the financial crisis.

Specifically, we will examine two subperiods of which one is labeled pre crisis and the other

is post crisis. To define pre and post crisis we use a key date which is the 15th

of September

2008, the day Lehman Brothers fell and which is seen as the day the financial crisis began.

Since this day displays abnormal returns, we will not include it into one of the subperiods. Pre

crisis will start three years earlier and will end the day before the 15th

September 2008 and our

period post crisis starts the day after the 15th

and will range for the following three years.

23

To summarize, we want to answer the following questions:

Is the Belgian stock market efficient as a whole? And what about the different size-

based compartments of the stock market?

Do we observe a difference in market efficiency between growth and value stocks?

Did the financial crisis marked by the 15th

of September 2008 as the starting date have

an effect on market efficiency?

We will carry out the tests, described in the methodology section on the next page, on the

different indices to answer these questions. That way in our conclusion, we will also be able

to compare results of different testing methods. But first we will run summary statistics on the

indices to help us make some expectations about our research results.

24

2. Summary statistics

Index Mean Standard

Deviation

Skewness Excess

Kurtosis

Minimum Maximum # Obs.

Daily Returns (Datastream, 19/02/1996 – 31/01/2014) – (MSCI, 14/02/1996 – 29/01/2014)

BAS 0,000182 0,011 -0,088 7,004 -0,087 0,103 4685

BEL-20 0,000121 0,013 0,013 5,838 -0,083 0,093 4685

BEL Mid 0,000293 0,011 0,506 28,201 -0,094 0,187 4684

BEL Small 0,000487 0,009 0,716 54,040 -0,118 0,142 4685

BAS Growth 0,000246 0,013 -0,235 4,957 -0,096 0,076 4685

BAS Value -0,000004 0,016 -0,680 16,216 -0,224 0,140 4685

Weekly Returns (Datastream, 16/02/1996 – 31/01/2014) – (MSCI, 14/02/1996 – 29/01/2014)

BAS 0,000888 0,026 -1,600 14,825 -0,280 0,111 938

BEL-20 0,000581 0,029 -1,202 8,972 -0,261 0,129 938

BEL Mid 0,001455 0,028 -1,216 15,982 -0,255 0,192 937

BEL Small 0,002413 0,020 -0,912 13,073 -0,186 0,131 938

BAS Growth 0,001232 0,029 -0,555 3,847 -0,164 0,123 938

BAS Value -0,000019 0,037 -0,724 9,158 -0,290 0,250 938

Monthly Returns (Datastream & MSCI, 1996:2 – 2014:1)

BAS 0,004048 0,053 -1,926 10,227 -0,369 0,114 216

BEL-20 0,002685 0,057 -2,131 12,004 -0,415 0,130 216

BEL Mid 0,006417 0,063 -0,822 5,597 -0,316 0,274 215

BEL Small 0,010653 0,048 -0,496 2,266 -0,213 0,153 216

BAS Growth 0,005525 0,055 -1,374 3,957 -0,253 0,145 217

BAS Value 0,000233 0,069 -1,715 6,470 -0,391 0,193 217

Quarterly Returns (Datastream, 1996: Q1 – 2013: Q4)

BAS 0,013439 0,100 -1,223 2,245 -0,372 0,183 72

BEL-20 0,009342 0,106 -1,517 3,404 -0,434 0,157 72

BEL Mid 0,018501 0,118 -0,960 3,867 -0,484 0,340 71

BEL Small 0,031685 0,089 -0,997 1,741 -0,310 0,184 72

Table 1 contains the summary statistics of the returns of all indices that we acquired for

different time frequencies. These returns are based on logarithmic prices. The total time

frame data are mentioned next to the frequency. Data was collected from two sources:

Datastream data contains: BAS, BEL-20, BEL Mid & BEL Small. And MSCI data is used for

the BAS growth & BAS value indices. Quarterly data is only available for the four

Datastream indices.

The All Shares index (BAS) has a positive mean return and standard deviation for all time

frequencies. The increase of the standard deviation could imply a random walk, for instance,

if we take a square root of five and multiply this with the standard deviation of daily returns

than this must be equal to the standard deviation of weekly returns in order to be market

efficient. However, conclusions will be made after a further analysis of our data. In each

frequency we detect a left skewness but for daily returns this can almost be neglected.

Kurtosis values make it clear that normal distributions are out of the picture although the data

with quarterly frequency is not far from the three-value of normal distribution. The longer the

frequency of the data, the smaller minimum and the larger maximum values tend to get.

25

The BEL-20 index also shows positive means and standard deviations for daily, weekly and

quarterly time frequencies. But just as with the BAS indices the increases of the standard

deviation may proof that the BEL-20 follows a random walk. Nevertheless, solid tests will

have to show whether this is also the case. For daily data the distributions are very slightly

skewed to the right as the others are skewed to the left. Kurtosis levels seem to increase over

daily and weekly data, whereas the level for quarterly data almost resembles that of a normal

distribution. When increasing the frequency, the minima get more extreme. Maximum values

do not display such remarkable increases when lengthening the frequency.

The summary statistics of the BEL Mid returns show positive means and standard deviations

for all frequencies. We observe increasing standard deviations when the time frequency is

longer. This increase corresponds with the assumption of a random walk in which variances

increase when frequencies become longer. However, we cannot make this conclusion out of

this table. For every time frequency, we find a right skewness except for daily data that is left

skewed. Weekly and quarterly data have lower kurtosis values. With a value of 3,87 , the

kurtosis of quarterly data even comes close to the value of three, which is found for normal

distributions. The fact that daily data is left-skewed is in line with the knowledge of negative

returns on the short term. On the other hand, weekly and quarterly data show positive returns

because these are long term. In line with our findings, the minima and maxima grow linearly.

Summary statistics of the BEL Small return index presents increasing means and standard

deviations; this logically implies increasing variances when the time frequencies become

larger as well. Similar to the BEL Mid returns, we find left skewness for daily data and right

skewness for other data. None of the kurtosis values come close to the value of three, which is

found for normal distributions. Furthermore, the kurtosis falls when the time frequency

becomes longer. We also find linear patterns in the minima and maxima of the returns. We are

faced with the same problem of few data points for quarterly data, as is also the case with the

other return indices.

26

3. Methodology

We will carry out weak-form tests to answer our questions. Below we will discuss the various

hypotheses that can be tested and move on to various testing possibilities to confirm or reject

the hypotheses, in order to establish their relevance in the context of market efficiency and to

explain how they work exactly.

The Random Walk hypotheses

First of all, we will classify the random walk hypotheses in three different kinds of models

with their own properties, in accordance to the overview given by Campbell et al. (1997).

Each model has at least one appropriate test to examine if the hypothesis is to be accepted or

rejected. We will then explain which model and corresponding tests we will follow to answer

our questions.

Campbell et al. (1997) distinguish three random walk models based on the (non)existing

dependence between returns of an asset on subsequent moments, t and t+k. Campbell et al.

(1997) use the following condition which they interpret as an ‘orthogonality’ condition, i.e.

without correlation:

Applies to all t’s and k≠ 0

If both functions f and g are not linear, thus unrestricted, we obtain the random walk 1 and 2

models. This means that both return functions are independent as well as their increments. If

both functions f and g are linear on the other hand, we have the third random walk model in

which the returns and increments are uncorrelated. Lastly one other case can be distinguished:

the martingale model, obtained when function f is unrestricted but in which function g is

linear.

This historical martingale (or fair game) model stems from the beginning of probability theory

and is in essence based on stochastic variables (Campbell et al., 1997). Stochastic variables

generate random values which are solely based on chance (McClave et al., 2007). In context

of asset pricing, this implies that today’s price will be the same as the one of yesterday when

only the historical prices are considered. When forecasting tomorrow’s price, the current price

is the best estimation but there is just as much chance for it to rise or to drop. This statement

already indicates that forecasting on base of the martingale model seems pointless. The

martingale model implies price changes (i.e. returns) to be uncorrelated no matter the lag size,

resulting in pointless linear forecasting models that use only historical return data indeed

(Campbell et al., 1997). On top of that, Campbell et al. (1997) also remark the absence of

compensation for the risk-return trade-off in the martingale model: the randomness and

unpredictability of a price change could be the needed compensation by investors for the risk

they take investing in the particular asset. This shows that the martingale is not all there is to

efficient markets and rational asset pricing but it sure is the foundation for the Random Walk

hypothesis.

27

Campbell et al. (1997) define the random walk 1 (RW1) as the model with independently and

identically distributed (IID) increments and Pt they define as follows:

with alpha as a drift (i.e. the expected price change) and the error term (i.e. increments)

independently and identical distributed. Campbell et al. (1997) note that because of the IID

condition of the increments, the random walk 1 is more solid than the martingale. Because of

the independence there is no correlation in the increments possible as well as in other

functions of the increments. Intuitively, Campbell et al. (1997) explain this formula by

looking at the expected Pt as the original Pt-1 added by the expected price change alpha over

time between t and t-1. Assuming a linear process of the means and variances over time in the

definition of the random walk 1 Pt series as defined here above, the random walk in the price

series will be nonstationary. This also applies to the random walk 2 and 3, although their

conditions to qualify as a random walk are less strong. Lastly, Campbell et al. (1997)

characterize the most common random walk 1 model as the one in which the increments

follow a normal distribution with mean 0 and after using natural logarithms of Pt in the

definition of the RW1 with IID here above. Normal distribution is in accordance to the

conditions that were just mentioned and it makes empirical processing more convenient.

Campbell et al. (1997) classify the random walk 2 (RW2) to deal with the problem of

identically distributed increments because this condition is not realistic for financial assets

over longer periods of time. Due to changes in markets and society of economical,

technological, environmental, legal and other natures, there also have been changes to stock

pricing. Though identical distributions are no longer necessary to get a random walk, the

increments still need to be independent. Campbell et al. (1997) speak of “independent but not

identically distributed (INID) increments” (p.33). This way the random walk 2 can also

examine a random walk in return series of which the increments are heteroskedastic.

Campbell et al. (1997) stress the importance of this characteristic as there is a lot of variation

over time in return series of financial assets and this variation is the main cause of

heteroskedasticity. Lastly, the random walk 2 remains valid although it does not account for

identical distributions as Campbell et al. (1997) state that it is still not possible to predict

future returns using whatever random past returns.

Finally there is also the random walk 3 model (RW3) which makes the independence

conditions less strong. RW3 is the same as the random walk 2 but the increments are allowed

to be dependent as long as they are uncorrelated. Campbell et al. (1997) move on illustrating

their third model with a case of return series of which the increments are uncorrelated but the

squared increments are not. Hereby, the correlation of the squared increments proves the

dependence of the increments.

Testing the Random Walk hypotheses

There are a lot of different tests to examine the three random walk models. Because of their

vastness we will not attempt to sum all of them up but just discuss the ones or their principles

needed to conduct our research in a solid way. Many tests examine the random walk 1 and 2

28

models which are great from a theoretical point of view but these hypotheses do not exactly

correspond fully with reality. However, these qualities can still be found in more newly

developed tests and thus show their indispensability (Campbell et al., 1997).

To test the random walk 1 model a lot of nonparametric tests can be used as is made clear by

Campbell et al. (1997), such as the Spearman rank correlation test, the Kendall τ correlation

test to mainly check correlations. Also sequences and reversals or runs can be tested to

determine a RW1. A run is a sequence of ongoing positive (or negative) returns in a data

series. The idea is that there is a positive run following a negative run and vice versa. The

number of runs then can be counted and compared in total of the examined dataset to test the

random walk. When examining random walks, we need to account for a possible drift (drift

0) that cannot be confused with a higher degree of predictability based on past prices. Also,

Campbell et al. (p. 35, 1997) state (based on the work of Cowles and Jones (1937)) that

whenever there is a drift, a trend is set in motion which results in a higher chance of

sequences than of reversals. However, in assumption of a martingale model this statement is

not valid. Campbell et al. (1997) remark that Cowles and Jones later on softened their

research statements as they are mostly theoretical and no attention was paid to transaction

costs.

To test the RW2, Campbell et al. (1997) notice the difficulty as the identical distribution is not

assumed under this hypothesis. And without assuming some kind of distribution it is very

hard to run statistical tests. Therefore two other kinds of tests were developed that are

perceived to have a lesser scientific value: filter rules and technical analysis. With filter rules

a filter is put on an asset when it is being bought because of a certain percentage increase in

price and the other way around when it is sold or shorted because of the fall of the price by

the exact same percentage as of the increase. Such filters are applied to portfolios and

compared to the portfolio following a buy-and-hold strategy. However, Campbell et al. (p. 42,

1997) refer to research of Fama and Blume (1966) that shows that the buy-and-hold strategy

pays off more than when applying filter rules. And even when filter rules seem more

profitable (when using small value filters), the result gets neutralized by the unavoidable

increase in transaction costs. This result is undoubtedly a good argument in favour of efficient

markets. Next to applying filter rules, there is also technical analysis based on charting to look

for potential patterns and trends in the graphs of past prices of an asset. Charting is usually

perceived as less scientifically. Therefore we will not put these techniques in scope of our

paper. However it may be interesting to mention that Campbell et al. (1997) notice that these

techniques may have significance to partly forecast future price movements in the very near

future.

