Anders Lönnquist 910707
Fall Term 2018
Master Thesis, 15 credits Department of Statistics
Örebro University School of Business
Supervisor: Farrukh Javed, Senior Lecturer, Örebro University Examiner: Stepan Mazur, Senior Lecturer, Örebro University
The efficiency of
the Swedish stock
market
Preface
Initially, I would like to extend a special thanks to my supervisor Farrukh Javed, Assistant Professor of Statistics at Örebro University. He has throughout this thesis project assisted me with insightful comments and guidance. I would also like to thank all my fellow students whom have taken their time to proof read and give constructive criticism.
Abstract
The purpose of this thesis has been twofold. Firstly, we have sought to evaluate if the Swedish stock market is efficient according to the weak for of the efficient market hypothesis. Secondly, we have tried to determine whether a stocks turnover, number of trades and trade volume has any significant effect on the probability of a stock being efficient. To achieve these purposes, runs test, variance ratio tests, autocorrelations tests and logistic regressions have been employed. The results of these tests suggest that some stock on the Swedish stock market might be efficient, whilst others clearly exhibits indications of inefficiency. As such, the market as a whole is not efficient according to the weak for of the efficient market hypothesis, whilst it is highly possible that individual stock might be. Furthermore, based on 12 logistic regressions we have been able to determine that a stocks turnover, number of trades and trade volume does not significantly affect a stocks probability of efficiency.
Content
1. Introduction and institutional background ... 1
2. Theoretical framework and previous studies ... 6
2.1. The weak form of market efficiency... 7
2.2. The semi-strong and strong form of market efficiency ... 8
2.3. Previous studies ... 9
3. Data ... 12
4. Method ... 15
4.1. Wald–Wolfowitz runs test ... 15
4.2. Variance ratio test ... 16
4.3. Ljung-Box test ... 18
4.4. General method and Logistic regression ... 19
5. Results and Analysis ... 22
5.1. Runs tests ... 22
5.2. Variance ratio tests ... 23
5.3. Ljung-Box test ... 25
5.4. Combined approaches ... 28
5.5. Logistic regressions... 29
6. Conclusions and Discussion ... 30
7. References ... 32
8. Appendix ... 36
8.1. Time series ... 36
1
1. Introduction and institutional background
In 1970, Eugene Fama published his seminal paper on the informational efficiency of financial markets. The papers contribution to the modern perception of market fluctuations is undeniable, and it has forever been enshrined in financial history by academic acknowledgements such as the Nobel prize1 (Nobel prize, 2013). Thus, the conclusions and theories put forth in this seminal paper are unquestionably of great importance and worthy of consideration. However, the intuitive conceptualization of efficient markets cannot be attributed to Fama alone, but rather to a series of events and individuals for without whom it is unlikely that the paper ever would have been published. Thus, a fitting introduction to the topic of market efficiency does not begin with Fama, but rather with the Dutch East India company, which in 1602 was instrumental in founding the world first stock market, namely the Amsterdam stock exchange (Petram and Richards, 2014). At the time, the exchange was nothing more than a building in which individuals could trade stocks and bonds associated company. However, due to the risk associated with seventeenth century sea travels, fortunes were quickly gained and lost (Ibid, 2014). Before long, the clear financial incentives inspired many to seek entry to the relatively aristocratic group of investors.
The concept of stock exchanges grew slowly throughout western societies, and it was not until the early nineteenth century that most civilised countries in the west had established some form of stock exchange (Smith, 1929). Additionally, by the early nineteenth century, the industrial revolution was in full swing and the wealth of nations was on the rise. Consequently, increased prosperity and financial surplus facilitated a surge of new prospects and investor (The Economist, 2018). In turn, this surge of investors increased the demand for more sophisticated and academic analyses of market fluctuations, a demand, that in part would come to be met by the French stockbroker Jules Regnault. In his seminal contribution, Regnault (1863) stated that… “the deviation of prices is directly proportional to the square root of time”, which has come to serve as a foundation of modern finance and efficient markets. At this time however, the observed pattern battled many contemporary scholars, for no coherent theory had yet been developed to explain the phenomenon. Today of course, we know that what the author had observed was the yet to be discovered random walk property of prices.
2 In subsequent years, authors such as Rayleigh (1879) and Venn (1888) would come to illustrate the theoretical foundation for Regnault (1863) observed patters, namely the theories of random walks and Brownian motion. Naturally, the concepts were still in their theoretical infancy, but the importance of the authors contributions should not be neglected, for both lie at the heart of not only efficient markets, but a vast literature belonging to a diversified range of sciences.
Two years after the pioneering works of Rayleigh (1879) and Venn (1888), Gibson (1889) declared a clear reference to the informational efficiency of financial markets whilst investigating the properties of the London, Paris and New York stock exchange. The author stated that “shares become publicly known in an open market, the value which they acquire may
be regarded as the judgment of the best intelligence concerning them”. Thus, what Gibson
described was the essence of what Fama would come to define some 81 years later. Consequently, Gibson (1889) can justifiably be regarded as one of the first to attempt to specify the efficient market hypothesis (EMH).
Although Gibson (1889) quite successfully captured the essence of the EMH, much of the theoretical and conceptual understanding was still absent. However, it would not take long before the French mathematician Louis Bachelier published his seminal doctoral thesis
“Théorie de la speculation”, in which he meticulously describes a mathematical model for
Brownian motion, and its utility in option pricing. More than this, Bachelier (1900) illustrated that …”the mathematical expectation of the speculator is zero”, a conclusion which has come to serve as a pillar stone of modern finance and efficient markets. With both these innovative contribution, it is often argued that Bachelier (1900) marks the dawn of modern finance and mathematical modelling in economics (Protter, 2008).
In subsequent years, much of Bacheliers (1900) contributions where ignored and therefore independently derived by authors such as Einstein (1905), Barriol (1908) and Langevin (1908) whom gain the praise of the contemporary academic establishment. Their contributions sparked a frenzy among financial scholars, which resulted in publications such as Dibblee (1912), Bachelier (1914), Taussig (1921) and Keynes (1923), whom famously proposed that risk, rather than superior information is the determinant of returns.
By the late 1920s the stock market was booming, the financial incentives peaked, and the academic interest was at an all-time high. However, on the 29 of October 1929, a date that will live in infamy, the stock market crashed, and with it, much of the preconceived notions of stock market fluctuations. Nonetheless, the notion of a random and efficient markets endured, which
3 is exemplified by, among others, Working (1934) and Keynes (1936) whom with regards to stock price fluctuations argued that they …”can only be taken as a result of animal spirits—of
a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities”.
By the mid-1930s, most of the theoretical foundation for the EMH and the random walk hypothesis (RWH) were derived. However, coherent, comprehensive and explicit theories as to why the stock market appeared random were scares. Thus, most of the empirical studies focused on testing the consequences of randomness, rather than the randomness itself. One such study was conducted by Cowles (1933), who investigated the performance of some 7500 stock recommendations between 1928-1932. In this study, the author concluded that professional recommendations yield no excess returns, thus being one of the first to empirically validify the existence of efficient markets.
During subsequent decades, a vast literature regarding the efficiency of financial markets started to emerge. However, the conclusions, definitions and viewpoints differed substantially, ranging from Friedman (1953) and Kendall (1953) positive endorsement to Cootners (1962) reserved scepticism. Thus, by the mid to late 1960s there was no academic consensus regarding how, and if, financial markets were efficient. Moreover, due to the lack of a coherent definition of what an efficient market was, the comparison between different viewpoints was problematic to say the least.
It was in this time of turmoil and academic dispute that Fama (1970) defined the previously ambiguous concept of market efficiency. He categorised it according to three different forms, each corresponds to the incorporation of different information sets into asset prices. These forms were defined as weak, semi-strong and strong, wherein the forms correspond to the incorporation of information regarding prices, all publicly available information and all available information, respectively. These definitions quickly became the norm and is to this day what is generally referred to when speaking of market efficiency.
