Martingale Difference Sequence Process Essay

Introduction

Graph 4: Proportion of countries on track to achieve the poverty reduction target

1. The Efficient Market Hypothesis (EMH)
1.1 Importance of Efficient Markets
1.2 Empirical Tests on the EMH

2. Literature Review

3. Financial Crisis and Market Efficiency

4. The Development of Sub-Saharan Stock Markets and Economic Growth

5. Characteristics and Trends of the Sub-Saharan Stock Markets

Table 1: Key figures of sub-Saharan stock markets

Graph 10: Performance of the Zimbabwe Stock Exchange

6. Data and Testing
6.1 Data
6.2 Descriptive Statistics
Table 3: Descriptive statistics of monthly returns for sub-Saharan Africa
6.3 The Model
6.4 Estimation Results

Summary and Conclusion

References

Appendix

Tables

Table 2: Key figures of emerging Asian stock markets

Table 4: Descriptive statistics of monthly returns for Asian developing countries

Table 5: Descriptive statistics of monthly returns for DAX and S&P

Table 6: Descriptive statistics of weekly returns for sub-Saharan Africa

Table 7: Descriptive statistics of weekly returns for Asian developing countries

Table 8: Descriptive statistics of weekly returns for DAX and S&P

Table 9: Augmented Dickey-Fuller Test (Nigeria)

Table 10: Augmented Dickey Fuller Test (Nigeria, 1st difference)

Table 11: Test for Weak-Form Efficiency: Sub-Saharan African Markets (monthly data)

Table 12: Test for Weak-Form Efficiency: Emerging Asian Markets (monthly data)

Table 13: Test for Weak-Form Efficiency: DAX and S&P 500 (monthly data)

Table 14: Test for Weak-Form Efficiency: Sub-Saharan African Markets (weekly data)

Table 15: Test for Weak-Form Efficiency: Emerging Asian Markets (weekly data)

Table 16: Test for Weak-Form Efficiency: DAX and S&P 500 (weekly data)

Table 17: Summary of tests for weak-form efficiency

Table 18: Test for Weak-Form Efficiency: Selected African countries (monthly data;current crisis effects excluded)

Graphs

Graph 1: Performance South Africa (1999-2009)

Graph 2: Performance S&P 500 (1999-2009)

Graph 3: Performance Ghana (1999-2009)

Graph 5: Time Path of efficiency, South Africa (1990-2001)

Graph 6: Time Path of efficiency, Nigeria (1994-2001)

Graph 7: Time Path of efficiency, Zimbabwe (1990-2001)

Graph 8: Time Path of efficiency, Mauritius (1990-2001)

Graph 9: Time Path of efficiency, Kenya (1990-2001)

E-views codes and general formulas

I. Estimated equation for RW

II. Estimated equation for RW

III. Q-statistic developments

IV. Performing the White Test in E-views

Introduction

Right now we are facing a huge crisis. Stock prices roughly dropped by half since their peak in October 2007 and only now are they showing some signs of recovery. The last financial crisis took place with the burst of the dotcom bubble in March 2000 and furthermore, we witnessed an external shock caused by the terrorist attacks in September 2001. The Johannesburg Stock Exchange (JSE) is the "regional leader" for the sub-Saharan stock markets (Wang et al., 2003, p. 532). By simply looking at its performance and comparing it to the S&P 500, the most representative index for the developed countries, I would conjecture that the impact of the crisis in 2000 and the following years is almost not existent in South Africa's index.[1]In contrast, today, within the world economic crisis we see massive losses in Africa, as well as the rest of the world. There are two possible explanations for this behavior. First, today's crisis is much bigger and therefore has a greater global impact. Second, today Africa's stock markets show a much higher degree of integration with the rest of the world and thus are more affected.[2]

The extent to which Africa's sub-Saharan stock markets are integrated in the global economy has important implications for their ability to attract capital from developed countries which is a primary goal in developing these stock markets (Jefferis, 1995). On the one hand, a stronger integration brings about a higher degree of publicity and in that way attracts more capital. On the other hand, however, a factor making the African stock markets attractive, at least for some investors, is its very low or even negative correlation with the rest of the world (Alagidede and Panagiotidis, 2009). This low correlation creates the opportunity of diversification strategies. With a stronger integration, the correlation is very likely to increase. Nevertheless, I believe the first factor to dominate, because Africa's stock markets will still remain among one of the markets in the world with the least correlation to the industrialized countries.