The RW3 is based on uncorrelated increments and this is best tested by examining if there is a

correlation between two data entries of returns at a different time. As the condition under the

RW3 is that all lags of the increments or first differences need to be uncorrelated, Campbell et

al. (1997) speak of serial correlation. Campbell et al. (1997) define the null hypothesis as

follows: “the increments or first-differences of the level of the random walk are uncorrelated

at all leads and lags. Therefore, under the null hypothesis the autocorrelation coefficients of

29

the first-differences at various lags are all zero.” (p. 44). This means that the autocorrelation

function has to be stationary in order for the coefficients all to be equal to zero. All tests

examining the random walk 3 that use autocorrelations consider this null hypothesis as a

starting point. Here there are also a lot of testing varieties, some tests use simply

autocorrelation coefficients, others use the sum of squared autocorrelations or in some tests

linearity is examined between autocorrelations to seek out deviations from the random walk

(Campbell et al., 1997)…

We will examine the random walk using techniques as explained by Koop (2006) and

Inghelbrecht (2013b), more specifically concerning general methods of regressions,

autocorrelation functions and autoregressive models. The model that leans closest to reality

and is being researched the most over the past years is the RW3, which we will follow as

well. That means that we will also take the implications of the null hypothesis defined by

Campbell et al. (1997) as our main guidance. We will take following steps in our research

method keeping the rules of Koop (2006) and Inghelbrecht (2013b) in mind: first we will test

if the return series (transformed by taking log differences of the price series) are stationary,

which they should be, to avoid problems of spurious regression when running regressions

with nonstationary time series. To test if the series is stationary we will carry out a Dickey-

Fuller test in the Gretl-software package which automatically supplies a t-statistic and p-value

to compare to the critical value of -3,45 if the model contains a deterministic trend or to -2,89

if the model does not have such deterministic trend. The null hypothesis, if the computed t-

statistic is bigger than the critical value (less negative), implies a unit root and thus

nonstationarity. The alternative hypothesis with a more negative t-statistic than the critical

value confirms stationarity. However, with stationary time series there can also be the

problem of residual autocorrelation which is not allowed and certainly not in a random walk.

Residual autocorrelation can be tested by comparing the absolute value of the Durbin’s h test

with the critical value of 1,96. The null hypothesis indicates no residual autocorrelation when

the absolute h-value is ≤ 1,96. The alternative hypothesis when |h| > 1,96, logically implies

residual autocorrelation. Residual autocorrelation makes coefficients inefficient, standard

errors potentially inappropriate and inflates the explanatory power of the regression model,

R². Naturally we will need to test our model for this phenomenon. Secondly, and before

handling residual autocorrelation, we will first run an autocorrelation function to give us a

first indication about the random walk. These values should all be equal to zero or at least

float very closely around zero. If deviations from zero occur, the negatives can counter the

positive deviations which should not be a reason to reject the random walk. Thirdly, we

estimate an autoregressive model containing sufficient (significant) lags, which will be tested

sequentially to find the optimal amount of lags. Depending the result of this autoregression

we can state if the autocorrelation coefficients are in fact equal to zero and thus imply a

random walk and with that market efficiency as well (Koop, 2006; Inghelbrecht, 2013b).

To conclude this part, we should take a moment to contemplate about unit root tests as

Campbell et al. (1997) also do. Although unit root tests (e.g. the well-known Dickey-Fuller

test) are often used to test the random walk, this method is not completely sound. Unit root

testing is not exactly meant to examine the predictability of returns, which is surely the case in

30

random walk testing. Instead Campbell et al. (1997) explain that the unit root tests examine

whether the dependent variable is difference-stationary (H0), in which case a shock to the

dependent variable is permanent in its influence on future values of the dependent variable

and thus when a stochastic trend is present (e.g. a change in Rt keeps having a certain effect

on all future Rt+…). Alternatively, if the dependent variable is trend-stationary (H1), a shock to

the dependent variable is deemed temporary as for example, the effect of Rt on future Rt’s

gets smaller when time passes by. In that way, Campbell et al. (1997) make it clear that unit

root tests examine shocks to dependent variables (sustainable or temporary) rather than trying

to forecast future changes in the dependent variable. In both hypotheses, the increments can

be zero-mean stationary and so there is ever a possibility to make forecasts, which is not

compatible with random walk testing for which a hypothesis of predictability and one of

unpredictability is necessary. So to end, no conclusive evidence on the random walk can be

delivered by accepting the null hypothesis of a unit root test, as the random walk is just a part

of the null hypothesis (Campbell et al., 1997).

The Mean Variance Ratio test

The mean variance ratio test, which is first mentioned in the paper of Lo and McKinley

(1988), is one of many tests to test the random walk hypothesis and more specifically in

context of the RW3 model. The test is based on the assumption that the variance of one period

must equal the variance of the sub period multiplied with the amount of time intervals. For

instance, the variance of one year must be twelve times the variance of one month. When this

assumption is met then markets are efficient, as a result, the random walk model is valid. The

simple intuition behind the test makes it very appealing to take a closer look at. The mean

variance ratio (VR), based on Lindemann et al (2005) can be written as:

Variance ratio =

The VR test is an efficient test for detecting autocorrelations (Lo and McKinley, 1988). When

the variance ratio is higher than unity, we can suggest positive autocorrelations. A ratio,

which is lower than one, displays negative autocorrelations. Therefore, the mean variance

ratio is equal to one under the null hypothesis of no autocorrelations, which implies a random

walk in the Belgian stock market when applied to our data. On the other hand, the presence of

autocorrelations suggest stock market inefficiencies which leads to rejecting the null

hypothesis. However, it is important to stress that autocorrelations are not immediate proof of

market inefficiency. Fama and French (1988) already noticed that autocorrelations are also a

phenomenon in efficient markets; he found positive as well as negative autocorrelations that

cancelled each other out. Consequently, this would lead to the acceptance of the null

hypothesis if the examined dataset is comprehensive enough over time.

Furthermore we need to pay more attention to mean reversion i.e. negative serial correlation

that cancels positive serial correlation out and vice versa. Mean reverting behaviour occurs

when an asset price increases or decreases and this movement is followed by a decrease or an

increase. This behaviour implies that an under or overreaction gets smoothened out by the

31

market. Although this behaviour sounds like a plausible and intuitive type of evidence, it also

can be proof against market efficiency. Fama and French (1988) argue that mean-reverting

behaviour makes returns predictable and especially in the case of smaller firms. They also

state that mean reversion can be either a statement or a counterstatement in favour of market

efficiency. Their paper provides two possible reasons to explain this paradox. The first reason

is that mean reversion is evidence of market efficiency when it is caused by changes in market

risk. For instance, when a stock price increases and a high risk due to (for example) a crisis is

present in the market. Investors will tend to cash their profits because the higher risk brings

more uncertainty about the next day’s return that has a higher probability of being negative.

This behaviour is efficient since there is a tradeoff between risk and expected return. The

second reason is that mean reversion is market inefficient when the price increase or decrease

is purely a deviation from the fundamental value; a deviation which cannot be explained by a

change in market risk. For example, Lakonishok et al. (1992) who found positive-feedback

effects in the stock markets when investors buy good performing stocks and sell bad

performing stock. In that way, investors are able to push the price away from its intrinsic

value. We will present our variance ratios in an x-y scatter diagram for analyzing mean-

reverting behaviour. When negative autocorrelations are followed by positive autocorrelations

or vice versa, this is called mean reversion. Despite having our variance ratios graphically

presented in order to look for mean reversion, this method does not say if the possible mean

reversion is a sign of market efficiency or not. That is why we will intuitively calculate a

mean value of our variance ratios in order to make assumptions if our found mean reversions

are evidence of the EMH or not. A mean value which varies from zero may be proof that the

mean-reverting behaviour is not market efficient. Nonetheless, we note that our variance

ratios represent the nature of short term returns and not represent long term returns. As a

result, our conclusion will be different of Fama and French (1988) who argued that strong

negative autocorrelations could be found on the long term. The formula that we use for

calculating the “mean variance ratio” will be:

The formula above explains how we will calculate our ‘mean variance ratio’. The μ stands for

the mean value of our four individual variance ratios. The n, in this case, is four e.g. equal to

the amount of individual variance ratios. The X is the value of each variance ratio separately.

It is important to make a distinction between homoscedastic time series and heteroscedastic

time series because each time series is based on a different assumption about their variance in

volatility. Homoscedastic time series imply that the volatility remains stable over time.

Whereas heteroscedastic time series have volatilities which change over time (Lo and

McKinley, 1988). Lo and McKinley (1988) use the z statistics under the assumption of

heteroscedasticity since this makes the results robust to heteroscedasticity. We will not take

every time series as a heteroscedastic time series, therefore, we will decide for each time

series if it is homoscedastic or heteroscedastic. Eventually, we will use the appropriate z

statistics. We use this method because the z statistics under the assumption of

32

homoscedasticity will provide the optimal results for homoscedastic time series. After all,

assuming heteroscedasticity in homoscedastic time series will make that we lose essential

information.

We test the random walk hypothesis where the affirmation of the absence of autocorrelations

forms the null hypothesis; while the alternative hypothesis will confirm our prediction that

stock markets are inefficient.

Ho: VR=1

Ha: VR≠1

We expect to obtain a variance ratio of unity for the Belgian All Shares (BAS) index and for

the BEL-20 index. The BEL-20 index is the most influential indicator for the Belgian stock

market, and is also responsible for a major part in changes of the BAS index (Euronext,

2014). The BAS index is value-weighted which results in a large weight, due to the large

market capitalizations, of the BEL-20 shares in the overall BAS index. Since the shares of

which the indices are composed are well-known and traded actively, we believe that the

stocks included will be well-analyzed and thus bring forth market efficiency. In the case of

the BAS index and the BEL-20 that we expect to be efficient, we believe that we will find

another less efficient outcome for the BEL Mid and the BEL Small. Since these two indices

contain smaller capitalization stocks, compared with the first mentioned indices, we expect to

find violations of the assumptions of the random walk and thus an indication of less

informationally efficient markets. Smaller indices are less liquid and are less analyzed than

larger stock indices and that is why these indices are less market efficient. We also predict

that the Belgian growth index will be the most efficient index in comparison with the Belgian

value index. We form our expectations on the publication of Clifford (1998) who researched

investing in growth and value stocks. He focuses on how investors look at both kinds of

stocks. He explains that investors which invest in growth stocks look at all the known

information about the growth stocks and participates on the underlying growth of the

company. On the other hand, investors which invest in value stocks put more weight to

finding any mispricing which makes a stock price more volatile. When profits decline, they

will sell the value stocks and will buy them when the profits increase. In other words, the

acceptance of our null hypothesis (i.e. the variance ratio is not different from one) is realistic

for the growth stocks, whereas, value stocks will provide proof against market efficiency.

33

IV. Results

In this section we will answer the questions we drafted in our research design. One by one, we

will treat them by running fitting tests that were explained in the methodology part (cf. supra).

Stock markets are efficient when the returns of today do not determine the stock returns of

tomorrow. In other words, this implies that autocorrelations do not exist and that stock returns

cannot be predicted. In this chapter we analyze the Belgian stock indices, namely the BEL-20,

BelAllShares, BEL Mid and BEL Small indices. The BEL-20 stock market index consists of

the 20 Belgian stocks which have the largest market capitalization. The Belgian All Shares

index is the largest index of the Belgian stock exchange. This index contains 137 shares and

represents Euronext Brussels (Trivano.com, 2014). The BAS is a large index because of its

large amount of stocks but is different from the BEL-20 because BAS index also contains

smaller stocks. The smaller indices, which are the BEL Mid and BEL Small, contain the 36

most representative companies, also based on market capitalization which displays the state of

the Belgian economy.

Is the Belgian stock market efficient as a whole? And what about

the different size-based compartments of the stock market?

1. The Random Walk

To start off with our main question, we examined the log returns of the BAS for the period

between February 1996 and the end of January 2014. The return series for all indices are

preferred over the price series for many reasons, of which the avoidance of multicollinearity

and the nature of many financial time series are the main ones. When using nonstationary

price time series in an autoregression, the lags will often show high (auto)correlation values

due to the shared trend with the dependent variable and with each other, which in turn also

brings multicollinearity issues. Due to potential multicollinearity, the regression will not be

able to keep apart the true effects that lags have in forecasting the dependent variable Pt. But

worst of all, we would not be in a position to make sound statements about the random walk

and market efficiency. Instead when we use return series (by differencing the log prices), the

series should be stationary. Then we can construct autoregressive models and call markets

efficient when no significance of the lags are found, which simply means that the lags do not

help explain/forecast the dependent variable (i.e. the return). However, when we do find a

significant lag in the regression model using returns, we can assume and conclude safely that

no random walk or positive market efficiency statement can be made (Inghelbrecht, 2013b;

Koop, 2006).

