Testing financial market efficiency

Since the birth of the financial literature until the 1970s, the efficient market hypothesis has been regarded as a central hypothesis. In the mid-1970s, there were theoretical and empirical evidence stating that the EMH seems untouchable. However, recently there has been an emergence of arguments doubting the EMH. The EMH implicitly indicates that stock prices can follow a random walk. Currently, financial theory has shown that stock prices do not follow a random walk. In this regard, our empirical study rejected the hypothesis of a random walk for 27 indices out of 28 studied. We confirm that the studied indices time series do not follow a random walk, and therefore we reject the financial markets efficiency hypothesis in its weak form. This result corroborates those of Fama and French (1992.993), DeBondt and Thaler (1985), Lo and MacKinlay (1991), Jagadeesh and Titman (1993) and Shleifer and Vishny (1997). Therefore, financial markets efficiency hypothesis in its weak form is also rejected. This result is logical given the limited capacity of the classical theory in explaining abnormal returns such as bubbles, crashes and excess volatility. of returns time ten following returns between (JKSE). between and


INTRODUCTION
predicts that these series are stationary (I (0)) assuming that the dynamic aspect of data has an ARMA structure. The ADF test is based on the estimation of the following regression: Where is a vector of deterministic terms (constant and slope). The lagged difference terms p Δ − are used to approximate the ARMA structure of errors and the p-value is configured such that errors are uncorrelated in a serial manner. The error term is assumed to be homoscedastic. The specification of the deterministic terms depends on the supposed behaviour of under the alternative hypothesis of stationarity of the trend. Under the null hypothesis, is I (1) which implies θ = 1. The t-statistic of the ADF and the standardized bias statistics are based on the least squares estimators of the regression equation above, given by: An alternative formulation of the regression of the ADF test is as follows: Where = − 1. Under the null hypothesis, Δ is I (0) which implies that = 0. The t-statistic of ADF is then the usual tstatistic to test = 0 and the standardized biased statistics of ADF is An important practical issue of implementing the ADF test is to specify lag length p . If p is very low, then the remaining serial correlation in the errors will bias the test. If p is very large, then the test power will suffer. Ng and Perron (1993 ) have suggested the following procedure for selecting the data -dependent lag length which results in stable sizes of the test with a minimum power loss. First, we determine an upper limit of p. Second, we estimate the regression of the ADF test with p = . If the absolute value of the t-statistic for testing the significance of the last lagged difference is greater than 1.6, then we set p = and we will run the unit root test. Otherwise, we will reduce lag length by one unit and we repeat the procedure.

B-Phillips-Perron Unit Root Tests:
Phillips-Perron (1988) developed a number of unit root tests that have become popular in financial time series analysis.
Phillips-Perron (PP) unit root tests primarily differ from those of ADF in how to deal with errors serial correlation and heteroskedasticity. PP tests regression is given by: Where, is I (0) and may be heteroscedastic. PP tests correct any errors serial correlation and heteroscedasticity using an OLS estimation and modifying test statistics =0 and .. These modified statistics denoted are given by: Since we used k lags in auto-covariances, the Newey-West estimator can be used to produce consistent estimates of variance parameters, Où, The estimated values of λ and its standard errors have been obtained from OLS of equation (5). Sample variance of the least squares residual û is a consistent estimator of σ ² and Newey-West estimator of long-term variance of u using û is a consistent estimator of ω ².
Under the null hypothesis which states that λ = 0, the statistics of PP test have the same asymptotic distribution as the ADF t-statistic and the standardized biased statistics. A comparative advantage of PP tests on ADF tests is that PP tests are robust to heteroskedasticity general forms in error terms ut . Another advantage is that the researcher is not forced to specify a lag length for the test regression.

C-Stationarity Tests:
More recently, DeJong et al (1992) and Diebold and Rudebusch (1991) found poor evidence against the standard ADF unit root and PP tests when the data exhibit a stable auto-regressive tendency with roots close to unit or when data are fractionally integrated. To circumvent this poor weak evidence, we will include in addition to unit root tests the stationarity test which checks the null hypothesis against the alternative of non-stationarity.
is I (0) and its initial value 0 is considered fixed and plays the same role of the constant term of the regression equation. Note that is a pure random walk with an innovation variance 2 .
The null hypothesis is that has a stationary trend formulated as follows: Implying that is constant.
KPSS test statistic is the Lagrange Multiplier test (LM) to check 2 = 0 against the alternative 2 > 0 and it is given by calculating the partial sum of the residuals ( ) generated in the regression, by fixing the constant and the time slope each time. Let 2 the error variannce estimator and the partial sum of residuals. We calculate the LM statistic as follows: is an asymptotically consistent estimator of 2 and is estimated as follows: Where ( , ) is an optional lag window. Kwiatkowski, Phillips, Schmidt and Shin (1992) used Bartlett window ( , = 1 − 1+ ) and showed that the test statistic in equation (10) has an asymptotic distribution equal to a Brownian Bridge function for the degree and trend of stationarity. For degree of stationarity, the distribution of equation (10) is shown as follows: is a Wiener process (Brownian movement). It should be noted that while testing stationarity of residuals in equation (10), we calculate residuals using the following subtraction: = − . For stationarity of the trend, the asymptotic distribution is given by:

D-The Variance Ratio Test:
To expose some elements of the theory of variance ratio test, let a stochastic process that satisfies the following recurrence relation: Where Where, deviation μ is an arbitrary parameter. The random walk hypothesis posits the restriction that errors are uncorrelated or that innovations are unpredictable from past innovations. Lo and MacKinlay (1988) developed the random walk test under two null hypotheses: the Gaussian increments are i.i.d and in general increments are uncorrelated but weakly dependent and possibly heteroscedastic.