Although the EMH was met with great enthusiasm among financial scholars, there were still those who found the paradoxical and counterintuitive market view preposterous. A clear example of this, is presented by Beinhocker (2007) whom, although in favour of the EMH, compares the theories practical implication with the following:
4 An old economist and a young economist walking down the street. The young economist looks down and sees a $20 bill on the street and says “Hey, look a twenty-dollar bill!” Without even looking, his
older and wiser colleague replies, “Nonsense. If there had been a twenty-dollar bill lying on the street, someone would have already picked it up by now” (Beinhocker, 2007. p 53).
Similarly, without resorting to analogies, the EMH suggests that excess returns are not attainable since prices should reflects all relevant attainable information. However, the EMH simultaneously suggests that markets can only be efficient if excess returns are achievable as a result of attainable information, for if this were not to be the case, there would be no private incentive to seek information to incorporate, thus creating an inefficient market. Consequently, it can be argued that the EMH rests on a non-arbitrage assumption. Clearly, this is a paradoxical conundrum with paramount practical implications, were perhaps the largest being the degree to which the billion-dollar industry of actively traded funds is financially justified.
Consequently, many empirical tests and studies have been conducted in order to established whether the EMH holds true (Zhang et al., 2012; Narayan et al., 2015; Simmons, 2012; Hårstad, 2014). However, these tests and their conclusions are inconsistent to say the least. Largely, the reason for this, is that the conclusions depend on which test is applied, which assumptions are assumed and a range of different market specific characteristics. As such, there exists no real consensus on whether the EMH holds true, especially with regards to lesser known markets for which the empirical investigation has been less extensive.
Therefore, in this thesis, we aim to investigate one such market, namely the Swedish stock market. More specifically, it is this thesis intent to try to answer two questions, namely
1. Is the Swedish stock market efficient according to the weak form of the EMH?
2. Is efficiency an increasing function of a stocks turnover, number of trades and trade volume, as is suggested in previous literature?
The answers to these questions might very well have paramount implications, for it the market is deemed efficient there is no financial justification for trading strategies based on historical prices (technical analysis). Furthermore, it could be of relevance not only to investors, but to regulatory agencies which seeks to maintain a fair and competitive marketplace. Moreover, if there is a positive relation between efficiency and a stocks turnover, number of trades and trade volume it could justify large capital flows to smaller markets, such as First North and the alternative stock market.
5 To answer the first question outlined above, historical daily prices of 30 stocks on the OMX30, as well as the volume weighted index, was collected. The data was collected for the period 2002-05-17 to 2017-12-29, which after some adjustment resulted in 30 price series with 3928 observations and one price series with 132 observations. Based on the logarithm of these price series, runs tests, variance ratio tests and autocorrelation tests were used to determine whether the null hypothesis of randomness could be rejected, and thus indicate the invalidity of the EMH. Moreover, the results from these tests are summarized in Table 1, which are mostly consistent with previous literature and suggest that the market as a whole is not efficient. Furthermore, two different approaches, which are further discussed and presented in the methodological and result section, were utilized to combine all tests.
Moreover, to answer the second question, differing assumptions were used to construct 12 binary variables that indicate weather individual stocks are efficient. These binary variables were then used as dependent variables in 12 different logistic regressions, where the stocks average turnover, number of trades and trade volume were used as covariates. In contrast to previous literature, all regressions fail to find any statistically significant effect of turnover, number of trades and trade volume on efficiency, thus indicating their irrelevance in the contest of efficiency.
In summary, the empirical findings suggest that some, but not all stocks on the OMX30 might be efficient according with the weak form of the EMH. Consequently, this suggests that some stocks, but not the market, might be efficient. Furthermore, the results suggest that the deterministic factors of efficiency are not turnover, number of trades and trade volume, as is suggested in previous literature. Lastly, there is always room for improvement, and in future studies it would be interesting to evaluate the EMH with a larger sample size, different tests, a broader range of evaluated stocks and methods which are less sensitive to differing assumptions.
Table 1: Summary results of the three different tests conducted to test the validity of the EMH.
Runs test Variance ratio test AC test 99 % confidence level Indication of efficiency 25 19 7 Indication of inefficiency 6 12 24 95 % confidence level Indication of efficiency 18 16 5 Indication of inefficiency 13 15 26 90 % confidence level Indication of efficiency 14 13 4 Indication of inefficiency 17 18 27
6
2. Theoretical framework and previous studies
The theoretical framework that underpins this evaluation is that of the EMH, notoriously suggested by Fama (1970). The author specified this previously ambiguous concept in such a manner that it became both concreate and most importantly, testable. Thus, the theoretical framework of this thesis will be restricted to the views and suggestions proposed by Fama (1970), which subsequently will be presented in the following sections.
In his seminal contribution, Fama (1970) suggests that saying…”in an efficient market prices
‘fully reflect’ available information” is not sufficient for any practical test of efficiency. Indeed,
the author argues, that to test and define efficiency, one must first begin with a model which govern the behaviour of prices and incorporates available information. According to Fama (1970) the most reasonable such model is one which is based on the presupposition that the expected price of a financial asset at time t+1 is a function of information and price at time t, namely:
𝐸(𝑝̃𝑗,𝑡+1|𝛷𝑡) = [1 + 𝐸(𝑟̃𝑗,𝑡+1|𝛷𝑡)] ∗ 𝑝𝑗,𝑡 , (1)
where 𝑝̃𝑗,𝑡+1 is the price of asset j at time t+1, 𝑝𝑗,𝑡 is the price of asset j at time t, 𝛷𝑡 is a general set of attainable information, 𝑟̃𝑗,𝑡+1 is the return of asset j at time t+1 and the tildes are meant to specify that 𝑟 and 𝑝 are random variables. Furthermore, the author argues, that if this model holds true, it has paramount implications, namely that no trading system based solely on information should be able to acquire excess returns. Mathematically, with regards to prices, Fama (1970) specifies this in the following manner:
𝑥𝑗,𝑡+1= 𝑝𝑗,𝑡+1− 𝐸(𝑝̃𝑗,𝑡+1|𝛷𝑡) ,
where 𝐸(𝑥̃𝑗,𝑡+1|𝛷𝑡) = 0 , (2)
or equivalently for returns
𝑧𝑗,𝑡+1= 𝑟𝑗,𝑡+1− 𝐸(𝑟̃𝑗,𝑡+1|𝛷𝑡) ,
where 𝐸(𝑧̃𝑗,𝑡+1|𝛷𝑡) = 0 , (3)
𝑥𝑗,𝑡+1 (𝑧𝑗,𝑡+1) is the excess market value (return) acquired from asset j at time t+1, thus constituting a fair game with respect to 𝛷𝑡. Alternatively, 𝑥𝑗,𝑡+1 (𝑧𝑗,𝑡+1) can be interpreted as the difference between the observed and predicted price (return) of asset j at time t+1.
Moreover, according to Fama (1970), the principle of a fair game can also be generalized to an arbitrary number of assets. This can clearly be illustrated by imagining a (any) trading system, that based on attainable information determines how much to invest in each asset (i.e. portfolio
7 weights). Formally, the author argues, that such a trading system can be defined in accordance with the following:
𝛼(𝛷𝑡) = [𝛼1(𝛷𝑡), 𝛼2(𝛷𝑡), 𝛼3(𝛷𝑡), … , 𝛼𝑛(𝛷𝑡)] (4)
where [𝛼𝑗(𝛷𝑡)] describes the portfolio weight of asset 𝑗. Moreover, with such a trading system, the excess return would simply be the sum of the weights multiplied with the excess return of each asset, i.e.:
𝑉𝑡+1 = ∑ 𝛼𝑗(𝛷𝑡) ∗ [𝑟𝑗,𝑡+1− 𝐸(𝑟̃𝑗,𝑡+1|𝛷𝑡)] 𝑛
𝑗=1
, (5)
which, in accord with the fair game principle suggested in the single asset example will be:
𝐸(𝑉̃𝑡+1) = ∑ 𝛼𝑗(𝛷𝑡) ∗ [𝑧𝑗,𝑡+1] 𝑛
𝑗=1
= 0 . (6)
Consequently, this is the principle on which the EMH stands, and equivalent to the statement that Fama argues is not enough, namely that …”in an efficient market prices ‘fully reflect’
available information”. As such, the author proposed an elaboration, namely the categorization
of different information sets. The author defined these categories as weak, semi-strong and strong, wherein each category differs with regards to which information that is captured in 𝛷𝑡.