During crises we usually observe very high volatility levels.[3]High volatility levels imply high chances for investors, especially in so called exotic markets. But Africa's sub-Saharan stock markets tend to be fairly small in terms of market capitalization, meaning that the associated low levels of liquidity raise questions regarding the efficiency of these markets and the process of price determination. Thus, "there is always the danger that thin markets become the objects to manipulation by insiders implying a loss for other investors" (Magnusson and Wydick, 2002). The results are a bad reputation and a lower flow of capital but this capital is urgently needed.

Nevertheless, over the past few decades, the world stock markets have bubbled up, above all, due to the boom of the emerging markets. The speed and extent of stock market development in developing countries so far have not been attained and gave way for fundamental changes both in the financial structures of less developed countries and in the capital flows from developed nations (Yartey, 2008). With increasing globalization and world-wide integration of financial systems, African stock markets received more attention. This trend is largely due to their low correlation with the rest of the world but is also due to the remarkable performance in the past. For instance, the Ghana Stock Exchange[4]was nominated the world's best performing stock exchange at the end of first quarter of 2004 with a year return of 144% in US dollars compared to 38.47% return by S&P 500 in the USA and 53.97% by DAX in Germany.[5]

In its February edition 2008 the Economist characterized Africa as globalization's final frontier for investors and asked them to "Buy Africa" (The Economist, 19/2/2008). Considering the issue that a well developed stock market can potentially support economic growth, this is a big chance for the development issues in sub-Saharan Africa. Hence, it is vital that the sub-Saharan stock markets are at least able to "pass the lowest hurdles of speculative efficiency" (Magnusson and Wydick, 2002, p. 2). In order to measure market efficiency, the Efficient Market Hypothesis (EMH) is applied. According to the EMH asset prices and returns are determined by the outcome of supply and demand in a competitive market due to rational traders (Alagidede and Panagiotidis, 2009). In its strongest form it is supposed in the EMH that by these rational traders in every period any known data is reflected fully, correctly and instantly in the current prices of traded assets (Fama, 1970). After briefly looking into the question what policies could be crafted in order to sustainably support Africa's stock market development, it is the aim of this study to test whether and in case, to which extent, the sub-Saharan African stock markets, namely Botswana, Cote d'Ivoire, Ghana, Kenya, Mauritius, Nigeria, South Africa and Zimbabwe fulfill these criteria. Additionally, I will carry out the same tests on some Asian developing countries (Indonesia, Korea, Taiwan and Thailand) in order to look at the differences in stock market efficiency as a possible reason, surely among others, for differences in the past economic development. One of the objectives of the Millennium Development Goals (MDGs), based on a $1 day poverty line is the aim to halve the proportion of people living below$1 a day from around 30% (of the developing world's population) in 1990 to 15% by 2015 (World Bank, 2009a and Vollan, 2009). East Asia has come close to halving the proportion in poverty over just eight years (1990-1998), 15 years ahead of schedule. This represents the largest fall in poverty ever witnessed in history and has led to be referred to a 'miracle' taking place in East Asia. In sub-Saharan Africa, instead, poverty rates remained stagnant (47.67% to 46.30%) whereas the absolute numbers in poverty even went up. Although China accounts for the bulk of this positive change in East Asia and the numbers for South Asia are rather bad as well (4 of 5 countries in South Asia with available data are not on track), the figure below shows the huge discrepancy between sub-Saharan Africa and Southeast Asia in terms of being on track to achieve the poverty reduction target or not even being close to it.

Graph 4: Proportion of countries on track to achieve the poverty reduction target

Abbildung in dieser Leseprobe nicht enthalten

From the examined Asian markets, Indonesia and Thailand are located in Southeast Asia, Korea and Taiwan in East Asia respectively. This thesis will analyze whether the efficiency levels of the respective stock markets reflect the differences in the past economic development progress, that is sub-Saharan Africa being least efficient, then Southeast Asia and East Asia performing comparably well. Thus, Taiwan and Korea, being part of the Four Asian Tigers and considered as role models for other developing countries, are expected to perform a lot better. As a benchmark of efficiency for the developing countries I will compare them further to the S&P 500 and the DAX.

