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### INTRODUCTION

Financial markets are important in an economy in that they involve lots of monetary funds in the capital markets. Many firms raise finance in the form of equities and debts in those markets as means to finance expansion or expenses. Hence they serve the intermediation process and also provide a means for investors to diversify their portfolio of assets. African stock markets have been subject to economic restructuration as well as stock exchange modernisation these recent years. They now face regional and global integration and so the need to investigate their returns characteristics.

Efficiency is an integral part of investment valuation. When markets are efficient, security prices are properly valued such that investors may not beat the market and make abnormally higher returns than others. Inefficiency leads to market prices deviating from actual value. Hence, those having reasonable level of expertise in the field of valuation will be able to spot and exploit above and under-valued stocks.

Recent years have witnessed the exploration of the concept of efficiency in developed countries though less attention has been devoted to less developed ones. These researches indicate the importance of developing stock markets for countries which are at appropriate stage of economic growth. Efficiency in equity markets is of significance to investors and policymakers in African markets. It indicates how efficiently those markets impound market information into security prices. Indeed, it is more convenient to test for weak form efficiency of market rather than testing for semi-strong or strong forms of efficiency due to lack of data and supervision pertaining to those markets.

This objective of the study is to examine the possibility of both short- and long-term memory in asset returns in selected African markets' stock indexes namely the Mauritius, Johannesburg and Morocco All Share indices and the EGX 30 for Egypt. Besides South Africa, all the other markets are still in developing state and this paper can provide some explanations as to whether market development and size can determine efficiency in these markets.

The rest of this paper is organised in the following way. Section 2 describes informational efficiency with emphasis on weak-form efficiency and random walk. Critics relating to the latter are then raised to emphasise on non-linearity and long-term dimensions. Section 3 provides a brief description of the characteristics of the selected African stock markets as well as their respective indices. We then proceed to the methodological discussion based on the different random walks and long-term analysis in the fourth section. Then tests, results and discussions are provided in section 5.

### LITERATURE REVIEW

### 2.0 Introduction

Efficient market hypothesis is one of the most debated and researched topics in the realm of the stock market. While most of the early studies have previously been centered on developed stock markets like USA, Japan and Europe, developing and emerging stock markets have been brushed aside. Before proceeding with a systematic and ordered approach, it might be useful to present a general review of the theory under study, which in turn aims at defining the main concepts and demonstrating familiarity with previous relevant findings concerning the same field of research.

### 2.1 Theoretical review

### 2.1.1 Market efficiency

Efficiency has various different contextual meanings but analysis of financial markets assumes an informational dimension. The attribute of those markets by virtue of which they respond to new information, is called informational efficiency. This implies that current market price reacts instantaneously to new information so that it incorporates all relevant information. Since, by definition, new information is unpredictable, it follows that change in stock price cannot be anticipated and thus move in a random manner.

Informational efficiency can be related to the hypothesis of random walk which assumes that prices do not exhibit predictive patterns over time and follow a random walk. Hence, prediction of future prices in absolute terms, based singly on information about historical price, will be unsuccessful. The theory had its roots from the early works of Bachelier (1900). In his own words, Bachelier argued that “past, present and even discounted future events are reflected in market price, but often show no apparent relation to price changes”. This emphasises the informational content of stock prices.

In his paper on the behaviour of stock and commodity prices[1], Maurice Kendall (1953) further supported the random walk theory. The findings, unexpectedly, showed that prices follow a random walk and not regular cycles. His conclusion was that the series appeared ‘wandering', ‘Almost as if once a week the Demon of Chance drew a random number from a symmetrical population of fixed dispersion and added it to the current price to determine the next week's price'

The random walk theory was also supported by Fama's thesis[2] where he reviewed previous works on stock price behaviour. His conclusion was that “it seems safe to say that this paper has presented strong and voluminous evidence in favour of the random walk hypothesis.” Indeed in a market where prices are determined rationally, only new information will cause them to change. Hence prices follow a random walk to reflect all current knowledge.

If price prediction were possible, this would have caused market inefficiency as prices don't incorporate all information. Fama (1965) was the first one who coined the term efficient market. He held that such a market is one constituting of a large number of competing rational and active profit-maximisers who try to predict individual values of securities. Information in those markets tends to be almost free. He argued that the essence of ‘instantaneous' adjustment in actual prices to new information is competition leading to efficiency in the market.

Later, the random walk theory was broadened into a concept called the efficient market theory. Based on the works of Samuelson (1965) and Roberts (1967), Fama (1970) developed a second landmark paper[3]. He distinguished between three levels of efficiency, as earlier initiated by Roberts (1967), based on three sets of information reflected in the price. He posited that a market is efficient in the weak-form if any information which might be contained in past price movements is already reflected in the security prices. It is semi-strong efficient when all relevant publicly available information is impounded in security prices while strong form efficiency suggests that security prices already reflect all available information, even private information.

In this stream of literature, Malkiel (1992) contribution is elaborated in his essay ‘Efficient market hypothesis' in the New Palgrave Dictionary of Money and Finance. He defines a capital market as efficient when it fully and correctly reflects all relevant information in security price determination. Hence, for some information set, Ωt, the market is efficient if security prices are unaffected by unveiling that information to market participants. Then it becomes impossible to make economic profits by exploiting the information set.

It can be deduced that both the random walk theory and the EMH is related to informational efficiency. Then the form of efficiency under consideration will depend upon the information set, Ωt, which determines the level of efficiency.

### 2.1.2 Weak Form Efficiency: Random walk and its critics

Weak-form efficiency focuses on the informational content of the previous sequence of stock price movements. An informational efficient market postulates that excess return cannot be realised from information contained in past prices. The rationale behind weak-form efficiency is that stock prices are the most publicly available information so that an investor may not be able to use information which is already available to others to beat the market.

A long considered necessary condition for an efficient asset market is the martingale process. Under market efficiency, the conditional expectation of future price changes, conditional on the price history, cannot be either positive or negative and therefore must be zero. In fact the martingale originated from gambling and the concept of fair game. Samuelson (1965) and Mandelbrot (1966) independently demonstrated that a sequence of prices of an asset is a martingale (or a fair game) if it has unbiased price changes. Danthine (1977), LeRoy (1976, 1989), Huang (1985) and Neftci (2000) held that if a security market can be equilibrium and for sure be a fair game, then the following equations must hold:

Ept+1Ωt=pt (1)

Ept+1-ptΩt=0 (1.1)

Where t denotes the price of an asset at date t and Ωt is a set of all past and current information regarding prices (pt, pt-1, pt-2,....). Hence, the directions of the future movements in martingales are impossible to forecast.

Equation (1) can be interpreted as tomorrow's price being equal to today's price given the entire history of the asset's price. Similarly, equation (1.1) asserts that the assets' expected price change or return is zero. Here investing activity is likened to gambling so that the ownership of the asset is viewed as participation in a fair game. According to the martingale asset price evolves in a random process so that the correlation coefficient between the successive price changes will be zero given information about current and past prices.

However, most assets are expected to yield a non-zero and positive returns. The martingale hypothesis does not take into account the trade-off between risk and return as pointed out in financial economics. The model implicitly assumes risk neutrality while investors are generally risk averse. In fact, an investor is likely to hold more risky assets provided they are compensated in terms of higher expected returns. In this case, knowledge of the riskiness of current information set implies some awareness about the expected returns. Hence the equilibrium model shall predict a positive price change in the assets price though the actual return is still unforecastable under market efficiency. Then an asset model, considering positive returns, may be formulated as Fama (1970). He suggested the sub-martingale process as a special case of the fair game model:

Or alternatively (1.2)

This states that the expected value of next period's price based on the information available at time t, Ωt, is equal to or greater than the current price. Equivalently, it stipulates that the expected returns and price changes are greater or equal to zero.

Market efficiency plus an equilibrium model for asset pricing normally produces a random character to asset prices or returns or excess returns. The equilibrium model generally shows how the assets' expected return varies with its risk and this can be closely related to Fama's submartingale model. However, the representative model for the asset uses log prices and the expected continuously compounded return.