2.1. The weak form of market efficiency
As previously mentioned, the difference between the different forms of the EMH concerns which information that is incorporated into asset prices, namely which information that is incorporated in 𝛷𝑡. With regards to the weak form of the EMH, the information that is assumed to be incorporated is that of historical prices. Thus, if the weak form of the hypothesis holds true, no excess returns should be attainable based on trading strategies that solely rely on historical prices (i.e. technical analysis). As such, one proposed way of testing the weak form of the EMH is to create models that, based on historical prices, generate risk adjusted excess returns. Subsequently, if such a model was discovered it could disprove the hypothesis (Metghalchi et al., 2008). However, the problems with approaches such as these are twofold. Firstly, if such a model was discovered, it would surely be used to earn money, thus incorporating the information it utilizes into the price of an asset, which subsequently would make the market efficient. Secondly, if no such models can be found, it does not prove that the EMH is true, it only proves the inadequacy of the models. Consequently, such tests are unreliable to say the least. So, how should the hypothesis be tested?
8 According to Dupernex (2007), the practical implication of the weak form of the EMH has come to be interpreted as equivalent to the concept of random price movements, i.e. that prices follows a random walk. This is a common held belief and supported by authors and arguments such as Malkiel (2003) whom argues that, if markets are efficient, only new and not previously attainable information should induce price fluctuations. Assuming this to be true, Malkiel (2003) argues that new information per definition is random, thus making price fluctuations random.
Consequently, a simple test of the weak form of the EMH is to test the randomness of prices. However, as argue by Fama (1970), this approach is not without its flaws, primarily due to the fact that there are several differences between the concept of an informational efficient market and a random walk. Nonetheless, Fama (1970) also suggests that approximating the weak form of the EMH with a random walk is not necessarily bad, for he argues that … “this procedure,
which represents a rather gross approximation from the viewpoint of the general expected return efficient markets model, does not seem to greatly affect the results of the covariance tests, at least for common stocks”.
As such, a category of adequate tests for the weak form of the EMH are tests that examines the randomness of prices, or equivalently, the white noise properties of returns. However, it is important to note that these are not direct tests of the EMH, but rather tests of an implied consequence of the hypothesis. Thus, when testing the weak form of the EMH with random walk tests, it is important to note that even if the random walk properties are rejected, it does not necessarily equate to a rejection of the EMH. Consequently, randomness can favourably be interpreted a sufficient and not necessary conditions of the weak form of the EMH (Korsvold and Jennergren, 1974). Nonetheless, it is this category of tests that will be applied and elaborated on further in the methodological section of this thesis.
2.2. The semi-strong and strong form of market efficiency
According to Fama (1970), the semi-strong and strong form of the EMH are based on the same conceptual premise as the weak form. However, the three forms differ with regards to the information that is incorporated into 𝛷𝑡. Whilst the weak form only assumes that historical prices is incorporated, the semi-strong form assumes that all publicly available information (i.e. stock splits, reports, news etc.) is incorporated, and the strong from assumes that even monopolistic (private/insider) information is incorporated into the price of an asset.
9 Naturally, a lot more can be said about the semi-strong and strong of the EMH and how they might be tested. However, since this thesis is restricted to investigating the weak form of the EMH, no further elaboration on the these from will be presented. For additional information on this topic, there are several studies such as Batista et al. (2018) and Gupta et al. (2008) whom both elaborate on and test the semi-strong and strong form of the EMH.
2.3. Previous studies
As previously mentioned, the literature investigating the efficiency of financial markets is vast, especially with regards to larger, more well-known markets, such as the Dow Jones, Nasdaq and Shanghai stock exchange. Consequently, a comprehensive global literature review is well beyond the scope of this paper. However, contrary to this, the literature concerning the Scandinavian and especially the Swedish stock market is relatively limited. Therefore, in subsequent sections, a combination of key international contributions with historical importance and a handful of papers investigating the Scandinavian stock markets will be presented.
After Fama’s (1970) seminal contribution, several books and papers endorsing the EMH quickly emerged. One of the most influential of these is Malkiel (1973), in which the author argues that short run price fluctuations are random, thus endorsing the weak form of the EMH. The author did however emphasize that the randomness of prices need not be true in the long run, thus diverging somewhat from the RWH but not necessarily the EMH.
Three years after Malkiel (1973) infamous book ‘a random walk down wall street’, Grossman (1976) provides a mathematical model which illustrated the paradoxical nature of the EMH, namely that, if markets are informationally efficient, there cannot exist any private incentive for the collection of information, thus making markets inefficient. Along the same lines, several authors such as Beja (1977), Lucas (1978) and Grossman and Stiglitz (1980) discussed and subsequently rejecting the hypothesis on the basis of theoretical intuition rather than empirical tests. However, Contrary to the views of Beja (1977), Lucas (1978) and Grossman and Stiglitz (1980), authors such as Jensen (1978) argued that the empirical evidence was overwhelmingly in support of the EMH. The author event went so far as to say that …” there is no other
proposition in economics which has more solid empirical evidence supporting it than the Efficient Market Hypothesis. The hypothesis has been tested and, with very few exceptions, found consistent with the data in a wide variety of markets: the New York and American Stock Exchanges, the Australian, English, and German stock markets, various commodity futures
10
markets, the Over-the Counter markets, the corporate and government bond markets, the option market, and the market for seats on the New York Stock Exchange”.
Furthermore, throughout the late twentieth century, the controversy regarding the validity of the EMH continued, with new tests and empirical evaluations. In favour of the hypothesis, were authors such as Eun and Shim (1989) whom found that the interdependence between national stock market, that had previously been pointed to as an example of inefficiency, actually was expected under the assumption of an informationally efficient market. Similarly, Richardson (1993) found that the autocorrelation of returns, that once again had been pointed to as an example of inefficiency, was well within the bound of reason given an informationally efficient market. Additionally, Metcalf and Malkiel (1994), much like Cowles (1933), investigated the performance of stock recommendations and found that so called “expert advice” yield no excess returns, thus supporting the EMH.
However, there were also scholars in opposition to the EMH. Two such scholars were Fama and French (1988) who found statistically significant negative autocorrelation in the returns of portfolios with an investment horizon of more than a year, thus contradicting the EMH. Two other sceptics were Lo and Mackinlay (1988) whom investigated the variance of returns and proposed that it was not in alignment with suggestions derived from the EMH.
Even more resent papers such as Malkiel (2003), (2005) and Simmons (2012) clearly point to a continued controversy regarding if, how, were and to which extent the EMH holds true. However, as suggested by Tóth and Kertész (2006), whom investigated the temporal change in cross correlation on the New York stock exchange, there does not seem to be any denying that financial markets are becoming increasingly efficient over time, even if they might not be efficient as of today. Consequently, what the authors suggests, is what many previous studies have failed to consider, namely the time heterogeneity of stock markets.
With regards to the Scandinavian markets, the literature is quite limited. However, there are some contributions that are worthy of consideration. On such contribution was written by Korsvold and Jennergren (1974) who investigated 30 stocks on the Swedish stock market between 1967 and 1971. With a total of 1254 trading days, the authors tested the random walk property of prices using a runs test, which consequently resulted in the rejection of the EMH. Building of the work of Korsvold and Jennergren (1974), Berglund et al. (1983) conducted a large study of the Helsinki stock exchange between 1970 to 1981. In this study, the authors utilized both a runs test and an autocorrelation test, which, in alignment with the results
11 presented by Korsvold and Jennergren (1974), resulted in the rejection of the EMH. Furthermore, the authors found that the autocorrelations of returns, in general, were only statistically significantly positive (negative) for lag 1-3 (4-9) and zero for subsequent lags. To explain this, the author suggests that the rate of which the autocorrelations tend to zero is dependent on total turnover of a stock. In other words, the authors argued that the degree to which a stock is efficient is dependent on the total turnover of a stock.