The rest of the paper is organized as follows. Section 1 presents the idea of the EMH, continued on a comprehensive literature review, covering the results and methods of the most influential papers concerning tests on the efficiency of African stock markets. After examining the possible effects of financial crises on efficiency in section 3, I will look into the case of the relation between stock market development and economic growth. In section 5 I will include an overview of the analyzed stock market's characteristics, past development and future trends. In the main part, section 6, I will explain the dataset, the methodology and the results of my study. That will bring me to the main conclusions of this work, a summary of the findings and its possible interpretation.

1. The Efficient Market Hypothesis (EMH)

If the EMH holds, the market prize is only affected by new information, which is immediately incorporated. We denote: [Abbildung in dieser Leseprobe nicht enthalten] with: - 6t denoting the full information set available at time t [Abbildung in dieser Leseprobe nicht enthalten] is the equilibrium expected prize

[Abbildung in dieser Leseprobe nicht enthalten]is the new information.

Thus, [Abbildung in dieser Leseprobe nicht enthalten]

It follows that the sequence of unexpected prize changes fet} is a "fair game" with respect to {Θ,} (Fama, 1970, p. 384). These simple calculations have strong implications. Excess market values/ returns are unpredictable, given the available set of information. Furthermore, it is impossible to make any abnormal returns by trading on own information.

The version described above is the most extreme form of the EMH which can be further categorized with respect to the available information set. The semi-strong form of the hypothesis states, that equity markets accurately process all publicly available information. The third, so called weak form of the EMH states, that past stock market information is irrelevant for predicting future movements in stock prices. According to Fama (1970) industrial countries satisfy the conditions for the weak and the semi-strong form of the EMH. Campbell et al. (1997) argue that perfect efficiency is not a realistic benchmark, since there are good reasons that markets do not support this hypothesis. As an example, the authors cite Grossman and Stiglitz (1980). They claim that perfect informationally efficient markets are impossible. The reason is that in case we have a perfectly efficient market, there is no return of acquiring any information since at the time you get the information it is already useless. In that way, we lack an incentive to trade at all and markets eventually collapse (Lim, 2008a). As a result, there must exist sufficient profit opportunities, namely inefficiencies which can compensate investors for the cost of trading and information gathering.[6]

The literature (including this study), however, concentrates on the weak-form of the EMH.

1.1 Importance of Efficient Markets

Market efficiency is, to a certain degree, an important characteristic of any stock market. Generally speaking, stock markets are an additional possibility to invest capital and can therefore reduce investment risk through diversification (Caprio and Demirguc-Kunt, 1998). Firms particularly benefit from efficient stock prices and yields since they provide a certain benchmark against which the cost of capital for and returns on investment projects can be judged (Green et al, 2005). Furthermore, efficient stock prices are always forward looking. In that way, they function as a unique record of shifts in investors' views about the future prospects of companies as well as the economy (Green et al, 2005).

Because of their lower liquidity levels (see section 5), developing countries are especially at risk of not satisfying the weak form of market efficiency. In contrast to the industrialized countries where a certain level of professional trading evolved within the long history, it is crucial for the developing countries to actively push forward the evolution of certain characteristics of an efficient market. Efficiency usually postulates fair pricing, almost no trading impediments, financial liberalization, a good regulatory framework, a certain degree of transparency and a quick and reliable clearing. In the case, that these characteristics are not met, interferences are caused which prevent efficient trading. Efficiency avoids arbitrage possibilities which can seriously compromise the stock exchanges themselves and moreover efficiency gives investors a reasonable level of safety and by that an incentive to invest money in the respective market.[7]

1.2 Empirical Tests on the EMH

There is a large empirical discussion on the EMH, since you can develop various ways of modeling the information set, which must be restricted somehow. The condition for the weak market efficiency that current prices fully reflect all available information implies that successive price changes/returns are independent. Further, successive price changes are identically distributed. "Together the two hypotheses constitute the random walk model" (Fama, 1970, p. 386). The random walk hypothesis states that the price of a financial asset is given by the following process:[Abbildung in dieser Leseprobe nicht enthalten]

, where μ can be seen as a drift term and P is the stock prize at time [Abbildung in dieser Leseprobe nicht enthalten] respectively. Depending on the assumption of et you can distinguish three types of a random walk (Cowles, 1960 and Campbell et al, 1997):

S RW1: iid (independent and identically distributed) increments S RW2: independent increments S RW3: uncorrelated increments.