Belgian All Shares (BAS)

Following our process as described before, we carried out Dickey-Fuller tests to search for

unit roots in the daily, weekly, monthly & quarterly BAS returns. These series had no unit

roots and thus are stationary, although the result of the quarterly series is very close to the

critical value (test statistic = -2,80; critical value = -2,89) and strictly would not pass as

stationary. However, the Gretl software shows significance and the graphical tests also seem

34

stationary and therefore we will assume stationarity in the quarterly series as well. Next step

in the process are the autocorrelation functions that also show patterns confirming stationarity

of the series. These graphs show which lags are significant up to twelve lags. In our

autoregressive models we do not always go as far back as these graphs indicate because of

either very small significance levels or the fact that the lag order would become too big in

order for it to be still of great explanatory value to the dependent returns variable. We will

maintain this decision throughout the discussion of all indices. Nevertheless, these

autocorrelation functions give valuable additional results when deciding if the (part of) market

is efficient.

ACF daily returns ACF weekly returns

ACF monthly returns ACF quarterly returns

Running the autoregressive models yielded the results here in the table directly below. Firstly,

no residual autocorrelation was found. This is a good stepping stone for the random walk 3

hypothesis, under which no correlation in the increments is allowed. Secondly, the alpha

values are insignificant and close to zero, eliminating the presence of a drift (i.e. the expected

price change based on firm-specific factors). Thirdly, the models based on daily and quarterly

returns both have significant important lags which consequently help forecast returns. An

additional correlation matrix for the daily returns gave correlation coefficients between Rt and

Rt-1 of 0,081 and between Rt and Rt-3 of -0,057. These coefficients are both quite low. In the

case of quarterly returns, we already mentioned the earlier stationarity doubts. Furthermore,

weekly and monthly returns show no significant lags up to respectively the eighth and fifth

order and the lag coefficients are relatively close around zero. Lastly, these autoregression

models have only a very low explanatory power as indicated by the R², which also limits the

economic relevance. The highest R² is found for the regression with quarterly returns which is

again open to doubts due to the close acceptance of the stationarity of the series and the nature

of these financial time series in which regressions with nonstationary series yield higher R²’s.

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

35

These arguments incline us to reject the random walk and market efficiency, especially when

using daily data and also the significant lags of a relatively high order when using weekly and

monthly data. We believe it to be logical that temporary behavioural elements could influence

daily returns which is not as obvious in our models when using longer frequencies in which

returns are more likely to be determined by rational elements. Furthermore, Lo and McKinley

(1988) already noticed daily data to be biased at times due to infrequent trading problems or

bid-ask spreads. This is not necessarily the case for the BAS but it is something to hold in

mind when moving on to smaller compartments. Besides, the smaller compartments are

included in the BAS and can influence this outcome. Nevertheless, we would like a second

opinion by calculating the variance ratio later on. But for now we cannot accept the BAS to be

efficient.

BAS Daily returns Weekly

returns

Monthly returns Quarterly

returns

α 0,0002

(1,087)

0,0009

(1,001)

0,0033

(0,915)

0,0100

(0,871)

Rt-1 0,0813

(5,570)***

-0,0040

(-0,123)

0,0947

(1,390)

0,2690

(2,332)**

Rt-2 -0,0119

(-0,814)

0,0302

(0,924)

0,0714

(1,049)

/

Rt-3 -0,0548

(-3,754)***

/ / /

R² 0,0098 0,0009 0,0156 0,0721

Residual

autocorrelation

No

(Durbin’s h=

1,95)

No

(DW= 2,00)

No

(Durbin’s h=

0,05)

No

(Durbin’s h=

0,26)

# observations 4685 938 216 72

Random Walk? No Partly Partly No Based on the basic autoregressive model: Rt= α + Rt-1 + ut ; number of lags chosen is based on significance of

the lags & the ACF’s; t-statistics between brackets underneath the coefficients; Partly indicates that the random

walk can be accepted up to a certain higher order, which makes the potential rejection of the RW less powerful.

BEL-20

For the BEL-20 index; daily, weekly and monthly returns are clearly stationary as no unit root

was found by the Dickey-Fuller test. Quarterly returns are in this case doubtful as well. The

test statistic of -2,86 is very close to the critical -2,89 and Gretl considers this result as

significant, i.e. absence of a unit root and therefore also stationary. Again, we will assume all

four indices to be stationary which is also confirmed by the patterns in the autocorrelation

functions.

36

ACF daily returns ACF weekly returns

ACF monthly returns ACF quarterly returns

For the BEL-20 we find similar results as was the case with the BAS. First of all, no residual

autocorrelation was found so this aspect of the RW3 (i.e. uncorrelated increments) is not

violated. Secondly, drifts can be neglected as the constants are insignificant in all four models.

Next, we find two relevant significant lags in the autoregression of daily returns, implying a

non random walk. In an additional correlation matrix we get weak correlation coefficients

between Rt & Rt-1: 0,079; between Rt & Rt-2: -0,015 and between Rt & Rt-3: -0,059.

Nevertheless, the correlations are present and are rejecting the random walk hypothesis. Here

we can also repeat the remarks made by Lo and McKinley (1988) about biases using daily

data, in analogy with the results of the BAS index. When lengthening the frequency to weekly

and monthly data, we find autoregressions confirming a random walk with insignificant lags.

Although in the monthly returns model Rt-3 is considered significant by Gretl, however the t-

statistic is still in bounds of the null hypothesis implying an insignificant lag. Also a fifth lag

is significant which makes us doubt the random walk when using monthly data anyway. The

model based on quarterly returns may be doubtful in question to stationarity of the returns and

so in the validity of the constructed model but it is also rejecting the random walk with one

significant lag. The overall regression models show only a very limited explanatory power, R²

and the same arguments can be made to the quarterly returns model as was the case for the

BAS.

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

37

BEL-20 Daily returns Weekly returns Monthly returns Quarterly

returns

α 0,0001

(0,657)

0,0006

(0,617)

0,0021

(0,527)

0,0070

(0,577)

Rt-1 0,0792

(5,430)***

-0,0350

(-1,072)

0,0648

(0,953)

0,2604

(2,257)**

Rt-2 -0,0166

(-1,131)

0,0300

(0,917)

0,0131

(0,193)

/

Rt-3 -0,0560

(-3,838)***

/ 0,1208

(1,779)(*)

/

R² 0,0098 0,0022 0,0199 0,0678

Residual

autocorrelation

No

(Durbin’s h=

0,85)

No

(DW= 2,00)

No

(Durbin’s h= -

0,24)

No

(Durbin’s h=

0,55)

# observations 4685 938 216 72

Random

Walk?

No Partly Partly No

Based on the basic autoregressive model: Rt= α + Rt-1 + ut ; number of lags chosen is based on significance of

the lags & the ACF’s; t-statistics between brackets underneath the coefficients; Partly indicates that the random

walk can be accepted up to a certain higher order, which makes the potential rejection of the RW less powerful.

BEL Mid

The initial unit root tests to the indices of the BEL Mid all yield convincing results of

stationary return series. In this case the quarterly results are well into the alternative

hypothesis of an absent unit root with a test statistic of -6,55. The daily, weekly and monthly

series also have the necessary more negative test statistics in comparison to the critical value,

accepting the alternative (stationary) hypothesis. The graphical output of the autocorrelation

functions paints a confirming picture.

ACF daily returns ACF weekly returns

ACF monthly returns ACF quarterly returns

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

38

The regressions show no sign of residual autocorrelation. Drifts can be neglected as well, no

coefficients are deemed significant and besides are closely to zero which makes them

economically insignificant too. The graphical output of the autocorrelation functions show

two significant lags that help explain returns for daily and weekly returns. For the

autoregression with daily data, the significance of the lags is higher and in a closer order (first

and second order). Opposed to the autoregression with weekly returns that results in lags with

smaller significance and to a higher order (second and eighth). The models using monthly and

quarterly returns seem market efficient, following a random walk. The sixth lag in the

monthly returns model is barely significant as is shown on the autocorrelation chart above and

could be neglected in the autoregression. The quarterly model follows a smooth random walk.

In the BEL Mid compartment, it seems that we get evidence of increasing market efficiency

as we lengthen the frequency of the data that is used to construct our models. The explanatory

power of the models also increase with the lengthening of time frequency, though the power

is still limited. A similar observation can be made with the previous BAS index and BEL-20

compartment, but this is not as obviously observable in the results as it is here for the BEL

Mid results.

BEL Mid Daily returns Weekly returns Monthly

returns

Quarterly

returns

α 0,0002

(1,594)

0,0013

(1,446)

0,0052

(1,205)

0,0177

(1,234)

Rt-1 0,1372

(9,392)***

0,0163

(0,499)

0,0656

(0,955)

0,1781

(1,480)

Rt-2 0,0381

(2,609)***

0,0689

(2,110)**

0,0860

(1,252)

-0,2050

(-1,705)

R² 0,0218 0,0051 0,0125 0,0630

Residual

autocorrelation

No

(Durbin’s h=

0,43)

No

(Durbin’s h= -

0,59)

No

(DW=2,01)

No

(Durbin’s h=

1,39)

# observations 4685 938 213 69

Random Walk? No No Partly Yes Based on the basic autoregressive model: Rt= α + Rt-1 + ut ; number of lags chosen is based on significance of

the lags & the ACF’s; t-statistics between brackets underneath the coefficients; Partly indicates that the random

walk can be accepted up to a certain higher order, which makes the potential rejection of the RW less powerful.

BEL Small

The daily, weekly, monthly and quarterly return series of the BEL Small are all stationary

with test statistics from the Dickey-Fuller test comfortably more negative than the critical

value of -2,89. Moving on to the autocorrelation functions, we also notice stationary patterns

in confirmation of the unit root tests.

39

ACF daily returns ACF weekly returns

ACF monthly returns ACF quarterly returns

The Durbin’s h values all indicate no residual autocorrelation as the absolute values are all

smaller than 1,96 and thus they categorize under the null hypothesis. The uncorrelated

increments assumption is met. For the first time the constants in the regressions are

significant, implying a drift in the random walk equation. This means that there is a part of the

returns that is being influenced by an expected change element that can help forecast returns.

However the coefficients are almost zero which makes them of no real use economically and

puts the predictive power also at zero. The significance of the drift does not have effect on the

acceptance or rejection of the random walk. In fact, rejecting the random walk because of a

significant drift would be a mistake (Campbell et al., 1997). However, the random walk will

have to be rejected due to the fact that all return series, regardless of their frequency, show

significant lags. The significant lags are visible in the graphical output of the autocorrelation

functions as well as in the autoregressions table here directly below. It seems that our

expectations of an inefficient small stocks compartment are just and that the result of Lo and

MacKinlay (1988), who found stronger autocorrelations for smaller stocks is confirmed by

our models as well. As to the explanatory power of the models, the R² values are very low

again with approximately two to three percent power for daily to monthly returns and

somewhat over ten percent when using quarterly returns. The coefficients of the lags in the

autoregression models are different from zero but overall, the limited influence they have on

the returns does not make them have a real groundbreaking economical forecasting value.

Lastly, when lengthening the time frequency, it seems that the number of significant lags

decreases which in turn can imply an increasing degree of market efficiency. Based on these

last remarks, Fama (1991) would not be convinced of market inefficiency but for now we will

conclude that the BEL Small compartment does not follow a random walk and is in fact

inefficient.

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

40

BEL Small Daily returns Weekly returns Monthly returns Quarterly returns α 0,0005

(3,611)***

0,0019

(2,915)***

0,0089

(2,709)***

0,0219

(2,066)** Rt-1 -0,0426

(-2,933)***

0,0887

(2,732)***

0,1680

(2,474)**

0,3217

(2,832)*** Rt-2 0,1172

(8,074)***

0,1117

(3,443)***

/ /

R² 0,0160 0,0223 0,0278 0,1028 Residual

autocorrelation No

(Durbin’s h= -

1,46)

No

(Durbin’s h= -

0,44)

No

(Durbin’s h= -

1,73)

No

(Durbin’s h=

1,90) # observations 4685 938 216 72 Random Walk? No No No No Based on the basic autoregressive model: Rt= α + Rt-1 + ut ; number of lags chosen is based on significance of

the lags & the ACF’s; t-statistics between brackets underneath the coefficients; Partly indicates that the random

walk can be accepted up to a certain higher order, which makes the potential rejection of the RW less powerful.