Another study worthy of mentioning in the context of empirical test was written by Frennberg and Hansson (1993). The authors used both a variance ration test and a test of autoregression of multiperiod returns to investigate the efficiency of the Swedish stock index between 1919 to 1990. The author concluded that the sufficient condition of randomness did no hold true during this period. However, the authors curiously fail to consider the time heterogeneity of stock markets, which consequently reduces the reliability of their conclusions.
In the late 1990:s, Chan et al. (1997) conducted a thorough analysed of the Swedish, and seventeen other national stock indices. In their evaluation, the authors utilized a unit root test on monthly data between 1961 to 1992 in an endeavour to evaluate the randomness of prices and thus the EMH. Contrary to the aforementioned studies, these tests revealed that all eighteen price series contained a unit root, which subsequently suggests that all stock indices could potentially be informationally efficient, thus validifying the weak form of the EMH.
Another study centred around the Swedish equity market was written by Metghalchi et al. (2008). Contrary to previous authors, Metghalchi et al. (2008) did not choose to test the randomness of prices, but rather if any conceivable trading strategy based on historical prices could generate risk adjusted excess returns. In this evaluation, the authors found several such trading strategies, even after adjusting for data snooping bias. Consequently, this would contradict the EMH. However, as previously mentioned, evaluations such as these are quite unreliable.
In summary, the literature investigating the EMH is vast, and even after decades of research no academic consensus have been reached. Even studies with similar methodology, time frame and assumptions seem to differ in their conclusions and interpretations of the results. Consequently, the verdict is still out on whether the EMH holds true in various markets.
12
3. Data
The data used in this thesis consists of historical prices of stocks listed on the Swedish stock index OMX30. More specifically, the data consists of the historical prices of the 30 stocks that as of the second quarter of 2017 constituted the OMX30, namely the once presented in Table 2. Additionally, historical prices of the volume weighted index2 has been included.
Table 2: Names of the thirty stocks that, as of the second quarter of 2017, constituted the OMX30
ABB Essity B SEB A
Alfa Laval FP Cards B Securitas B
Assa Abloy B Getinge B Skanska B
Astra Zeneca HM B SKF B
Atlas Copco A Investor B SSAB A
Atlas Copco B Kinnevik B Swedbank A
Autoliv Nordea Bank Swedish Match
Boliden S.H Banken A Tele2 B
Electrolux B Sandvik Telia Company
Ericsson B SCA B Volvo B
Moreover, these stocks, as well as the index, were selected so to represent the Swedish equity market. Naturally, it is a vast generalization to state that these stocks represent the entire market. However, it is arguably the best proxy available for the roughly 320 publicly traded companies on the Stockholm stock exchange. That being said, it would clearly be preferable to include all publicly traded companies in a less restrictive study, and thus facilitate a more robust conclusion with regards to the probability of a stock being efficient. However, this was not plausible in this thesis, and is therefore left as a recommendation for future studies.
Nevertheless, the data used in this thesis has exclusively been gathered from Nasdaq OMX Nordic (2018a; 2018b), where adjusted daily closing prices were selected so to account for stock splits, dividends, rights offerings and similar interventions that affect the nominal market value of a stock. The period that has been investigated is between 2002-05-17 to 2017-12-29, which was chosen so to minimize the adverse effects of missing values and incoherence between stocks.
For 29 of the selected price series (stocks and index), all daily data was readily available, thus represented by 3928 daily observations. However, two stocks did not confirm to this unity, namely Fingerprint Cards B and Essity B. With regards to Fingerprint cards, their discrepancy from the norm was due to company specific trading halts that resulted in missing values. In total, seven such missing values were observed, which was dealt with by the use of linear interpolations, thus allowing for 3928 observations. This is by no means an optimal solution in
13 the context of financial data. However, since the number of missing values were quite low, the specific method of imputation was deemed adequate due to its ease of use and negligible impact on the results. With regards to Essity B, the company did not go public until mid-June of 2017, and reliable daily data was not available before the 2017-06-26. Consequently, only 132 observations were available and thus used.
Furthermore, the price data was not directly applied in testing the EMH. Instead, the logarithmic prices and returns were used. Naturally, there are several test-specific reasons for this, which will be discussed in the methodological section of this thesis. However, more than this, according to Hudson and Gregoriou (2015), there are several general advantages with using logarithmic time series in financial data. Among other things, the authors argue, that in the context of financial time series, logarithmic returns reduce the influence of extreme values, approximate the normal distribution and facilitate continuous compounding (Zaimovic, 2013). Consequently, the returns used in this thesis is defined in accordance with the following:
𝑥𝑗,𝑡 = 𝑙𝑛 𝑦𝑗,𝑡− 𝑙𝑛 𝑦𝑗,𝑡−1 , (7)
where 𝑙𝑛 𝑦𝑗,𝑡 is the logarithmic price of asset 𝑗 at time 𝑡 and 𝑥𝑗,𝑡 is the logarithmic return of asset 𝑗 at time 𝑡.
Moreover, data concerning the stocks turnover, number of trades and trade volume have been utilized in order to establish their effect on efficiency. In accordance with previous data, information regarding turnover, number of trades and trade volume during the period 2002-05-17 to 202002-05-17-12-29 was exclusively gathered from Nasdaq OMX Nordic (2018a), where daily data was readily available. With regards to Fingerprint cards, the same method of imputation was used for the seven missing values, and with regards to Essity B, only 132 observations were used.
Moreover, a stocks turnover, number of trades and trade volume is defined in accordance with the following:
Number of trades: The number of trades that has occurred on any given day. Trade volume: The number of shares that has been trades on any given day. Turnover: A measure of liquidity defined in accordance with equation (8).
𝑇𝑢𝑟𝑛𝑒𝑜𝑣𝑒𝑟 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑗 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡
14 Furthermore, in order to convey a comprehensive understanding of the data used in this thesis, some descriptive statistics for each stocks turnover, number of trades, trade volume and logarithmic returns are presented in Table 3. Additional illustrations of time series plots and autocorrelations can be found in the appendix.
