Financial markets are important in an economy as they involve lots of monetary funds in the capital markets. These funds enable firms to raise finance in the form of equities and debts 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. As regards African Stock markets, there has been attracting institutional and international investors as means to diversify risks these last years. Subject to the process of economic restructuration as well as stock exchange modernisation, these markets 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 as they absorb all information at each point of time. This leads to optimal allocation of private and social resources. Moreover, investors may not beat the market and make abnormally higher returns than others, based on information asymmetry. Conversely, inefficiency leads to market prices deviating from actual value so that investors having reasonable level of expertise in the field of valuation will be able to spot and exploit above and undervalued stocks. Hence, the price of a security is an unbiased estimate of its true value in an efficient market.
Limited studies on efficiency have been performed on African markets but the concept has been widely applied to developed countries; some have evidenced in favour of efficiency (Fama, 1965), (Shiguang & Barnes, 2001), (Malkiel, 2003) but others have been contradicting to suggest the presence of patterns in stock prices. These predictable patterns present trends in form of anomalies like January effect, day of the week effect and size effect among others. However, the proponents of the market efficiency consider these trends as temporary and shortlived so that investors may not continuously beat the market.
With the growing importance of African markets, it is worth testing their efficiency. Generally lack of data and supervision are inherent to those markets and so it is more convenient to test for weakform efficiency of the market rather than testing for semistrong or strong forms of efficiency. Then rejection of weakform efficiency will automatically imply rejection of stronger forms of market efficiency.
The objective of this study is to examine the possibility of both short and longterm memory in asset returns in selected African markets' stock indexes. Besides South Africa, all the other markets are still in developing states so that efficiency can be gauged on basis of market development and size. The paper is organised as follows:
Efficient market hypothesis (EMH) is one of the most researched topics in the realm of the stock market. The question is whether security prices on stock markets reflect all available information pertaining to the prospect of the concerned stock. While many investors and technical analysts consider it as the efficient market hypothesis bunk, others do believe in their presence, considering excess returns earned as either luck or temporary. The hypothesis is applied to African markets but a general review of the theory is first presented. This aims at defining the main concepts and demonstrating familiarity with previous relevant findings concerning the same field of research.
This section starts with a formal view of the EMH with emphasis on weakform efficiency and the random walk hypothesis. Starting with the martingale model, necessary assumptions are made to develop a model consistent with Lo and McKinlay (1997) model specification. Under different assumptions about the model's increments, a formal presentation of the different linear random walks is made and criticised to allow for nonlinearity.
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, 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.”
In his thesis, "Behaviour of stock market prices", Fama (1965) supported the random walk theory where he reviewed previous works on stock price movements. He concluded 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 profitmaximisers 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 paper: "Efficient capital markets: A review of theory and empirical work." 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 weakform if any information which might be contained in past price movements is already reflected in the security prices. It is semistrong 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.
Hence, both the random walk theory and the EMH are related to informational efficiency. Then the form of efficiency under consideration will depend upon the information set, Ωt, which determines the level of efficiency.
Weakform 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 weakform 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) considered a security market to be in equilibrium provided the following equations hold:
Ept+1Ωt=pt (1)
Ept+1ptΩt=0 (1.1)
Where t denotes the price of an asset at date t, Ωt is a set of all past and current information regarding prices pt,pt1,pt2….. and pt+1pt=rt. Hence, it may not be possible to forecast the directions of the future movements in martingales.
If pt is a martingale in equation (1), the best forecast of pt+1 that could be derived on basis of current information Ωt, equals pt. For equation (1.1), rt is a fair game if the forecast is zero for any possible value of Ωt. Then pt is a martingale only if rt is a fair game. In this case, 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 nonzero and positive returns. The martingale hypothesis does not take into account the riskreturn tradeoff. It implicitly assumes risk neutrality while investors are generally risk averse. In fact, an investor is likely to hold more risky assets provided they are rewarded in terms of higher expected returns. Hence, knowledge of the riskiness of current information set implies some awareness about the expected returns. Then the equilibrium model shall predict a positive price change in the assets price though the actual return is still unforecastable under market efficiency. An asset model (submartingale process), considering positive returns, may be formulated as Fama (1970):
Ept+1⃓Ωt≥pt or alternatively Ert+1⃓Ωt≥0 (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 stipulated, 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, rt+1.
