طبقه بندی چند بعدی ناهنجاری های بازار: شواهدی از 76 شاخص قیمت

This paper makes the first attempt to present explicit empirical evidence that market inefficiency can be multi-dimensional. Testing the Efficient Market Hypothesis (EMH) over 76 stock indices using 17 best established indicators (e.g. runs test), we show that most indices exhibit some type(s) of anomaly and that indicators differ from each other in terms of statistical power and/or the type of anomaly detected. A principal components analysis (PCA) demonstrates that indicators group along orthogonal dimensions, and hence a market can exhibit short-term memory, long-term memory and/or calendar effects, which are all distinct sources of possible inefficiency. This research presents statistical evidence on the extent and nature of market inefficiency, offers possible explanations for conflicting previous findings, and provides new insights into studying market efficiency.

This article shows that inefficient markets, like Tolstoy's unhappy families, may be inefficient in several different ways.1 This perspective not only offers a possibility to reconcile some conflicting findings regarding market efficiency in the literature, but also provides new insights into portfolio investment and future research on weak-form market efficiency. The efficient market hypothesis (EMH) has been a dominant organising principle in financial markets research for several decades. To test EMH, researchers have sought out new data from new markets and over extended time-spans; have analysed at the micro level of minute-by-minute prices or of individual trader transactions; have devised new statistical techniques; or/and have made insightful use of new explanatory variables. Yet there is still disagreement about the central proposition of EMH: are markets efficient? Research still emerges that resolutely rejects EMH, or just as resolutely defends EMH. For instance, Worthington and Higgs (2006) claimed: “The serial correlation and runs tests conclude that all of the [15 Asian] markets are weak-form inefficient”; whereas Lean and Smyth (2007), using a subset of these same markets, claimed the opposite: “The overwhelming conclusion is that stock prices in the [eight] Asian markets studied are characterized by a random walk.” Several principal reasons explain how conflicting findings might arise. First, there are different understandings on how to define market efficiency. Some researchers claim that markets are inefficient if they find irregularities in market movements; while others define it more restrictively, saying that a market is inefficient only if exploitable opportunities constantly exist. Second, researchers tend to examine too few markets that are too homogeneous. Understandably this gives research a focus, but it limits the generality of any findings. Third, researchers tend to assess market inefficiency using a very limited set of indicators (used in this paper to mean statistical tests of EMH), often only one. Different indicators may have different statistical power to detect inefficiency, and the greater the power disparity among indicators used in different studies, the greater the risk of different conclusions. Fourth, most previous studies have tacitly assumed that markets may be inefficient in only a single way, and by implication, that all indicators are tests of this uni-dimensional inefficiency. However, if markets are inefficient in a variety of ways, with different indicators tapping different dimensions of inefficiency, then the need to sample a sufficient variety of both markets and indicators becomes crucial.2 Past literature has never before explicitly tested these possibilities. As argued by Lo (1997), even after decades of research and thousands of journal articles, economists have not yet reached a consensus about whether financial markets are efficient or not. We are not naively claiming that this paper could resolve the debate and it is not our key aim to be one of many testing whether markets are efficient or not. Instead, the main objectives of this paper however lies in: to benchmark indicators as to their ability to detect anomalies (Objective 2); and to explore whether multi-dimensionality of anomalies is a robust phenomenon (Objective 3). To achieve these objectives, it is the necessary first step for us to test whether markets exhibit certain statistical anomalies (Objective 1), which for the convenience, we call market inefficiency3 in this paper. We selected a large number of diverse stock markets (more strictly, 76 price indices), and 17 indicators that have been widely used and well-accepted in the finance literature. The inefficiency of each index was measured by each indicator in turn, producing a 76 × 17 matrix of either z-scores or Chi-square scores, depending on the indicator/test. To clarify the scope of this paper, note that most previous research stops processing after analysing a small subset of such a matrix. By contrast, obtaining the matrix is just the starting point for further analyses here. The information in the 76 × 17 matrix was synthesized, e.g. through principal components analysis (see Section 3), to achieve our three research objectives. Our results show there is a wide difference in the statistical power of the indicators, which explains why researchers who use different indicators may come to different conclusions. Across the globe, markets are found to exhibit certain anomalies. Moreover, these anomalies can be categorized orthogonally: i.e. market inefficiency can be multi-dimensional. In the next section, we review the literature on market inefficiency and give a synoptic survey of the indicators we use. In Section 3 we explain the indicators, indices, data, and how we analyse the data. Section 4 presents and discusses the results. In Section 5 we illustrate how any given new method of testing market efficiency can augment our analysis; and Section 6 concludes the paper. 