The RW1 assumes that st is independent and identically distributed with mean zero and variance σ2, i.e. et~iid (0, σ2). Under RW2 the increments are independent but not identically distributed which allows for unconditional heteroskedasticity. "Unconditional heteroskedasticity occurs when heteroskedasticity of the error variance is not correlated with the independent variables in the model" (De Fusco et al, 2007, p. 348). The RW3 is the weakest form and assumes that the Cov(st,st_k) = 0 Vk Ψ 0, thus, independent but not identically distributed. Consequently, it allows for conditional heteroskedasticity. In this case the error variance is correlated with (conditional on) the values of the independent variables in the regression.

The thesis focuses on all three forms of the weak version of the EMH and runs tests on them in sequence.

2. Literature Review

It is possible to use different kinds of tests in order to assess market efficiency. Besides, data differs in terms of the time span, frequency (daily, weekly, monthly), currency, type of index, etc..

From very early tests Fama (1970) draws the conclusion that it seems fair to say that for industrialized countries the results for the weak and semi strong form of the EMH are in strong support. For example, Kendall (1953) considered the serial correlation in United Kingdom stock prizes to be so weak that it, in his opinion, it was impossible to predict stock price movements, concluding the weak form of the EMH. Later economists started to become more askance, triggered by the work of Lo and MacKinley (1988) carefully distinguishing the iid random walk.

Dickinson and Muragu (1994) find evidence consistent with the random walk in Nigeria and Kenya respectively. Thompson and Ward (1995) examined a wide range of literature concerning empirical test on the efficiency of the Johannesburg Stock Exchange (JSE). They conclude that different approaches give different results, so that an ultimate result is hard to figure out. However, prediction is considered useless for the JSE since any autocorrelation was too small. Olowe (1999) rejected the random walk in Nigeria. Jefferis and Okeahalm (1999) made an event study on the efficiency on the stock markets of South Africa, Botswana and Zimbabwe. They tried to find out how quick and accurate the individual stocks react to information announcements in the pricing process. Their result is, that only the JSE acts according to the weak form of market efficiency whereas the other two markets were found to be inefficient.

Smith et al. (2002) used weekly data and a joint variance ratio test. They found the South African stock market followed an iid random walk over the period from 1990 to 1998. Nevertheless not one of the other examined markets, namely Botswana, Egypt, Kenya, Mauritius, Morocco, Nigeria and Zimbabwe followed an iid random walk nor a martingale difference sequence. A Martingale is a stochastic process whose expected value, given all anterior observations, is equal to the value of the last observation (Wang, 2003, p. 4):

Abbildung in dieser Leseprobe nicht enthalten

The martingale hypothesis states that the price of a financial asset is a martingale. Thus, written as a difference it satisfies the following condition:

Abbildung in dieser Leseprobe nicht enthalten

Thus, the martingale difference (price change) is a fair game. One can conclude that every random walk [Abbildung in dieser Leseprobe nicht enthalten] without a drift term (μ) is a martingale difference but not the other way round, remembering that the expected value of the error term is equal to zero.

Magnusson and Wydick (2002) used the random walk model described in section 1.3, which at the same time will be the basis for this paper's testing. Using monthly data they found out that Botswana, Ghana, Nigeria (only in local currency) and Zimbabwe are inefficient, while Côte d'Ivoire, Kenya, Mauritius and South Africa are weak form efficient. None of the markets is considered to follow an iid random walk. The results do not show that the African countries can pass the same high hurdles of efficiency as the US market, but in their study they compare favorably to some Latin American and Asian markets. The authors conclude that this presents evidence, that the sub-Saharan markets are no more an object of manipulation than any other emerging market.