Ert+1Ωt=pt+1pt (1.3)
Under the efficient market hypothesis, investors cannot earn excess profits based on the information set, other than by chance. This is in line with Jensen (1978) who characterises 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:
zt+1=rt+1Ert+1⃓Ωt (1.4)
Since market efficiency implies that all information is already impounded in stock prices, the following applies:
Ezt+1⃓Ωt=0 (1.5)
Under the assumption that the equilibrium model determining asset prices in (1.3) is constant over time, the deduction is that expected return does not depend on the information available at time t such that:
pt+1pt=Ert+1⃓Ωt=Ert+1=r (1.6)
Therefore market efficiency produces a result conducive to changes in asset prices following 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):
pt+1= μ+pt+ εt+1 (1.7)
rt= μ+αrt1+ εt (1.8)
If the stock price index follows a random walk, then, α = 0 so that (1.8) is equivalent to (1.7). Generally, if stock prices and returns are unpredictable then time series have the ‘characteristic' of random walk and white noise to validate the EMH
Within the random walk hypothesis, three successively more restrictive subhypotheses with sequentially stronger tests for random walks exists (Campbell et al. 1997); Random Walks (RW) 1, 2 and 3. Based on their extensive research, the orthogonality condition for the random walk is:
covfrtgrt+k=0 (1.9)
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 (1.9) 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), hereby LM, the variance ratio tests figure 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.
However, the random walk models are do not conform to nonlinear structure as evidenced by the application of nonlinear dynamics and chaos theory to financial series. Exclusion of nonlinear analysis in financial series could lead to inappropriate deductions as regards weakform efficiency. In practice, 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 nonlinear models in the ‘shortrun'.
Further departures from the random walk hypothesis exist in the longrange 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. Their tests generally obey nonlinear dynamics and include the rescaled statistic, GPH estimates and the ARFIMAFIGARCH among others. They categorise the long and shortterm memory based on the estimated values of their statistics and degrees of fractional difference.
Studies of testing weakform efficiency started on developed markets which were mostly in favour of efficiency. On the other hand, research findings of developing and less developed markets are mixed and controversial too. Some previous research aiming at testing the weakform efficiency of a particular group of stock markets are presented below.
Central European transition economies' equity markets (the Czech Republic, Hungary, and Poland) were subject to tests of weakform efficiency by Gilmore and McManus (2001). Using different approaches comprising of univariate, multivariate tests as well as the modelcomparison approach for the period July 1995 to September 2000 different conclusion were drawn. While the serial correlationbased tests largely support a conclusion that these markets are weakform efficient, the results of comparing forecasts of alternative model provide evidence against the random walk hypothesis.
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 ChowDenning 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.
As regards the Gulf Cooperation Council (GCC) stock markets, Elango and Hussein (2008) tested whether daily returns series are an approximation of normal distribution or not. Dubai, AbuDhabi, Saudi Arabia, Qatar, Kuwait, Oman and Bahrain stock market indices were examined using the KolmogorovSmirnov test, Runs test, Autocorrelation Function and Partial Autocorrelation Functions. 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.
The random walk hypothesis was investigated by Urrutia (1995) in four latin amearicn countries. Employing monthly data from December 1975 to March 1991 for Argentina, Brazil, Chile and Mexico, he made use of the Varianceratio tests and the runs tests. While results of the variance ratio estimates<
reject the random walk hypothesis, runs tests specify controversial results Latin American equity markets. These findings indicate the ineffectiveness of developing trading strategies that would allow domestic investors to earn abnormal returns.
Revisiting the empiricism of random walk hypothesis in eight emerging markets in the Middle East and North Africa (MENA), AlKhazali, Ding and Pyun (2007) used LoMacKinlay Variance ratio, Wright's rank and sign VR and the standard runs tests. These countries included Bahrain, Egypt, Jordan, Kuwait, Morocco, Oman, Saudi Arabia, and Tunisia, all of which rejected the hypothesis of random walk based on Wright's (2000) rank and sign VR test. However, once data were reconciled for distortions from thinly and infrequently traded stocks, these stock markets did 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 AutoCorrelation 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 research which focuses on African markets was that of Jefferis and Smith (2005). It covers seven African stock markets: South Africa, Morocco, Egypt, Zimbabwe, Nigeria, Kenya and Mauritius using a GARCH approach with timevarying parameters to detect changes in weakform efficiency through time. They emphasised on RW 3 model with volatilities changing over time and found that Johannesburg stock market was weakform 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 weakform efficient towards the end of the period. The results for Kenya, Zimbabwe and Mauritius, however, showed tendency towards efficiency and rejected the hypothesis of weakform efficiency.