2. Review of indicators and inefficiency The efficient market is a central concept in modern finance, which has had its adherents (Malkiel, 1973) and its detractors (Lo and MacKinlay, 1999). Consistent with the importance of this debate, the literature describes many indicators to test whether a market is inefficient, e.g. total runs, variance ratios and calendar effects. Empirical findings that are inconsistent with EMH are called anomalies. It is natural to ask two questions of these indicators. First, do some indicators have greater statistical power than others to detect inefficiency, and if so, to what extent? Second, do different indicators detect the same or different aspects of market inefficiency (i.e. types of anomalies)? Understandably, researchers want to use the best tests; and to be able to assess the results of others bearing in mind the power of the tests they used. Consequently, the idea of benchmarking one test against another occupies a definite place in the study of market efficiency. For instance, using Monte Carlo simulations, Lo and MacKinlay (1989) compared their Variance Ratio test against the Dickey Fuller test and the Box-Pierce test, concluding that Variance Ratio was generally more powerful than the other two tests. Giraitis et al. (2003) compared their new rescaled variance (V/S) test against the rescaled range (R/S) test ( Lo, 1991), and the KPSS test ( Kwiatkowski et al., 1992). The raison d’être for power comparisons is well accepted because any new method can justify its existence by being more powerful than similar rivals. But the piecemeal practice of comparison creates in the literature a disjointed set of preference orderings, each covering a small number of similar indicators. This paper presents a more comprehensive comparison of the relative power of disparate indicators, which therefore avoids the limitations of previous comparisons. The second question has been less clearly articulated in past research. Nonetheless, distinctions made in the literature do suggest the possibility of a multi-dimensional view of inefficiency. For instance, short-term effects have sometimes been contrasted with the market's long memory (Fama and French, 1988 and Lo, 1991—see Section 2.2.4; Chow et al., 1995); and persistence is often contrasted with anti-persistence.4 Theoretically, degree of persistence and length of memory may occur in any combination within a market. Other indicators search for evidence of inefficiency in external sources of influence on current returns, which should be irrelevant to an efficient market. Examples of such sources are: weather effects ( Hirscheifer and Shumway, 2003 and Chang et al., 2008; though see Jacobsen and Marquering, 2008; and the ensuing debate in Kamstra et al., 2009 and Jacobsen and Marquering, 2009); national success or failure in sporting events ( Edmans et al., 2007 and Kaplanski and Levy, 2010); and calendar effects (see Section 2.2.5 for a brief review). However, weather and sport are not well suited to global comparisons because the phenomena are intrinsically localised (blizzards in Toronto; typhoons in Shenzhen). For this reason we examine only calendar effects as external sources of influence. The above discussion suggests that although the literature has provided hints that markets may be multi-dimensionally inefficient, it has provided neither a thorough theoretical classification (of inefficiencies), nor an empirically grounded one. We attempt to fill this research gap. To sum up, there are a number of ways in which an indicator may have value. The two emphasised in this paper are (i) that it is better (Objective 2) and (ii) that it is different (Objective 3). In the Darwinian evolution of EMH indicators, those that have the greatest overall ability to detect inefficiency should survive. But a less powerful indicator should also survive in a niche if it detects a different kind of inefficiency to other indicators. It is important to note that if inefficiency is multi-dimensional, then we need to qualify our interpretation of the power analysis in terms of power within a category or dimension, instead of power per se. An indicator that appears weak may be so because the dimension of inefficiency that it detects is itself weak rather than that it is intrinsically a poor indicator. There are other grounds by which to value an indicator. It may make different assumptions to other indicators in its category: especially valued are indicators that make fewer assumptions. It may be more robust to violations of its starting assumptions than other indicators. It may simultaneously detect many sources of inefficiency, which is necessary in a portmanteau test. Alternatively, it may detect only one source of inefficiency with no contamination from other sources, which is important in a highly focussed test. Finally, it may simply provide independent confirmation of a contentious or ambiguous indicator of inefficiency. Although important in their own right, we leave these criteria for future research. In the next few subsections we review a number of indicators that have appeared prominently in the literature. We choose only well-established indicators, most of them described in Chapter 2 of Campbell et al. (1997). The real value added by this paper is not in the novelty of the indicators, or in the sophistication of the techniques that we use to synthesise this data, but in the additional synthesising steps in order to build new, large scale maps that chart the diversity of market inefficiency, and the relative power of indicators. Note also, that by restricting ourselves to standard tests, we establish a minimum position for the dimensionality of market inefficiency – more advanced versions of these tests or more recent innovations in testing can only add to, but never subtract from this diversity.