Ert+1Ωt=pt+1-pt (1.3)

Where (pt+1-pt) represents return and rt+1 is continuously compounded return.

Under the efficient market hypothesis, investors cannot earn abnormal profits on the available information set other than by chance. This is in line with Jensen (1978) who defines a market as efficient with respect to the information set, Ωt, if it not possible to make economic profits on the basis of this set of information. Hence, defining excess returns as zt+1:

(1.4)

Since market efficiency implies that all information is already impounded in stock prices, the following applies:

(1.5)

Under the assumption that the equilibrium model determining asset prices in (1.3) is assumed to be constant over time, the deduction is that expected return does not depend on the information available at time t such that:

(1.6)

Therefore market efficiency produces a result that implies that the changes in asset prices follow a random walk. The appropriate model would then be a random walk with drift where the arbitrary drift parameter, reflects how prices change on average to provide returns to holding the asset over time. The following equation sets the random walk model similar to the one defined by Lo and MacKinlay (1997):

(1.7)

rt= μ+αrt-1+ εt

where rt = pt - pt-1 and pt is the natural logarithm of a stock price index. If the stock price index follows a random walk, then, α = 0.

Generally, if stock prices and returns are unpredictable then time series have the properties of martingale, fair game, random walk and white noise implying the validity of EMH. Thus, given an equilibrium model for asset pricing, the test for weak-form efficiency is that of random walk tests of market efficiency. Ko and lee (1991) maintained that “If the random walk hypothesis holds, the weak form of the efficient market hypothesis must hold, but not vice versa. Thus, evidence supporting the random walk model is the evidence of market efficiency. But violation of the random walk model need not be evidence of market inefficiency in the weak form”. Depending on the restrictions put on the increments,εt+1, different forms of the random walk are tested.

Within the random walk hypothesis, three successively more restrictive sub-hypotheses with sequentially stronger tests for random walks exists (Campbell et al. 1997). These are range from the most restrictive form of Random Walk 1 (RW1) to the least restrictive one which is the Random Walk 3 (RW3). Based on their extensive research, the orthogonality condition for the random walk is:

covfrtgrt+k=0 (1.8)

Where frt and grt+k are two arbitrary functions and rt and rt+k refers to the returns for period t and t+k respectively. If 1.8 holds for all functions frt,grt+k this corresponds to RW1 and RW2. The former is the most restrictive version of random walk model implying it is not possible to predict either future price movements or volatility based on past prices. It states that returns are serially uncorrelated with independently and identically distributed increments with mean, zero and variance, σ2. Under RW2, the returns are serially uncorrelated, corresponding with a random walk hypothesis with increments that are independent but not identically distributed. In case frt,grt+k are arbitrary linear functions, the RW3 applies so that it is not possible to use information on the basis of past prices to predict future prices. Hence, returns in a market conforming to this standard of random walk are serially uncorrelated, corresponding to a random walk hypothesis with dependent but uncorrelated increments.

The foundation of traditional tests of random walk rests on the assumption of IID. The most famous tests remain the sequences and reversals test proposed by Cowles and Jones (1937) and the runs test. Tests of RW2 and RW3 encompass the variance ratio tests and unit root tests which are more recent tools. Developed by Lo and MacKinlay (1988), the variance ratio tests out that the variance of the innovations pertaining to a random walk model is linear functions of time. This popular test does not restrict only to the RW1 but also to the RW2 and RW3.

More recently, the application of non-linear dynamics and chaos theory to financial series has shown that they evidence non-linear structure. Hence exclusion of non-linear analysis in financial series could lead to inappropriate deductions as regards weak-form efficiency. In practice, the IID normal assumption does not hold. Returns distributions exhibit leptokurtic behaviours as opposed to normal distribution. They often reflect volatility clustering thereby the level of volatility in the next period tends to be positively correlated with its current level. Then it may be possible for information on the variance of past prices to predict the future volatility of the market. Indeed, share price movements could be unpredictable when using linear models but forecastable under non-linear models in the ‘short-run'. This contradicts the use of linear models for testing the efficient market hypothesis.

Further departures from the random walk hypothesis exist in the long-range dependence. This is analogous to high autocorrelation structure in a series so that there is persistent dependence between distant observations. In this case covfrtgrt+k does not tend to zero at higher lags. As regards market efficiency, persistence implies that past data contain useful information for prediction so that long memory violates the concept. Several tests have been developed for this purpose including the rescaled statistic to test for long-term ‘randomness' of the market series and the ARFIMA-FIGARCH which categorises the long- and short-term memory based on the estimated value of the fractional difference.

### 2.2 Empirical Review

Following the work of Fama (1965) “Random walk in stock prices” arguing for random walk hypothesis, a multitude of research has been performed throughout the world. While most of the well developed markets were found to be efficient, research findings of developing and less developed markets are mixed and controversial too. Most of the less developed market encounters the problem of thin trading. Besides, it is easier for large traders to manipulate small markets. Though emerging markets are generally assumed to be less efficient, empirical evidence does not always support the idea. Some previous research aiming at testing the weak-form efficiency of a particular group of stock markets are presented below.

A research that aims at testing weak-form market efficiency in the equity markets of the three main Central European transition economies (the Czech Republic, Hungary, and Poland) is that of Gilmore and McManus (2001). Using different approaches comprising of univariate, multivariate tests as well as the model-comparison approach for the period July 1995 to September 2000 different conclusion were drawn. While the serial correlation-based tests largely support a conclusion that these markets are weak-form efficient, the results of comparing forecasts of alternative models are consistent in rejecting the random walk hypothesis.

Examining the existence of weak-form efficiency in European stock market, Worthington and Higgs (2003) used daily returns for sixteen developed markets (Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and the United Kingdom) and four emerging markets (Czech Republic, Hungary, Poland and Russia) to perform a number of testing procedures of random walk. They started with the serial correlation coefficient test and the runs test, and found that Netherlands and Germany do follow a random walk while the United Kingdom, Ireland and Portugal were efficient under one test or the other. All remaining markets were weak form inefficient. The unit root tests (ADF, PP statistics and KPSS) as well as the multiple variance ratio tests rejected the presence of random walk in most of the markets. While in the developed markets only the United Kingdom, Portugal, Ireland, Sweden and Germany satisfied the most stringent random walk criteria, in emerging markets only Hungary did so.

Weak-form efficiency for emerging equity markets were also tested by Chang, Lima and Tabak (2003). They deduced that random walk hypothesis is not consistent with Asian equity markets while left apart Chile, Latin American indices resemble a random walk. Using daily prices from January 1992 to December 2002, multivariate variance ratios using heteroscedastic robust bootstrap procedures and test trading rules using trading range break (TRB) levels were employed. Using the US and Japan as yardsticks, they were not able to reject the random walk hypothesis.

Another study considering a group of selected Asian markets; Kim and Shamsuddin (2008) argues that market efficiency varies with the level of stock market development. Using new multiple variance ratio tests based on the wild bootstrap and signs as well as the conventional Chow-Denning test, they found that the Hong Kong, Japanese, Korean and Taiwanese markets adhere to the martingale property while Indonesia, Malaysia, Philippines markets are inefficient. Besides, the results revealed evidence that the Singaporean and Thai markets followed a random walk after the Asian crisis.

In their study, Elango and Hussein (2008) tested whether daily returns series of Gulf Co-operation Council (GCC), stock markets are an approximation of normal distribution or not. Dubai, AbuDhabi, Saudi Arabia, Qatar, Kuwait, Oman and Bahrain stock market indices were examined by him in his study. Kolmogorov-Smirnov test, Runs test, Autocorrelation Function and Partial Autocorrelation Functions were applied by them to test for randomness. The results revealed that the distribution of daily returns on these markets deviated from the normal distribution during the study period. Also, the runs test rejected the hypothesis of random walk for all seven markets.