2. The Variance ratio

Daily returns

This paragraph contains our results with respect to daily returns of our four Belgian stock

indices. Since the volatility changes over time, we use the z statistics under heteroscedasticity.

We found that all our time series of daily stock returns are heteroscedastic via the White’s test

which is the appropriate test for determining the behaviour of the variances. In other words,

the volatility changes over time for daily returns.

Table 1a provides the results of our variance ratio for daily data based on our dataset which starts from 23

februari 1996 till 31 january 2014. The first value is the value of the variance ratio which is followed by the Z-

statistics. The Z-statistic is the value under heteroscedasticity.

Type of

index

Number of observations

2 4 8 16 Mean VR

BAS 1,077*

(2,729)

1,076

(1,420)

1,031

(0,709)

1,042

(0,351) 1,057

BEL-20 1,076*

(2,891)

1,068

(1,390)

0,995

(-0,066)

0,977

(-0,210) 1,029

BEL Mid 1,142*

(4,177)

1,272*

(4,541)

1,345*

(3,932)

1,411*

(3,441) 1,293

BEL Small 0,950

(-0,630)

1,055

(0,429)

1,163

(1,010)

1,351

(1,895) 1,130

Table 1a presents the values of the variance ratios for certain holding periods which are 2,4,8

and 16. We find significant variance ratios for all stock indices except for the BEL Small

which has not a single z statistic that is larger than our critical value of 1,96. In the first row of

table 1a, we find a significant value of 1,077 for a number of two base observations. The

value of the variance ratio can be divided into 1 + the amount of approximate serial

correlation. In this case, we find a serial correlation of 7,7 % which is rather small because an

41

increase of 1 % of the share price the day before will cause an increase of 0,077 % the day

after. This small autocorrelation is still proof that the BAS is market inefficient. The

remaining variance ratios of the BAS are insignificant and the z statistics halve when the

amount of base observations doubles. The values of the variance ratio, even if they are

insignificant, show a declining pattern. However, we see a surge of the variance ratio at 16

base observations. The variance ratios of the BEL-20 are presented in the second row of table

1a. Our findings are similar to the BAS since we also find declining variance ratios, one

significant value at a base observation of two and declining z statistics over time. The

significant variance ratio of the BEL-20 shows a serial correlation of 7,6 %. This

autocorrelation comes close to the 7,7 % that we find for the BAS as well. This could be due

to the fact that the BAS is a weighted stock index where the BEL-20 stocks have a major

influence on this index. The first two variance ratios show positive autocorrelations of 7,6 and

6,8 %, where the last two variance ratios show negative autocorrelations of -0,005 and -0,023

%. Although the majority of the variance ratios are insignificant, a mean-reverting pattern can

be analyzed. The third row of table 1a gives the variance ratios of the BEL Mid and its z

statistics. We see strong z statistics of 3,441 and higher. We analyze a declining z statistics

over time and an increase in the variance ratios. Our findings state that the variance ratios of

the BEL Mid are strongly significant. When we compare the serial correlation with the

preceding indices, we conclude that BEL Mid variance ratios show large autocorrelations.

These autocorrelations are between 14 % and 41 %. A serial correlation of 41 % signifies a

strong predictability of the BEL Mid stock returns. The fourth row displays the variance ratio

of the BEL Small index. The results clearly show that the z statistics increase over time just

like the variance ratios. The variance ratio of two base observations contains a negative serial

correlation of 5 %, whereas the variance ratio of 4,8 and 16 observations have positive serial

correlation with a maximum of 35,1 %. Once again, this pattern is, like the literature says,

proof of mean-reverting behaviour.

Graph A presents the variance ratios of daily data

All variance ratios are visualized in graph A because this makes it easier to look for mean-

reverting behaviour in daily returns. We know that mean-reverting behaviour is not per se

proof against market efficiency. However, it could still provide additional value to our

research. When we analyze graph A, we find similarities between the BEL-20 and the BAS

and we also find similarities between the BEL Mid and the BEL Small. It seems that this

cohesion is not by chance, since we can group these indices by size. Mean reversion is found

0

0,5

1

1,5

2 4 8 16

BAS

BEL-20

BEL Mid

BEL Small

42

for the BEL-20 and the BEL Small but this reversion process happens differently for each

index. The BEL-20 index starts with a positive serial correlation which reverses into a

negative serial correlation, where the BEL Small has exactly the opposite working of the

mean-reverting process. We also find that the first-order correlations of the BEL-20 do not

intensely deviate from zero. For instance, the autocorrelations deviate between -2,3 and +7,6

%. The mean variance ratio is +2,9 % which is a very low amount of autocorrelation. The

negative serial correlation practically compensates the positive serial correlation which

demonstrates that the mean reversion of the BEL-20 is market efficient. On the contrary, the

serial correlations of the BEL Small depart more strongly from zero. The autocorrelations of

the BEL Small lie between -5 and +35,1 %. The mean variance ratio of the BEL Small

indicates an overall positive autocorrelation of 13 % that is rather strong evidence of

inefficient mean-reverting behaviour.

To conclude the variance ratios calculated with daily returns, we expected to find a strong

significance for the BEL Small variance ratios and a less strong significance for the BEL Mid

variance ratios. On the other hand, we expected the larger indices to provide insignificant

results. However, our study contradicts our thoughts about daily returns because the reality

seems to be paradoxical. We are able to reject the random walk for all Belgian indices except

for the BEL Small index. We accept the null hypothesis for the BEL Small at a significance

level of five percent. The BEL Small index does not have one significant value which proof

that this index is market efficient, thereby, follows a random walk. It is absolutely impossible

to predict daily stock returns of our smallest stock index. When we look at the remaining

indices, we can only conclude that they do not follow a random walk. They contain variance

ratios, which are different from one that is why we reject the null hypothesis. Despite having

some insignificant values, we find mean-reverting behaviour for our BEL-20 and our

BelSmall index.

Weekly returns

This paragraph contains our results with respect to weekly returns of our four Belgian stock

indices. We found that the BAS and the BEL Mid index have changing volatilities i.e.

heteroscedasticy the other two indices are homoscedastic.

Table 1b provides the results of our variance ratio for weekly data based on our dataset which starts from 23

februari 1996 till 31 january 2014. The first value is the value of the variance ratio which is followed by the Z-

statistics. The Z-statistic is the value under heteroscedasticity/homoscedasticity .

Type of

index

Number of observations

2 4 8 16 Mean VR

BAS 0,989

(-0,237)

1,017

(0,206)

1,052

(0,394)

1,187

(0,971) 1,061

BEL-20 0,955

(-1,366)

0,955

(-0,728)

0,972

(-0,289)

1,041

(0,282) 0,981

BEL Mid 1,020

(0,757)

1,105

(0,961)

1,251

(1,651)

1,418*

(2,008) 1,199

BEL Small 1,099*

(3,026)

1,271*

(4,416)

1,496*

(5,109)

1,741*

(5,131) 1,402

43

The first row of table 1b contains the variance ratios that we find for the BAS index. All

variances are found to be insignificant since the z statistics are not larger than the critical

value of 1,96. When we analyze the z statistics and the variance ratios, we find that the z

statistics increase over time like the value of the variance ratios do. A pure interpretation of

the variance ratios, this is without keeping significant values in mind, show that the BAS

include a mean-reverting pattern. The variance ratio for a base observation of two has a

negative serial correlation of 1,1 % where the other variance ratios all have positive serial

correlation with a maximum of 18,7 %. The second row of table 1b presents the variance

ratios found for the BEL-20. Like the BAS, we do not find any significant variance ratios.

However, mean-reverting behaviour can be found once again. The first three variance ratios

have negative autocorrelation due to the values smaller than one. On the contrary, the

variance ratio at a base observation of 16 exist of 1 + a positive serial correlation of 4,1 %.

We also find a surge in the value of the variance ratios over time and an incline in the z

statistics. The third row of table 1b consists of variance ratios of BEL Mid trading data. Like

the previous indices, we find an upturn of the z statistics and the variance ratios there selves.

The first three variance ratios are insignificant since the z statistics are not larger than the

critical value of 1,96. Yet, we find a significant variance ratio of 1,418 when we look at a base

observation of 16. The variance ratio displays a strong serial correlation of 41,8 %. The BEL

Mid can be easily predicated. In contrast to the preceding indices, no mean-reverting

behaviour can be found. The fourth row of table 1b contains our variance ratios of the BEL

Small index which are 1,099 , 1,271 , 1,496 and 1,741. The z statistics and the values increase

for all variance ratios and they are all significant with z statistics of higher than 3,026. We

even argue that this significance is very strong because of the z statistics which are much

larger than our critical value. Like the BEL Mid, we are not able to find a pattern of mean-

reverting behaviour.

Graph B presents variance ratios of weekly data

Graph B is a graphical representation of our variance ratios from weekly market data. All

indices show an identical image where the value of the variance ratios grows over time. The

lines in the graph of the BAS and the BEL-20 begin under one (i.e. the value of a variance

ratio under a random walk) and when the amount of base observations gets larger, the

variance ratios become larger than one. Even both indices look similarly; we found one

difference which is that one of four variance ratios of the BAS is less than one, whereas three

out of four variance ratios of the BEL-20 are smaller than one. The mean variance ratio of the

BAS accounts for 6,1 %, while the mean variance ratio of the BEL-20 has a negative value of

0

0,5

1

1,5

2

2 4 8 16

BAS

BEL-20

BEL Mid

BEL Small

44

1,9 %. Both mean autocorrelations are weak and are no signs of market inefficient mean

reversion due to a well-balanced compensation between positive and negative correlations.

In conclusion using weekly data, we analyzed variance ratios build on weekly stock market

data in table 1b and our findings come close to our predications. Our predictions were that the

smaller indices showed any significance and that the larger stock indices would show the

opposite of that. The largest stock indices, which are the BAS and the BEL-20, do not contain

any significant z statistics. As a result, the null hypothesis is accepted for our two large sized

indices. However, we find mean-reverting behaviour for the BAS and the BEL-20. On the

contrary, the BEL Mid index and the BEL Small index are both proof against the random

walk hypothesis since any significance can be found. The BEL Mid presents one significant

variance ratio at a holding period of 16 which rejects the null hypothesis. The BEL Small

contains only significant z statistics so that the variance ratios are strong evidence against the

efficient market hypothesis and its random walk.

Quarterly returns

This paragraph contains our results with respect to quarterly returns of our four Belgian stock

indices. We found that all indices are homoscedastic i.e. do not have changes in volatility.

Table 1c provides the results of our variance ratio for quarterly data based on our dataset which starts from the

first quarter of 1996 and ends at the fourth quarter of 2014. The first value is the value of the variance ratio

which is followed by the Z-statistics. The Z-statistic is the value under homoscedasticity .

Type of

index

Number of observations

2 4 8 16 Mean VR

BAS 0,956

(-0,367)

1,004

(0,018)

1,036

(0,097)

1,566

(1,028) 1,141

BEL-20 1,146

(1,225)

1,307

(1,352)

1,723

(1,954)

1,873

(1,587) 1,512

BEL Mid 1,139

(1,167)

1,053

(0,232)

1,177

(0,477)

0,976

(-0,044) 1,086

BEL Small 1,343*

(2,956)

1,484*

(2,194)

1,694*

(1,990)

1,582

(1,058) 1,523

The variance ratios of the first row in table 1c represent our findings of the BAS index. We do

not find any statistically significant values and the z statistics, which increase over time, are

much smaller than the critical value of 1,96. Although the variance ratios are insignificant in a

statistical way, they are economically valid. For instance, we find for two base observations

that the variance ratio displays a negative serial correlation of 0,044. On the contrary, we find

positive autocorrelation when observing the other variance ratios. We even find an

autocorrelation of 5,66 %. The change from negative to positive serial correlation is proof that

the BAS is subject to mean-reverting behaviour. The second row of table 1c displays the

variance ratios made of BEL-20 trading data. We do not find any significant z statistics,

though the z statistic for a base observation of eight comes close to a rejection of the random

walk. The value of 1,954 is not high enough to reach the critical value of 1,96. This high

value of 1,954 implies that the BEL-20 index comes close to market inefficiency.

Additionally, the insignificant variance ratios all show positive autocorrelation which also

45

increases through time. An impressive value of 1,873 can be found for 16 base observations.

In the third row, we find our findings about the BEL Mid variance ratios. Clearly, all variance

ratios are insignificant and they also display a drop in the z statistics. We see positive serial

correlation for the first three variance ratios, while the last variance ratio indicates a negative

autocorrelation. Mean-reverting behaviour can be found like we found for the BAS index. The

fourth row of table 1c contains the variance ratios of the BEL Small. Three out of four

variance ratios seems to be significant with a minimum z statistic of 1,99 , whereas, the

variance ratio with a base observation of 16 becomes insignificant. The third variance ratio of

the BEL Small comes near to insignificancy which assumes that the BEL Small changes

towards market efficiency. The z statistics of the row decline when the amount of base

observations gets larger. Together with a fall of the z statistics, the values of the significant

variance ratios increase. The serial correlation that we found is positive and between 34,3 %

and an astonishing 69,4 %.

