Weak form Efficiency and Calendar Anomalies: Comparison between Developed and Developing Equity Markets
Syed Zulfiqar Ali Shah Assistant Professor-Finance, Department of Business Administration Faculty of Management Sciences, International Islamic University Islamabad E-mail: zulfiqar.shah@gmail.com Muhammad Husnain Ph.D Scholar (Finance) Mohammad Ali Jinnah University Islamabad Email: Husnain_ctn@yahoo.com Abstract Financial economists have continuously questioned the efficient market hypothesis especially in last decade. Major part of discussion is whether the equity markets are efficient and if not then up to what extent one can forecast the meaningful future movement of equity prices. On one side there are believers of random walk and contrary there are followers of chartist theories. Those who negate the random walk suggested that there exist anomalies in the equity markets and hence are not perfectly efficient. The major objective of this study is to check the weak form of efficiency and presence of calendar anomalies in equity markets of developing and developed countries. On the basis of most recent and relatively longer horizon (14 Year) data on daily basis and a range of powerful econometrics this study suggested that in broader sense both of developed and developing equity markets are weak form inefficient. Hence there is no remarkable difference in term of market efficiency in equity markets of developed and developing countries. Hence one can reject the random walk hypothesis and therefore presence of markets efficiency is again a matter of theory not as much practical. Key Word: Weak form efficiency, Random Walk, Calendar Anomalies, EGARCH JEL classification: G12, G14, G15

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Introduction The role of capital market in the economy of a country is to channelize the flow of resources from one party to other having different interest efficiently. These involved parties are generally more affluent when efficient capital markets are used to channelize the flow of resources (Akintoye 2008). In late twentieth century, the concept of efficient capital markets by Fama (1970) has been of great interest even in academia and practice. The idea of efficient market hypothesis comes in the literature of finance in 1960s by E. Fama. According to this investors are not in a position to beat the markets and hence one cannot earn abnormal returns. This is mainly according to Fama (1970) was that stock prices fully reflect all available information at any specific time. The idea at the back is when any information arises then without any impediment this set of information is fully incorporated into price of security. This leads the fact that neither the follower of technical analysis nor fundamental analysis are in a position to earn abnormal returns. Followers of efficient market hypothesis have firm believed that it is absolutely futile exercises to locate undervalue stock and investors are winning on the basis of chance. The idea of efficient market hypothesis is linked with random walk model. Random walk suggested that prices of stocks move in random and unpredictable fashions. It is assumed that prices solely depend upon arrival of new information and of course the process of arrival of new information is random and hence movement in prices is erratic. On the bases of skills and good planning, one cannot beat the abnormal returns. The theory of efficient market hypothesis has been under immense debate. The counter argument against it widely based upon empirical research. It is argued that there exist consistent trend in the movement of equity prices and one can forecast the future of equity prices. In real life there are portfolio managers whose past performance are much better than competitors. Now it is generally believed that equity markets are not perfectly efficient but there may be some imperfection in the equity markets. Hence one can forecast the future based upon various factors (Malkiel 2003). Due to this the utility of technical analysis, fundamental valuation of stocks, psychological and behavioral factors are getting much importance in literature. In existence literature there is considerable review of presence of predictable patterns. Presence of short term momentum in equity markets by Cootner (1964), Lo and MacKinlay (1999) and Shiller (2000), the long run return reversals by Fluck et al. (1997), the presence of calendar anomalies by Keim (1983), Haugen and Lakonishok (1988) and Ariel (1990), the expected trend based upon different valuation parameters by Fama and French (1988), Bagwell and Shoven (1989), Campbell and Shiller (1998). Other also identified like size effect by Fama and French (1993) and value stocks by Ball (1978) and Basu (1983). Hence efficient market hypothesis may not be perfectly accepted in its unadulterated form. There are three classification of efficient market hypothesis namely weak form of efficiency, semi strong form of efficiency and strong form of efficiency. A market is weak form of efficient only if it reflects all historical information in the prices of securities. If it is the case then technical analysis is out of law and no one earn abnormal returns based upon historical information. The major objective of this study is to check the weak form of efficiency of equity markets. It aims at
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a comparison in term of weak form of efficiency of equity markets of developing and developed countries. Furthermore we also analyze the presence of calendar anomalies in term of day of the week effect in the markets of developing and developed countries. The present study is up to an extent an effort to comments upon the applicability of technical analysis in the equity markets of developing and developed countries. Hence on one side it provide the comparison on the basis of weak form efficiency among developing and developed countries equity markets and simultaneously it helps the technician whether they can earn abnormal return based upon past information. Furthermore it also provides direction in a sense that which of the markets is supposed to be more lucrative for them. Moreover this study uses most recent and relatively longer horizon data on daily basis and a range of powerful econometrics which ultimately strengthen the recommendation of this study for investors and fund managers. The rest of the paper is structured as follow. Next section elaborates the theoretical support to the study which is followed by the data description and methodology. Finally there is an empirical result which is further followed by conclusion. Review of related literature Every individual wants to forecast the future with great accuracy. As long as capital markets have existed, people with varying interest might try to predict them. People have expectation that it is the right forecasting which ultimately bring great fortunes. Investor put their investment in the equity market with the hope of not only getting positive return on their investment but also wishes to beat the market. A stockbroker Regnault (1863) concluded that changes in the price of security may end with abnormal results only if it was hold for considerable time duration. In other words, this change in price was directly related to holding time period. When anyone talks about the market efficient then it is accepted that a market is efficient only if it is able to price the security quickly and accurately. Markets are efficient in the context of absorbing the new information. The notion of efficiency depends on the precise definition of information (Copeland and Weston 1988). An information structure may be defined as a message about various events which may happen. There is relationship between the release of information and market price of security. Holbrook Working (1934) investigated that movement in return of equity markets was just like the numbers from a game of chance. Further Working (1949) concluded that the successful prediction of security price would be impossible even for the professionals. In the same vein, F. MacCauley(1925) observed that if someone throws the dice, the movement of this was actually the true reflection of movement in equity markets. The believers of random walk emphasized the fact that price of any security follow a random pattern and hence negate the existence of any investment strategy that consistently beat the market. In the literature, there is mixed evidence regarding the random walk. According to this future prices depends upon the information and the process of information is random hence prices move randomly. According to Cowles (1933), equity market forecasters were not in a position to accurately forecast the future price of securities. The stock market behavior is associated with random walk hypothesis by Godfrey et al. (1964). Again the evidence in the
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favor of random walk model is attributed to Kendall (1953). He examined the twenty two (22) different price series data and confirmed the presence of random walk. On the contrary there exists also reasonable evidence to reject the random walk model. Cootner (1962) and Osborne (1962) reject the hypothesis regarding the presence of random walk. Alexander (1961) revealed that one can not reject the random walk but later on in 1964 he negates previous results. By examining the S&P industrial, Alexander (1964) found that there is no more random walk model. In the same sense Jegadeesh (1990) and Steiger (1964) argued that one can predict the future movement of any stock which ultimately leads toward the rejection of random walk. However the remarkable ‗efficient market hypothesis‘ by E.Fama (1970) suggested that stock prices fully reflect all available information at any specific time. According to him any market can be labeled an efficient market only if all information is fully reflected in security prices. Further it is also suggested that all the market participants have equal information hence no one in a position to beat the market or earn abnormal returns. Prices reflect all available information and precisely this set of information is not confined to any specific set of financial information. Reflected information may be any financial news and activities, any economic or non-economic event, political related issues or any social and worthwhile events. The decision regarding the accuracy of information may be less important but important is how investor and fund manager analyze the information. Maynard Keynes (1923) observed that reward in the financial market is independent of knowing better than other but it dependent upon the risk bare. According to Malkiel (2005), the fund managers are not in a position to outperform the market and hence concluded that prices reflect all available information. But in 2003, Malkiel again concluded that markets are efficient and their predictability powers are less as suggested by recent researchers. But this is not the end. The obvious phenomena of consistent patterns are quite clear from the analysis of real world equity markets. As according to EMH one cannot beat the market but there are people who outperform the markets. Hence the accuracy of efficient market hypothesis is still debated. Among others Bernstein (1999) also criticized the EMH. There are numerous studies which reject the existence of efficient markets. LeRoy and Porter (1981) concluded that equity markets are not efficient in context of information and further revealed that stock markets exhibits ‗excess volatility‘. Similarly Lo and Mackinlay (1988) examined the random walk model and strongly reject this for weekly equity returns. Investor in the markets can be influenced from arrival of news. This effect of news on the movement of equity returns was explored by Cutler et al. (1989) and suggested that news had no significant impact on the equity returns. . Bondt and Thaler (1985) confirmed that equity markets are not weak form efficient. In the same footing, Lehmann (1990), Roll (1994), Lee et al. (2010) and Malkiel (2003) concluded that stock markets are not efficient.