In his paper investigating the random walk hypothesis, Urrutia (1995), used monthly data from December 1975 to March 1991 for four Latin American equity markets: Argentina, Brazil, Chile, and Mexico to observe whether they are weak-form efficient. He made use of the Variance-ratio tests and the runs tests. While results of the variance ratio estimatespixel

rejects the random walk hypothesis, runs tests specify that Latin American equity markets are weak-form efficient. These empirical findings suggest that domestic investors might not be able to develop trading strategies that would allow them to earn excess returns.

Using Lo-MacKinlay Variance ratio, Wright's rank and sign VR and the standard runs tests; Al-Khazali, Ding and Pyun (2007) revisited the validity of random walk hypothesis in eight emerging markets in the Middle East and North Africa (MENA): Bahrain, Egypt, Jordan, Kuwait, Morocco, Oman, Saudi Arabia, and Tunisia. When assessed by Wright's (2000) rank and sign VR test, all the markets rejected the hypothesis of random walk. However, once data are reconciled for distortions from thinly and infrequently traded stocks, all eight stock markets do follow a random walk.

African countries were investigated in the paper ‘How Efficient are Africa's Emerging Stock Markets' by Magnusson and Wydick (2002). Testing procedures considered monthly data for eight African markets in comparison with nine other developing countries in Latin America and Asia. Distinguishing among the three types of random walk models, they started by testing the RW 3, by investigating the Partial Auto-Correlation Function(PACF) of the historical series and examining whether they are statistically different from zero. Markets in Botswana, Cote d'Ivoire, Kenya, Mauritius and South Africa did conform to the RW3 while those of Ghana, Nigeria and Zimbabwe were rejected. Proceeding with the RW2, excluding Botswana, results did not change. However none of the African Markets were conform to the RW1 White test for heteroscedasticity. They conclude that African countries do conform quite favourably to some regions of the developing world.

Another piece of research that focuses specifically on African markets was that of Jefferis and Smith (2005). It covers seven African stock markets: South Africa, Egypt, Morocco, Nigeria, Zimbabwe, Mauritius and Kenya and use a GARCH approach with time-varying parameters to detect changes in weak-form efficiency through time. They focused on RW 3 model with volatilities changing over time and found that Johannesburg stock market was weak-form efficient with no tendency to change like many other developed markets. On the other hand, the stock markets of Egypt, Morocco and Nigeria showed changing levels of inefficiencies to become weak-form efficient towards the end of the period. The results for Kenya, Zimbabwe and Mauritius, however, showed tendency towards efficiency and rejected the hypothesis of weak-form efficiency.

More recently, McMillan and Thupayagale (2009) in their paper “The efficiency of African equity markets” examined long memory effects of both equity returns and volatility for eleven African countries, taking the UK and US as reference. They made use of unit roots test and the GARCH(1,1) models before proceeding with ARFIMA-FIGARCH and ARFIMA-HYGARCH models. They ended up with mixed results. The ARFIMA-FIGARCH models provide evidence for long term memory in African equity markets with the exception of Mauritius, Morocco, Botswana and Nigeria where the results were unpredictable. Also, the US stock return volatility was marked by long memory process while the UK was non-stationary. These results were further supported by the ARFIMA-HYGARCH models.

### 2.3 Conclusion

During the course of the literature review, limited evidence on weak form efficiency of African markets was found. These countries have attracted significant investment these last years and are of much importance to portfolio managers. Univariate time series analysis might be important tool for technical analysts in trying to outperform these markets. Indeed, the battery of econometrics software now paves the way for investigation of the random walk hypothesis based on different sets of assumption. A preliminary analysis of the African markets shall provide us with an insight to efficiency based on their attributes and consultation of previous works.

### GENERAL OVERVIEW OF THE AFRICAN STOCK MARKETS

### 3.0 Introduction

African stock markets, following in the wake of the surge in the world stock markets over the few decades, are starting to take off. Indeed, recognizing the importance of stock markets in economic development, several African countries launched stock exchanges during the past two decades. In fact, the African Stock Exchange Association (ASEA) has been set up in 1993 in a view to promote the development of the stock market. Prior to 1989, there were just five stock markets in Sub-Saharan Africa and three in North Africa. Today, Africa has about 20 active stock markets, with some exchanges more established than others, depending on when they were established. Alongside the rapid expansion of stock markets in the continent, there has also been a significant growth in market capitalization and the number of listed companies. However, with the exception of the well established markets, stock markets in Africa remain thin and illiquid. This study covers four African stock markets namely South Africa, Mauritius, Morocco and Egypt over periods for which data is available.

### Mauritius Stock Exchange

Since its start of trading on the 5th July 1989 under the Stock Exchange Act of 1988, the Mauritius Stock Exchange (SEM) has come a long way. From a pre-emerging market with trading taking place only once a week, the SEM has emerged as one of the leading exchanges in Africa. It operates two markets namely the Official and the Development and Enterprise market (DEM), established in August 2006 to replace the over-the-counter market. The exchange is regulated by the Financial Services Commission. As the second sub-Saharan stock exchange member of the World Federation of Exchanges, SEM operates in line with international standards. In addition, its developing institutional and retail investor base make it an attractive investment destination for foreign investors. The SEM offers quite a limited range of products to its investors and the aim for the next few years would be to increase the range of products offered. The three main indices of the official market are namely the SEMDEX, SEM-7 and the SEMTRI. As at 30 June 2009, some 40 companies, with a market capitalisation of Rs 130.77 bn, are listed on the Official market and 52 companies, with a market capitalisation of Rs 45.41 bn, are listed on the Development and Enterprise Market (DEM).

The SEM maintained an upward momentum, amidst typical market fluctuations, until the end of February 2008. The total market capitalization of the Official Market and the DEM was Rs 173.1 bn at end 2007. This is in line with the levels observed in well-established emerging stock markets. However, like other exchanges, the SEM experienced market volatility since the start of the financial crisis in September 2008. The main pillars of the Mauritian economy were adversely affected and this reflected on hotels and banks stocks listed on the SEM. The market then picked-up by mid-March 2009 on the back of interest rate cuts and stimulus packages put forward by the Government of Mauritius.

### Johannesburg Stock Exchange

The Johannesburg Stock Exchange (JSE), regulated by the Financial Services Board under the Securities Services Act 2004, is the largest exchange in Africa and among the top twenty largest in the world in terms of market capitalisation. JSE Securities Exchange existed since November 1887 and was incorporated as a public limited company on 1st July 2005, pursuant to its demutualization. Since then, the JSE has evolved from a traditional floor based equities trading market to a modern securities exchange providing fully electronic trading, clearing and settlement in equities, financial and agricultural derivatives and other associated instruments and has extensive surveillance capabilities. Technical agreement with the London Stock Exchange (LSE) enables dual primary listings on both exchanges since 2001. Between the listed entity and its trusted trading platforms the South African economy becomes an active hub of activity where expansion is encouraged, businesses are enhanced, performance is driven and shareholder value is created. The JSE currently operates four boards for the equities market and the South African bond market is a leader among emerging-market economies. The main market indices are Top 40, Industrial 25, All Share, Oil and Gas Index.

As the gateway to Africa's economy, the JSE provides the link between international markets and the continent. In 2008, a daily average of 334 million shares was traded on the JSE. At year-end, there were 992 listed securities on the JSE with a total market capitalisation of R4,514 billion compared to R5,696 billion in 2007.

### Casablanca Stock Exchange

The sixth largest in the Arab World and the third largest Bourse in Africa, the Casablanca Stock Exchange(CSE) founded in 1929 in Morocco is relatively modern, having experienced reform in 1993. The exchange is well regulated by the Conseil Deontologique des Valeurs Mobilieres (CDVM). Originally, CSE had the Index de la Bourse des Valeurs de Casablanca (IGB) but this was replaced on January 2002 by two indexes: MASI (Moroccan All Shares Index) which comprises all listed shares, allows to follow up all listed values and to have a long-term visibility and MADEX (Moroccan Most Active Shares Index), comprising of most active shares listed continuously with variations closely linked to all the market serves as a reference for the listing of all funds invested in shares. Of its 77 listed securities, around 25 are traded on a daily basis, most of which are listed on the continuous market. On the alternate markets namely the Marché Croissance and Marché Développement orders are cleared only twice during the 5 1/2 hour trading session.