Graph C with quarterly data variance ratios

Graph C shows the variance ratios which are based on quarterly returns in a graphic way,

therefore, makes it easier to look after mean reversion in stock markets. No similarities are

found between our four indices; none of them follow the same pattern. We only see that the

BAS and the BEL Mid which contain a mean reversion pattern when we analyze quarterly

data points. A positive first-order correlation of the BAS can be found for the first variance

rate, where the other ratios contain positive autocorrelations. The autocorrelations of the BAS

range between -4,4 and +56,6 %. The BEL Mid has positive first-order correlations except for

the last variance ratios which becomes negative with 2,4 %. The mean variance ratio of the

BAS indicates a total positive autocorrelation of 14,1 % which forms strong evidence of

inefficient mean-reverting behaviour since there is an overcompensation of positive serial

correlation. We find similar results for the Bel Mid that has a global autocorrelation of 8,6 %

which is less strong proof of inefficiency.

Before concluding the quarterly results, we want to remark that the study of quarterly stock

market data via variance ratios is hardily studied. This is why we could not make any

predications based on what we already know. Therefore, we were not certain if insignificant z

statistics could be found because quarterly data are separated by large periods of time. We

were surprised by the findings. The BAS, the BEL-20 and the BEL Mid appear to be in line

with the random walk hypothesis. We could not find any significant values which could be a

stronghold for rejecting the null hypothesis i.e. variance ratios which are different than one.

Yet, we are able to reject the random walk for BEL Small quarterly market data because of

0

0,5

1

1,5

2

2 4 8 16

BAS

BEL-20

BEL Mid

BEL Small

46

three significant z statistics out of four. Against all expectations, we even find any

predictability in quarterly data.

Comparison with Variance ratio literature

The variance ratios build on daily stock returns of the Belgian stock market provide an image

were the size of a market index is not the main factor for market inefficiency. We show that

our smallest index, which is the BEL Small, is pure evidence for the efficient market

hypothesis while the larger indices are overall market inefficient. These rather strange results

contradict the paper of Lo and MacKinlay (1988) who researched the US stock market. They

found that the size of a market index was a determinant for the rejection of the null

hypothesis. The acceptance of the null hypothesis would mean that returns become

unpredictable which is in line with the theory of Fama. Lo and MacKinlay (1988) stated that

smaller indices turned to be market inefficient, whereas, larger indices became more efficient.

However, they never used daily data due to the several possible biases that daily returns can

have. Fratzscher (2002) investigated stock returns of all the countries which are members of

the European Monetary Union. His data was, like our paper, build on daily stock returns. He

found that the European stock returns are predictable but found that this predictability

decreases over time due to the European integration. There is also the paper of Charles and

Darné (2009) where five emerging stock markets in Latin America where studied. All data

were based on daily returns and showed many rejections of the market efficiency hypothesis

for Latin America. For instance, the financial markets of Argentina, Chile and Mexico are all

found to have predictable return patterns.

Variance ratios of weekly returns are a beautiful compromise between a minimum reduction

of biases and a sufficient amount of observations (Lo and MacKinlay, 1988). That is why Lo

and MacKinlay (1988) use especially weekly time frequencies. Like already mentioned, they

analyzed portfolios composed off different kind of stocks which varied from large to small

stocks. Our research mentions considerable significant variance ratios. The two smallest

Belgian stock indices, which both contain significant values, support the rejection of return

unpredictability. This unpredictability is in line with Lo and MacKinlay’s paper (1988) where

they argue that portfolios consisting of smaller stocks are less market efficient compared with

portfolios constructed of larger stocks. Our results are also similar to those of Lo and

MacKinlay (1988) because they found insignificant variance ratios for larger indices,

implying that a larger index tend to be more market efficient. We note that the variance ratios

show positive autocorrelations just as Lo and Mackinlay (1988) found.

Though it is popular to use daily or weekly time frequencies, we also make use of quarterly

data. Quarterly data is not hardily used in the field of market predictability; therefore, it would

be an enrichment for the literature to use this type of returns. Our BEL Small index shows any

significance and contradicts the null hypothesis of market efficiency. This is strong evidence

knowing that we have a maximum minimization of biased returns. That is, the BEL Small

returns are predictable even when we use large spans of periods. We know that the paper of

Ayadi and Pyun (1994) studied quarterly data for the Korean stock market and found no

results which were able to reject the random walk.

47

The remaining question after discussing the results is: What causes differences in the rejection

of a random walk between daily, weekly and quarterly data? When daily data is used, it could

be biased by non-synchronous trading. Smaller stock indices have a higher likelihood of

having non-synchronous trading biases than larger indices. As a result, the random walk will

be rejected for daily variance ratios and especially for smaller stocks/indices. However, our

findings prove that the Belgian stock indices are not likely to have infrequent trading because

of the insignificant variance ratios that we found for the BEL Small. On the contrary, the

larger indices have significant values but are liquid enough so that they have no non-trading

days. When we use weekly data, it will not be affected by biases like non-trading and bid-ask

spreads. In the case of weekly returns, our two smallest indices do not follow a random walk.

This rejection could not be due to biases which we already discussed. If we use quarterly data

then it is certain that no biases will be found. Consequently, differences between our time

frequencies cannot be explained by biases like non-trading behaviour but we can explain these

differences by serial correlation which is also called spurious correlation. Spurious

correlation, like Ayadi and Pyun (1994) researched, is a phenomenon where stock information

is made publicly but not immediately picked up by most investors which results in a lag. This

lag explains why some Belgian stock indices have predictable returns.

Do we observe a difference in market efficiency between growth and value stocks?

To answer this question we examine MSCI data covering a period of almost thirty years, from

01/01/1975 to 01/04/2014 with three different frequencies: daily, weekly and monthly data.

We examine two types of indices, the one consists out of returns of growth stocks (BAS

Growth) and the other out of value stocks (BAS Value), both indices together form the BAS.

This is very interesting for our research as the BAS is divided into two separate indices

having other characteristics. Value stocks pertain to well-founded stable companies that can

be regarded as ‘safe bets’ with a stable dividend policy. The growth stocks index includes

high-potential companies characterized by fast growth for investors seeking a maximum

return on their investment and are willing to take a risk on a company of which the profits are

precarious. The big question: is there a difference in efficiency observable when comparing

growth & value stocks?

1. The Random Walk

BAS Growth index

Just as with the Datastream indices, we transformed the MSCI data into suitable return series.

So before constructing the autoregressive random walk models, we tested all different return

indices on stationarity. All indices containing growth and value stocks over all frequencies

rejected the null hypothesis of unit root presence, thus all of them are stationary and ready to

be used in autoregressive models. Here below are the graphs reflecting the autocorrelation

functions up to twelve lags of the BAS Growth index.

48

ACF daily returns ACF weekly returns

ACF monthly returns

Immediately, we notice from the autocorrelation functions that there are significant lags, so

evidence to reject a random walk is already present. The same result is given by the

autoregressions in the table below except for the regression using daily data. In this case,

significant lags only show in the sixth and eighth order. Also a drift is present, albeit

economically insignificant and no reason to reject the random walk hypothesis. But based on

the weekly and monthly series we should reject the random walk, especially in the monthly

series in which three significant lags are found and the lag coefficients are different from zero

and mostly positive making the case for positive autocorrelation. The uncorrelated increments

condition of the RW3 is fulfilled based on the Durbin tests. Furthermore the models do not

have high predictive power as indicated by the R². So economically seen, the models do not

have much meaning which also depowers the rejection of the random walk but nevertheless

the RW3 is rejected.

-0,03

-0,02

-0,01

0

0,01

0,02

0,03

0 2 4 6 8 10 12

lag

ACF for ld_MSCI_BELGIUM__G___PRICE_IN

+- 1,96/T^0,5

-0,03

-0,02

-0,01

0

0,01

0,02

0,03

0 2 4 6 8 10 12

lag

PACF for ld_MSCI_BELGIUM__G___PRICE_IN

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

ACF for ld_MSCI_BELGIUM__G___PRICE_IN

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

PACF for ld_MSCI_BELGIUM__G___PRICE_IN

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_MSCI_BELGIUM__G___PRICE_IN

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_MSCI_BELGIUM__G___PRICE_IN

+- 1,96/T^0,5

49

BAS Growth Daily returns Weekly returns Monthly returns

α 0,0002

(2,012)**

0,0010

(1,724)

0,0040

(1,663) Rt-1 0,0096

(0,976)

-0,0330

(-1,503)

0,1828

(3,971)*** Rt-2 -0,0089

(-0,903)

0,0279

(1,272)

-0,0521

(-1,116) Rt-3 / -0,0229

(-1,044)

-0,1104

(-2,355)** Rt-4 / 0,0477

(2,178)**

0,0995

(2,169)** Rt-5 / 0,0686

(3,127)***

/

R² 0,0002 0,0093 0,0522 Residual

autocorrelation No

(DW= 2,00)

No

(Durbin’s h= -0,43)

No

(Durbin’s h= 1,40) # observations 10238 2042 467 Random Walk? Partly No No

Based on the basic autoregressive model: Rt= α + Rt-1 + ut ; number of lags chosen is based on significance of

the lags & the ACF’s; t-statistics between brackets underneath the coefficients; Partly indicates that the random

walk can be accepted up to a certain higher order, which makes the potential rejection of the RW less powerful.

BAS Value index

We already stated that the return series of both growth and value stocks are stationary which

is also the case for the patterns of the autocorrelation functions here below, before moving on

to the autoregressions.

ACF daily returns ACF weekly returns

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0 2 4 6 8 10 12

lag

ACF for ld_MSCI_BELGIUM__V___PRICE_IN

+- 1,96/T^0,5

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0 2 4 6 8 10 12

lag

PACF for ld_MSCI_BELGIUM__V___PRICE_IN

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

ACF for ld_MSCI_BELGIUM__V___PRICE_IN

+- 1,96/T^0,5

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0 2 4 6 8 10 12

lag

PACF for ld_MSCI_BELGIUM__V___PRICE_IN

+- 1,96/T^0,5

50

ACF monthly returns

We expected value shares to be more efficient but in reality they are less efficient than growth

stocks. When looking at the graphs we notice more significant lags and when running the

autoregressions we find that those lags have higher significance based on the t-statistics

between brackets, in comparison to the results of the regressions for growth indices (cf.

supra). With the value stocks regressions we can reject the random walk for all three models

using different frequencies. Again the coefficient values of the lags are not that high and the

R² is very limited, which makes the economical implications of our models in real life not that

valuable for predictive purposes. The only fulfilled condition of the RW3 are the uncorrelated

increments for the three models. In conclusion, the random walk is rejected for the value

stocks out of the BAS.

BAS Value Daily returns Weekly returns Monthly returns

α 0,0002

(1,590)

0,0009

(1,275)

0,0030

(1,097) Rt-1 0,0548

(5,547)***

-0,0595

(-2,695)***

0,2021

(4,379)*** Rt-2 0,0035

(0,349)

0,0382

(1,730)

0,0021

(0,045) Rt-3 -0,0490

(-4,967)***

0,0157

(0,711)

-0,0410

(-0,870) Rt-4 / 0,0777

(3,523)***

0,1232

(2,674)*** Rt-5 / 0,0468

(2,117)**

/

R² 0,0054 0,0129 0,0550 Residual

autocorrelation No

(Durbin’s h= -0,34)

No

(Durbin’s h= -0,35)

No

(Durbin’s h= 0,33) # observations 10237 2042 467 Random Walk? No No No

Based on the basic autoregressive model: Rt= α + Rt-1 + ut ; number of lags chosen is based on significance of

the lags & the ACF’s; t-statistics between brackets underneath the coefficients; Partly indicates that the random

walk can be accepted up to a certain higher order, which makes the potential rejection of the RW less powerful.

Growth vs. Value

Both indices have rejected the random walk hypothesis using daily, weekly and monthly

return data. However daily data of the growth index seems to accept the random walk up to

the sixth lag order. Furthermore the lags of the BAS Value are more significant than those of

the BAS Growth (based on t-statistics). Probably the most interesting observation we can

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_MSCI_BELGIUM__V___PRICE_IN

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_MSCI_BELGIUM__V___PRICE_IN

+- 1,96/T^0,5

51

make is that when comparing these two indices to the tests we ran on the whole BAS, the

BAS is more efficient and more likely to accept the random walk based on lag significance

amongst others. So although we find inefficient growth and value compartments, the BAS in

its entirety is more efficient when examining the entire index.