Table 3: Descriptive statistics of the logarithmic returns, trade volume, number of trades and turnover, of
all stocks listed on the OMX30 as well as the index itself
Stock Nr. Obs Median return Mean return Std. dev Average total
volume Average turnover
Average trades ABB 3928 0,000 0,000 0,029 3225660,847 326035430,368 2239,429 Alfa Laval 3928 0,000 0,001 0,022 2211773,670 192775609,876 2094,656 Assa Abloy B 3928 0,000 0,000 0,020 5490986,179 295913948,866 2503,929 Astra Zeneca 3928 0,000 0,000 0,016 1290273,873 451168115,557 2168,350 Atlas Copco A 3928 0,001 0,001 0,022 6328578,031 587752987,370 3777,352 Atlas Copco B 3928 0,001 0,001 0,022 1619171,033 147408401,020 1614,760 Autoliv 3928 0,000 0,000 0,018 363135,677 160098837,997 1371,462 Boliden 3928 0,000 0,001 0,029 3452281,088 350090435,099 3692,585 Electrolux B 3928 0,000 0,000 0,022 3118918,037 347185475,023 2919,541 Ericsson B 3928 0,000 0,000 0,027 20202293,126 1660709078,705 6964,619 Essity B 132 -0,002 0,000 0,013 1226763,098 287857382,215 4176,504 FP Cards B 3928 0,000 0,000 0,052 3760396,211 137425264,597 2320,530 Getinge B 3928 0,000 0,000 0,019 792414,723 111685887,811 1692,590 HM B 3928 0,000 0,000 0,016 4594359,238 824696511,547 5254,961 Investor B 3928 0,000 0,000 0,017 1892597,218 281630933,448 2360,627 Kinnevik B 3928 0,000 0,001 0,021 675386,446 102483864,632 1623,100 Nordea Bank 3928 0,000 0,000 0,021 11228959,394 735871358,631 4083,660 Sandvik 3928 0,000 0,000 0,021 6833730,280 555309054,036 4009,668 Securitas B 3928 0,000 0,000 0,020 2164600,508 186011830,221 1720,057 SEB A 3928 0,000 0,000 0,024 7695491,395 501269281,076 3707,374 Skanska B 3928 0,000 0,000 0,019 1762284,064 204805893,477 2135,593 SKF B 3928 0,000 0,000 0,020 3824810,879 426824143,067 3304,952 SSAB A 3928 0,000 0,000 0,026 2726687,957 191731389,676 2863,117 SCA B 3928 0,000 0,000 0,030 2561481,242 307634305,777 2583,210 S.H Banken A 3928 0,000 0,000 0,018 6524616,012 434360877,485 2840,888 Swedbank A 3928 0,000 0,000 0,023 4571272,634 526455257,665 3665,140 Swedish Match 3928 0,000 0,000 0,015 1370636,818 194653626,541 1960,365 Tele2 B 3928 0,000 0,000 0,020 2600550,321 240009432,837 2360,495 Telia Company 3928 0,000 0,000 0,018 12814892,100 564515953,066 3847,866 Volvo B 3928 0,000 0,000 0,022 9397260,802 730019308,821 5190,691 omx30 3928 0,001 0,000 0,014 135034831,286 11777319681,774 87012,500
15
4. Method
The general methodological premise of this thesis consists of runs tests, variance ratio tests, autocorrelation tests and logistic regressions. Consequently, these are all tests that are aimed to evaluate the implied consequence of the weak form of the EMH, namely randomness. As such, the hypothesises of these tests can loosely and generally be defined in accordance with the following:
𝐻0: 𝑃𝑟𝑖𝑐𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 𝑓𝑜𝑙𝑙𝑜𝑤 𝑎 𝑟𝑎𝑛𝑑𝑜𝑚 𝑤𝑎𝑙𝑘
𝐻𝐴: 𝑃𝑟𝑖𝑐𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 𝑑𝑜 𝑛𝑜𝑡 𝑓𝑜𝑙𝑙𝑜𝑤 𝑎 𝑟𝑎𝑛𝑑𝑜𝑚 𝑤𝑎𝑙𝑘 (9)
However, according to Korsvold and Jennergren (1974), it is important to note that randomness is merely a sufficient and not a necessary condition for the EMH. Thus, a rejection of the null hypothesis does not necessarily equate to a rejection of the EMH. Therefore, in this thesis, a rejection of the null hypothesis will simply be interpreted as an indication and not a proof of an informationally inefficient market. Conversely, failure to reject the null hypothesis will be interpreted as an indication that the EMH holds true.
Furthermore, in order to convey a comprehensive understanding of the test used in this thesis, the subsequent sections will be devoted to a thorough description of the said tests.
4.1. Wald–Wolfowitz runs test
Although several authors such as Wishart and Hirschfeld (1936), Kermack and McKendrick (1937), Mood (1940) and Olmstead (1940) conducted extensive research and contributed to the development of the so-called runs test, the general test is often attributed to Wald and Wolfowitz (1940) who in their seminal publication proposed a multi-sample test for the evaluation of whether two different samples are drawn from the same population. Since the authors pioneering contribution however, has the literature concerning runs tests grown vast and its applicability wide. One such application, is tests that evaluated whether the data generating process of a series of observations is random. Consequently, it is such an application that has been employed in this thesis.
According to Siegel (1956), the run tests, so called because is consider the number of runs in a string of data, is a non-parametric test which can be used to determine serial dependence. According to the author, the test archives this by comparing the number of runs in any given data set with the expected number of runs under the null hypothesis, were a run is defined as a …”succession of identical symbols which are followed and proceeded by different symbols or
16 Furthermore, according to Siegel (1956), under the null hypothesis of no serial dependence (i.e. random stock price fluctuations) the expected number of runs is defined in the following manner:
𝐸(𝑅) = 2 ∗ 𝑁1∗ 𝑁2
𝑁 + 1 , (10)
and the standard error is defined as:
𝜎𝑅= √
2 ∗ 𝑁1∗ 𝑁2∗ (2 ∗ 𝑁1∗ 𝑁2− 𝑁)
(𝑁2) ∗ (𝑁 − 1) , (11)
where 𝑁1 is the number of runs above a certain threshold (often assumed the mean or median,
which allows for a drift component), 𝑁2 is the numbers runs below the same threshold and 𝑁 is the total number of runs. With both the number of runs, the expected number of runs and the standard error of runs, Siegel (1956) argues that the following asymptotically normally distributed test statistic can be constructed:
𝑍 = 𝑅 − 𝐸(𝑅)
𝜎𝑅 , (12)
where R in the number of runs, E(R) is the expected number of runs and 𝜎𝑅 is the standard error of runs. Consequently, with such a test statistic simple p-values can be derived for the rejection of the null hypothesis and thus the EMH.
With regards to its application in this thesis, the runs test has been applied to the logarithmic returns. Furthermore, the threshold value was set to zero, which corresponded to the median for all stocks except Atlas Copco A, Atlas Copco B, Essity B and OMX30, which deviated negligible from zero and is thus of no great concern.
Lastly, it is often argued that the parametric Kolmogorov–Smirnov (K-S) test is more powerful than the runs when evaluating randomness (Hawaldar et al., 2017). However, although the power of K-S test is greater, the required distributional assumptions makes the K-S test inadequate in the context of stock returns, for the test gives no account of weather it might fail due to the existence of randomness or distributional assumptions.
4.2. Variance ratio test
The second test employed in this thesis, is the parametric, heteroscedastic robust, variance ratio test proposed by Lo and Mackinlay (1988). The test is based on the premise that stock prices can be described in accordance the following:
17 where, 𝑦𝑡 is the price of a stock at time t, 𝜇 is an unknown drift parameter, 𝑦𝑡−1 is the price of a stock at time t-1 and 𝜀𝑡 represent error terms which needs not be, neither independent nor identically distributed. Thus, the test is based on the premise that stock prices follow a martingale process.
According to Amélie and Darné (2009), the assumption of a martingale process is key, for it suggests that the variance of k-period returns should be equal to the one-period variance multiplied by k. In other words, it suggests that the variance of returns should be linearly increasing and directly proportional to the length of the time interval. That is, the same observed pattern that was described by Regnault (1863). Consequently, this is the property of returns that Lo and Mackinlay (1988) suggests can be tested by considering the variance ratio described by the following equation:
𝑉(𝑘) = 𝑉𝑎𝑟(𝑥𝑡+ 𝑥𝑡−1+ ⋯ + 𝑥𝑡−𝑘−1)
𝑉𝑎𝑟(𝑥𝑡) ∗𝑘 , (14)
where 𝑥𝑡 is the return of a stock at time t.
Now, if the variance is directly proportional to the length of the time interval, the ratio should be equal to one, which consequently is the expected value of the ratio under the null hypothesis of randomness. Naturally however, the true variances, as described in equation (14) are not known. Therefore, the empirically testable ratio is defined in accordance with the following:
𝑉𝑅(𝐾) = 𝜎̂2(𝑘) 𝜎 ̂2(1) , (15) where 𝜎̂2(1) = 1 𝑇−1∗ ∑ (𝑥𝑡− 𝜇̂) 2 𝑇 𝑡=1 , (16) 𝜎̂2(𝑘) = 1 𝑘 ∗ (𝑇 − 𝑘 + 1) ∗ (1 − (𝑘𝑇)) ∗ ∑(𝑦𝑡 − 𝑦𝑡−𝑘− 𝑘 ∗ 𝜇̂)2 𝑇 𝑡=𝑘 , (17)
and k is the length of the interval, T is the total sample size and 𝜇̂ is estimated as a simple average, namely as:
𝜇̂ =1
𝑇∗ ∑ 𝑥𝑡
𝑇
𝑡=1
. (18)
Furthermore, Lo and Mackinlay (1988) suggests that the ratio described in equation (15) can be tested by the following heteroscedastic robust and normally distributed test statistic:
18 𝑀2(𝑘) = 𝑉𝑅(𝑘) − 1 √ϕ∗(𝑘) , (19) where ϕ∗= ∑ [2(𝑘 − 𝑗) 𝑘 ] 2 ∗ 𝛿(𝑗) 𝑘−1 𝑗=1 , (20) and 𝛿(𝑗) =∑ (𝑥𝑡− 𝜇̂) 2∗ (𝑥 𝑡−𝑗− 𝜇̂) 2 𝑇 𝑡=𝑗+1 [∑𝑇𝑡=1(𝑥𝑡 − 𝜇̂)2]2 . (21)
With the test statistic derived, much like in the case of the runs test, simple p-values can be calculated for the rejection of the null hypothesis. That is, for the rejection of randomness and the indication of whether the EMH holds true.