The CSE currently has 16 members with a total market capitalization of 531.7 billion dirhams as of end of year 2008 compared to 586.3 billion dirhams at the end of 2007. This fall of 9.31% was partly due to the fall in the number of IPO's and various public offering operations.

### Egyptian Exchange

Egypt's Stock Exchange recently renamed Egyptian Exchange (EGX), is one of the oldest stock exchange in the Middle East. It comprises of two exchanges: Alexandria which was established in 1883 and Cairo established in 1903, both governed by the same board of directors and sharing the same trading, clearing and settlement systems. Between 1961 and 1992 the exchange suspended operations due to socialist policies and central planning by the government. A change in economic reform in the 1990's, recognizing the development of equity markets and the financing of capital formation as long term growth prospects, however, enabled the revival of the stock exchange. A new law[4] enforced the regulatory framework and the Capital Market Authority (CMA) as an independent regulatory agency for the securities agency enhanced confidence of investors and ensured proper financial disclosure requirements. The CMA was recently replaced (effective as from 1st July 2009) by the Egyptian Financial Supervisory Authority (EFSA) responsible for supervising the non-bank financial instruments and markets.

The number of listed securities declined throughout the period under consideration mainly due to the delisting of rarely traded securities or those not complying with listing requirements. As at end 2008, there were 373 listed companies on Egyptian Exchange. Market capitalization declined from LE 768 bn at end of 2007 to LE 474 bn at end of 2008.

### 3.1 Index Analysis

The markets studied in this paper are South Africa, Mauritius, Morocco and Egypt. Daily frequency indices of Mauritius (SEMDEX), Morocco (MASI), Egypt (EGX 30) and South Africa (JSE All Share) were collected from their respective stock markets websites. The tenure of the data would be from 1 January 2000 to 28 Dec 2009, with the number of observations varying due to missing prices on holidays in the respective markets. Before proceeding with the data analysis, a graphical analysis is conducted to observe whether there is any apparent pattern of the stock returns.

The plots of the series exhibit upward but not linear trend in all cases with persistent fluctuations around it. There are also increasing variability as the levels of the series increase. Such behaviour justifies the logarithmic transformation such that the trend is eliminated by the first difference of the log prices (returns).

The SEMDEX, amidst typical fluctuations drifts upwards until February 2008 then starts falling to take off again as from end of March 2009. The downward trend of the stock price was mainly due to insecurity pertaining on the stock market caused by the global recession that followed the so called global financial crisis. This downward spiral in 2008 was also reflected in the other three stock indexes. The collapse of Lehman Brothers and the emergent buyout of Merrill Lynch by the Bank of America in September 2008 brought about a fall of 4.5% in the JSE All share index in just 2 weeks. As regards the earlier part of the series, the JSE All share index was affected by monetary policies directed towards keeping inflation rate in the range of 3-6%. This caused an appreciation in the rand leading to a decline of the index in local currency terms. Besides, the index felt the impact of stock market crashes in year 2000 to 2002 associated mainly with the tech bubble and the 11th September terrorist attack.

Though Morocco imports primarily oil and foodstuffs, sound policies enable the combat of unprecedented oil price and foodstuff upsurges in year 2006 as reflected by the growing ‘trend' of the MASI. Before this period, that is 2000 to 2002, the index fell continuously due to lack of institutional investors and transparency. As promised economic reforms did not materialise, investors further lost confidence to deepen the downward trend in 2002. However, the index lost around 30% from mid-March 2008 till January 2009 due to sub-prime crisis and psychological factors. The Egyptian stock market felt the real effect of the crisis around the end of September 2008 when international stock market and the seven stock markets in the Gulf experienced losses. Indeed, EGX30 lost about 70% from May 2008 to February 2009. In addition, the index also faced falling prices from February 2006 to June 2006 due to global inflation fear which created uncertainty on investor sentiment, causing negative shocks in many stock markets including that of Egypt. The falling index in periods 2001 and 2002 was due to its macroeconomic environment and the geo-political tensions which prevailed in the Middle-East but ameliorated after this period.

### 3.2 Conclusion

Most African countries have been subject to various reforms regarding their stock markets in recent years. They are looking for new ways to develop that engine which is the backbone of growth in many economies. Beside technology and financial products, laws governing these markets have been constantly revised to account for greater transparency as they reinforce efficiency and stimulate private investment. To gauge the ‘effects' of such innovations on market efficiency, it is important to test for the weak-form efficiency as its rejection implies refusal of the stronger-forms of efficiency. We proceed to a methodological analysis before performing these tests.

### METHODOLOGY

### 4.0 Introduction

This section of the paper provides the methodological settings for testing the market efficiency of five African stock markets in the weak form. Several parametric and non-parametric tests are used to examine whether the stock returns are weak-form based on the three notions of random walks. Long-term tests are nowadays receiving much attention in general academic researches. We use these to check for the presence of persistence, anti-persistence and random walks in returns as well as volatility.

### 4.1 Empirical Methodology

### Tests of Random Walk

This paper uses continuously compounded returns for testing efficiency in the selected African markets. Natural log of relative price are taken such that rt=lnptpt-1, where pt and pt-1 represent the stock index at time t and t-1 respectively. This practice is common rather than use discrete compounding. The rest of the paper uses this procedure except from the unit root tests which use log prices for levels and the variance ratio tests with Lo and MacKinlay (1988) model specification.

### 4.2 Test of Serial Independence

### 4.2.2 Runs test

Weak form efficiency implies that successive price changes are independent and follow a random walk. RW1 implies that there should be no serial correlation. The runs test is a solid alternative to parametric serial correlation tests in which distributions are assumed to be normally distributed. Ignoring the distribution of data, the null hypothesis of the test is that the observed series is a random series. A run is a sequence of successive changes in log prices bearing the same sign and may be positive, negative or zero. By comparing the number of runs in the data with the expected number of runs under RW1, a test of IID can be performed. Under the null hypothesis that successive outcomes are independent, the total expected number of runs is distributed as normal with the following mean, µ, and standard deviation σµ:

μ=NN+1-i=13ni2N

σμ=i=13i=13ni2+N(N+1)-2N(i=13ni3-N3)N2(N-1)12

The test for serial dependence is carried out by comparing the actual number of runs in the price series to the expected number μ. If there are too few or too many runs as compared to the expected number of runs as in a random series, this is analogous to non-randomness. If there are too few runs, it would mean that the stock returns in the time series do not change signs frequently, thus indicating a positive serial correlation and in turn, may imply that the price changes do not follow a random walk model. Similarly, if there are too many runs, they may suggest negative autocorrelation.

### 4.3 Unit root tests

For the purpose of testing RW2, unit root tests are used to test for non-stationarity. If the log price series is non-stationary and the first difference of the series (returns) is stationary, the series contains a unit root. The Augmented Dickey-Fuller (ADF) test is performed on both log prices (level) and return (difference) to check for random walk.

### 4.3.1Augmented Dickey-Fuller) test

The ADF test uses an ordinary least squares (OLS) regression of the first differences of the series against the series lagged once, as well as lagged difference terms, with optional constant and time-trend terms:

∆pt=a0+a1t+γpt-1+βi∆pt-i+1+εt

In this equation Δ is the first difference operator, a0 is an intercept, a1t is a linear time trend, et is an error term, and i is the number of lagged first-differenced terms such that et is white noise. The test for a unit root has the null hypothesis that γ = 0.

### 4.4 Variance ratio test

The variance and unit root tests are often considered as complementary but we use both of them to get a higher robustness of the conclusions. The lack of power of the unit root tests and even its failure to detect departures from the random walk nature of time series led to the development of the variance ratio test. Developed by Lo and MacKinlay (1988), hereby LM, the variance ratio statistic is derived from the assumption of linear relations in observation interval regarding the variance of increments.The test statistic is a weighted-sum of estimated autocorrelation coefficients, with the weights declining in return horizon. While the homoscedastic assumption tests for the Gaussian i.i.d assumption (RW1), the heteroscedastic assumption applies for RW2 and RW3.