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In equity markets the presence of anomalies which may be calendar, fundamental or technical etc. are quite possible. For example the existences of day of the week-effect, the January effect etc. Different researcher identified different anomalies. The evidence regarding the calendar anomalies were observed by Poterba and Summers (1988) in equity markets. In the context of day of the week effect, French (1980) observed the presence of significant day of week effect in the equity returns. The presence of predictable patterns near end of month i.e. month of the year effect was well documented by Lakonishok and Smidt (1988) and existence of patterns near holidays in the equiy markets had been observed by Ariel (1990). Evidence regarding the calendar anomalies have also been documented by various researchers like Keim 1983) and Haugen and Lakonishok (1988). According to EMH prices are moving in random and unpredictable pattern and it may be over or undervalued and finally converge to mean. The term market efficiency has no link with the phrase that in all situation prices are equal to intrinsic value. Hence the possibility of deviation of actual results form expected is still there in the presence of EMH. The believer of efficient market hypothesis firmly believed that equity markets are completely unpredictable and move in random pattern. Hence it is impossible to beat the market because the markets are efficient. The whole game is fair. One cannot forecast the future path of equity prices and there is no room for technical analysis. According to Fama (1970), equity market can be divided into three categorized based upon information. It may be weak form, semi-strong form and finally strong form efficient. Hence for very first time efficiency was in the shoulder of information. The Weak form of efficiency suggests that all past information regarding prices are incorporated into prices of stocks. One cannot forecast the future path of equity prices based upon past information and there is no room for technical analysis. According to Hassan, Abdullah and Shah (2007), there exists the possibility to predict the future movement in the stock prices. There are mixed nature of evidence in the literature regarding the efficiency of markets. So there is still need to analyze the efficiency of equity markets. In the context of comparison between the equity markets of developing and developed countries there require still considerable attentions. This study is an effort to check the market efficiency and calendar anomalies in the equity markets of developing and developed countries. Data Description and Methodological Issues The major objective of this study is to test the weak form efficiency along with calendar anomalies in equity market of developing and developed countries. For this we have selected Pakistan and India as developing countries whereas USA and UK as developed markets. The total data period for this study is 14 years which starts from August 1997 to July 2011 which corresponds to 3532 daily observations. This study selected KSE-100 index and BSE indices for the proxies of the equity markets of Pakistan and India whereas FTSE-100 and S&P500 for the equity markets of UK and USA. The data regarding the respective indices have been collected from the website of Yahoo finance which is considered as a reliable source for such type of data.
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The continuous compounded return series is generated for all the four selected equity markets indices by using the given below formula.