The test described above is straight forward enough, however the choice of which, and how many different values of k (time periods) that should be tested is not, and the relevant literature gives little justification for specific values. However, according to Deo and Richardson (2003) and Amélie and Darné (2009), the standard approach is to consider k = {2,5,10,30} for daily data. Thus, this is what has been done in this thesis, which consequently corresponds to four different variance ratio tests for each of the evaluated price series.
4.3. Ljung-Box test
The third and last test used in this thesis, is the portmanteau autocorrelation test proposed by Ljung and Box (1978). Naturally, there are several different portmanteau tests that could have been applied. However, the test proposed by Ljung and Box (1978) is arguably the most popular, due to its superior properties and power when dealing with finite samples.
Nonetheless, the test proposed by Ljung and Box (1978) is designed to evaluate whether the joint autocorrelation up to lag m is statistically significantly different from zero. Thus, the tests hypothesis can be described in accordance with the following:
𝐻0: 𝜌(1) = 𝜌(2) = ⋯ = 𝜌(𝑚 − 1) = 𝜌(𝑚) = 0
𝐻𝐴: 𝑆𝑒𝑟𝑖𝑎𝑙 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 , (22)
where 𝜌(𝑚) is the autocorrelation at lag m. Consequently, this hypothesis is equivalent to the general hypothesises described in equation (9), which, as previously mentioned, can be interpreted as a test of the EMH. Furthermore, the means by which Ljung and Box (1978) argues that this hypothesis should be tested, is by the use of the test statistic described below:
19 𝑄𝑚′ ≡ 𝑇(𝑇 + 2) ∗ ∑ 𝜌2(𝑘) 𝑇 − 𝑘 𝑚 𝑘=1 , (23) where 𝜌2(𝑘) = ( 𝐶𝑜𝑣(𝑥𝑡, 𝑥𝑡+𝑘) √𝑣𝑎𝑟(𝑥𝑡)√𝑉𝑎𝑟(𝑥𝑡+𝑘) ) 2 , (24)
T is the total sample size, m is the lag to which the test is specified, 𝑥𝑡 is the return of a stock at time t and 𝜌2(𝑘) is the squared autocorrelation at lag k. Consequently, the test statistic proposed
by Ljung and Box (1978) is merely a sum of squared autocorrelations, which as the authors show, follows a χ2 distribution with m degrees of freedom. Thus, this is a straight forward test,
which allows for easily accessible p-values for the rejection of the null hypothesis.
However, as in the case of the variance ratio test, it is rather unclear which, and how many lags that should be tested. That is, which values of m that should be evaluated. Furthermore, based on the relevant literature, the specific choice of lags seems quite arbitrary (given certain restrictions, such as m<41). Therefore, in the name of consistency and comparability, lag m =
{2,5,10,30} have been tested. Consequently, four different Ljung-Box tests have been
conducted for each of the evaluated price series.
4.4. General method and Logistic regression
The tests described in previous sections gives insight to whether the EMH holds true, and their result can and will be evaluated independently. However, viewing the results independently, inevitably leads to quite contradictory conclusions, namely because the tests do not covey the same truths. Therefore, in addition to viewing the results independently, two approaches have been utilized to combine the runs tests, the variance-ratio tests and the autocorrelation tests. Thus, combining all nine tests into one, in the hope of generating more robust and less contradictory conclusions.
The first approach used to combine the results, is a simple averaging of p-values, thus similar to Edgington's method for additive probability values (Edgington, 1972). As such, the first approach can be summarized in accordance with equation (25) and (26), where the average p-values are evaluated on a 95 % and 90 % confidence level.
𝐼𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑜𝑛_𝑜𝑓_𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝐼𝑂𝐸 (5%) = { 1, 𝑖𝑓 ∑𝑛𝑖=1𝑃𝑣𝑎𝑙𝑢𝑒(𝑖) 𝑛 > 0,05 0, 𝑖𝑓 ∑ 𝑃𝑣𝑎𝑙𝑢𝑒(𝑖) 𝑛 𝑖=1 𝑛 ≤ 0,05 (25)
20 𝐼𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑜𝑛_𝑜𝑓_𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝐼𝑂𝐸 (10%) = { 1, 𝑖𝑓 ∑𝑛𝑖=1𝑃𝑣𝑎𝑙𝑢𝑒(𝑖) 𝑛 > 0,1 0, 𝑖𝑓 ∑ 𝑃𝑣𝑎𝑙𝑢𝑒(𝑖) 𝑛 𝑖=1 𝑛 ≤ 0,1 . (26)
Thus, two binary variables were created in this fashion, which allowed for a straight forward analysis of the validity of the EMH, for each price series.
The second approach, are decision rules that are based on a row sum of binary variables, which indicate whether a stock is efficient or not. Consequently, such binary variables were initially created based on each individual test, for both a confidence level of 95% and 90%. Thus, these binary variables can be illustrated in accordance with Table 4,
Table 4: Illustration of the binary variables created for the second approach used to combine all results
Stock Runs test Variance ratio test Autocorrelation test
k=2 k=5 k=10 k=30 m=2 m=5 m=10 m=30 stock 1 1 1 1 1 1 0 0 0 0 stock 2 1 1 1 0 0 1 0 0 0 . . . . . . . . . . . .
where “1” and “0” suggests an informationally efficient and inefficient market, respectively. As such, nine binary variables were created for each of the considered confidence levels. Subsequently, was the legitimize of the EMH evaluated by the five decision rules described below: 𝐼𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑜𝑛_𝑜𝑓_𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝐼𝑂𝐸 = { 1, 𝑖𝑓 ∑ 𝐵𝑖 𝑛 𝑖=1 > 𝑧 0, 𝑖𝑓 ∑ 𝐵𝑖 𝑛 𝑖=1 ≤ 𝑧 , (27) where 𝑧 = {4, 5, 6, 7, 8} ,
𝐵𝑖 represent the nine binary variables for each of the considered confidence levels and z refers to the critical values described by the five decision rules. Furthermore, this approach was deemed adequate mainly thanks to its unquestionable ability to encompasses a wide variety of assumptions regarding the certainty with which the EMH should be rejected.
Consequently, with both approaches described above, six different binary variables were created for each confidence level. As such, in total, 12 binary variables were created, which allowed for a straight forward interpretation regarding the validity of the EMH and the evaluation of the effect of varying assumptions.
21 Moreover, the 12 binary variables allowed for 12 logistic regressions that were used to evaluate whether a stocks turnover, number of trades and trade volume affects the probability of efficiency. Consequently, it is by this mean the marginal effect of a stocks turnover, number of trades and trade volume has been evaluated.