The single variance ratio test of the random walk hypothesis tests the null that the variance ratio equals one at all horizons of q>1. Non rejection of the null hypothesis implies random walk and thus market efficiency. While positive serial correlation is reflected by the existence of variance ratios greater than one, negative correlation applies for variance ratios less than one.

If in a finite sample, the times series of returns follows a random walk then the increments in the variance are linear in the observation interval. This implies that the variance is proportional to the sample interval. Hence, the variance of monthly return should be equal to about twenty times the variance of daily returns. Under the hull hypothesis of white noise, the variance ratio statistic, VR(q), is defined as:

VRq=σc2σa2qq=1 eq(1)

where σ 2c (q) is an unbiased estimator of 1/q of the variance of the q-differences and σ 2a (q) is an unbiased estimator of the variance of the first differences.

The formulas for calculating σ 2c (q) and σ 2a (q) are given below in equations (2) and (3):

σc2q=1mt=q+1nq+1pt-pt-q-qμ2 eq(2)

and

σa2q=1nq-1t=2nq+1pt-pt-1-μ2 eq(3)

where

m=qnq-q+11-1n

μ=1nqpnq+1-p1

For testing the RW1, the standard normal test statistic under the hypothesis of homoscedasticity, Z(q), is:

Zq=VRq-1∅q12~N0,1 eq(4)

where ∅(q) =[2(2q −1)(q −1)]/3q(nq), which is the asymptotic variance of the variance ratio under homoscedasticity.

As variances of most stock returns are conditionally heteroscedastic with respect to time, LM (1988) also derive a refined statistic, Z * (q), which adjusts for heteroscedasticity. Hence under the RW2 and RW3, the heteroscedasticity-robust standardized variance ratio is given by:

Z*q=VRq-1∅*q12~N0,1 eq(5)

where ∅ * (q) is the heteroscedasticity-consistent asymptotic variance of the variance ratio, given by:

∅*q=j=1q-12q-jq2δj

δj=t=j+2nq+1pt-pt-1-μ2pt-j-pt-j-1-μ2t=2nq+1pt-pt-1-μ22

We use one-day as the base observation interval and calculate variance ratio estimates VR(q), asymptotic variances of the variance ratio ∅ ( q ) and ∅ * (q) and variance ratio test statistics Z( q ) and Z * (q) for an upper bound approximating q=T so that q = 2, 4, 8, 16, 32 and 64 for each cases. These are then compared to the critical values obtained from the normal table.

### 4.5 Long-term dependence

### 4.5.1Rescaled statistics

Persistent series are normally characterised by distinct but non-periodic cycles. Proposed by Hurst (1951), Mandlebrot (1972) developed the ‘classical' rescaled statistic for testing long memory. The test measures the range of the partial sums of deviations from its mean rescaled by its standard deviation, as follows;

Qn=1σnqMaxj=1krj-rn-Minj=1krj-rn

and

σnq=n-1j=1nrj-rn21/2

The first bracketed term denotes the maximum of the partial sums of the first k deviations of rj from its mean which is nonnegative. As for the second term, it represents the minimum of the same sequence of the partial sums and is nonpositive. Hence the difference between the two quantiles is positive, that is Qn≥0.

The classical rescaled statistics aims at finding a value for the Hurst parameter H for a long-range dependent process. Hurst's empirical evidence model the relation EQn~cnHas n→∞, where H displays the long memory property of the series. In order to estimate the value of H, we run an OLS regression of the form:

logQn=α+βlogn+ε

Where β is the estimated value for H. For H=0.5, the series exhibits random walk while 0.5<h<1 indicates persistency. Conversely, 0<H<0.5 displays anti-persistency, analogous to negative dependence. To test for long memory, the null hypothesis is that of no long-term dependence (H=0.5).

A usual criticism pertaining to the R/S statistic is its sensitiveness to short range dependence such that the discrepancy between the data and the predicted behaviour of the R/S statistic under the null hypothesis of no long-range dependence may not be the result of long range dependence but simply of be a symptom of short-term dependence. Hence the use of the modified statistic provided by Lo (1991) can be a better alternative as it incorporates short-run dependence into its denominator, that is,

σn2q=1nj=1nrj-rn2+2nj=1qωiqi=j+1nri-rnri-j-rn=

σn2q=σr2+2j=1qωjqγj

ωjq=1-jq+1

Where γj, j=1,2,...,q, representing the autocovariance of rj and ωjq is the Bartlett window weight. The value of the truncation lag, q plays an important role as it should account for short range memory dependence while a too large one can alter the finite distribution of Qn. Andrews (1991) suggests the following rule for selecting q:

q=kn

kn=3n2132γ11-γ1223

Where kn is the greatest integer less than or equal to kn and γ1 is the first order autocorrelation. Also the weights above are now changed to:

ωj(q)=1-jkn

Lo (1991) showed that in the presence of long-term memory,Vn(q)≡ Qn.T-1/2 weakly converges to the range of a Brownian motion with the probability distribution:

Fv=1+2k=1∞(1-4k2v2)e-2(kv)2. The critical values shown in table 1 in the appendix are the fractiles of the limiting distribution of the statistic which are used to test the null hypothesis of no long-term dependence (H=0.5) against long-term dependence alternative (0.5<H<1).

### 4.5.2Dual long-memory: ARFIMA-FIGARCH

Much of financial time series consider the order differencing, d, to be either one or zero. Though stationary, these series tend to exhibit dependence between distant observations. This gives rise to the concept of persistence which can be used to test for long-term memory. Persistence is often detected in both conditional mean and conditional variance justifying our preference for the ARFIMA-FIGARCH. The methodology used is to first check the autocorrelation functions for the returns and squared returns for the returns. The latter shall infer an idea about any long-memory in volatility present in the markets. This is further investigated using the generalized autoregressive conditional heteroscedasticity (GARCH).

The GARCH process as proposed by Bollerslev (1986), is widely applied in financial time series analysis as it allows for a time variant conditional variance and nonlinearities in the generating mechanism. In this study, we restrict to a simple GARCH(1,1) adhering to Brook and Burke (2003) who suggest that the lag order is sufficient to capture all volatility clustering in the data. This model is run on raw data and can be set as:

ht=ω+λεt-12+θht-1 εt=htϵt

Where ϵt is a sequence IID random variables with mean 0 and variance 1. ω>0,λ≥0,β≥0. The GARCH (1, 1) is weakly stationary if α+β<1, μ is the mean, εt-12 is the information about volatility from the prior period (the ARCH term), and ht-1 the conditional variance is the previous period forecast variance (the GARCH term) and must be nonnegative.

Based on the results of the GARCH(1,1), we then proceed to the estimation of dm and dv for an ARFIMA(p,dm,q)-FIGARCH(m,dv,s) and make appropriate deductions about long-term persistency. For the time being, we lay the methodological framework for the study of dual memory as developed by Baillie, Han and Kwon (2002).

∅L1-Ldm(rt-μ)=θ(L)εt (1)

εt=ξt2ht (2)

⋋L(1-L)dvεt2=ω+1-βLυt (3)

Where dm and dv captures the long memory behaviour in the mean and variance respectively. L is the lag operator and ∅L, θ(L), ⋋L and β(L) are the polynomials in the lag operator. The innovation is υt≡εt2-ht2 with ξt~iid0,ht and Eεtεs=0 for s≠t. To ensure stationarity, the roots ∅L, θ(L), ⋋L and 1-β(L) must lie outside the unit root circle.

The long-memory operator can be expanded as a hypergoemetric function :

1-Ld=k=0∞Γk-dΓk+1Γ(-d)Lk=k=0∞⋋kLk

Where Γ(.) represents the gamma function with Γk+1=k!=k×Γk and ⋋k=k-d-1k⋋k-1. The process is stationary and ergodic for d<0.5. When d=0 implies stationarity while d∈(-0.5, 0) implies short-term memory for negative autocorrelations. For d∈(0, 0.5) is analogous to long-memory due to positive autocorrelations which decay hyperbolically. The variance of rt is infinite so that the process is non-stationary but is still mean-reverting for d∈(0.5, 1).