Here stands for the return of the respective index at day‗t‘ and , stands for the value of respective index for day ‗t‘ and ‗t-1‘ respectively. Autocorrelation and Q-Statistics Autocorrelation can be simply defining as the correlation between the elements of a data series. To check the existence of autocorrelation at a particular lag, the autocorrelation function is considered a more fruitful tool. Autocorrelation plot can be generally applied to test the randomness of data. The Ljung-Box which checks the overall randomness is considered a better tool for testing autocorrelation at more than one lags jointly. The null hypothesis which is tested is the data are random against the alternative hypothesis of data are not random. The test statistics is written as: ∑ In the above equation, m denotes the sample size, is the autocorrelation at lag ‗i‘ and ‗k‘ stands for the tested number of lags. The null hypothesis is rejected if . Kolmogorov-Smirnov (KS) Test KS is a test of goodness of fit. Basically ‗Goodness of fit (GoF)‘ is employed to check whether a specific data series confirm the expected distribution or not. A GoF in fact examines that how well the specific distribution fit the data. A K-S test is a nonparametric test. It simply means that a test which is not relying on any explicit assumption regarding the distribution. It (KS) checks whether the data points of any series sensibly appear from any specific distribution and this specific distribution possibly be normal, uniform, Poisson etc. A KS test carefully stands on empirical cumulative distribution function (ECDF). Let there is sample of ‗m‘ points then ECDF can be written as

The K-S test statistics can be equation as: ( ) ( )

Here ‗F‘ denotes the theoretical cumulative distribution of what distribution is going to be checked with the assumption that it is continuous. The null hypothesis which is that desired data follows a specific distribution is tested. It is rejected if KS test statistics ‗D‘ is more than critical value. Run Test Run test is applied to test the serial dependency of a data series. The major purpose of this run test is to identify the non randomness. A run test is a nonparametric test. It simply means that a test which is not relying on any explicit assumption regarding the distribution. Run is the series
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of ascending values or descending values. The length of run is actually the total number of increasing or decreasing values. The probability of any arbitrary value ‗m‘ (Say), in a set of random data, is higher or smaller than ‗m-1‘th value follow binomial distribution is actually provide the foundation to the run test. Under the run test the null hypothesis is that the particular data series will flow in random manner. The test statistics for run test is given as:

In the above equation, ‗R‘ denotes the observed number of runs, ‗ ‘ denotes the expected number of runs in the data series and denotes the standard deviation of number of runs. The value of ‗R‘ and ‗ ‘ can be calculated as follow:

Where ‗a‘ and ‗b‘ shows the counting of positive and negative values of data set. For a relatively large sample, if the test statistics is larger than the │1.96│ (at 5% sig. level) then reject the null hypothesis of randomness. Unit Root Test A unit root test is applied to test the stationarity of the equity indices. As a necessary condition for random walk is presence of non stationarity in the equity indices. Proposition in the simple unit root test is that the value of autoregressive (AR) parameter is 1 in an AR model. Augmented Dickey Fuller (ADF) is considered a powerful tool for the testing the stationarity of the time series data especially for larger sample. The equation for ADF test is presented below: ∑ Here denotes the vector of deterministic term i.e. Constant, Constant and Linear Trend etc. q denotes the lags and is the desired time series to be tested. The null hypothesis which is is integrated to order zero hence . The test statistics for the ADF is simple t statistics for testing the . To describe ‗trend properties‘ of the series of data, the suitable selection of both the null and alternate hypothesis is decisive right before the application of unit root test. The appropriate form of the regression under the alternate is based upon the trend properties. This study will apply the common trend cases that are constant, constant and trend and finally none. Under the constant only, the regression equation is: The hypotheses here are: Null Hypothesis Alternate Hypothesis But for the case of constant and trend, the regression equation can be written as:
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The hypotheses here are: Null Hypothesis Alternate Hypothesis
Autoregressive (AR) Model If in the regression analysis, dependent variable is regressed against its past value then such type of model is called autoregressive model. Alternatively, in AR model the lag value of the explained variable is treated as an explanatory variable. When there is significant impact of dependent variable on independent variable in AR model then it proves the fact that future of any series can be predicted based upon its historical information. A general AR model with ‗j‘ lag can be written as: This particular study uses the following AR (2) model; In the above equation (A) , and represents the equity return of a particular equity index series at time ‗t‘, its first and second lag respectively, stands for intercept, represents the slope or sensitivity of specific explanatory variable toward regressed and represents the error term at time ‗t‘.