Formally, the logistic regressions that were evaluated were specified in the following manner: 𝐼𝑂𝐸 = 𝛽0+ 𝛽1∗ 𝑇𝑢𝑟𝑛𝑒𝑜𝑣𝑒𝑟𝐴𝑉𝐺+ 𝛽2∗ 𝑇𝑟𝑎𝑑𝑒𝐴𝑉𝐺+ 𝛽3∗ 𝑉𝑜𝑙𝑢𝑚𝑒𝐴𝑉𝐺 , (28) where 𝐼𝑂𝐸 is a vector of binary indicators of whether the EMH holds true for individual stock (i.e. the 12 binary variables derived from approach 1 and 2), 𝐴𝑉𝐺_𝑡𝑢𝑟𝑛𝑜𝑣𝑒𝑟 is a vector of the average daily turnover, 𝐴𝑉𝐺_𝑇𝑟𝑎𝑑𝑒𝑠 is a vector of the average daily number of trades and 𝐴𝑉𝐺_𝑣𝑜𝑙𝑢𝑚𝑒 is a vector of the average daily trade volume. With these models, beta coefficients, standard errors and thus simple p-values were easily derived. Subsequently, these p-values allowed for evaluation of the statistical significance of the marginal effect of the covariates. In turn, these tests allowed for assumption robust conclusion with regards to the covariates effect on efficiency.
22
5. Results and Analysis
In order to present an appealing and comprehensive overview of the results derived from the aforementioned tests, the results have been divided into five different subsections. In each of first thee subsections, the results from the different tests are presented and analysed independently. In the fourth subsection, the results from the two combining approaches is presented and in the fifth section the logistic regressions are analysed.
5.1. Runs tests
In total, 31 runs tests were conducted, namely one for each of the price series evaluated. The results of these tests, which are presented in Table 5, suggests mixed results. Thus, to blatantly, and indiscriminately reject the EMH for the entire Swedish stock market does not seem warranted.
Table 5: Results from the runs tests.
Stock N (return<=0) N (returns>0) Obs N (runs) Z Prob>|z|
ABB 1990 1938 3928 1925 -1,270 0,210 Alfa Laval 2005 1923 3928 1995 0,990 0,320 Assa Abloy B 2032 1896 3928 2052 2,860 0,000* Astra Zeneca 2022 1906 3928 1944 -0,620 0,540 Atlas Copco A 1943 1985 3928 2031 2,110 0,030** Atlas Copco B 1955 1973 3928 2019 1,720 0,080*** Autoliv 1983 1945 3928 1901 -2,040 0,040** Boliden 2006 1922 3928 1979 0,480 0,630 Electrolux B 2049 1879 3928 2043 2,610 0,010* Ericsson B 2018 1910 3928 1981 0,560 0,580 Essity B 78 54 132 67 0,390 0,690 FP Cards B 2277 1651 3928 1822 -3,050 0,000* Getinge B 2037 1891 3928 2041 2,520 0,010* HM B 2016 1912 3928 2007 1,390 0,170 Investor B 1977 1951 3928 1995 0,960 0,340 Kinnevik B 1973 1955 3928 1924 -1,310 0,190 Nordea Bank 2025 1903 3928 2055 2,940 0,000* Sandvik 1977 1951 3928 1967 0,070 0,950 Securitas B 2058 1870 3928 2015 1,740 0,080*** SEB A 2032 1896 3928 2039 2,440 0,010* Skanska B 2013 1915 3928 1987 0,740 0,460 SKF B 2001 1927 3928 2019 1,750 0,080*** SSAB A 2025 1903 3928 1876 -2,780 0,010* SCA B 2039 1889 3928 2018 1,790 0,070*** S.H Banken A 2052 1876 3928 2029 2,170 0,030** Swedbank A 2000 1928 3928 1999 1,110 0,270 Swedish Match 2053 1875 3928 2100 4,450 0,000* Tele2 B 2031 1897 3928 1995 1,030 0,300 Telia Company 2065 1863 3928 2063 3,300 0,000* Volvo B 1967 1961 3928 1993 0,890 0,370 omx30 1945 1983 3928 2059 3,010 0,000*
23 Furthermore, when viewing the results presented in Table 5, it is important to remember that a rejection of the null hypothesis corresponds to the equivalence of a rejection (indication) of the EMH. Consequently, a low p-value in Table 5 suggests an informationally inefficient market. As such, it is apparent that any conclusions drawn with regards to which price series that are inefficient, are highly dependent on the meticulousness of the confidence level. Therefore, are the results from Table 5 summarized in Table 6, in which the cumulative number of inefficient and efficient price series is presented for different confidence levels.
Consequently, on a 99 % confidence level, merely six of the evaluated price series allowed for the rejection of randomness, which suggests that the EMH might hold true for most of the stocks listed on the OMX30. However, on a 95% confidence level, the number of price series that allowed for the rejection of randomness increased to 13. Clearly, this is an increase from the six stocks for which randomness was rejected on a 99% confidence level. However, it is still less than half of the evaluated stocks, which at least suggest the possibility of a relatively efficient market. Furthermore, on a 90 % confidence level, 17 of the evaluated price series allowed for the rejection of randomness and thus the strong indication of the EMH invalidity.
In summary, these results are quite ambiguous, and assumptions regarding confidence levels are of paramount importance. However, there seems to be no denying several price series does not conform to the runs tests standards of randomness. As such, whilst several individual stock do seem to exhibit ample indications of randomness, the market as a whole does not, thus indicating some form of inefficiency.
5.2. Variance ratio tests
As previously mentioned, four different periods were chosen for the variance ratio tests, namely
k = {2,5,10,30}, thus constituting four separate tests. As such, four different variance ration
tests were conducted for each of the 31 evaluated price series. Consequently, in total, 124 variance ratio tests were conducted in order to establish whether the EMH holds true. The results from these tests are presented in Table 7, where a low p-value suggest an informationally inefficient market, as described by the weak form of the EMH.
Table 6: Summary of the results derived Table 5.
99% 95% 90%
Number of stock that indicate efficiency 25 18 14 Number of stock that indicate inefficiency 6 13 17
24
Table 7: Results from the heteroscedastic robust Lo-Mackinlay variance ratio tests.
Stock k= 2 k= 5 k= 10 k= 30 N VR p>|z| VR p>|z| VR p>|z| VR p>|z| ABB 3928 1,064 0,386 1,088 0,514 0,977 0,899 0,913 0,783 Alfa Laval 3928 0,984 0,526 0,903 0,101 0,788 0,019** 0,638 0,017** Assa Abloy B 3928 0,910 0,000* 0,830 0,001* 0,704 0,000* 0,624 0,005* Astra Zeneca 3928 1,031 0,186 0,991 0,853 0,929 0,343 0,850 0,253 Atlas Copco A 3928 0,957 0,066*** 0,841 0,001* 0,742 0,001* 0,630 0,009* Atlas Copco B 3928 0,952 0,035** 0,824 0,000* 0,721 0,000* 0,594 0,003* Autoliv 3928 1,016 0,533 1,049 0,365 1,042 0,599 1,029 0,833 Boliden 3928 1,014 0,572 0,999 0,991 1,016 0,844 1,218 0,140 Electrolux B 3928 0,974 0,181 0,895 0,015** 0,821 0,008* 0,781 0,077*** Ericsson B 3928 1,045 0,174 0,982 0,792 1,019 0,852 1,015 0,931 Essity B 132 0,904 0,244 0,927 0,709 0,889 0,723 1,117 0,819 FP Cards B 3928 1,011 0,715 1,022 0,705 1,015 0,851 1,119 0,321 Getinge B 3928 0,956 0,030** 0,922 0,064 0,892 0,089*** 0,821 0,103 HM B 3928 0,956 0,055*** 0,873 0,009* 0,775 0,002* 0,657 0,007* Investor B 3928 1,000 0,999 0,960 0,467 0,883 0,157 0,822 0,215 Kinnevik B 3928 1,039 0,120 1,075 0,167 1,101 0,213 1,195 0,173 Nordea Bank 3928 0,954 0,106 0,852 0,013** 0,746 0,006* 0,699 0,065*** Sandvik 3928 0,988 0,590 0,905 0,056*** 0,788 0,006* 0,750 0,068*** Securitas B 3928 0,995 0,816 0,935 0,188 0,837 0,027** 0,751 0,048** SEB A 3928 1,009 0,810 0,911 0,233 0,777 0,050*** 0,724 0,172 Skanska B 3928 0,979 0,368 0,939 0,225 0,869 0,090*** 0,738 0,058*** SKF B 3928 0,947 0,014** 0,841 0,001* 0,730 0,000* 0,644 0,006* SSAB A 3928 1,024 0,317 0,998 0,972 0,937 0,438 0,979 0,889 SCA B 3928 0,975 0,135 0,959 0,188 0,943 0,166 0,912 0,144 S.H Banken A 3928 0,942 0,029** 0,837 0,005** 0,748 0,006* 0,615 0,021** Swedbank A 3928 0,974 0,422 0,942 0,411 0,856 0,186 0,997 0,988 Swedish Match 3928 0,885 0,000* 0,726 0,000* 0,628 0,000* 0,521 0,000* Tele2 B 3928 0,972 0,191 0,910 0,062*** 0,881 0,098*** 0,825 0,171 Telia Company 3928 0,940 0,066*** 0,805 0,004* 0,718 0,004* 0,612 0,013** Volvo B 3928 1,022 0,366 0,976 0,645 0,899 0,197 0,925 0,595 omx30 3928 0,966 0,156 0,867 0,014** 0,766 0,005* 0,720 0,064***
*, **, *** Indicate a rejection of the EMH on a 1 %, 5%, 10% confidence level, respectively.