Smith (2008) points out possible reasons for the differences in results obtained by Smith et al. (2002) and by Magnusson and Wydick (2002). "First, tests using monthly series tend to reject the iid random walk and mds [martingale difference sequence] null hypotheses less frequently because they cannot detect serial correlation at weekly frequencies" (Smith, 2008, p. 162). Second, the dataset becomes smaller with monthly data, assuming we observe the same time span, meaning the applied tests lack power in small samples. As a consequence Magnusson and Wydick (2002) tend to fail to reject the corresponding hypothesis, when it is false. Since the authors rejected the iid random walk more often (e.g. for South Africa), these reasons, at first, seem a little surprising considering that they are a contradiction to the statement. But they do make sense for the martingale difference sequence which is rejected more often in Smith's paper from 2002. The problem is that Smith (2008) defines the martingale difference hypothesis as "a less restrictive version [of the EMH] which allows for conditional heteroskedasticity" (Smith, 2008, p. 162f.). He compares his results concerning the martingale difference hypothesis from 2002 with Magnusson's and Wydick's RW2 (allowing for unconditional heteroskedasticity) instead of RW3 which is the one that allows for conditional heteroskedasticity (recall 1.2). In case he really meant the RW3, which Magnusson and Wydick reject more often, this leads the arguments pointed out by Smith (2008) ad absurdum. The matter is that Smith et al. (2002) used variance ratio tests, that have the advantages that you do not have to assume returns to be normally distributed and they permit general forms of heteroskedasticity. Not having to pay particularly attention to the form of heteroskedasticity I reckon Smith's reference refers to Magnussons' and Wydick's RW2 model allowing for unconditional heteroskedasticity which would be consistent with his argumentation.

Mlambo et al. (2003) performed a test on the iid random walk hypothesis for four African countries (Egypt, Kenya, Morocco and Zimbabwe). It is worth mentioning that they calculate the returns in a way that makes it possible to adjust for thin trading which is very often the case in such small markets. In about half of the daily returns series ranging from 1997 to 2002 significant serial correlation was detected and interpreted as rejecting the iid random walk hypothesis.

An interesting approach was made by Jefferis and Smith (2005). They argue that especially for the new emerging markets it is implausible that the parameters stay the same, so efficiency rather emerges over time. Hence, the authors have a look into the changing efficiency of the African stock markets. Using a GARCH (generalized autoregressive conditional heteroskedasticity) approach with time-varying parameters, a test of evolving efficiency was implemented.[8]Expressing it in simplified terms they tested the following model:

Abbildung in dieser Leseprobe nicht enthalten

If we have a random walk, ßi will be equal to zero. Thus,

Abbildung in dieser Leseprobe nicht enthalten

Graphs 5-9 in the appendix show the results for the markets of South Africa, Nigeria, Zimbabwe, Mauritius and Kenya from 1990 until 2001. The figures show the time-paths of the estimated ßit coefficients together with their 95% confidence intervals. For South Africa (Graph 5) one can observe constant parameters on an efficiency level. That means according to Jefferis and Smith (2005) South Africa was efficient throughout the monitored time period. The result is unique in their sample and comparable to many developed stock markets, as the London Stock Exchange reported in Zalewska-Mitura and Hall (1999). Nigeria (Fig. 5), however, exhibits changing levels of inefficiency followed by changes towards weak form efficiency. In contrast, the results for Zimbabwe and Mauritius (Graph 7 and 8) both reject the hypothesis of weak form efficiency with time-varying ßlt. Kenya (Graph 9), instead, almost doesn't fluctuate and remains around the value of 0.379 implying no weak form efficiency and neither a tendency towards it.

Lim (2008a) reviews a concept of relative market efficiency, introduced by Evans in 2006. Evans measures the efficiency of one market against another. Conventional efficiency studies are mostly criticized because the empirical tests focus on the so called "all-or- nothing notion of market efficiency" (Lim, 2008a, p. 2), meaning that the tests only lead to the result that a market either is weak-form efficient or it is not. There is no information on market improvement in efficiency, the degree of efficiency or a quantitative comparison.[9]But besides the work of Evans (2006), not many pursued this approach.[10]Jefferis and Thupayagale (2008) use a different technique. They look for long memories in the stock market data since indicating a significant autocorrelation and giving it a predictable component. Therefore, long memory is seen as evidence against the weak form of the EMH. They reason that Botswana has such a component whereas in South Africa it is rather small and not statistically different from zero. Returns in the Zimbabwe and US markets show an anti-persistent process, with the effect in the former case of not being statistically significant.