The Exponential GARCH model Among others, the assumption of homoskedasticity has great importance in the regression analysis. Practically, it is generally believed that for the equity prices there are period of high volatility which are followed by the periods of low volatility (volatility clustering). So it requires the use of such models which simultaneously deal with return as well as volatility. Engle (1982) come with the idea of ARCH model. Later on it generates discussion and labeled as ARCH family model. A major assumption behind the ARCH /GARCH model is that they are symmetrical. By symmetry we mean that both positive and negative shocks have same impact on the volatility of equity markets. In reality, it is generally believed that bad news has more impact on the volatility of equities than good news of the same magnitude. The EGARCH model was introduced by Nelson (1991) and its main target is to capture the asymmetries. The mean equation is: And the variance equation for EGARCH model is:

∑

√

∑

√

∑

The parameters are to be estimated. Symbol denotes the variance and therefore on the left hand side the value of is always non negative. As the major concern of EGARCH model is to capture the asymmetric effect and hence parameter of concern is . A model is symmetric when and when and significant then bad news has more impact on the volatility of equity indices than of good news. To capture the day of the week effect, this study uses the EGARCH model along with day dummies in both of the mean and variance equation. Moreover, Q-Q plot is used to check the asymmetries in the return series. Variance Ratio Test
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The predictability nature of equity markets has been generally a topic of great interest. One more well-liked way to probe this is Lo and MacKinlay (1988, 1989) variance ratio test. In their proposed methodology the predictability nature of equity returns are analyzed by evaluating the variances of differences of series over diverse intervals. In a series of data, if the variance of specific ‗p‘ (say) period difference is ‗p‘ times the variance of one period difference then that particular series will follow the unpredictable pattern. Let us assume be a finite series of equity prices at specific time then for random walk the variance of is actually the 1/m times the variance of . Assume this time series fulfilling the equation where represents any arbitrary drift parameter. The important is to test the and for Lo and MacKinlay (1988) come up with two different tests for the confirmation of the predictable patterns in the series of data. These are based upon the assumption regarding the . These are Heteroskedastic and homoskedastic random walk hypothesis. The estimator for the mean of 1st difference and scaled variance of p-th difference can be written as: ̂ ∑

̂ The variance ratio is:

∑

̂

⁄̂ ̂ The variance ratio Z-statistics suggested by Lo and MacKinlay (1988) is ⁄ ̂ which is asymptotically normally distributed with zero mean and variance equal to one. Under the heteroskedastic random walk hypothesis the value of ̂ is: ̂ In the last equation the value of ̂ is: ̂ ∑ ̂ ̂ ⁄ ∑ ̂ ∑ ̂

The variance ratio test is true for all and hence it can be applied for different value of p. Chow and Denning (1993) also suggested a way that analyzed the maximum absolute value of set of MVR statistics. Hence this study evaluates equity series in both perspective i.e. joint and individual test.

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Empirical Findings: The below is the line graph of the natural logarithm of the selected equity indices of the developing and developed countries. It is clear that up to some extent the equity markets of Pakistan and India showing the same pattern in the long run. On the other hand the equity markets of UK and USA are also moving parallel. Moreover, apparently both of these equity markets are also exhibiting the same patterns in the long run.

Figure 1: Trend in the equity indices

10.0 9.6 9.2 8.8 8.4 8.0 7.6 7.2 6.8 6.4 500 1000 1500 Ln BSA Ln KSE-100 2000 2500 3000 3500

Ln FTSE-100 Ln SP-500

Below table 1 uncovers the descriptive statistics of the daily return time series of selected equity markets of developing and developed countries. It is cleared from this table that the equity markets of developing countries (Pakistan and India) offer a higher daily return respectively 0.05% and 0.04% to their investor as compare to the equity markets of developed countries (UK and USA). But following the risk and return relationship, developing countries‘ equity markets are more risky than those of developed countries. Among developing countries, Pakistani equity market offer comparatively higher return having relatively low risk than Indian equity market. The value of skewness is not equal to zero showing that the equity returns are not normally distributed. Moreover, the return series of all the equity markets are negatively skewed. The value of kurtosis is also greater than three confirming that the return series are leptokurtic instead of mesokurtic. The results of Jarque-Bera test also reported in the table below. It can be used to comments upon the normality of the data. The P value of all the return series is even less than 0.01 confirming the rejection (at 1% significance level) of the null hypothesis which is equity market indices exhibiting the normal distribution. On the basis of skewness, Kurtosis and JarqueBera it is pretty visible that the equity markets of selected developing and developed countries
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are not normally distributed. Hence it can be concluded that return of indices of equity markets are not random. Table 1: Descriptive Statistics Pakistan India UK 0.0005 0.0004 0.0001 Mean 0.001 0.0011 0.0004 Median 0.1276 0.1599 0.0938 Maximum -0.1321 -0.1181 -0.0926 Minimum 0.0171 0.0174 0.0129 Std. Dev. Measure of Normality -0.351 -0.1059 -0.1049 Skewness 8.2834 8.2161 8.2818 Kurtosis 4110.869[0] Jarque-Bera[P] 4179.365[0] 4009.606[0]

USA 0.0001 0.0006 0.1096 -0.0947 0.0134 -0.1746 10.2663 7785.958[0]

Autocorrelation and Q-Statistics Table 2 provides the results of autocorrelation and Ljung-Box statistics up to 10 lag. As stated, when the markets are efficient then one cannot reject the null hypothesis. It is cleared that there exist significant pattern in the return series of the indices of all the equity markets under studied. As probability value is less than 0.01, one can reject the null hypothesis of randomness. The significance level is at 1%. As the value of autocorrelation coefficient are non zero suggesting that there exist relationship between today‘s and past return of the equity markets. Gupta and Basu (2007) also confirmed the presence of autocorrelation in the equity markets. Hence it is concluded that future value of return series of the equity indices can be forecasted based upon the past data which confirms the deviation of the equity markets from the weak form efficiency. Table 2: Autocorrelation and Q-Statistics Country 1 2 3 4 5 6 7 8 9 10 AC 0.088 0.02 0.031 0.05 0.033 -0.003 0.035 0.01 0.057 0.007 Pakistan Q-Stat 27.442 28.835 32.193 40.884 44.736 44.779 49.109 49.483 60.988 61.169 Prob. 0 0 0 0 0 0 0 0 0 0 AC 0.053 -0.062 0.006 0.04 0.003 -0.048 -0.015 0.011 0.041 0.053 Q-Stat 9.8902 23.502 23.613 29.258 29.301 37.378 38.126 38.532 44.556 54.352 India Prob. 0.002 0 0 0 0 0 0 0 0 0 AC -0.035 -0.053 -0.084 0.075 -0.052 -0.044 0.009 0.056 -0.002 -0.01 Q-Stat 4.313 14.059 38.894 58.735 68.471 75.427 75.736 86.762 86.774 87.117 UK Prob. 0.038 0.001 0 0 0 0 0 0 0 0 AC -0.075 -0.065 0.026 -0.006 -0.04 0 -0.038 0.017 0.001 0.025 Q-Stat 20.088 34.814 37.178 37.327 43.027 43.027 48.011 49.065 49.069 51.218 USA Prob. 0 0 0 0 0 0 0 0 0 0
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Kolmogorov-Smirnov (KS) Test Results of K-S test are presented at the below table 3. Here the null hypothesis of all the equity market is rejected at 5% significance level. Hence it can be carefully said that the return of daily equity market indices of all the countries don‘t follow the normal distribution. The result is identical when the test distribution is uniform. As for as Pakistani equity market is concern, our result are also consistent with Hassan,Abdullah and Shah (2007). Table 3: Kolmogorov-Smirnov Test* Pakistan India UK USA 0.0980 0.0583 0.0647 0.0721 Absolute 0.0694 0.0556 0.0625 0.0721 Positive -0.0980 -0.0583 -0.0647 -0.0714 Negative 5.8258 3.4666 3.8470 4.2871 Kolmogorov-Smirnov Z 0.0000 0.0000 0.0000 0.0000 Asymp. Sig. *Test distribution is Normal Run test The following table 4 presents the results of run test. Here if the value of test statistics is larger than the │1.96│ (at 5% sig. level) then reject the null hypothesis of randomness. As the ‗P‘ value for Pakistan, India and USA is less than 0.05 leads towards the rejection of the null hypothesis. It is concluded that there exists non-randomness in the return series of the equity indices of Pakistan, India and USA. The run test also revealed that the return of equity market of UK follow the random walk and it accepts the null hypothesis. These results also hold when someone applies the run test at N equal to zero. On the basis of run test, it is concluded that future value of return series of the equity indices (Except UK) can be forecasted based upon the past data. These results are also consistent with the study of Anand Pandeyy (2003) and Abraham et al. (2002).

Table 4: Run Test
Mean Return = N N Cases < N Cases ≥ N Total Cases Number of Runs Z Asymp. Sig. Pakistan 0.0005 1706 1825 3531 1644 -4.061 0* India 0.0004 1699 1832 3531 1666 -3.303 0.001* UK 0.0001 1723 1808 3531 1786 0.691 0.4894 USA 0.0001 1687 1844 3531 1878 3.879 0.0001*

*Significance level 1% Unit Root Test
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The below table 5 evident the results of unit root test. The stationarity of the equity markets indices are examined by employing the most desirable test for unit root i.e. ADF test. The ADF is applied by the inclusion of constant, constant and trend and without any term as regressor in the model. From the table it is clear that the ADF test statistics for Pakistan (-0.4479) is less than the critical value (Mackinnon) even at 1% significance level in absolute term which suggests that one can not reject the null hypothesis of a series has unit root. Hence it is concluded that Pakistani equity market has unit root and hence non stationary. The result remains hold as one assumes different ‗trend properties‘. The same pattern of Pakistani equity market is also followed by the rest of the selected markets. But the indices of all the markets are integrated at their first difference I (1). Table 5: Unit Root Test
ADF (Level) Constant Pakistan India UK USA 1% level 5% level 10% level -0.4479 -0.3439 -2.0442 -2.2046 -3.4320 -2.8622 -2.5671 Constant, Linear Trend -1.8699 -2.4036 -2.0449 -2.1976 -3.9606 -3.4111 -3.1274 None 0.8968 0.8669 -0.0288 0.1563 -2.5656 -1.9409 -1.6166 Constant -52.3030* -55.3771* -38.7102* -46.2549* -3.4320 -2.8622 -2.5671 ADF (First Difference) Constant, Linear Trend -52.3004* -55.3824* -38.7048* -46.2490* -3.9606 -3.4111 -3.1274 None -52.275* -55.357* -38.714* -46.257* -2.5656 -1.9409 -1.6166

Test critical values

*Significance level 1%
Autoregressive (AR) Model

The below table 6 presents the result of AR model up to two lags for all the four selected equity markets return. From this model, one can reject the weak form efficiency of a specific equity markets only if the regressed is being explained significantly from the set of explanatory variable. From the table below, F-statistics (for all the models) is significant at 1% significance level confirming the overall fitness of the model. In the context of Pakistan there exists a significant positive impact (with non zero coefficient) of previous period returns on the current returns. Hence it can be said that historical returns can be used to forecast the future equity returns for Pakistani markets. In the same footing, there exists reasonable evidence that future of all equity return series can be forecasted on the basis of its own past information. Therefore it is reasonable to document that equity return series are dependent (depending upon its lag), non random and leads toward the rejection of weak form efficiency for the entire data period.
Table 6: Autoregressive (AR) Model Country Variable Coefficient 0.0005 Pakistan 0.0865 t-Stat[Prob.] 1.62[0.11] 5.14[0]** F-stat[Prob.] 13.9[0]**

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India

UK

USA
**

0.0122 0.0004 0.056 -0.065 0.0001 -0.0368 -0.0538 0.0001 -0.0807 -0.0706

0.73[0.47] 1.38[0.17] 3.33[0]** -3.87[0]** 0.26[0.8] -2.19[0.03]* -3.2[0]** 0.49[0.62] -4.81[0]** -4.2[0]**

12.39[0]**

7.28[0]**

18.97[0]**

Significance level 1%

Day of the Week

The following table 6 shows the risk and return patterns of the selected equity markets throughout the week. It represents the mean return and standard deviation of each of the working days i.e. Monday, Tuesday, Wednesday, Thursday and Friday. Mean return are varying in all the equity markets as one moves from Monday to Friday. There exists also variation in the value of standard deviation which is a measure of risk. Pakistani equity markets offers a negative mean return (-0.05%) to their investor on Monday with a standard deviation of 2.06%. On the basis of this risk and return patterns, it can be revealed that there exists evidence of day of the week effect which further leads toward market inefficiency which are consistent with Solnik and Bousquet (1990) and Poshakwale (1996). It is also an indicator of the presence of anomalies especially calendar anomalies in the equity markets of the selected countries. To further analyze the day of the week effect and weak form efficiency, we also applied the more sophisticated tool called EGARCH model.

Table 7: Day of the Week Pattern
Day of the Week Mean Return Pakistan Standard Deviation Mean Return India Standard Deviation Mean Return UK Standard Deviation Mean Return USA Standard Deviation Monday Tuesday -0.0005 0.0004 0.0206 0.0163 0.0012 0.0001 0.0206 0.0156 0.0006 0.0002 0.0147 0.0123 0.0001 0.0003 0.0148 0.0140 Wednesday 0.0015 0.0172 0.0010 0.0160 -0.0007 0.0123 0.0002 0.0127 Thursday 0.0005 0.0151 0.0000 0.0158 -0.0002 0.0127 0.0001 0.0134 Friday 0.0009 0.0159 -0.0003 0.0180 0.0004 0.0126 -0.0002 0.0119

The EGARCH model The results of EGARCH (1, 1) are presented in table 8. From the result of Q-Q plot which are presented at appendix 1 clearly revealed that there exist asymmetries in the return of all the equity market indices. To capture these asymmetries in term of positive and negative news along
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with day dummies, we applied the EGARCH (1, 1) model. Generally, it is believed that investors are not only concern with returns but also with risk (standard deviation). Form the table one can find the impact of day of the week not on the return but also on the variance of return of equity indices. If coefficient is significant and negative then we can said that there exist asymmetries and bad news has more impact than good news of the same magnitude. Hence it is pretty visible that in these equity markets negative news has more impact than positive news. Result evident that all the day of the week has significant impact on the return of Pakistani equity market except Wednesday. Similarly there exist a reasonable day of the week effect in the return of selected equity markets. To comment upon the day of the week effect in variance of the equity markets this study also incorporate the day dummies as a variance regressor. These results are presented at lower half of the table. As on the left hand side of variance equation there is a measure of risk and all the day dummies are added on the right hand as explanatory variable. Hence if the coefficient of any day dummy is significant and positive (negative) then this particular day has an exciting effect (calm effect) on the volatility of equity market. For Pakistan day Monday and Friday have an exciting effect hence it significantly increases the volatility of equity return in contrast to Tuesday and Wednesday. The same can be interpreted for rest of the markets. Therefore on the basis of EGARCH model, it is concluded that there exist significant day of the week effect in all the equity markets. Moreover the day of the week has its dominancy in both i.e. return and risk of all equity markets which ultimately leads toward market inefficiency.
Table 8: EGARCH (1, 1) model to capture the Day of the Week effect Mean Equation Variable Pakistan India UK Lag(-1) 0.0671*** 0.0995* -0.0217 Monday -0.0022*** 0.0013** -0.0001 Tuesday -0.0013** -0.0004 -0.0010** Wednesday -0.0006 0.0003 -0.0009* Thursday -0.0013** 0.0003 -0.0007 Friday 0.0014*** -0.0003 0.0007* Variance Equation Variable Pakistan India UK -0.0650*** -0.11056*** -0.10583*** Monday 0.4627*** 0.313565*** 0.13045 Tuesday -0.1851*** -0.35106*** 0.040993 Wednesday -0.2090*** 0.122405* 0.119244 Thursday -0.0147 -0.02972 0.133999 Friday 0.0994** 0.06747 -0.07985
Note: the significance level at 10%, 5%, 1% are denoted by *, **, *** respectively

USA -0.0397** 0.0009* -0.0001 0.0009** 0.0005 -0.0005 USA -0.1194*** 0.0699 0.1967*** -0.1451* 0.0718 -0.0477

Variance Ratio Test

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The predictability nature of equity markets has been tested by applying one more well-liked way i.e. Lo and MacKinlay (1988, 1989) variance ratio test. The results of variance ratio test along with the assumption of Heteroskedasticity are given in the table 8. ‗Joint test‘ tests the joint null hypothesis for all periods and its result is given table 8(a). On the other side ‗Individual test‘ is the variance ratio test examined to individual period and results are given table 8(b). From the table 8(b) the Chow-Denning Max │Z│value for Pakistan, India, UK, and USA are respectively 3.7676, 1.9723, 2.7164, and 3.1723. For Pakistan this maximum value linked from individual test at period 16 and p-value which is attained with studentized maximum modulus (SMM) is 0.0025. Hence on the basis of this one can strongly reject the null hypothesis and concluded that Pakistani equity market don‘t follow the random walk. Similarly we can also reject the null hypothesis of random walk for USA (at 5% significance level), UK (at 10% significance level) but one can‘t reject this null for Indian market. These results are also confirmed by the individual test which is given at table 8(b). Hence variance ratio test concludes that there exists random walk only in the Indian market while there are evidence to reject the weak form efficiency for Pakistan, UK and USA. Hence future movements can be predicated based upon the past data.

Table 8 (a): Variance Ratio Test (Joint Test)
Country Pakistan India UK USA Joint Tests Value 3.7676 1.9723 2.7164 3.1723 df 3531 3531 3531 3531 Probability 0.0025 0.5262 0.0945 0.0224

Max │Z│ Max │Z│ Max │Z│ Max │Z│

Table 8 (b): Variance Ratio Test (Individual Test)
Country Pakistan Period Var. Ratio Z-Statistic Prob. Var. Ratio Z-Statistic Prob. Var. Ratio Z-Statistic Prob. Var. Ratio Z-Statistic Prob. 2 1.0804 2.8077 0.0050 1.0509 1.9723* 0.0486 0.9650 -1.2458 0.2128 0.9243 -2.6018 0.0093 3 1.1180 2.6197 0.0088 1.0262 0.6881 0.4914 0.9185 -1.9119 0.0559 0.8557 -3.1723* 0.0015 4 1.1507 2.6178 0.0089 1.0198 0.4213 0.6735 0.8525 -2.7164* 0.0066 0.8358 -2.8292 0.0047 8 1.2881 3.1740 0.0015 1.0302 0.4185 0.6756 0.7710 -2.6029 0.0092 0.7595 -2.5673 0.0102 12 1.3970 3.5400 0.0004 1.0572 0.6384 0.5232 0.7579 -2.1778 0.0294 0.7243 -2.2997 0.0215 16 1.4851 3.7676* 0.0002 1.0880 0.8488 0.3960 0.7441 -1.9743 0.0484 0.7283 -1.9236 0.0544 16

India

UK

USA

* Max │Z│ Conclusion According to Fama (1970), markets are labeled as efficient in the context of information only if the prices ‗fully‘ reflect all the available information. The world ‗fully‘ is challenging perquisite for above definition resulting in the favor that there exists no real market which is efficient (Sewell, 2011). But from the existing literature there is mixed level of empirical viewpoint regarding the EMH. Especially in the context of weak form of efficiency there is great debate. The major objective of this study is to test the weak form efficiency along with calendar anomalies in equity market of developing and developed countries. For this we have selected the Pakistan and India as developing countries whereas USA and UK as developed markets. The line graph of selected equity indices of the developing and developed countries cleared that up to some extent the equity markets of developing countries showing the same pattern in the long run. On the other hand the equity markets of developed countries are also moving parallel. The equity markets of developing countries (Pakistan and India) offer a higher daily return to their investor as compare to the equity markets of developed countries (UK and USA). On the basis of skewness, Kurtosis and Jarque-Bera it was pretty visible that the equity markets of selected developing and developed countries were not normally distributed. Hence it can be concluded that return of indices of equity markets were not random. Autocorrelation and LjungBox statistics for all the selected markets evident that future value of return series of the equity indices could be forecasted based upon the past data and was consistent with Gupta and Basu (2007). Results of K-S test carefully uncover that return of daily equity market indices of all the countries don‘t follow the normal distribution which were consistent with Hassan, Abdullah and Shah (2007). On account of run test, it was concluded that future value of return series of the equity indices (Except UK) can be forecasted based upon the past data. Moreover, all the indices of equity market have unit root and hence non stationary. Autoregressive model suggested that there exists reasonable evidence that future of all equity return series could be forecasted on the basis of its own past information. Therefore it was reasonable to document that equity return series were dependent (depending upon its lag), non random and leads toward the rejection of weak form efficiency for the entire data period. On the basis of EGARCH model, it was concluded that there exist significant day of the week effect in all the equity markets. Moreover the day of the week has its dominancy in both i.e. return and risk of all equity markets which ultimately leads toward market inefficiency. Variance ratio test concluded that there exists random walk only in the Indian market while there are evidence to reject the weak form efficiency for Pakistan, UK and USA. Hence future movements could be predicated based upon the past data. Surprisingly but based upon empirical evidence this study concludes that there is no remarkable difference on the basis of market efficiency in the markets of both the developed and developing countries. It is generally believed that equity markets of developed countries are supposed to be more efficient than that of equity markets of developing countries. But based upon a range of different econometrics test it is inferred that both of developed and developing equity markets
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are weak form inefficient. Hence one can reject the random walk hypothesis and therefore presence of markets efficiency is again a matter of theory not as much practical. So investor should be vigilant regarding the use of past information for meaningful and better future predictions.

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References Ariel, Robert A. (1990). ―High Stock Returns Before Holidays: Existence and Evidence on Possible Causes.‖ Journal of Finance. December, 45:5, pp. 1611–626. Basu, Sanjoy. (1983). ―The Relationship Between Earnings‘ Yield, Market Value and the Returns for NYSE Common Stocks: Further Evidence.‖ Journal of Financial Economics. June, 12:1, pp. 129–56. Bernstein, P. L. (1999), A new look at the efficient market hypothesis, The Journal of Portfolio Management 25(2), 1– 2. Chow. K.V., and K. Denning, (1993), ―A simple multiple variance ratio test‖, Journal of Econometrics,58, 385-401. Cootner, Paul, ed. (1964). The Random Character of Stock Market Prices. Cambridge, Mass.: MIT Press. Cowles, 3rd, A. (1933), Can stock market forecasters forecast?, Econometrica 1(3), 309–324. Dickey, D. A., and W.A. Fuller, (1981), ―Likelihood Ratio Statistics for Autoregressive Time Series with Unit Root‖, Econometrica, 49, pp. 1057-1072. Errunza, N. and E. Losq, (1985), ―The Behavior of Stock Prices on LDC Markets‖ Journal of Banking and Finance, p. 561 – 575. Fama, E., (1970), ―Efficient Capital Markets: A Review of Theory and Empirical Work‖ Journal of Finance, 25, p. 383 – 417. Fama, E., (1991), ―Efficient Capital Markets: II. ―Journal of Finance, 46, p. 1575 – 1617.

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Fama, Eugene. (1998). ―Market Efficiency, Long-Term Returns, and Behavioral Finance.‖ Journal of Financial Economics. 49:3, pp. 283–306. Fama, Eugene and Kenneth French. (1988). ―Permanent and Temporary Components of Stock Prices.‖ Journal of Political Economy. 96:2, pp. 246–73. Hassan, Shah and Abdullah, (2007), ―Testing of Random Walks and Efficiency in an Emerging Market‖ The Business Review Cambridge, Volume 9, Nov 1 Kendall, M. G. (1953), The analysis of economic time-series—Part I: Prices, Journal of the Royal Statistical Society. Series A (General) 116(1), 11–25. Lakonishok, Josef and S. Smidt. (1988). ―Are Seasonal Anomalies Real? A Ninety-Year Perspective.‖ Review of Financial Studies. Winter, 1:4, pp. 403–25.

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Lo, A. and C. Mackinlay, (1988), ―Stock Market Do Not follow Random Walks: Evidence From a Simple Specification Test‖ Review of Financial Studies,1, p. 41 – 66. Malkiel, B. G. (2003), The efficient market hypothesis and its critics, The Journal of Economic Perspectives 17(1), 59–82. Samuelson, Paul. (1965). ―Proof that Properly Anticipated Prices Fluctuate Randomly.‖Industrial Management Review. Spring, 6, pp. 41– 49. Shleifer, Andrei. (2000). Inefficient Markets: An Introduction to Behavioral Finance. New York: Oxford University Press. Working, H. (1934), A random-difference series for use in the analysis of time series, Journal of the American Statistical Association 29(185), 11–24.

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Appendix 1:

Quantile-Quantile Plot

INDIA
.08 .08

Pakistan

Quantiles of Normal

.00

Quantiles of Normal
-.2 -.1 .0 .1 .2

.04

.04

.00

-.04

-.04

-.08

-.08 -.15

-.10

-.05

.00

.05

.10

.15

Quantiles of INDIA

Quantiles of PAKISTAN

UK
.06 .04 .06 .04

USA

Quantiles of Normal

.02 .00 -.02 -.04 -.06 -.10

Quantiles of Normal
-.05 .00 .05 .10

.02 .00 -.02 -.04 -.06 -.10

-.05

.00

.05

.10

.15

Quantiles of UK

Quantiles of USA

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