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 ARFIMAFIGARCH and ARFIMAHYGARCH models. They ended up with mixed results. The ARFIMAFIGARCH 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 nonstationary. These results were further supported by the ARFIMAHYGARCH models.
During the course of the literature review, we found limited evidence of African markets which focused on weak form efficiency. Hence, univariate time series analysis might be important tools 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.
African markets, following in the wake of the surge in the world stock markets over the few decades, are starting to take off. Stock markets are important in economic development and several African countries have recognised this and launched their exchanges. Today, Africa has about 20 active stock markets compared to pre1989 when there were only five stock markets in SubSaharan Africa and three in North Africa. However, with the exception of the well established ones, these markets face problems regarding liquidity. A brief description of the four African stock markets considered in this study as well as their respective index analysis over periods for which data is available.
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 preemerging 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 overthecounter market. The exchange is regulated by the Financial Services Commission. In addition to operating in line with international standards, SEM has a developing institutional and retail investor base which makes 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, SEM7 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 figure is in line with the levels observed in wellestablished 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 pickedup by midMarch 2009 on the back of interest rate cuts and stimulus packages put forward by the Government of Mauritius.
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, it has evolved remarkably from a traditional floor based equities trading market to a modern securities exchange offering 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 emergingmarket 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 yearend, there were 992 listed securities on the JSE market capitalisation totalling R4,514 billion compared to R5,696 billion in 2007.
Founded in 1929, the Casablanca Stock Exchange (CSE) in Morocco is relatively modern, having experienced reform in 1993. The exchange is well regulated by the Conseil Deontologique des Valeurs Mobilieres (CDVM). CSE now comprises of the Moroccan All Shares Index (MASI) and the Moroccan Most Active Shares Index (MADEX) which replaced the Index de la Bourse des Valeurs de Casablanca (IGB) on January 2002. MASI which comprises all listed shares permits the follow up of all listed values as well as longterm visibility. On the other hand, MADEX encompasses most active shares listed continuously with variations closely linked to all the market, hence acting as a reference for the listing of all funds invested in shares. Out of the 77 listed securities, only about 25 of them are traded frequently (daily), most of which are listed on the continuous market. On the alternate markets namely the Marché Croissance and Marché Développement, clearance of orders occur 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.
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 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 nonbank 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.
The markets studied in this paper are based upon availability of data and include the 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. We used EGX30 although it comprises only of the best 30 companies as the EGX70 or EGX100 were introduced in year 2008 and 2006 respectively, thus providing insufficient data. 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 started 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 global financial crisis. This downward spiral in 2008 was also reflected in the other three stock indexes. As regard the JSE All Share index, 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 just 2 weeks. The earlier decline in the series was the result of monetary policies directed towards keeping inflation rate in the range of 36%. Causing an appreciation in the rand, this led to a fall in 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.
In period 20002002, the MASI 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 general upward trend thereafter can be attributed to sound policies which also enabled the combat of unprecedented oil price and foodstuff upsurges in year 2006, albeit Morocco imports primarily oil and foodstuffs. From midMarch 2008 till January 2009, the index lost around 30% due to subprime 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. The EGX30 lost about 70% from May 2008 to February 2009. From February 2006 to June 2006, global inflation 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 geopolitical tensions which prevailed in the MiddleEast but ameliorated after this period.
Most African countries have been subject to various reforms regarding their stock markets during these last years. For instance, Egypt's trading system was upgraded to XStream in 2007 while Zambia, Ghana and Uganda joined Mauritius, Namibia and Botswana in 2008 by introducing electronic trading systems. Beside technology and financial products, regulatory frameworks have been constantly revised to account for greater transparency as they reinforce efficiency and stimulate private investment. Such innovations may have important effects on market efficiency. We proceed to a methodological analysis before testing of random walk for our selected countries.