2. The Variance ratio

Table 1a provides the results of our variance ratio on the dataset which starts from the 1st January 1975 untill 26

March 2014. The first value is the value of the variance ratio which is followed by the Z-statistics. The Z-statistic

is the value under heteroscedasticity.

Value Number of q base observations

2 4 8 16 Mean RV

Daily 1,075*

(3,059)

1,095*

(2,128)

1,046

(0,664)

1,087

(0,866) 1,076

Weekly 0,919

(-1,509)

0,960

(-0,411)

0,985

(-0,100)

1,107

(0,500) 0,993

Table 1a, which handles the returns of all Belgian value stocks, presents significant values for

daily returns. We find any significance for two and four base observations. The values of

these significant variance ratios are larger than one at a significance level of five percent. In

other words, we find positive serial correlation. Based on daily data, we conclude that the

Belgian value index contains market inefficiency. An autocorrelation of 7,5 % is found for

two base observations and we even find a higher one for four base observations which is 9,5

%. We reject the null hypothesis that our variance ratios are equal to one. When we analyze

weekly data, we find no significance. All z statistics are smaller than our critical value of 1,96

and that is why we accept our null hypothesis. To summarize, the value index for weekly data

confirms the efficient market hypothesis where past returns do not predict future returns.

Graph A presents the value ratios of the value and growth index for daily returns

Graph A compares the variance ratios of the value and growth indices which is based on daily

returns. We do not find any conformity between the value and growth index since the growth

index has negative autocorrelation and the value index has positive autocorrelation. However,

we do not find any mean-reverting behaviour for both periods.

0

0,2

0,4

0,6

0,8

1

1,2

2 4 8 16

Value

Growth

52

Table 1b provides the results of our variance ratio on the dataset which starts from the 1st January 1975 untill 26

March 2014. The first value is the value of the variance ratio which is followed by the Z-statistics. The Z-statistic

is the value under heteroscedasticity.

Growth Number of q base observations

2 4 8 16 Mean RV

Daily 0,993

(-0,380)

0,977

(-0,682)

0,924

(-1,489)

0,886

(-1,437) 0,945

Weekly 0,943

(-1,576)

0,909

(-1,303)

0,965

(-0,324)

1,033

(0,212) 0,963

No variance ratios different from one are found for the Belgian growth index. Moreover, both

daily and weekly data show that the price of the growth stocks is not subject to mispricing and

that the index has an overall market efficiency. We are able to accept the null hypothesis and

argue that growth stocks follow a random walk in comparison with daily returns of value

stocks.

Graph B presents the variance ratios of the valueand growth index for weekly returns

Graph B Shows the patterns of the variance ratios from weekly returns that we found for the

value and growth indices. Both indices show a mean reversion in the values of the variance

ratios, thereby following almost an identical path. Their path is not completely identical

because the growth index has higher negative correlation. Weak correlations can be found

which are between -9,1 and 3,3 %. On the other hand, the value index has a higher positive

correlation at the end which is 10,7 %. The negative serial correlations of these indices are

very alike, however, the positive autocorrelation differ a lot. For instance, the positive

correlation of the value index of 10,7 % is a three-fold of the positive correlation of the

growth index, which is 3,3 % for 16 base observations. The mean variance ratios of weekly

data for the value and growth indices are both negative and close to zero. The value index has

a negative mean of -0,007 % and the growth index has a value of 3,7 %. The mean value of

the value index implies that the found negative and positive serial correlations are equivalent,

that is why we state that mean reversion for our value index is market efficient. On the

contrary, the growth index gives weak evidence of inefficient mean-reverting behaviour with

more positive autocorrelations than negative ones.

Comparison with Variance ratio literature

We compared growth and value stocks via the usage of variance ratios in order to study if

differences in market efficiency could be found. Our results show that Belgian value stocks do

not follow a random walk when investigating daily returns. When we study weekly returns,

0

0,2

0,4

0,6

0,8

1

1,2

2 4 8 16

Value

Growth

53

we conclude that Belgian value stocks seem to be market efficient. No evidence could be

found which proof that Belgian growth stocks do not follow a random walk. Our overall

conclusion says that the growth indices follow a random walk even on a daily basis and that

the value index for weekly returns is also market efficient. Yet, the results of our value index

based on daily returns reject the random walk. Our findings are in line with Clifford (1988)

who argued in his paper that value stocks are more exposed to speculation than growth stocks.

The returns of these stocks show a different behaviour for every type of stock i.e. value or

growth stock. Clifford (1988) explains these differences in market efficiency by examining

the perspective of stock investors. He states that value investors mainly look for arbitrage

opportunities, whereas, growth investors try to get all the information about the growth

opportunities, instead of looking after arbitrage opportunities, to form a reasonable price.

Did the financial crisis marked by the 15th of September 2008 as the starting date have an effect on market efficiency?

With this question we want to examine two important things: the first one is to find out

whether shorter time intervals could be efficient in comparison with our whole dataset of

Datastream indices going from 1996 until 2014. The second aspect is to examine if the

occurrence of the 2008 financial crisis was a turning point in market efficiency. That would

have to require markets to be efficient for the shorter three-year interval before 15th

September 2008. Practically, we work with two intervals of three years before and after the

key date: 16/09/2005 – 12/09/2008 & 19/09/2008 – 16/09/2011. To avoid biases from

abnormal returns we will not include 15th

September 2008 in the intervals. Lastly, for this

question we use daily and weekly data in the construction of our tests.

1. The Random Walk

Belgian All Shares (BAS)

Both return series using daily data are stationary for both intervals 2005-2008 & 2008-2011.

On top of that the graphical autocorrelation function show no significant lags for the first

interval and several significant lags for the second (post crisis) interval.

2005-2008 ACF daily returns 2008-2011 ACF daily returns

The autoregressions confirm the graphs, pre crisis we can accept a perfect random walk with

insignificant lag coefficients close to zero and no residual autocorrelation. Post crisis, we

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

54

reject the random walk due to a significant lag Rt-2, and other lags of higher orders as shown

in the autocorrelation graph. However the explanatory power of the models is very low not to

say economically meaningless.

BAS – Daily returns Pre crisis Post crisis

α -0,0001

(-0,169)

-0,0003

(-0,4527) Rt-1 -0,0359

(-1,000)

0,0635

(1,780) Rt-2 / -0,0961

(-2,696)***

R² 0,0013 0,0126 Residual autocorrelation No

(DW= 1,99)

No

(Durbin’s h= -0,56) # observations 781 781 Random Walk? Yes No

The same tests carried out with weekly data yield the same results. The series are stationary

and the autocorrelation functions and autoregression models are similar to those applied onto

daily data. However, for the post crisis interval the random walk holds up to the eighth lag

order. In this case it is much harder to reject the random walk, especially since no residual

autocorrelation is present and the lag coefficients are very close to zero.

2005-2008 ACF weekly returns 2008-2011 ACF weekly returns

BAS – Weekly returns Pre crisis Post crisis

α -0,0003

(-0,143)

-0,0018

(-0,597) Rt-1 -0,0163

(-0,202)

0,0029

(0,036) R² 0,0003 0,0000

Residual autocorrelation No

(DW= 2,00)

No

(DW= 1,99) # observations 157 157 Random Walk? Yes Partly

We can conclude that in a three-year period before the crisis marked by 15 September the

BAS was efficient, i.e. following a random walk. In the following three years after the crisis

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

55

outbreak, market efficiency decreased to a lesser degree although the inefficiency is less

severe when using weekly data.

BEL-20

The daily and weekly return series of the BEL-20 index are all stationary which makes them

fit for our tests. The results of the BEL-20 yield the same outcome as the tests of the BAS.

2005-2008 ACF daily returns 2008-2011 ACF daily returns

BEL-20 – Daily returns Pre crisis Post crisis

α -0,0001

(-0,197)

-0,0003

(-0,546) Rt-1 -0,0429

(-1,197)

0,0753

(2,106)** Rt-2 / -0,0696

(-1,949)* R² 0,0018 0,0098

Residual autocorrelation No

(DW= 1,99)

No

(Durbin’s h= -1,93) # observations 781 781 Random Walk? Yes No

Both autocorrelation functions as autoregressions accept the random walk pre crisis and reject

it for the post crisis interval. Pre crisis there are no significant lags as opposed to post crisis

for which a couple of significant lags can be found, indicating autocorrelation in breach of a

random walk. The increments are uncorrelated which is necessary for the RW3. Again the R²

is very low at best.

Weekly returns paint the same picture as is shown here below, no significant lags in the three

years prior to the financial crisis outbreak and only two significant lags of a higher order (8th

& 9th

) for the three years following the outbreak. These results are in concurrence with the

BAS results: market efficiency pre crisis and signs of inefficiency post crisis. However, the

inefficiency is not of a severe economical size, in fact evidence is weak due to low R²’s and

significance of only two higher lag orders of the weekly BEL-20 data set.

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

56

2005-2008 ACF weekly returns 2008-2011 ACF weekly returns

BEL-20 – Weekly returns Pre crisis Post crisis

α -0,0003

(-0,179)

-0,0023

(-0,710) Rt-1 -0,0169

(-0,210)

-0,0328

(-0,408) R² 0,0003 0,0011

Residual autocorrelation No

(DW= 1,99)

No

(DW= 1,99) # observations 157 157 Random Walk? Yes Partly

BEL Mid

As is the case for the BAS and BEL-20, the BEL Mid also brings the same results but with

some differences in the way we can reject the random walk for post crisis intervals. To start

with the daily data, of which the indices are stationary, we show the autocorrelation graphs

and autoregression estimation results.

2005-2008 ACF daily returns 2008-2011 ACF daily returns BEL Mid – Daily returns Pre crisis Post crisis

α 0,0000

(0,103)

0,0001

(0,141) Rt-1 0,0227

(0,634)

0,1415

(3,982)*** R² 0,0005 0,0199 Residual autocorrelation No

(DW= 2,00)

No

(Durbin’s h= -0,22) # observations 781 781 Random Walk? Yes No

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0,25

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0,25

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,1

-0,05

0

0,05

0,1

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

57

As far as daily returns go, everything is in line with our previous results of the two previous

indices. During three years before 15th

September 2008, markets were efficient and following

a random walk. After the crisis, significant lags can be found with coefficients different from

zero which is sufficient to reject the RW3 and efficiency. Now, moving on to the tests based

on weekly data for the BEL Mid that is also stationary as examined with a Dickey-Fuller unit

root test.

2005-2008 ACF weekly returns 2008-2011 ACF weekly returns

A first glance at the autocorrelation function graphs would lead us to believe that weekly

returns indicate market efficiency three years prior and after the crisis. However only the

period prior to the crisis can keep this conclusion, when estimating autoregressions an

interesting phenomenon ensures the rejection of the random walk 3 post crisis. As shown in

the table with the regression results directly below, the Durbin’s h value is highly significant,

accepting the alternative hypothesis of residual autocorrelation, which breaches a core

assumption of RW3. In this case the R² can be inflated, which is probably the case in this

model with an explanatory increase of roughly five percent and coefficients of the lags are

also inefficient as a consequence. On top of that some of the lags are somewhat significant

which also leads to random walk rejection.

BEL Mid – Weekly returns Pre crisis Post crisis

α 0,0002

(0,128)

-0,0001

(-0,044) Rt-1 0,0704

(0,880)

-0,0441

(-0,554) Rt-2 / 0,1518

(1,898)(*) Rt-3 / 0,0027

(0,033) Rt-4 / -0,0210

(-0,262) Rt-5 / 0,1789

(2,222)** R² 0,0050 0,0578

Residual autocorrelation No

(Durbin’s h= -1,65)

Yes

(Durbin’s h= -3,67) # observations 157 158 Random Walk? Yes No

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

58

BEL Small

The BEL Small compartment is the outsider in compartment classes when it comes to the

results of our tests to answer this third research question. The random walk is rejected pre and

post crisis, using daily as well as weekly data. The daily and weekly returns have stationary

indices pre and post crisis, fit for testing.

2005-2008 ACF daily returns 2008-2011 ACF daily returns

Graphs showing significant autocorrelations post and pre crisis and the regression results with

significant t-statistics, prove what we stated before: the BEL Small is the compartment’s

misfit having significant lags in all intervals. Markets comprising the Belgian small caps were

inefficient before the crisis broke out and they remain inefficient for the following three years

after the outbreak, we found several significant lags. On top of that, post crisis the Durbin’s h

value is only just in range of the null hypothesis implying no residual autocorrelation.