Smith (2008) performs tests on efficiency for eleven African stock markets (Botswana, Cote d'Ivoire, Egypt, Ghana, Kenya, Mauritius, Morocco, Nigeria, South Africa, Tunisia and Zimbabwe) using joint variance ratio tests over the period from the year 2000 to 2006 with weekly and monthly data. The iid random walk hypothesis is rejected in each of the markets but the weekly return of Egypt, Nigeria, Tunisia and South Africa weekly conform to a martingale difference sequence, supported by evidence from tests on monthly returns. For Mauritius, Morocco and Zimbabwe the martingale difference sequence is rejected for weekly but not for monthly data.

Alagidede and Panagiotidis (2009) perform a test on the random walk model for Egypt, Kenya, Nigeria, Morocco, Tunisia, South Africa and Zimbabwe. They reject the random walk as an adequate characterization of returns in their sample. This rejection of the random walk, however, does not imply rejection of weak form efficiency in these markets. This is because a test of the random walk hypothesis is a joint test of both weak form efficiency and constancy of expected returns. Alagidede and Panagiotidis run a variety of tests on the iid properties of the data which all reject the null of iid for the residuals of the RW1. There is no testing on weaker forms though. But the authors continue testing the presence of nonlinearities in the series which could imply evidence of return predictability.[11]For instance, they apply the form of a GARCH model. A GARCH model tries to describe the volatility process of an asset return assuming the mean equation to be adequately described by an ARMA (autoregressive moving average) (Tsay, 2001, p. 93). The underlying idea is that with the correct specification of the volatility the squared standardized residuals should be unpredictable on the basis of observed variables. They only find a few significant p-values. The overall results from all the models, however, denote that there is no remaining structure in the data. Thus, they did not find evidence to reject weak form efficiency for these markets.

Nowadays, many people tend to believe that it is at least to some extent possible to predict stock prices, a view that is consistent with the behavior of many market participants (Smith, 2008).

Due to the existence of severe crises in the dataset, below I will elaborate on the relationship between financial crises and market efficiency.

3. Financial Crisis and Market Efficiency

While there is extensive literature on tests of the weak-form EMH and on financial crises per se, the effects of certain financial crises on stock market efficiency did not receive much attention. E. g. during the Asian emerging market crisis in 1997 the massive stock market losses were very often quoted, with only a few formal analyses of the possibly related changes in market efficiency (Lim, 2008b). In this context Lim (2008b) cites Cheong et al. (2007) who divide their sample data into pre-crisis (January 1991 - December 1996), crisis (January 1997 - August 1998), USD pegged (September 1998 - December 2000) and post-crisis (January 2001 - April 2005) periods. The authors report that the highest inefficiency occurs during the crisis period, followed by pre-crisis, post-crisis and US dollar pegged periods. Furthermore, Hoque et al. (2007) examine the weak-form efficiency of emerging Asian stock markets in the pre-crisis (1990-1997) and post-crisis (1998-2004) periods. They demonstrate that the crisis has no significant effect on the degree of efficiency in most of the examined cases. The exception is Taiwan, which became more efficient over time. Kim and Shamsuddin (2008) also did not find much of a change in the level of market efficiency. But there are two cases, Singapore and Thailand, which exhibit a higher degree of efficiency after the crisis. Lim (2008b) himself, examining several sectors in Malaysia, exemplifies that the highest inefficiency occurs during the crisis period for almost all economic sectors.

The results, presented by Lim et al. in the same year, showed that the crisis lowered the efficiency levels of most Asian stock markets but also shows that they recovered again shortly after.

In conclusion there are different findings that show how the markets reacted after the Asian financial crisis in 1997. Findings of higher inefficiency during the crisis should not be too surprising considering the following facts. In general when stock markets begin to drop dramatically, investors tend to panic. People do not only overreact to local news, but also to events happening in other markets (Lim, 2008b). Consequently, trading is not dominated by rational investors any more, but by sentiments which in turn leads to inefficiency.[12]The mere existence of a 'produced' phenomenon, called a bubble, automatically contradicts the theory of rational prizes. Thus, it depends on the public intervention whether they are able to calm the investors/ the mass or not. Since the industrialized countries tend to have 'stronger' governments due to their higher influence and greater set of support strategies, it is assumed that the efficiency of the developing countries is affected to a higher extent.