The first step of this paper was to explore whether markets were efficient or not, roughly defined as whether nor or not they exhibit statistical anomalies. Our results show that generally, most of the markets are statistically inefficient, exhibiting one type or the other of anomalies. However, we have to point out that our results should be cautiously interpreted. Finding statistical anomalies or regularities does not immediately reject EMH. Richard Roll, among others, points out that a market is only inefficient when investors can properly exploit it in a systematic way (Malkiel, 2000). To address this, transaction costs and other factors need to be taken into consideration, which is too ambitious and is beyond the scope of this paper.13 We leave this for future studies, which might be possible if narrowing down the indicators and markets. However, even with this limitation, investors or researchers can use our findings as a starting point to explore opportunities. For example, some markets are more efficient than the others. Investors can start with those markets that show strongly significant statistical anomalies and examine whether a certain market is exploitable or not with their knowledge of current transaction costs, etc. Also, the paper makes clear that inefficiency is as much in the index as in the market – the FTSE100, for instance, being among the most efficiently traded indices, but the FTSE SmallCap being amongst the least efficiently traded. The finding has important implications for market participants and policy makers, such as in building investment portfolios and setting market regulations across the markets. The more innovative and valuable findings of the paper relate to our second and third objectives. The second objective was to benchmark indicators. We have found a remarkable disparity in the power of different indicators to detect anomalies among even the same market. Therefore, whether a market is observed to be inefficient depends to a large extent on the indicator used. The third objective was to explore whether market inefficiency is a multi-dimensional concept. We found that various anomalies load on several dimensions; i.e. market inefficiency, if a market is inefficient, can be multi-dimensional. Therefore, whether a market is observed to be inefficient depends on whether it is inefficient along a particular dimension, and whether the indicator is tuned to detect inefficiency along that dimension. Thus, to better understand market efficiency, both conditions need to be present. This paper provides a more sophisticated understanding of market inefficiency and the indicators. It offers possible explanations for why researchers often come to opposite conclusions, even when examining the same data. For instance, Lean and Smyth (2007) used tests that are not only likely to be less statistically powerful than those used by Worthington and Higgs (2006), as argued in Section 4.3; but also tap into different kinds of anomalies. Moreover, this paper provides important insights for future research. Methodologically, the paper shows that samples and indicators are important when testing for EMH. Theoretically, the paper suggests a multi-dimensional view of market inefficiency. The existence of multiple dimensions suggests that, while markets are to some degree globally integrated, there is plenty of local variety in need of explanation. Future studies that investigate the origins of the different inefficiencies should eventually help build up a more complete picture of how markets operate, the nature of inefficiency, where and when it emerges, and what causes it to disappear or evolve. Hints have been provided in the literature to look at different trading behaviours, e.g. individual vs. institutional investors, or foreign vs. domestic traders (Venezia and Shapira, 2007, Barber et al., 2008 and Chen et al., 2009). Being the first empirical study explicitly devoted to multi-dimensionality, we have focused only on the best-established indicators and methods of synthesis that are applicable using index-level data. But every new indicator has the potential to reveal a new dimension, or to re-evaluate the importance of an old one. Future research could extend the coverage of data, methods of synthesis, and indicators. Prima facie, those most likely to unearth novel inefficiencies are ones that exploit aspects of the price index process we have not explored, for instance the contingency of volatility or extreme daily returns ( Bali et al., 2008); or that use external, non-index data, such as weather at the stock exchange ( Hirscheifer and Shumway, 2003). Unfortunately, whereas historical data always includes the closing price index at an exchange, other useful price-related data may not always be available (e.g. in Datastream only 36 of our 76 prices indices had a full set of daily high and low for the index). Even when external data is available, as presumably it is for daily weather in any of the world's cities, it may be difficult to interpret in a local context – what counts as fine weather in Singapore versus Toronto? These difficulties may limit the kinds of indicators that could supplement the PCA, as presented in this paper, to ones that process the price index itself. To conclude, the paper is the first to grasp the nettle of the multi-dimensionality of market inefficiency, i.e. efficient markets are all alike; however, for those inefficient markets, they can be inefficient in their own way. Although the multi-dimensional perspective makes life more difficult, it also makes it more realistic and more interesting. Researchers could use this paper, e.g. the index × indicator scores matrix, as a starting point for their own analyses. In this way, even if we cannot directly alleviate the “family unhappiness” of inefficient markets, we may at least augment the family therapist's stock of knowledge.