This section of the paper provides the methodological settings for testing the market efficiency of four African stock markets in the weak form. Several parametric and nonparametric tests are used to examine whether the stock returns are weakform based on the three notions of random walks. Longterm tests are nowadays receiving much attention in general academic researches. We use these to check for the presence of persistence, antipersistence and random walks in returns as well as volatility.
This paper uses continuously compounded returns for testing efficiency in the selected African markets. Natural log of relative price are taken such that rt=lnptpt1, where pt and pt1 represent the stock index at time t and t1 respectively. This is a more common practice as it easier to derive timeseries properties of additive processes. 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.
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 used to this effect, as 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 having a null hypothesis that successive outcomes are independent. Following a normal distribution, the total expected number of runs has the following mean, µ, and standard deviation σµ:
μ=NN+1i=13ni2N
σμ=i=13i=13ni2+N(N+1)2N(i=13ni3N3)N2(N1)12
To perform the test for serial dependence, the actual number of runs in the series is compared to the expected number μ. Nonrandomness occurs when there are too few or too many runs as compared to the expected number of runs as in a random series. Too few runs would mean that the stock returns in the time series do not change signs frequently, thus indicating a positive serial correlation while too many runs may suggest negative autocorrelation.
For the purpose of testing RW2, unit root tests are used to test for nonstationarity. If the log price series is nonstationary and the first difference of the series (returns) is stationary, the series contains a unit root. The Augmented DickeyFuller (ADF) test is applied on both log prices (level) and return (difference) to check for random walk.
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 timetrend terms:
∆pt=a0+a1t+γpt1+βi∆pti+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 firstdifferenced terms such that et is white noise. The test for a unit root has the null hypothesis that γ = 0.
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. 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. It follows 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 qdifferences 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+1ptptqqμ2 eq(2)
and
σa2q=1nq1t=2nq+1ptpt1μ2 eq(3)
where
m=qnqq+111n
μ=1nqpnq+1p1
For testing the RW1, the standard normal test statistic under the hypothesis of homoscedasticity, Z(q), is:
Zq=VRq1∅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 heteroscedasticityrobust standardized variance ratio is given by:
Z*q=VRq1∅*q12~N0,1 eq(5)
where ∅ * (q) is the heteroscedasticityconsistent asymptotic variance of the variance ratio, given by:
∅*q=j=1q12qjq2δj
δj=t=j+2nq+1ptpt1μ2ptjptj1μ2t=2nq+1ptpt1μ22
We use oneday 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 country. These are then compared to the critical values obtained from the normal table.
A weakly stationary process is longmemory provided there is a real number H and a finite constant C such that the autocorrelation function bears the following rate of decay:
ρk~Cτ2H1 as τ→∞ C≠0 1<d<12
The parameter, H, called the Hurst Exponent, characterises the longmemory property of the series. For a long memory series which is fractionally integrated of order d, the latter's relationship with the parameter H is as follows:
d=H12
Persistent series are normally characterised by distinct but nonperiodic 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=1krjrnMinj=1krjrn
and
σnq=n1j=1nrjrn21/2
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 longrange 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 antipersistency, analogous to negative dependence. To test for long memory, the null hypothesis is that of no longterm 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 longrange dependence may not be the result of long range dependence but simply of be a symptom of shortterm dependence. Hence the use of the modified statistic provided by Lo (1991) can be a better alternative as it incorporates shortrun dependence into its denominator, that is,
σn2q=1nj=1nrjrn2+2nj=1qωiqi=j+1nrirnrijrn=
σn2q=σr2+2j=1qωjqγj
ωjq=1jq+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=3n2132γ11γ1223
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)=1jkn
Lo (1991) showed that in the presence of longterm memory,Vn(q)≡ Qn.T1/2 weakly converges to the range of a Brownian motion with the probability distribution:
Fv=1+2k=1∞(14k2v2)e2(kv)2. The critical values shown in table A6 in the appendix are the fractiles of the limiting distribution of the statistic which are used to test the null hypothesis of no longterm dependence (H=0.5) against longterm dependence alternative (0.5<H<1).
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 longterm memory. Persistence is often detected in both conditional mean and conditional variance justifying our preference for the ARFIMAFIGARCH. 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 longmemory 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) following 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+λεt12+θht1 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, εt12 is the information about volatility from the prior period (the ARCH term), and ht1 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 longterm persistency. For the time being, we lay the methodological framework for the study of dual memory as developed by Baillie, Han and Kwon (2002).