Moreover an integration order dv=0 implies the reduction of a FIGARCH to a GARCH model while dv=1 is equivalent to an IGARCH model. As per Baillie (1996) et al. 0≤dv<1 implies a long-memory behaviour so that a shock on the conditional variance of the FIGARCH(p,q,d) processes decrease at a hyperbolic rate. Thus dv=0 encompasses long-term dynamics of volatility and GARCH considers short-term ones.

### 4.6 Conclusion

The aim of this study is to use many econometric tools to investigate the random walk theory and hence, weak-form efficiency. Ranging from earlier tests like the runs tests, the paper adopts more recent and powerful ones like the variance ratio test. Moreover, long-memory has been given little attention as regards African stock markets. The rescaled statistics and ARFIMA-FIGARCH address the issue for our selected markets. The results are being discussed in the next section and provide an insight to the development of the stock markets of the selected African markets.

### FINDINGS AND ANALYSIS

### 5.0 Introduction

This section shows the results for the various tests undertaken in the methodology section. Before proceeding to the weak efficiency tests, graphical analysis and descriptive statistics are presented to provide an insight to the distribution and volatility pertaining to the markets under consideration. The remaining part is then divided into the results for the different random walks and making appropriate deductions about long-term memory.

### 5.1 Graphical Analysis

The SEMDEX returns shows that volatility was quite low and near zero throughout the period from the start of the year 2000 to mid of year 2006 then volatility started to increase and the year 2008 received the highest volatility ever. A closer look at the daily return volatility during 2008 indicate that the values ranged from more about 0.06 to -0.06 reaching high of nearly 0.08 at end of March 2009 depicting the recovery from the crisis. The series for JSE All Share index fluctuates randomly around its mean level with a concentration of most of the values ranging from 0.04 to -0.04. However, volatility was higher during the year 2008 with highest negative returns of nearly -0.08 in October 2008. Also, values in the series tend to be close the previous value thus indicating positive autocorrelation. As for the MASI returns, it shows some wide fluctuations but most of the values are likely to range from 0.02 to -0.02. Although EGX 30 returns does not show much wide fluctuations, the daily returns series appear to be highly volatile ranging mostly from 0.05 to -0.05. Volatility can be gauged by looking at the descriptive statistics hereby shown.

### 5.2 Descriptive Statistics

One of the basic assumptions underlying weak-form efficiency is the normality of the distribution of the return series. Table 1 represents a summary of descriptive statistics of the returns for all four countries indexes in order to test the distribution of the returns series.

### Table 1 Summary statistics for daily returns

SEMDEX

JSE ALL SHARE

MASI

EGX30

Mean

0.000539

0.000475

0.000307

0.000669

Median

0.000361

0.000801

0.000249

0.001022

Maximum

0.076546

0.06834

0.055637

0.18377

Minimum

-0.063827

-0.076892

-0.050167

-0.17986

Std. Dev.

0.007879

0.013552

0.008589

0.018605

Skewness

0.24

-0.176566

-0.11884

-0.27623

Kurtosis

20.47115

6.280669

8.821614

12.03395

Jarque-Bera[5]

31705.38

1133.206

3513.586

8403.363

Probability

0

0

0

0

Sum

1.343739

1.186247

0.762471

1.64647

Sum Sq. Dev.

0.154572

0.458606

0.18317

0.851905

Studentized Range[6]

17.81609

10.71665

12.31855

19.54475

Observations

2491

2498

2484

2462

From table 1 it can be seen that mean stock returns are positive and close to zero as expected for the returns of the time series. Standard deviation is relatively lower for SEMDEX and MASI, indicating low volatility in returns for these two indexes. This can be due infrequent trading of many listed stocks or the lack of frequent price fluctuations. It is worth noting that the most politically stable country, which is Mauritius, has less volatile returns followed by Morocco.

Generally, values for skewness and kurtosis of zero and three respectively represents that the observed distribution is perfectly normally distributed. The statistics shows that SEMDEX exhibits positive skewness while the others are negatively skewed. Positive skewness of the returns suggests that the weights of large positive returns dominate their negative counterparts. Besides, returns for all for indexes display excess kurtosis indicating that the distributions are leptokurtic, that is, their distribution have fatter tails than a normal distribution. This can be interpreted as “large shocks are more common than expected statistically''. Thus, skewness and leptokurtic distribution of the stock return series indicates that the distribution is not normal.

Non-normality is further supported by the Jacque-Bera statistics which, based on skewness and kurtosis, tests for the joint hypothesis that S=0 and K=3. For a 99% confidence interval, if the p value of the JB test is more than 0.05 the null hypothesis is accepted in the favour of normally distributed series. Here, the p-value of all indices is less than 0.05 suggesting that the null hypothesis can be rejected.

Moreover, Fama (1965) provided another test known as the studentized range to determine the extent to which the data deviates from normality. Under the null hypothesis, the data follows a normal distribution and it is rejected if that range is greater than 6. It is observed from table 1 that all the values are greater than 6 thus indicating that stock returns series deviates from the prior condition of random walk, that is, returns are normally distributed.

### 5.3 Results for testing RW1, RW2 and RW3

### Table 2: Results for Runs Test

SEMDEX

JSE ALL SHARE

MASI

EGX30

Test Valuea

0.000539

0.000474879

0.000307

0.000669

Cases < Test Value

1300

1215

1254

1213

Cases >= Test Value

1191

1283

1230

1249

Total Cases

2491

2498

2484

2462

Number of Runs

951

1172

952

1066

Z

-11.771

-3.087

-11.676

-6.683

Asymp. Sig. (2-tailed)

0.000

0.002

0.000

0.000

### a. Mean value

From table 2, the estimated z-values are significant at the 5% level for all markets since all p-values are less than 0.05. The negative z-values for all markets indicate that the actual numbers of runs are fewer than expected under the null hypothesis of return independence. This is conducive to positive serial correlation and indicates the market's overreaction to information such that there is an opportunity of making excess profit. SEMDEX, MASI and EGX 30 have z-values of much higher than 5.0 while JSE All share index has the smallest z-value to affirm a relatively more efficient stock market. The large z-values further reveal some sort of persistency, thus violating the weak-form efficiency of the markets though there is no clue about its magnitude and direction. This is formally investigated later.

The results prove that the selected African markets do not follow random walks or at least in the most restrictive form of random walk, that is RW1. The next testing procedure is based on RW2 to further the investigation of weak form efficiency based on unit root tests. The results are hereby being disclosed.

### Table 3: Unit roots test result

Test

SEMDEX

JSE ALL SHARE

MASI

EGX30

intercept

intercept and trend

intercept

intercept and trend

intercept

intercept and trend

Intercept

intercept and trend

ADF- Levels

0.32

-2.1

-0.54

-2.02

-0.1

-2.09

-0.24

-1.62

p-value

(0.98)

(0.54)

(0.88)

(0.59)

(0.95)

(0.55)

(0.93)

(0.78)

ADF-Diff

-73.12

-73.11

-83.60

-83.58

-68.89

-68.88

-76.15

-76.14

p-value

(0.0001)

(0.0001)

(0.0001)

(0.0001)

(0.0001)

(0.0001)

(0.0001)

(0.0001)

Table 3 reports the results for ADF statistics for intercept and intercept with trend together with their respective p-values. The null hypothesis for those tests is that the series have a unit root (non-stationary). The results of ADF test computed for the statistics with and without trend, fail to reject the null hypothesis at 5% significance level for all four indexes; they have significant p-values. This suggests non-stationary for the log price series. The conclusion is reinforced by the results for the first difference. Returns need to be stationary and this is confirmed by the high values of their statistics for each country. The null hypothesis is rejected as they reject the critical values for the test which is the case at even 1% significant level. The results affirm that the levels for those countries are I(1), that is, they need to be differenced one time to become stationary while the returns are I(0). Hence the Augmented Dickey Fuller test do not reject the null hypothesis at levels but reject it for the first difference for all countries so that the necessary condition for RW2 is attained.