D-1 The Null Hypothesis of Gaussian i.i.d:
Let the null hypothesis which denotes the case where innovations are normally, randomly and identically distributed variables with variance ² and we assume that we have nq+1 observations ( 0 , 1 , … , ) where n and q are integers greater than the unit. Consider the following estimators of the unknown parameters and ² : The estimator is simply the sample variance of the first difference . Consider the variance of the qth differences of , which is under the null hypothesis 1 is q times the variance of the first differences. Dividing by q, we obtain the estimator 2 ( ) which also converges to 2 under 1 where: The estimator 2 ( ) is written as a function of q to highlight the fact that the distinct alternative estimator of 2 can be formed for each q. Under the null hypothesis of the Gaussian random walk et 2 ( ) should be almost equal. However, the random walk test is performed by calculating the difference = 2 − 2 and checking its proximity to zero.
Alternatively, a test may also be based on the = 2 2 − 1 ratio which converges to zero probability. Lo and Mackinlay (1988) showed that has the following limit distribution under the null hypothesis 1 :

D-2 The Heteroscedastic Null Hypothesis:
Under the conditions that enable a variety of heteroscedasticity forms by including ARCH processes, Lo and Mackinlay (1988) showed the limit distribution of variance ratio as an approximate linear combination of autocorrelation where: And ( ) are estimators consistent with the heteroskedasticity of the asymptotic variance of autocorrelation ∆ defined as, The test of the null hypothesis of heteroscedasticity (equation (21)) under the standardized variance ratio 2 may be defined as follows: Also, the null hypothesis of homoscedasticity (equation (21)) under the standardized variance ratio may be specified as follows:

2-1 Presentation of data
We will consider 28 market indices across three main regions: the Americas, Europe and Pacific Asia. The following table shows the different indices by region: We will run different tests on indices returns. The data frequency is daily and all time series are extracted from the Yahoo website! Finance.

2-2 The Hypotheses:
Our empirical validation aims at testing the following hypotheses: • Hypothesis 1: Market indices returns follow a random walk, • Hypothesis 2: Markets do not follow a random walk.
In what follows, we will, first, describe of the characteristics of our data, and second, we will perform market efficiency tests to, finally, accept or reject our hypotheses.

3-1 Time Series Descriptive Statistics:
A

. Descriptive Statistics of The Americas Time Series:
The table below reports the descriptive statistics of market indices time series of the American region:  Kurtosis values are all greater than 3 and, therefore, all are leptokurtic distributions.
Descriptive statistics of Asia and the Pacific time series: The  The statistics of returns time series of the ten market indices in the Asia and the Pacific region leads to the following observations. Mean returns range between -0.000582 Descriptive statistics of the time series in Europe: The table below shows the descriptive statistics of the time series of the European market:

A. The unit root test A-1 The American region :
The table below reports the two unit root tests, the ADF and PP, for the American market indices. The ADF and PP statistics reported in Table (5) have absolute values greater than the critical values (at the 1%, 5% and 10% levels). This implies that the two tests reject the null hypothesis of unit root and then market indices time series do not follow a random walk.

A-2 The Asia and Pacific Region:
The table below reports the results of the two tests:

A-3 Europe:
The table below reports the statistics of the two unit root tests for the European indices:   Table 9 reports KPSS test results for the Asian and Pacific time series.

B-3-Europe :
KPSS test results for the European time series are reported in the following table:

C-1 American Region :
The

C-2 Asia and The Pacific:
The

C-3 Europe :
The

D.
Variance Ratio Test:

D-1 The American Region :
The variance ratio test is the most important phase of this empirical validation. It allows for directly testing the null hypothesis of a random walk.      The null hypothesis of a random walk is rejected for all time series of the different European indices and for all lags because the probabilities are all inferior to 1%. Therefore, the European indices do not follow a random walk.

D-3 Europe:
Given the weakness in the unit root tests robustness, we run, in addition to ADF and PP unit root tests which test the hypothesis of random walk, the KPSS stationarity test which also tests the hypothesis of random walk. We found that the null hypothesis of unit root is rejected for all the considered 28 indices, while the KPSS test was not conclusive because it gave different results for the indices.
The null hypothesis of stationarity has been accepted for 18 indices and rejected for 10. However, checking stationarity of time series confirms the presence of predictable components and rejects the hypothesis of random walk. Among other things, the 10 indices for which we could not accept the hypothesis of stationarity do not necessarily follow a random walk.
the variance ratio test of Lo and Mackinlay (1988) was successful and conclusive: All time series of the studied 28 indices do not follow a random walk.
Moreover, we performed Breusch-Pagan/Cooke-Weisberg heteroscedasticity test to examine errors evolution. We concluded that the null hypothesis of errors homoscedasticity is rejected for 27 indices and accepted for one. This finding confirms that errors are independent variables, and consequently, there are predictable components in errors. Hence, we confirm that the 27 indices do not follow a random walk consistent with Blaga (2012) and Aga and Kocaman (2011).
Our empirical study rejects the hypothesis of random walk for all the studied indices. This rejection implies that successive price changes can be predicted from historical values. The main causes behind rejecting a random walk can be mainly lack of transparent and asymmetrical information.
Against these results, we reject our first hypothesis and accept the second. We confirm that the studied indices time series do not follow a random walk, and therefore we reject the hypothesis of financial markets efficiency in its weak form. This