∅L1Ldm(rtμ)=θ(L)εt (1)
εt=ξt2ht (2)
⋋L(1L)dvεt2=ω+1βLυt (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≡εt2ht2 with ξt~iid0,ht and Eεtεs=0 for s≠t. To ensure stationarity, the roots ∅L, θ(L), ⋋L and 1β(L) must lie outside the unit root circle.
The longmemory operator can be expanded as a hypergoemetric function:
1Ld=k=0∞ΓkdΓk+1Γ(d)Lk=k=0∞⋋kLk
Where Γ(.) represents the gamma function with Γk+1=k!=k×Γk and ⋋k=kd1k⋋k1. The process is stationary and ergodic for d<0.5. When d=0 implies stationarity while d∈(0.5, 0) implies shortterm memory for negative autocorrelations. For d∈(0, 0.5) is analogous to longmemory due to positive autocorrelations which decay hyperbolically. The variance of rt is infinite so that the process is nonstationary but is still meanreverting 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 longmemory 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 longterm dynamics of volatility and GARCH considers shortterm ones.
The aim of this section is to explain the econometric tools necessary to investigate the random walk theory and hence, weakform efficiency. Ranging from earlier tests like the runs tests, more recent and powerful ones like the variance ratio tests are adopted. Since longmemory has been given little attention as regards African stock markets, rescaled statistics and ARFIMAFIGARCH are used to address the issue. The results are being discussed in the next section and provide an insight to the development of the stock markets of the selected markets.
This section shows the results for the various tests undertaken in the methodology section. Before proceeding to the weak efficiency results, 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 longterm memory.
Daily return for SEMDEX during 2008 ranged from more about 0.06 to 0.06 reaching high of nearly 0.08 at end of March 2009. At the start of year 2000, volatility was relatively low till mid of year 2006 when it started to increase and peaked in year 20082009. 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. As for the MASI, returns show some wide fluctuations but most of the values were likely to range from 0.02 to 0.02. Although EGX 30 returns did not show much wide fluctuations, the daily returns series appear to be highly volatile ranging mostly from 0.05 to 0.05. In all cases, signs of positive autocorrelation were suggested by close values between consecutive periods.
One of the basic assumptions underlying weakform 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.
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 
0.05017 
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 
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] 
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. Such difference in skewness can be attributed to greater impacts of financial crisis on the other stock markets as well as macroeconomic fundamentals and political problems faced by them as discussed in section 3. These resulted in higher volatility in the markets and negative shocks as opposed to SEM which became more volatile in the mid2006 only.
Besides, returns for all for indexes display excess kurtosis indicating that the distributions are leptokurtic so that their distributions have fatter tails than a normal distribution. Nonnormality is further supported by the JacqueBera 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 pvalue 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.
Hence, these countries are characterised by high volatility but relatively lower returns and most of them have negatively skewed distributions. The above graphical illustrations depict nearzero mean returns for these countries but large fluctuations. They are thus characterised by bouts of returns and volatility patterns implying that investors and portfolio managers should enter and exit the markets in timely manners else they could stand as losers.
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. (2tailed) 
0.000 
0.002 
0.000 
0.000 
From table 2, the estimated zvalues are significant at the 5% level for all markets since all pvalues are less than 0.05. The negative zvalues 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. In absolute form, SEMDEX, MASI and EGX 30 have zvalues of much higher than 5.0 while JSE All share index has the smallest zvalue, affirming a relatively more efficient stock market.
This inefficiency can be assigned to nonsynchronous trading so that each day's returns values tend to follow each other in the smaller markets. Indeed thin trading causes inefficiency as the market does not adjust for price on daily basis but rather over longer periods of time, for example a month or so. In this case, positive returns tend to follow positive ones which also applies for negative returns to cause positive autocorrelation. The ‘smaller' value for the JSE is due to more frequent trading in a relatively more active market. As for Egypt, it has lower absolute value than Mauritius and Morocco as the EGX 30 considers the most liquid firms but still the market is characterised by thin trading as judged by relatively higher absolute value for z. These large values further reveal some sort of persistency, thus violating the weakform efficiency of the markets but there is no clue about its magnitude and direction yet.