The African countries under consideration tend to display characteristics weak-form efficiency as regards the less constrained random walk (RW2). However, unit root tests suffer from low power, that is, they are likely to accept the null of unit root when none exists. Hence, it becomes crucial to use more sophisticated tests to confirm the results. Variance ratio tests are applied to this effect. The results are given below.

### Table 4: Variance ratio test results

VR(q)

Q

2

4

8

16

32

64

Mauritius

1.2525

1.7737

1.6939

2.1746

2.4075

2.9618

South Africa

1.0599

1.0668

0.9764

0.9960

0.9548

0.9802

Morocco

1.3191

1.5412

1.6760

1.9456

1.9839

2.0975

Egypt

1.1889

1.2872

1.3900

1.5917

1.6765

2.0557

Assuming homoscedasticity z(q)

Q

2

4

8

16

32

64

Mauritius

12.6023

20.6453

11.7099

13.3216

11.0150

10.7274

South Africa

2.9938

1.7851

-0.3996

-0.0456

-0.3542

-0.1083

Morocco

15.9035

14.4165

11.3905

10.7367

7.6876

5.9914

Egypt

9.3730

7.6185

6.5421

6.6700

5.2626

5.7379

Assuming heteroscedasticity z*(q)

Q

2

4

8

16

32

64

Mauritius

2.7685

6.2272

16.7403

19.0443

15.7468

15.3357

South Africa

1.1885

0.8368

-0.3478

-0.0397

-0.3087

-0.0945

Morocco

5.3263

20.8124

30.5213

9.3513

20.6005

16.0588

Egypt

1.9097

2.1946

2.6634

-3.8155

1.5113

1.9430

The variance ratio tests presented in table 4 show that all countries have a VR(q) which differs from zero. With the exception of South Africa, the VR(q) increase with higher values of q. They are greater than one suggesting positive autocorrelation for each and every lag. JSE All share index has VR(q) nearing one for all lags indicative of weak-form efficiency.

Under LM's derivation, ρ1≈VR2-1. Hence, the first order autocorrelation for Mauritius, South Africa, Morocco and Egypt amount to about 25%, 6%, 32% and 19% respectively. Regressing the returns on a constant and first lag, we observe that the R2 of the regression is the square of the coefficient of the slope. The latter is simply the first-order autocorrelation. Therefore, an autocorrelation of 25% for Mauritius implies that 5.4% of the variation in daily SEMDEX is predictable based only on the previous day's return. For South Africa, Morocco and Egypt, the s are 0.35%, 10.1% and 2.9% respectively. Thus, JSE bears the lowest predictability based on previous day's return (See table A5 in appendix for results).

Under the homoscedastic assumption, the null hypothesis is rejected only at lag 2 for South Africa but rejected at all lags for the other countries. The values for their z(q) are much greater than the asymptotic critical values of 2.576 at 1% significance level. This clearly rejects the assumption of white noise for these three stock markets but Egypt displays relatively lower values than Mauritius and Morocco.

The rejection of the random walk hypothesis for Mauritius, Morocco and Egypt may be due to autocorrelation or heteroscedasticity. However, their heteroscedasticity-consistent estimates are greater than 2.576 for all lags at 1% significance level rejecting the RW2 and RW3 hypothesis. South Africa is the sole country which confirms the results obtained in the unit root tests as the variance ratio tests fail to reject the null hypothesis. The fact that strictest random walk hypothesis is rejected for JSE but not for the relaxed version could be the result, or at least the partial result of heteroscedasticity in the data and not singly to autocorrelation. The reported z*(q) values are much lesser than the critical values for the JSE suggesting the presence of RW2 and RW3.

The persistence in the markets, as doubted by result for the runs test, is confirmed by non-randomness for Mauritius, Morocco and Egypt. For these countries, the values of VR(q) display increasing trends for higher values of q. It should also be noted that the variance ratio test provides evidence for the low power of unit root tests as it does not validate the RW2 for the other countries.

### 5.4 Long-term dependence

### Table 5: Rescaled statistics results

Country

Hurst exponent

Modified R/S Statistics

Mauritius

0.7343036*

0.7869677

South Africa

0.5020860*

0.8293697

Morocco

0.7289226*

0.6537525

Egypt

0.8203611*

1.5068207

*consult A1 in appendix

Table 5 shows that the Hurst exponents for all the countries are between 0.5 and 1 indicating persistent behaviour. Then an increase in returns at time t is likely to be followed by an accompanying upsurge in returns in the next period. Similarly a decrease tends to follow a decrease. Mauritius, Morocco and Egypt have quite high H estimates implying the presence of strong trends and greater possibility of future returns predictions. However, South Africa displays a nearing 0.5 indicating a noisier series and less defined trend. This is consistent with random walk so that there is about 50% probability that future returns will go either up or down.

The results are consistent with the modified R/S statistic computed at lags 11, 2 and 13 for Mauritius, South Africa and Morocco respectively. From table A6 in the appendix, the acceptance for a two-tailed test a 95% significance level is (0.809,1.862). South Africa and Egypt (computed for lag 5) have modified R/S statistic which fall within the acceptance region implying that the null hypothesis of no long-term persistence cannot be rejected. There is a controversy regarding the results for the two tests as regards Egypt. However, Teverovsky, Taqqu and Willinger (1999) hold that Lo's rescaled R/S test can be too severe. They numerically showed that Lo test cannot reject the null hypothesis of short-term dependence for a long-memory time series bearing a moderate Hurst exponent of 0.6. Based on this argument, we reject the null hypothesis due to the relatively high value of H for Egypt. In short, all indices show signs of fractional Brownian motion or, biased random walks except the JSE All Share index.

To further our examination of the long memory property of the respective markets stock returns and volatility (using returns squared as a proxy), we inspect the diagrammatic schema of their respective autocorrelation functions over 900 lags.

### Egypt Morocco

From 1, we notice that all the markets depict more or less the same pattern centered on zero. The square returns in 2 depict autocorrelations functions for stock return volatility. Mauritius and South Africa have volatility decaying more or less at hyperbolic rates indicating that the series are strongly correlated up to long lag but other countries do not seem to be consistent with the characteristic of long memory behaviour. However there exists the clustering effect so that the any day's volatility depends on previous days' values justifying our use of GARCH.

### Table 6: GARCH(1,1) results

Variance Equation

Mauritius

South Africa

Morocco

Egypt

C

2.97E-07

2.91E-06

5.50E-06

4.95E-06

(-1.79E-08)

(8.09E-07)

(3.28E-07)

(8.59E-07)

λ

0.069109

0.101349

0.318405

0.123656

(0.00288)

(0.010778)

(0.017469)

(0.007751)

θ

0.930453

0.883252

0.646992

0.870223

(0.002633)

(0.012715)

(0.010421)

(0.00817)

The results for the GARCH(1,1) meet the positivity constraints confirming the existence of time-varying conditional variance. The θ estimates are considerably larger than the λ implying that large market surprises result in only small adjustments in future volatility. Furthermore, the sum of the parameters ( λ+θ ) indicates that high persistence of volatility clusters all the markets. Indeed, Mauritius has a sum of 0.999 while South Africa, Morocco and Egypt have approximate sums 0.984, 0.965 and 0.993 respectively which are very close to one. There is also a marked tendency for the rate of decay of the response function to shocks, on a daily basis, to die slowly; for an initial shock, about 0.930 and 0.9180 of the impact remain after one and six months respectively. Hence, evidence of high volatility persistence and long-memory implies that FIGARCH may be appropriate for the data. Dual-memory can now be examined by looking at the fractional difference for dm and dv in the ARFIMA(0, dm,0)-FIGARCH(0, dv,0) model.