BEL Small – Daily returns Pre crisis Post crisis

α 0,0001

(0,627)

0,0001

(0,178) Rt-1 0,1259

(3,517)***

0,1475

(4,153)*** Rt-2 0,0951

(2,648)***

0,1343

(3,780)*** Rt-3 0,0709

(1,979)**

/

R² 0,0377 0,0466

Residual autocorrelation No

(Durbin’s h= 1,32)

No

(Durbin’s h= 1,95) # observations 781 781 Random Walk? No No

With weekly data the tests yield the same results, though the rejection power to the random

walk seems weaker when compared to the test results using daily data. Less significant lags

can be found, t-statistics are weaker and the Durbin’s h values also are closer to accepting no

residual autocorrelation. In fact, with tests using weekly data, the BEL Small seems more

efficient after the crisis. Only one significant lag post crisis remains and this one even seems

abnormal when viewing the autocorrelation graph directly below. The absolute value of the

Durbin’s h has decreased as well post crisis. The improvement of market efficiency for

smaller caps post crisis is an interesting and unexpected result.

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

59

2005-2008 ACF weekly returns 2008-2011 ACF weekly returns

BEL Small – Weekly returns Pre crisis Post crisis

α 0,0006

(0,586)

0,0001

(0,045) Rt-1 0,1968

(2,522)**

0,0208

(0,265) Rt-2 / 0,2245

(2,858)***

R² 0,0394 0,0510

Residual autocorrelation No

(Durbin’s h= -1,32)

No

(Durbin’s h= 0,39) # observations 157 157 Random Walk? No No

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,1

0

0,1

0,2

0 2 4 6 8 10 12

lag

ACF for ld_PI

+- 1,96/T^0,5

-0,2

-0,1

0

0,1

0,2

0 2 4 6 8 10 12

lag

PACF for ld_PI

+- 1,96/T^0,5

60

2. The Variance ratio

Daily returns Table 2a provides the results of our variance ratio for daily data based on our subsamples. The first value is the

value of the variance ratio which is followed by the Z-statistics. The Z-statistic is the value under

heteroscedasticity.

Type of

index

Number of observations

BAS 2 4 8 16 Mean RV

Non-crisis

period

0,951

(-0,990)

0,984

(-0,171)

1,023

(0,152)

1,114

(0,515) 1,018

Crisis period 1,062

(0,916)

0,981

(-0,151)

0,985

(-0,077)

0,932

(-0,233) 0,990

BEL-20

Non-crisis

period

0,945

(-1,131)

0,952

(-0,524)

0,984

(-0,108)

1,076

(0,346) 0,989

Crisis period 1,076

(1,237)

1,041

(0,350)

1,031

(0,165)

1,003

(0,010) 1,038

BEL Mid

Non-crisis

period

1,018

(0,412)

1,121

(1,388)

1,204

(1,413)

1,219

(1,010) 1,141

Crisis period 1,139*

(2,061)

1,225

(1,762)

1,232

(1,174)

1,195

(0,696) 1,198

BEL Small

Non-crisis

period

1,146*

(2,670)

1,402*

(3,811)

1,710*

(4,054)

2,043*

(4,177) 1,575

Crisis period 1,173*

(2,611)

1,418*

(3,092)

1,598*

(2,800)

1,838*

(2,818) 1,507

When we see the results of the BAS in table 2a, we do not find any significant values for our

variance ratios. Both subsamples show that the BAS stock index was market efficient during

these periods. We accept the null hypothesis at a significance level of five percent. Although

both subperiods show market efficiency, there are some differences between these periods.

We state that during the non-crisis period, negative serial correlation of maximum 0,049 can

be found at the beginning but reverses into positive serial correlation of 2,3 and 11,4 %. The

follow up of negative and positive autocorrelation is dubbed as mean-reverting behaviour.

The crisis period show a positive correlation of 6,2 % at the start but changes into a negative

correlation which are between 1,9 and 6,8 %. Both periods show mean-reverting behaviour,

though, this behaviour happens differently for every period. Table 2a also provides more

information about the variance ratios for the BEL-20 index. We have almost the same

conclusions in comparison with our conclusion of the BAS index. The BEL-20 index also

contains mean-reverting behaviour. However, we have to note that the index is market

efficient before the crisis as well as after the crisis and that none of the variance ratios are

statistically significant. We accept for both periods the null hypothesis that the variance ratios

are equal to a value of one. In table 2a, variance ratios for the BEL Mid are displayed. We

find one significant value of 1,139 for the variance ratio of the crisis period for a number of

two observations. Additionally, we find a z statistic of 2,061 which is larger than our critical

value of 1,96. The value of 1,139 imply that for the crisis period, we find a large

61

autocorrelation of 13,9 %. If we analyze the insignificant variance ratio, no mean-reverting

behaviour can be found. We only find positive correlations between our returns of the BEL

Mid stock index. Because we are able to find autocorrelation for the BEL Mid index, we

argue that this index is market inefficient when the crisis broke out. Yet, the BEL Mid index

is market efficient before the crisis of 2008. The BEL Small index of table 2a is strong market

inefficient since we find for every variance a significant value. The minimum value we find is

a value of 2,611 and we find a maximum value of 4,177. Both periods provide evidence

against the market efficiency hypothesis; therefore, we reject the null hypothesis that the

variance ratios show no autocorrelation. These autocorrelations, we find, differ from 14,6 %

till an astonishing 104,3 %. If we compare both periods than we find that the serial correlation

gets stronger through time. We also find that the non-crisis period ends with a larger

autocorrelation in comparison with the crisis period, however, at the beginning the largest

autocorrelation is found for the crisis period. The z statistics of the BEL Small show stronger

market efficiency before the crisis compared with our subperiod after the Fall of Lehman

Brothers.

Graph A presents variance ratios of the BAS index for the non-crisis and the crisis period

Graph B presents variance ratios of the BEL-20 index for the non-crisis and crisis period

0,8

0,85

0,9

0,95

1

1,05

1,1

1,15

2 4 8 16

Non-crisis

Crisis

0,85

0,9

0,95

1

1,05

1,1

2 4 8 16

Non-crisis

Crisis

62

Graph C presents variance ratios of the BEL Mid index for the non-crisis and crisis period

Graph D presents variance ratios of the BEL Small for the non-crisis and crisis period

Graph A shows the patterns that the variance ratios of the BAS follow for two sub periods of

which is before the crisis of 2008 and one is during the crisis of 2008. During the non-crisis

period, we see a mean reversion in the variance ratios. The two first variance ratios are under

one (i.e. negative first-order correlation) and our study ends with the two variance ratios

which are larger than one. The serial correlations lie between -4,9 and 11,4 % which is rather

small. The variance ratios of the crisis period show an opposite pattern. This pattern starts

with one positive variance ratio and ends with three negative variance ratios. Autocorrelations

of -6,8 and 6,2 % are found and these are quit small. The mean variance ratios of the BAS

indicate serial correlations of 1,8 % for the non-crisis period and 0,011 % during the crisis

period. We argue that these means are weak because of the low correlations of 1,8 and 0,01 %

that we find. It is even remarkable to find a mean of 0,01 % that is near zero for the crisis

period, indicating market efficient mean reversion. The graphic presentation of the variance

ratios of the BEL-20, which can be found in graph B, is clearly showing differences between

our subperiods. Before the crisis, mean-reverting behaviour is detected but mean reversion

disappears when the crisis occurs during our crisis period. Serial correlations between -5,5

and 7,6 % are found but they remain small once again. The mean variation ratio of the BEL-

20 is close to zero with a ratio of 0,011 %. The negative and serial correlations balance each

other out, therefore, we state that this mean reversion is market efficient. Graph C contains the

variance ratios of the BEL Mid index for both sub periods. We do not find any mean

reversion in our results. Graph D contains all variance ratios calculated of daily returns. We

0

0,2

0,4

0,6

0,8

1

1,2

1,4

2 4 8 16

Non-crisis

Crisis

0

0,5

1

1,5

2

2,5

2 4 8 16

Non-crisis

Crisis

63

see that both sub periods look very similar but we do not find any signs of mean-reverting

patterns as well.

Weekly returns Table 2b provides the results of our variance ratio for weekly data based on our subsamples. The first value is the

value of the variance ratio which is followed by the Z-statistics. The Z-statistic is the value under

heteroscedasticity.

Type of

index

Number of observations

BAS 2 4 8 16 Mean VR

Non-crisis

period

0,979

(-0,244)

1,103

(0,612)

1,353

(1,323)

1,094

(0,250) 1,132

Crisis period 0,986

(-0,197)

0,762

(-1,655)

0,667

(-1,274)

0,551

(-1,066) 0,742

BEL-20

Non-crisis

period

0,983

(-0,198)

1,119

(0,714)

1,339

(1,287)

1,042

(0,114) 1,121

Crisis period 0,944

(-0,588)

0,780

(-1,246)

0,721

(-0,968)

0,582

(-0,930) 0,757

BEL Mid

Non-crisis

period

1,066

(0,789)

1,136

(0,847)

1,305

(1,183)

1,483

(1,262) 1,248

Crisis period 0,924

(-0,875)

0,854

(-0,820)

0,889

(-0,378)

0,751

(-0,544) 0,855

BEL Small

Non-crisis

period

1,193

(1,824)

1,456*

(2,435)

1,644*

(2,319)

2,001*

(2,645) 1,574

Crisis period 1,026

(0,311)

1,096

(0,643)

1,157

(0,569)

0,954

(-0,103) 1,058

Table 2b displays the variance ratios of the BAS which are all statistically insignificant for

both subperiods. However, the variance ratios can be economically significant and provide

useful results. The non-crisis period, for instance, shows mean-reverting behaviour because

we find a negative autocorrelation of 1,4 % for two base observation which changes into a

positive autocorrelation when the amount of base observations gets larger. The crisis period is

not similar at all and displays no mean-reverting behaviour since we only find negative serial

correlation between 1,4 and 44,9 % which becomes stronger over time. Still, we argue that the

results of the BAS do not reject the random walk that stock returns are unpredictable. The

variance ratios of the BEL-20 index in table 2b are very similar to those of the BAS. We do

not find significant variance ratios at a significance level of five percent and find mean

reversion for the non-crisis period. In comparison with the crisis period of the BAS, the crisis

period consists of negative autocorrelation that grows over time. These conforming results are

due to the fact that the BAS index is strongly influenced by the BEL-20 shares since the BAS

index is a weighted index and that BEL-20 shares have a large market capitalization. Table 2b

also displays our results of the BEL Mid index. Like all previous indices, we are not able to

find statistically significant variance ratios. The highest z statistic for the non-crisis period

amounts for 1,262 which is not even close to our critical value of 1,96. Moreover, no mean-

reversion is found in our data. Although that the non-crisis period does not contain any

64

significant variance ratios, they seem to be positive serial correlated which correlations grows

when the amount of base observations gets larger. Like the non-crisis period, we do not find

mean reversion but we do find negative serial correlation for all variance ratios. Table 2b

displays the variance ratios of the BEL Small and, compared with the other indices,

provides us with significant values. Yet, we only find significant values for the non-crisis

period for a base observation of 4,8 and 16. The autocorrelation goes from 45,6 % to a

remarkable 100 %. The z statistics also vary from 2,319 to 2,645. When we analyze the crisis

period, we find no significant values but we find mean-reverting behaviour. We find a

negative autocorrelation for a base observation of 16 and we find positive serial correlation

for the remaining base observations. To sum up, the non-crisis period of the BEL Small index

is proof against the random walk and that is why we reject the null hypothesis that the

variance ratios are different from one. Yet all other indices and their subperiods support the

theory that stock returns are random, we find mean reversion for the BAS and the BEL-20

indices.

Graph A present variance ratios of the BAS for the non-crisis and crisis period

Graph B presents variance ratios of the BEL-20 for the non-crisis and crisis period

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

2 4 8 16

Non-crisis

Crisis

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

2 4 8 16

Non-crisis

Crisis

65

Graph C presents variance ratios of the BEL Mid for the non-crisis and crisis period

Graph D presents variance ratios of the BEL Small for the non-crisis and crisis period

Graph A displays the variance ratios of the BAS index for weekly stock returns. Both periods

show a graphical divergence where the non-crisis period contains mean-reverting behaviour

and the crisis period does not. The non-crisis period starts with a negative serial correlation of

-2,1 % which is very small. In other words, a very small return predictability is found. The

following variance ratios all contain positive serial correlation with a maximum of 9,4 %. The

mean variance ratio of 13,2 % of the BAS index is large enough to conclude that the mean

reversion is market inefficient. The positive serial correlations are stronger than the negative

ones. The crisis period does not have a mean-reverting pattern in its variance ratios. Graph B

looks similar to graph A because of the identical patters. The autocorrelations that we find are

between -1,7 and 9,4 %. An overall mean correlation of 12,1 % can be found. This is rather

large and strong enough to show that the mean reversion of the BEL-20 is inefficient. When

we study graph C, we cannot state that the BEL Mid is subject to mean reversion. Graph D

indicate mean reversion when the line of the BEL Small goes under the value of one. We find

a correlation interval of 15,7 and -4,6 %. We find a total correlation of 5,8 % that is half the

correlations that we find for the BAS and the BEL-20 indices. There is a small imbalance of

negative and positive serial correlation where the positive correlation has slightly the upper

hand but this is not enough to reject market efficient mean-reverting behaviour.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

2 4 8 16

Non-crisis

Crisis

0

0,5

1

1,5

2

2,5

2 4 8 16

Non-crisis

Crisis

66

Conclusion Variance ratios

Without looking at the results of the BEL Small, we conclude that the Belgian stock indices

follow a random walk during our subperiods. The daily returns of the BEL Small, however,

do not follow a random walk during the crisis and non-crisis period where the BEL Small is

only market inefficient during the non-crisis period when we analyze weekly returns. We

conclude that our smallest index is more weak-form efficient for daily data and even becomes

weak-form efficient for weekly returns during the financial crisis. Why weekly returns are

different from daily returns is hard to tell due to the ample explanations that are found during

a crisis. Though we are not able to find foundations on which we can base this theory, we can

carefully assume it. The theory, which is based on the positive-feedback effects of

Lakonishok et al. (1992), sounds that a proportion of the investors were noise traders which

bought increasing stocks and sold losing stocks. This investor sentiment could explain the

serial correlation in our results. The crisis broke out, thus noise traders moved out of the

market after losing on their investments. Eventually this resulted in more market efficiency

for daily returns and complete market efficiency for weekly returns.