The above 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.
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 
pvalue 
0.98 
0.54 
0.88 
0.59 
0.95 
0.55 
0.93 
0.78 
ADFDiff 
73.12 
73.11 
83.6 
83.58 
68.89 
68.88 
76.15 
76.14 
pvalue 
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 pvalues. The null hypothesis for those tests is that the series have a unit root (nonstationary). 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 pvalues. This suggests nonstationary for the log price series. As for returns, they need to be stationary and this is confirmed by the high values of their test 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 necessary conditions for RW2 are attained for all countries.
The African countries under consideration tend to display characteristics weakform efficiency as regards the less constrained random walk (RW2). However, unit root tests suffer from low power so that they are likely to accept the null of unit root when none exists. Then it becomes crucial to use more sophisticated tests like the variance ratio tests to confirm these.
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. Except for 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 weakform efficiency.
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 heteroscedasticityconsistent 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.
Hence there is evidence that the VR(q) are statistically different from one in the smaller stock markets. Under LM's derivation, ρ1≈VR21, so that 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 firstorder autocorrelation. Therefore, an autocorrelation of 25% for Mauritius implies that 5.4% of the variation in daily SEMDEX is predictable based on the previous day's return only. For South Africa, Morocco and Egypt, the figures 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).
The higher significant VR(q) for SEMDEX, MASI and EGX30 at higher lags also confirm the persistence doubted by the result for the runs test. 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 these countries.
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 figure 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 figure 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 truncation lags 11, 2 and 13 for Mauritius, South Africa and Morocco respectively. From table A6 in the appendix, the acceptance for a twotailed 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 longterm 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 shortterm dependence for a longmemory 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 black noise 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.
Mauritius
SouthAfrica
Egypt
Morocco
Mauritius
South Africa
Morocco
Egypt
From Figure 1, we notice that all the markets depict more or less the same pattern centered on zero. The square returns in Figure 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 whereby any day's volatility depends on previous days' values justifying our use of GARCH.
Variance Equation 
Mauritius 
South Africa 
Morocco 
Egypt 
C 
2.97E07 
2.91E06 
5.50E06 
4.95E06 
(1.79E08) 
(8.09E07) 
(3.28E07) 
(8.59E07) 

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 timevarying 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. Moreover, the rate of decay of the response function to shocks, on a daily basis, tends 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 implies that FIGARCH may be appropriate for the data. Dualmemory 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.
Country 
dm 
pvalue 
dv 
pvalue 

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. They are all within the range of 0 and 0.5 confirming longmemory processes. Besides South Africa, the dm estimates are all significant, consistent with the results from the rescaled statistic. For the JSE, the estimate is not statistically different from zero even at 10% significance. While 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 longterm predictable component. This is consistent with the results depicted for the GARCH(1,1) model above.
The differences in the degrees of persistence between these markets could be attributed to the differences in their institutional organisation and trading. Moreover, fluctuations of national output may have tendencies to display longterm persistence (Kuznets and Kondratiev). This longmemory in output could manifest itself in the equity returns so that the degrees of persistence in returns could depend on the extent to which output is longterm dependent.
In short, the use of dualmemory test is to provide 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. Longmemory in mean is synonymous to predictable patterns in stock prices which is inconsistent with the weakform 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.
Hence all longmemory tests presume longmemory in returns for Mauritius, Morocco and Egypt. Such persistence can be dangerous as long periods of calmness in pricing may be disturbed by unexpected and large discontinuities and drawdown. As regards the Hurst exponent, Egypt's is closer to H=1 indicating relatively higher risk of large and sudden changes. But still there exist chances to earn positive returns when H nears one. Indeed Neely et al. (1997) found that combining technical trading rules with a genetic programming algorithm resulted in significant outsample excess returns.