As is suggested in Table 7, the results from the variance ratio tests are far from consistent. For several stocks, such as Alfa Laval, Electrolux, Getinge, Nordea Bank, Sandvik, Securitas, SEB, Skanska, Tele2 and the OMX30 index, the null hypothesis of randomness is rejected for some, but not all time-periods. Furthermore, if the stocks were informationally efficient, we’d expect not to be able to reject the null hypothesis for any time-periods. Thus, in consensus with previous literature, if randomness is rejected for at least one period, the EMH is rejected the entire price series.
Understandably, the inconsistency in Table 7 might at first glance be difficult to comprehend. Therefore, a summary contingency table is presented below, in which the number of inefficient stocks is subdivided according to confidence level and time-period.
25
Table 8: Contingency table in which the number of inefficient stocks is subdivided according to confidence
level and time-period.
99% 95% 90%
m=2
Number of stock that indicate efficiency 29 25 22 Number of stock that indicate inefficiency 2 6 9
m=5
Number of stock that indicate efficiency 23 20 17 Number of stock that indicate inefficiency 8 11 14 m=10
Number of stock that indicate efficiency 19 17 13 Number of stock that indicate inefficiency 12 14 18
m=30
Number of stock that indicate efficiency 25 21 16 Number of stock that indicate inefficiency 6 10 15 m =2,5,10,30
Number of stock that indicate efficiency 19 16 13 Number of stock that indicate inefficiency 12 15 18 Based on Table 8, the variance ratio tests suggest that 19, 16 and 13 price series could be efficient, assuming 99%, 95% and 90 % confidence levels, respectively. Consequently, this is a reduction in the number of suggested efficient stocks compared to the runs tests, which would indicate that the runs test might be the more conservative of the two. However, the results from the tests are quite similar, and have identical test results in 70,9% of all price series, across all confidence levels. Arguably, this increases the validity of both tests and the conclusion that some, but not all of the evaluated stocks could be informationally efficient according to the weak form of the EMH.
5.3. Ljung-Box test
As in the case of the variance ratio test, four different Ljung-Box tests have been applied for each of the evaluated price series, thus constituting 124 tests. The results from these tests, which are presented in Table 9, paint a rather different picture than previous tests, and suggests that the EMH might not hold true for most of the evaluated price series. However, as pointed out by Fama (1970), the existence of statistically significant autocorrelation is not necessarily proof of the EMH invalidity. The author argues that with a large enough sample size, even the smallest of autocorrelations will be statistically significantly different from zero, whilst economically insignificant and therefore not in opposition to the EMH.
26
Table 9: Results from the Ljung-Box autocorrelation tests
Stock m= 2 m= 5 m= 10 m= 30 N Q p>Q Q p>Q Q p>Q Q p>Q ABB 3928 16,478 0,000* 50,103 0,000* 107,266 0,000* 206,612 0,000* Alfa Laval 3928 4,595 0,101 21,960 0,001* 29,714 0,001* 67,032 0,000* Assa Abloy B 3928 33,088 0,000* 64,468 0,000* 67,347 0,000* 111,860 0,000* Astra Zeneca 3928 6,660 0,036** 14,369 0,013** 15,912 0,102 52,827 0,006* Atlas Copco A 3928 13,440 0,001* 30,347 0,000* 32,971 0,000* 79,724 0,000* Atlas Copco B 3928 16,178 0,000* 32,969 0,000* 36,961 0,000* 85,568 0,000* Autoliv 3928 1,440 0,487 3,878 0,567 9,636 0,473 30,912 0,420 Boliden 3928 4,427 0,109 7,667 0,176 23,783 0,008* 72,731 0,000* Electrolux B 3928 4,796 0,091*** 17,460 0,004* 25,588 0,004* 59,025 0,001* Ericsson B 3928 13,015 0,002* 33,511 0,000* 53,831 0,000* 111,313 0,000* Essity B 132 1,212 0,545 5,043 0,411 6,871 0,738 29,673 0,482 FP Cards B 3928 0,862 0,650 1,874 0,866 7,949 0,634 36,279 0,199 Getinge B 3928 7,640 0,022** 8,193 0,146 19,107 0,039** 28,297 0,555 HM B 3928 9,135 0,010* 20,297 0,001* 26,632 0,003* 50,693 0,011** Investor B 3928 0,087 0,958 14,198 0,014** 15,981 0,100*** 54,761 0,004* Kinnevik B 3928 7,385 0,025** 14,019 0,016** 17,092 0,072*** 45,653 0,034** Nordea Bank 3928 17,452 0,000* 27,906 0,000* 35,563 0,000* 101,700 0,000* Sandvik 3928 4,015 0,134 28,013 0,000* 31,759 0,000* 74,351 0,000* Securitas B 3928 0,554 0,758 16,076 0,007* 25,126 0,005* 45,429 0,035** SEB A 3928 3,351 0,187 40,829 0,000* 44,758 0,000* 172,178 0,000* Skanska B 3928 2,181 0,336 19,052 0,002* 34,693 0,000* 57,339 0,002* SKF B 3928 14,255 0,001* 38,599 0,000* 48,907 0,000* 88,992 0,000* SSAB A 3928 2,909 0,234 23,845 0,000* 32,547 0,000* 66,515 0,000* SCA B 3928 2,419 0,298 3,299 0,654 4,255 0,935 9,242 1,000 S.H banken A 3928 23,070 0,000* 42,184 0,000* 54,396 0,000* 156,235 0,000* Swedbank A 3928 4,526 0,104 25,916 0,000* 31,286 0,001* 81,813 0,000* Swedish Match 3928 64,955 0,000* 68,643 0,000* 85,540 0,000* 110,832 0,000* Tele2 B 3928 4,728 0,094*** 10,466 0,063*** 14,328 0,159 41,379 0,081*** Telia Company 3928 22,712 0,000* 33,183 0,000* 52,530 0,000* 82,592 0,000* Volvo B 3928 2,176 0,337 18,101 0,003* 18,800 0,043** 58,532 0,001* omx30 3928 8,134 0,017** 26,894 0,000* 36,245 0,000* 95,636 0,000*
*, **, *** Indicate a rejection of the EMH on a 1 %, 5%, 10% confidence level, respectively.
As is suggested in Table 9, many of the evaluated price series allowed for the rejection of randomness. However, the results are once again rather inconsistent, in the sense that randomness is rejected for some, but not all lags. Therefore, the same approach as for the variance ratio test have been applied, namely, that if randomness and thus the EMH is rejected for one lag, it is rejected for the entire price series.
Furthermore, to convey a comprehensive understanding of the results presented in Table 9, they are summarized in Table 10, which suggests that 24, 26 and 27 of the evaluated price series justified the rejection of randomness on a confidence level of 99 %, 95% and 90%, respectably. Consequently, according to Table 10, only 7, 5 and 4 price series suggest that the EMH might hold true.