67

V. Conclusion

To end our paper, investigating Belgian stock market efficiency we will put forward the most

interesting conclusions that our research yielded of the random walk autoregressions and the

variance ratio tests.

1. Conclusions Random Walk autoregression models

Between 1996 and 2014, the random walk 3 hypothesis is rejected when using daily and

quarterly returns for the entire Belgian stock market reflected by the BAS. However, in both

cases issues arise. For daily data, biases can be at the base for the rejection of the random

walk such as infrequent trading and bid-ask spreads that are elements of the nonsynchronous

trading effect (Lo and MacKinlay, 1988, 1990). With the quarterly returns we assumed the

return series to be stationary when in fact weak proof of unit root presence was available. Use

of weekly and monthly data for the tests results in random walk acceptance up to higher lag

orders. Nevertheless, we should reject the random walk and brand the Belgian stock market as

inefficient, in accordance to the result of Lee et al. (2005) who found inefficiency between

1999 and May 2007, as mentioned in the literature overview. Our research of the BEL-20

yielded mainly the same results. Judging by Fama’s past statements, he would probably not

agree with our conclusion as the explanation power of the models are fairly low (almost non-

existing), the presence of a number of insignificant lags with coefficients close to zero that

can imply mean-reversion and the questionable stationarity of the quarterly data for the BAS

and BEL-20. For the same 1996-2004 period the BEL Mid is also inefficient when using daily

and weekly returns. However with monthly and quarterly returns we can make a good case for

random walk acceptance and efficient markets. With the quarterly returns dataset all RW

conditions are met. The random walk for BEL Small data does not apply. None of the daily,

weekly, monthly or quarterly BEL Small returns meet the random walk conditions.

To answer our fist research question, we cannot conclude that the Belgian stock market is

efficient over a period between 1996 and 2014. However the BAS and BEL-20 are efficient

up to a higher lag order when using weekly or monthly data. So even though we cannot fully

and strictly accept the efficient markets hypothesis, we take comfort in the fact that the overall

market is efficient to some degree, whereas the compartment for small caps is not efficient at

all. The BEL Mid positions somewhere in the middle when it comes to market efficiency.

When lengthening the time frequency of returns to construct autoregressive models, the

autocorrelation significance becomes less powerful which implies easier outcomes to accept

the random walk.

Under our second research question, we divided the BAS into growth and value indices. But

no matter the frequency of the returns, all used indices come up rejecting the random walk

and thereby also market efficiency. Interesting fact is that the growth indices seem more

eligible to accept the random walk in comparison with value indices. This seems strange

68

intuitively as value indices include stable firms with a good financial reputation, which would

give us an ex ante impression that value stocks are priced more efficiently. In fact, the

opposite is proven by our autoregressions and autocorrelation functions. A last remarkable

observation is that when examining the entire BAS instead of the divisions into growth and

value indices, the undivided BAS is more efficient and accepts the random walk to a certain

degree with weekly and monthly return data.

In our third research question we examine the influence of the financial crisis outbreak and we

use two three-year period intervals pre and post crisis outbreak, resulting in a non-crisis and

crisis period. To recap, these are the precise periods by date: 16/09/2005 – 12/09/2008 and

19/09/2008 – 16/09/2011. In our analysis we used daily and weekly data and the results are

truly remarkable. The BAS followed a random walk and thus was market efficient in the non-

crisis period. Furthermore, when using weekly data, the post crisis interval seems to partly

follow the random walk, only with rejections due to high order autocorrelation. We can

conclude that the Belgian stock market as a whole was efficient, three years before the crisis

broke out and became less efficient after the outbreak due to a (sometimes less powerful and

partly) rejection of the random walk. The exact same is to be said from the BEL-20 results.

For the BEL Mid we get the same result as well but when using weekly returns we have to

reject the random walk post crisis due to residual autocorrelation, which violates the RW3

assumption of uncorrelated increments. Ending with the BEL Small, we get different results.

The BEL Small did not follow a random walk before the crisis outbreak as well as three years

after. Although, and this is remarkable, post crisis the autocorrelating lags in our models

became less significant and the amount of significant lags diminished up to the twelfth order.

This could imply a higher degree of market efficiency for the small caps compartment after

the crisis, albeit still in an inefficient part of the market.

The results of our third and last research question are in consensus with the research of Kim et

al. (2011) that we discussed earlier in the literature section. They found that market efficiency

varies over time and when examining subperiods of a dataset, these subperiods may be

efficient even though the entire dataset has an inefficient outcome which is the case in our

research. Their research also found lower explanatory power, R² of their regressions when the

market crashed. Our regressions had the opposite effect, in the subset after the market crash,

the R² explanatory power increased although it remained at a limited low value.

We can answer our third research question simply, the financial crisis did have an effect on

market efficiency. For the BAS, the BEL-20 and the BEL Mid we found efficient markets pre

crisis, in a normal period of economic and financial tranquility. After the crisis broke out, we

found signs to reject the random walk in the BAS, the BEL-20 and the BEL Mid which

implies inefficient markets. The BEL Small is the exceptional case, pre crisis this

compartment of the market was inefficient due to random walk rejection. And in the crisis

period starting from 19/09/2008, we found improvements in market efficiency.

69

2. Conclusions Variance ratios

The variance ratios show that the Belgian stock market is inefficient since most indices do not

follow a random walk. The variance ratios reject the random walk hypothesis since daily

returns are predictable for the BAS, BEL-20 and the BEL Mid. The BEL Small on the other

hand contrasts our expectations since the variance ratios suggest a random walk for daily data.

The variance ratios of weekly returns are different from daily returns. Biases like bid-ask

spreads and the nonsynchronous trading effect could be the reason for this divergence Lo and

MacKinley (1988). More interesting results come from weekly data where the smallest

indices, the BEL Mid and the BEL Small, do not follow a random walk. In contrast to the

smallest indices, the returns of the largest indices are not predictable i.e. follow a random

walk. Lo and MacKinley (1988) get similar results in their paper where a smaller index tends

to be less market efficient than a larger index

By analyzing the Belgian value and growth indices, we gathered more insights about the

distinction in market efficiency between these two indices. Weekly returns show that the

value index and the growth index both follow a random walk. Nevertheless, daily returns

provide a different conclusion, support to reject the random walk is found for the value index

whereas the growth index accepts the random walk. Therefore, we conclude that value stocks

indices are less efficient when we take daily and weekly returns into account.

Comparing return data influenced by a crisis in the financial markets with data of prosperous

times in the financial markets, brings us interesting results. In the third research question, we

used daily and weekly returns as they count the largest number of data entries. Daily variance

ratios imply that all four Belgian indices follow a random walk during both subperiods.

Moreover, weekly variance ratios of the subperiods also show that the BAS and the BEL-20

follow a random walk. The BEL Mid index becomes inefficient due to the crisis. The BEL

Small is market efficient before the crisis but becomes inefficient after the outbreak of the

financial crisis.

3. Differences and matches between the RW autoregression models & Variance ratios

Differences between the results of both tests can be explained by how both tests analyse the

market data. Our random walk autoregression models look beyond the first lag order and this

makes our regressional random walk model more sensitive for predictability in returns than

the variance ratio test that only takes first-order correlation into account. This should make us

prefer to choose the test results of the autoregression before the variance ratio test when the

outcomes contradict, although we must admit that both tests complete each other very well.

The random walk autoregression as well as the variance ratio test both hold a general rejection

of market efficiency in the Belgian stock market for the period between 1996-2014. Our tests

are able to predict some future returns of Belgian stock indices based on the past returns as is

shown by the significant autocorrelations. However, for the BAS and BEL-20, the RW

rejection is of a lesser degree, implying less market inefficiency, albeit still inefficient. In our

70

research of growth and value indices we get the same outcome by both tests, growth stocks

seem to be more efficient than value stocks, although they are both inefficient. In the answer

of our third research question, there are differences when comparing the tests. Where the

regressions show a clear turning point in market efficiency after the crisis outbreak for daily

and weekly data, the variance ratio yields slightly contradicting results. However the market

efficiency conclusion for the non-crisis interval for the BAS, the BEL-20 and the BEL Mid is

in accordance. The two main differences are that with the variance ratio the turning point (i.e.

the effect of the crisis) is only observable for the BEL Mid and the other difference is that the

BEL Small results differ as well. Nevertheless, we prefer and trust our test results from the

autoregression models as we explained that these are more reliable since they take also higher

correlation orders into account.

4. Fama vs. Shiller, who fits the Belgian stock market?

To end we take a moment to remember our title. We do not want to point out any theories or

Nobel Prize winners as absolute victors but it is remarkable that our research, that strictly

viewed rejects market efficiency for the last eighteen years, could be simply disregarded by

Fama. As we already mentioned, our autoregression models have limited explanatory power

and the significant lags have limited coefficients close to zero which also implies a limited

forecasting ability and no real economical use. On top of that, some statistical issues arise as

well such as questionable stationarity of some quarterly return series combined with limited

data entries in all quarterly data sets. Then the question of the ability to test market efficiency

and the joint-hypothesis problem as explained before in the literature review remains open for

discussion between academics. So an absolute efficiency statement will likely never be

correctly made. Nevertheless we attempted to make a statement of relative efficiency in which

different markets or market compartments are compared to each other as suggested by

Campbell et al. (1997). We believe to have succeeded in this task and more importantly to

have showed that market efficiency is a complex and dynamic process in which efficient

periods are possible even though an entire dataset comprising eighteen years of data does not

seem efficient. Another referral to Kim et al. (2011) seems well in order, who also focus on

the dynamic process that is market efficiency. Concluding that market efficiency is dynamic

we would also like to recommend active portfolio management for investors. When inefficient

periods occur it is logical that the seasoned investor or investment firms should be able to

make additional profits and benefit from this market state. However, this is not as simple as

we make it out to be. Surely, Lo (2004, 2012) the founder of the adaptive markets hypothesis

and also many behavioural finance academics would agree that such opportunities will not be

correctly exploited due to errors in human behaviour and investment decision. Additional

research in this field will thus surely be justified.

71

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75

VII. Appendix

Testing of the full sample for heteroskedasticity/homoskedasticity used for the variance ratios.

Type of index Daily returns Weekly return Quarterly returns

BAS 0,000* 0,00* 0,235

BEL-20 0,000* 0,672 0,623

BEL Mid 0,000* 0,011* 0,320

BEL Small 0,002* 0,178 0,520 *indicates that the data is heteroskedastic

Dickey-Fuller test-statistics for the examined indexes before running autoregression models.

Index Test-statistic

BAS daily returns -21,985

BAS weekly returns -7,722

BAS monthly returns -4,896

BAS quarterly returns -2,792 (!)

BEL-20 daily returns -22,172

BEL-20 weekly returns -8,001

BEL-20 monthly returns -4,386

BEL-20 quarterly returns -2,856 (!)

BEL Mid daily returns -17,762

BEL Mid weekly returns -9,068

BEL Mid monthly returns -3,402

BEL Mid quarterly returns -6,548

BEL Small daily returns -19,795

BEL Small weekly returns -10,303

BEL Small monthly returns -3,952

BEL Small quarterly returns -5,970

BAS Growth daily returns -35,685

BAS Growth weekly returns -18,468

BAS Growth monthly returns -10,458

BAS Value daily returns -27,438

BAS Value weekly returns -13,421

BAS Value monthly returns -6,199 The values marked by (!) do no strictly pass the critical value. All test statistics have to be compared

to critical value -2,89 as no deterministic trends were significant.