From the above results, we can deduce that there is a tendency for South Africa to exhibit weakform efficiency both in the short and longterms. Irrespective of the ‘time' period, 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 ALMAGHRIB, 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 20052009. 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 weakform inefficient. These are the size and development, as measured by capitalisation, number of listed companies and capitalisation/GDP. The JSE 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 nonrandom walk prevailing in ‘small' stock markets. Turnover ratio can be used as a proxy to thin market/liquidity. Indeed, high liquidity makes markets stable resulting in closeness of fair and actual prices. But low liquidity can present trading opportunities. To assess liquidity, we use the mean for the period 20002008 which amounted to 47.9% for South Africa but a meagre figure 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 prerequisite to informational efficiency is that price should instantaneously incorporate new information. Hence weakform efficiency should prevail if the appropriate ‘framework' is set. Contrary to the other countries, the JSE provides a realstock exchange news services. 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 satellitebased 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.
The prevailing efficiency in South Africa could be the consequence of the replacement of the exchange's trading and information systems with that of the London Stock Exchange (LSE). A sophisticated system connecting the JSE was remotely 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.
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 welldiversified 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 weakform efficiency. These have important implications in the modern financial economics.
Presence of shortterm or longterm 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 follows the usual perception that longhorizon 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. Conversely, a market that displays antipersistence reverses itself in the shortterm, implying that investors would buy and sell securities consistently to beat the market.
Longmemory 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 longterm memory.
This paper presents an indepth analysis of the efficiency of selected African markets observing both short and long memory dependence of asset returns in order to provide the best conclusions. Following the tests all the other stock markets, besides JSE, did not adhere to the random walk hypothesis. Longterm persistence was found for Mauritius, Morocco and Egypt which was confirmed using dual longmemory tests. South Africa was the only market to be consistent with market efficiency.
The analysis supported the claim that larger and more liquid stock markets tend to be more efficient. African stock markets are becoming increasingly sophisticated in pricing, isolating and transferring risks but most of them are still facing various obstacles that constraints them from attaining efficiency. The main ones include thin trading, liquidity problems, no proper market diversification, limited financial instruments offered and inadequate regulatory supervision. In fact, the number of listings on the African stock markets has not experience a significant growth in recent years. While the number of securities listed on the CSE rose by 24 from year 2000 to 2008, those listed on the SEM fell by 2 reaching 40 at year ended 2008. This number is incredibly low as compared to the 992 securities listed on the JSE.
The stock market needs specific attention when it comes to the threats and challenges prevailing in the future. The various policies that can be implemented to further lead to the development of African stock markets are:
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This paper is centered on market efficiency. However, given the excessive literature that exists in this field, it is beyond the scope this study to review all the previous works related to the study. We therefore provide only a brief discussion on the main findings associated to the weakform efficiency to provide a general overview of the paper. Besides, the main limitation of this paper is that we restrict to the weakform efficiency using time series analysis. Consequently, the statistical tests are only used to test for market efficiency excluding any transaction costs adjustment such as the bidask spread. Finally, we use daily data for the analysis though it may lead to possible biasness in the observations. According to Lo and MacKinlay (1988), though daily series provides a large number of observations, “the biases associated with nontrading, the bidask spread, asynchronous prices …” presents significant problems. Also, missing data due to holidays was not replaced as proposed by Lo & MacKinlay (a non random walk down Wall Street). We believe that using a longer time period would help to reduce this problem.
Tests for the evolving efficiency of the countries could be undertaken. The aim would be to test whether there has been improved efficiency over time. Using daily data, three successive time frames could be taken. For instance, pre, post and subprime crisis periods could be used. Traditional tests of random walks alongside longmemory tests could be undertaken to check for persistence in the markets and deduce whether there was more predictability during the subprime crisis. Moreover, forecasting techniques, linear or nonlinear depending on appropriate volatility results, could be used to deduce whether these models could outperform the naïve model (random walk) based on Theil's U statistic.
African countries were among the last ones to have switched from traditional to modern exchanges. This process has been gathering pace these last years with economic and regulatory reforms, attracting more and more foreign investors. The world economic downturn has also provided impetus to international investment in many of these countries like Nigeria, Kenya, Zambia and Ghana among others. This could have revived investors' appetites for frontier markets. But still, reforms need continuation to boost investors' confidence which can subsequently result in better valuation of shares and increase liquidity. Indeed markets do not become efficient by themselves; it is the actions of investors through bargains and schemes which make them efficient and this can be brought about in highly liquid markets.
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Law No.95/1992
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JB=n[s26+K32/24]
[Studentized Range]
Studentized range is Max Rt  MinRtσRt
Turnover ratio measures the trading of domestic equities on an exchange relative to market size
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