On the other hand, we face very high frequency of trading in extreme situation which greatly raises the turnover in stock markets for a short time. This phenomenon and the possible overall positive tendency in terms of efficiency in certain markets might explain the few paradox results of higher efficiency seen above.

[...]

[1]See graph 1 and 2 in the appendix. There is a drop after September 9/11 though, but generally, during this time, the JSE did not follow the S&P 500's downward trend. You could observe a similar behavior of the sub- Saharan stock markets during the global emerging market crisis from 1997-1998. The degree of global integration of African stock markets was very limited at that time. For a detailed study with cointegration analysis see Wang et al. (2003).

[2]You have to take into consideration that neither the collapse of internet-based companies nor the subprime crisis has a lot to do with the African economy itself, at least at first hand.

[3]For example, S&P 500 daily volatility, as measured by the daily return standard deviation for the previous 30 days, averaged 3.3% from October 27, 2008 to April 27, 2009. But it averaged only 0.6% in the time from January 1, 2007 to June 30, 2007.

[4]See graph 3

[5]Compare to Adjasi and Bipke (2006), plus own calculations with data from Datastream.

[6]An intuitive argument against this reasoning is the existence of gambling casinos. There is no predictability at all, hence, no compensation for 'trading' and information gathering. Nevertheless, people gamble.

[7]Surely, arbitrage possibilities are also a strong incentive for traders seeking to outperform the market. But this work does not focus on risky traders but on long-time capital investments in the respective market which need a reasonable level of safety with respect to efficiency.

[8]Emerson et al. (1997) and Zalewska-Mitura and Hall (1999) have developed a test for evolving efficiency which detects changes in weak form efficiency through time, e.g. applied by Jefferis and Smith (2005) as seen above.

[9]The above discussed model of Jefferis and Smith (2005) are able to say something about improvements in efficiency. Moreover, in the statistical tests used in this paper the degree of rejection gives an idea about the level of the respective efficiency.

[10]Lim (2008) only cites Kellard et al. (1999) and Ma (2004).

[11]There is not a lot of literature available on the weak form of market efficiency where nonlinearities are taken into account. Only compare to Brooks (2007), Lim et al (2008) and Panagiotidis (2005) and the references therein.

[12]The rise of behavioral economists, like Richard H. Thaler and Robert J. Shiller, showing that mass psychology, herd behavior and the like can have an enormous effect on stock prizes, indicates that perhaps the market in general is not quite so efficient after all (Nocera, 2009).

A stochastic process $\{X_t\}$ is called a martingale if

$$\operatorname{E}[X_{t+1} \mid X_{t}, \ldots, X_1\} = X_t$$

That is, the expectation of the future conditional on the past is the present.

Fumio Hayashi's Econometrics defines a process $\{Z_t\}$ as white noise if $\operatorname{E}[Z_t] = 0$ and for any $j \neq 0$ $\operatorname{E}[Z_tZ_{t+j}] = 0$.

Let process $\{Y_t\}$ be a series of independent flips of a fair coin where $Y_t = 1$ if heads and $Y_t = -1$ if tails. Observe that:

• $\{Y_t\}$ is white noise
• $\{Y_t\}$ is NOT a martingale. If we flip a coin heads, we don't expect the next flip to be heads! The conditional expectation of $Y_t$ is always zero, not $Y_{t-1}$.

Perhaps what you're thinking? (or what your Prof is leading you to...)

A process $\Delta_t$ is called a martingale difference sequence if the conditional expectation of $\Delta_t$ given past information $\mathcal{F}_{t-1}$ is zero, that is, $\operatorname{E}[\Delta_t \mid \mathcal{F}_{t-1}] = 0$. Consequently a white noise process is a martingale difference sequence. Why is $\Delta_t$ called a martingale difference sequence? Define $X_t = X_{t-1} + \Delta_t$. Then $X_t$ is a martingale.

(Note also that a martingale difference sequence need not be white noise.)