### Table 7: Results for order of integration: ARFIMA-F1GARCH

Country

dm

p-value

dv

p-value

Mauritius

0.2242

0.0000

0.2954

0.0000

South Africa

0.0204

0.2727

0.2294

0.0000

Morocco

0.1779

0.0000

0.3139

0.0000

Egypt

0.1325

0.0000

0.2702

0.0000

In table 7, the size of the fractional difference parameter, dm, is examined when dealing with market efficiency. The fractional difference parameter for all the countries are within the range of 0 and 0.5 confirming long-memory processes. Besides South Africa, the dm estimates are all significant, consistent with the results from the rescaled statistic. For the JSE All Share index, the estimate is not statistically different from zero even at 10% significance level but still shows relatively less persistency. An interesting point to note is that whereas the ADF test conceives an I(0) process for returns, the ARFIMA suggests a fractionally differenced process to emphasise the low power of the former. As regards long memory in volatility, they are within the theoretical value indicating long-term predictable component. This is consistent with the results depicted for the GARCH(1,1) model above.

The use of dual-memory test is to provide for higher robustness to the test, for example, as opposed to the ARFIMA. The property is present in the return and volatility of those markets except for South Africa. Long-memory in mean is synonymous to predictable patterns in stock prices which is inconsistent with the weak-form efficient market hypothesis while long memory in volatility indicates that risk needs consideration when modelling data in those countries. Thus, future volatility is a function of its past values and is predictable.

### Possible reasons for in/efficiency in the selected African markets

From the above results, we can deduce that there is a pronounced tendency for South Africa to exhibit weak-form efficiency both in the short- and long-terms. Irrespective of the ‘time' periods, all the countries showed signs of positive autocorrelation though JSE proved to be less predictable. This can be attributed to infrequent or nonsynchronous trading causing large errors for the relatively smaller markets. As held by Poterba and Summers (1988), positive autocorrelation in stock index is the result of infrequent trading of some securities. For countries like Mauritius, Morocco and to a lesser extent, Egypt, large stocks are subject to more trading as opposed to smaller ones. As a result, new information is first impounded into large stocks' prices but with lags for smaller stocks. Such lags provoke positive serial correlation.

Indeed, the Mauritius Commercial Bank Ltd and the State Bank of Mauritius were among the largest in terms of market capitalisation for Mauritius in 2008. They accounted for about 20.3% of total volume traded during that year suggesting the high concentration of investment in large firms. On the other hand, Plastic Industry (Mtius) Ltd and Mauritius Stationary Manufacturers Ltd were among the smallest capitalised firms accounting for about 0.7% of total trade. As for Morocco, there was still high concentration with the top four companies, including ITISSALAT AL-MAGHRIB, ATTIJARIWAFA BANK, BMCE BANK and CGI holding about 51% of total volume traded. Among the smallest capitalized firms was REBAB COMPANY with around 0.006% of trade taking place. Similarly, Egypt had a mean approximating 77% in terms of number of traded companies as a percentage of listed companies for periods 2005-2009. The lowest was 59.27% in 2005. Hence, it is likely that new information will be incorporated into less traded stocks with lags to cause autocorrelation.

There are further institutional characteristics of the stock exchanges which can make them weak-form inefficient. These are the size and development, as measured by capitalisation, number of listed companies and capitalisation/GDP. The Johannesburg Stock Exchange is often related to developed ones and is much larger than the other selected African markets with respect to the number of listed companies, market capitalisation and traded value. Indeed, the mean capitalisation/GDP for Mauritius, South Africa, Morocco and Egypt, based on available data, was 41.1%, 207.4%, 50.8% and 50.9% respectively. Hence, market size can provide some justification to efficiency.

Thin market is often cited as a reason for the non-random walk prevailing in ‘small' stock markets. Turnover ratio[7] can be used as a proxy to thin market. The calculated mean for the period 2000-2008 amounted to 47.9% for South Africa but a meager of 6.6% is found for Mauritius. As for Morocco and Egypt, this stood at 42.4% and 32.7% respectively which reflect relatively less thin markets. Moreover the JSE had a turnover approximating 72%, comparable to the larger developed markets in 2008. Hence, the argument of thin trading cannot be brushed aside for the inefficient markets.

A pre-requisite to informational efficiency is that price should instantaneously incorporate new information. Hence weak-form efficiency should prevail if the appropriate ‘framework' is set. Contrary to the other countries, the Johannesburg Stock Exchange provides a real-stock exchange news services which was launched in 1997. This leads to higher transparency in the market and boosts investor confidence. The listing requirement necessitates listed companies to publicise price sensitive information on the service before any other form of media. Obviously, this impacts on the price as they are more responsive to new information regarding prices. These technologies are unavailable in Mauritius and Morocco explaining inefficiency but the Egyptian Stock Market is now undertaking massive progress as regards information dissemination. Since this year, information (economic, financial and company) is transmitted through satellite-based systems and TV broadcasting. The first phase includes national dissemination and will expand to Middle East and North Africa in the second phase. This could result in higher efficiency in the future.

Another possible reason for the prevailing efficiency in South Africa is that the exchange's trading and information systems were replaced with that of the London Stock Exchange (LSE). A sophisticated system connecting the JSE remotely was established. This enables access to more than 1,500 traders and information users. Trade information of instruments listed on the JSE is disseminated by the LSE to many terminals around the world to enhance international recognition and investors' confidence. Indeed frequent trade can cause price to move in an unpredictable manner as new prices are determined by the reaction of a large number of traders in the market.

### Implications:

Generally, investors buy and sell securities under the assumption that there is an opportunity to beat the market. However, if the market is informational efficient then, buying and selling of shares would be a game of chance rather than skills. The random walk model thus suggests passive investment strategies which advocate buying and holding assets in a well-diversified portfolio without trying to look for profitable opportunities. Yet, results of short memory as well as long memory component in asset returns are not supportive of weak-form efficiency. These have important implications in the modern financial economics.

Presence of short-term or long-term dependence in asset prices would allow investors and portfolio managers to make predictions about future price and to adopt speculative strategies designed to make extraordinary gains. They would thus adopt active investment strategies and choose the best portfolio by investing less in stocks when they have climbed above trend and less when they have plunged behind trend. Samuelson (1992) inferred that it is better to have more equity exposure with long investment horizon than short horizon. This is in line with the usual perception that long-horizon investors will be willing to endure more risks to reap higher returns. Besides, long memory persistence in asset returns could infer investors to buy and hold securities after a downward spiral in a market exhibiting an upward trend. On the other hand, a market that displays anti-persistence reverses itself in the short-term, implying that investors would buy and sell securities consistently to beat the market.

Long-memory in asset returns also indicates that investors can no more use traditional tests of capital asset pricing model or arbitrage pricing theory since the usual forms of statistical inference do not apply to time series exhibiting such persistent statistical dependence. Similarly, pricing of derivatives such as options and futures with martingale methods may not be appropriate, since the class of continuous time stochastic processes most commonly employed is inconsistent with long-term memory.

### CONCLUSION

This paper presents an in-depth analysis of the efficiency of selected African markets observing both short and long memory dependence of asset returns. It was believed that a combination of both could provide the best conclusions. Following the tests for the various random walks, we find out that apart from South Africa, all the other three stock markets did not adhere to the random walk hypothesis. Furthermore, the Hurst exponent indicated long-term persistence for Mauritius, Morocco and Egypt. South Africa was the only market to be consistent with market efficiency. Finally, these results were confirmed using dual long-memory, vis-a-vis, ARFIMA-FIGARCH.

The above analysis shows that the African Stock Exchanges hold both promise and peril. They have the potential for remarkably high returns, while simultaneously harboring substantial risks. Undoubtedly, the African countries are working towards reforming and deepening their financial system through the expansion of its stock markets in order to improve their ability to mobilize resources. It is often argued that it may not be feasible for all African markets to promote stock markets given the huge costs and the poor financial structures. However, if appropriate measures are adopted, stock markets can surely help in promoting economic development and growth.

[1] M. G. Kendall, “The Analysis of Economic Time Series, Part I. Prices,” Journal of the Royal Statistical Society 96 (1953), pp. 11-25.

[2] E.Fama, "Behavior of stock market prices." Journal of Business (1965)

[3] E.Fama, "Efficient capital markets: A review of theory and empirical work. Journal of Finance (1970)

[4] Law No.95/1992

[5] JB=n[s26+K-32/24]

[6] Studentized range is Max Rt - MinRtσRt

[7] Turnover ratio measures the trading of domestic equities on an exchange relative to market size