The Efficient Market Hypothesis often called Random Walk Theory is one of the mist considerable discussed and debated notions in financial market theory. While a random walk does not imply that a market cannot be exploited by insider traders. It does imply that excess returns are not obtainable through the use of information contained in the past movements of prices (Borges 2008).

In the current research three versions of the Efficient Market Hypothesis exists; weak-from, semi-strong and strong form of market efficiency. The weak form suggests that the information used to determine the prices include only past prices or returns. Hence, past rates have no effect on future prices. The semi-strong form follows the belief that for the calculation of current stock prices all public information are considered. Therefore, only market participants with additional private or insider information can outperform the general market. However, the strong-form of market efficiency include this information. Thus, both publicly and any not publicly available information are accounted for in the asset price. In that case, no information can give an investor a superior result.

The Efficient Market Hypothesis is usually applied to the return on stocks as follows:

R_(t+1)= R_(t+1)^e+ ?_(t+1)

The random walk hypothesis (RWH) state that stock prices following a random walk meaning that they are independent from price change and developments in the past. Thus, prices always fully reflect information available and no profit can be made based on information trading (Lo and MacKinley, 1999). The Random Walk exists in three versions:

RW1: ?_t is independent and identically distributed (iid) (0,?^2)

RW2: ?_t is independent (allows for heteroscedasticity)

RW3: ?_t is uncorrelated (allows for dependence in higher moments)

In this paper, I examine the DAX and Russell2000. I maintain the hypothesis that the Russell2000 consisting of 2000 small-cap U.S. companies reveals less predictability according to the RWH. On the contrary, I assume that the DAX reveals higher predictability because it only consists of the 30 largest German publicly-listed companies.

Literature Review

Since Fama (1970), several financial economists and statisticians including Fama and French (1988) and Lo and MacKinley (1988), among others, researched that market theory. The random walk theory is not entirely focussing on equity markets but also foreign exchange market. Ajavi and Karemera (1996) examine the predictability of daily and weekly industrial currency dynamics with the variance ratio test introduced by Cochrane (1988). The results show that the Random Walk Hypothesis is not consistent with the dynamics in the majority of these currencies. Furthermore, Borges (2008) looks into the RWH of global equity markets such as Latin America, Africa and Asia. Several markets reporting inconsistency for the RWH in equity markets. In more detail, Huang (1995) tests whether the RWH in developed and emerging Asian equity markets holds. Both market types reveal rejection and non-rejections. Thereby, indicating that neither developed nor emerging Asian equity markets are generally prone to follow a random walk. Similar results are also confirmed by Worthington et al. (2004) in 16 European equity markets that emerging markets are unlikely to be associated with random walks – required for the assumption of weak-form market efficiency. However, Hungary is the only emerging market that does satisfy this criterion.

Data sample

The data used in this report consists of daily closing indices prices from 01. January 1997 to 31. December 2017 for the German DAX and U.S. Russell 2000 Index. The data was collected from Bloomberg Data Service. The DAX is the major German index consisting of the 30 largest companies based in Germany. The DAX is a value weighted index. The Russell 2000 Index is a U.S. small-cap index consisting of 2000 firms. I generate from the daily values weekly and monthly log-values and log-returns and separate the whole sample into two subsamples. The first subsample ranges from 01. January 1997 to 31. December 2007 and the second subsample from 01. January 2008 to December 2017.

Table 1: Descriptive statistics

Methodology

To test for the random walk in equity index time series data, I use the procedure introduced by Lo and MacKinlay (1988). They test the Random Walk Hypothesis by examining the null-hypothesis that the variance ratio is as follow:

VR(n)= 1/n (Var(S_t-S_(t-n)))/(Var(S_t-S_(t-1)) )=1

If the ratio is below unity it reveals negative serial correlations whereas above unity reveals positive serial correlation. The initial intuition of that methodology is to compare the variance of different subsets of time series data in early period with variances in late periods.

Secondly, I will use the autocorrelation function (ACF). The existence from serial correlation which is examined by rhw ACF function proves in inefficiency among stock prices and thereby testing the weak form of market efficiency. Between a time-series from t to s the ACF function is defined as:

?(s)= (?(s))/(?(0))= (E(X_(t+s)- ?)(X_t- ?))/(E(X_t- ?)^2 )

Additionally, I use a third test the Ljung-Box Q-test to test on serial correlation which provides a better fit to chi-square distribution:

Q_LB=n(n+2) ?_(t=1)^m?(? ?^2 (t))/(n-t)

My fourth test is the augmented Dickey-Fuller (ADF) test which is the common unit-root test besides the Philips Perron test. The ADF test is in particular suitable for a larger and more complicated time-series data set. It tests the existence of a unit root in a particular time series of stock returns. The test statistic results into a negative number which indicates the more negative the results becomes the stronger the rejection of the null hypothesis. The ADF test is defined as follows:

?P_t= ?_0+ ?_1 t+ ?_0 P_(t-1)+?_(i=1)^q???_i ?P_(it-i)+?_it ?

My fifth test is the Chow Denning test. Even though the Lo and MacKinley test statistic is a powerful test for homoscedasticity and heteroscedasticity nulls, it does not replace a multiple variance ratio approach first introduced by Hochberg (1974) because it ignores the joint nature of testing for the RWH. The multiple variance ratio test allows individual variance ratios while covering the entire sample rather than certain subsample. To test the joint null hypothesis, the Chow-Denning (1993) test statistic is defined as: