The empirical research for market efficiency investigates if there is past available information which can help to predict future returns profitably.

University of Sunderland in London

PGBM144 – International Financial Markets and Econometrics

An Assignment Support Note on Efficiency of Financial Markets

Recapitulating Some Basics:

ü  The empirical research for market efficiency investigates if there is past available information which can help to predict future returns profitably.

ü  We have to employ statistical and econometrics methods to test the independence of prices data and see whether the stock prices are predictable or not by exploring serial dependence of stock returns.

ü  Efficient Market Hypothesis (EMH) implies that the future price of a stock is unpredictable with respect to currently available information.

ü  EMH assumes that share price adjust rapidly for any new information consequently, the current prices fully reflect all available information’s and should follow a random walk process, which means sequential stock price changes (returns) are independently and identically distributed (IID).

ü  Weak form of efficient market hypotheses assumes that the current stock prices fully reflect all historical market information such as: (historical sequence of prices, trading volumes, and any market generated information).

ü  Weak form efficiency is the very basic form of efficiency. The information set includes only the history of prices or returns themselves.

ü  A capital market is said to satisfy weak-form efficiency if it fully incorporates the information in past stock prices.

ü  The weak form of market efficiency popularly known as random walk theory of stock market in which prices reflect all information found in the record of past prices and volumes.

ü  A review of theory and empirical work” stated that a market is said to be efficient if not even a single investor has competitive advantage as far as right to use to information and deployment of that information is concerned.

ü  The weak form of EMH is the lowest form of efficiency that defines a market as being efficient if current prices fully reflect all information contained in past prices.

ü  This form implies that past prices cannot be used as a predictive tool for future stock price movements.

ü  Therefore, it is not possible for a trader to make abnormal returns by using only the past history of prices.

ü  You can test the weak-form efficient market hypothesis of a stock market by hypothesising normal distribution and random walk of the return series.

ü  Semi strong-form efficient market hypotheses, assumes that the current stock prices reflect not only the historical information but also new publicly available information such as: dividend announcements, and economic and political news.

ü  Strong-form efficient market hypotheses assume that stock prices reflects all information from both public and private sources, so that no one investor can reap abnormal rate of return.

The Random Walk Theory:

ü  The Random Walk Model (RWM) is the model which assumes that subsequent price changes are sovereign and homogeneously distributed random variables and changes in future prices cannot be forecasted through historical price changes and movements.

ü  The Random Walk Model is generally used to testify the weak-form Efficient Market Hypothesis. Parametric and non-parametric methods can be applied to test the random walk hypothesis (RWH).

ü  According to the random walk theory we are unable to predict the future stock price by analysing historical information.

ü  Abnormal return is generally not possible to achieve on a continuous basis.

ü  Therefore, technical analysis does not work in that particular scenario.

ü  You can test the weak-form efficient market hypothesis of a stock market by hypothesising normal distribution and random walk of the return series.

Your focus:

The specific objectives of the study are:

ü  To study the randomness of stock market.

ü  To test the weak form of efficiency in stock market.

ü  To test whether the equity markets are weak form efficient.

ü  To test the weak form efficiency of the different sectoral indices of LSE. 

Empirical Tests of Financial Market Efficiency:

Commonly, the following analytical tools appropriate to test market efficiency, (which are all powerful tools):

a) Run test

b) Unit root (Stationary test) -ADF, PP and KPSS.

c) Durbin-Watson test (Autocorrelation/Serial correlation test)

d) Jarque-Bera test (Normality test)

e) Kolmogorov-Smirnov test

f) Variance ratio test (Heteroscedasticity test)

g) Ljung-Box Q test (Autocorrelation/Serial correlation test)

h) Autoregressive Conditional Heteroscedasticity Lagrange multiplier (ARCH-LM) Test

i) Autocorrelation function (ACF) test

Please get your stock market data ready in an Excel file.

What Software?

ü  Eviews or Excel (NumXL) are my suggested/recommended software.

ü  The two packages are very user-friendly as you can also plot your graphs with the software packages.

Unit Root Test (ADF and PP):

✓ Unit root tests are among widely statistical tests used to examine the randomness of the return series.

✓ Basically, the test is done to investigate the presence of a unit root, that is, non-stationary of the return series.

✓ Unit root test can be applied for testing the efficiency of markets.

✓ An efficient market needs to be in the random form because market efficiency demands randomness in the daily returns of commodities and unit root test investigates whether the time series is non-stationary or not.

✓ As the efficient market hypothesis propounds that release of new information are instantaneously reflected in the stock prices, so with these information future stock prices cannot be predicted.

✓ As stock prices cannot be predicted according to efficient market hypothesis so the daily return series should be non-stationary.

✓ As a random walk has a unit root, so, if the variable in question follows a random walk, it is therefore not stationary. Therefore testing to determine stationarity of the variable, is said to be testing for a ‘unit root’.

✓ If the test statistic is more negative or smaller than the critical value (Mackinnon tabulated value) then the null hypotheses will be rejected which means data is not non-stationary.

✓ There are various types of unit root tests, only two types of unit root tests namely: (i) the Augmented Dickey-Fuller (ADF) and (ii) the Phillips-Perron Test (PP) are usually employed to investigate the randomness behaviour of the stock market return series.

✓ Efficiency in the weak-form requires that the return series are non-stationary at level, hence the use of ADF unit root test to know if the stock market all share index for the period is stationary at level or not.

✓ The series containing unit root is said to be non-stationary, that is, behaving in random fashion which supports the weak form efficiency hypothesis.

✓ Market efficiency demands randomness (i.e. non stationary) in behaviours of price of the security.

✓ So, if the test results show the return series are stationary then the market cannot be regarded as efficient.

Random walk of a time series requires the time series to contain a unit root. No unit root implies stationary and returns/prices not random.

✓ If stock market return series of does not contain unit root, absence of unit root point towards the stationarity in the data, hence predictability is possible in calculating future returns. So, price/returns do not follow random walk.

✓ If a market is non-stationary, then it is unpredictable or cannot be forecasted. In a stationary situation, a market can be predicated beforehand and there is a possibility of achieving arbitrage profit.

✓ The daily returns have no unit root, thus implying that the null hypothesis was rejected, meaning

that index’s daily return series was stationary, i.e., not random.

✓ The non-stationarity of data at level series indicates that the behaviour of stock market all share indexes conforms to and is consistent with the weak form efficiency of the market, which states that financial time series behave as random walks.

✓ Both ADF and PP tests use the following null and alternative hypotheses in unit root tests:

𝐻0 = The series does contain a unit root (non-stationary) – there is random walk (returns/prices are random).

𝐻1 = The series does not contain a unit root (stationary) - there is no random walk (returns/prices are not random).

✓ According to the ADF test, null hypothesis assume that daily return series has a unit root. A series having unit root means it is non-stationary in nature.

✓ If the return series are stationary, the series can be modelled and hence predictions of future movements are possible, implying that the stock market is not efficient in the weak form and security prices do not reflect all past information and it is possible to earn super-normal gain by utilising past information as share prices do not adjust instantaneously in response to any new information release in the market.

Notes on Interpreting the Unit Root Test:

✓ For daily return series, if calculated ADF test statistic negatively go above from the MacKinnon tabulated value and p-value is also smaller than alpha (i.e. 0.05), it leads to the rejection of null hypothesis.

✓ If the ADF test statistic for negatively exceeds the MacKinnon tabulated value and the p-value is well below 0.05 (5% significance level), reject null hypothesis that daily return series contains unit root, implying that the return series are stationary and does not exhibit randomness in nature.

✓ Analysing the data from a period, with almost zero probabilities indicates that, it rejects the null hypothesis of random walk.

✓ Watch the p-value. In general, a p-value of less than 5% means you can reject the null hypothesis that there is a unit root.

If t* > ADF crtitical value, ==> accept null hypothesis, i.e., unit root exists.. mean data is non stationary If t* < ADF critical value, ==> reject null hypothesis, i.e., unit root does not exist. mean data is stationary

Variance Ratio Test - Heteroskedasticity assumption:

✓ Variance ratio test has been employed to examine the predictability of asset returns proposed by Lo and Mackinlay (1988).

✓ The single variance ratio test, proposed by Lo and Mackinlay (1988), demonstrates that the variance ratio test.

✓ The test uses the fact that if a series of stock prices follows a random walk, then the increments are said to be serially uncorrelated and that the variance of those increments should increase linearly in the sampling intervals.

✓ The test is based on the assumption that the variance of increments in the random walk series is linear in the sample interval.

✓ Specifically, if a series follows a random walk process, the variance of its q-differences would be q times the variance of its first differences.

✓ According to this test, variance of difference of time series has compared over dissimilar intervals.

✓ In a random walk of a time series, variance of p periods must be p times the variance of single period difference.

✓ Variance ratio test statistics are used to examine the random walk behaviour of time series under homoskedastic and hetroskedastic assumption with the help of asymptotic distributional.

✓ The variance ratio test assesses the null hypothesis that a univariate time series is a random process.

The variance ratio test hypothesis are stated thus:

H0: Data series follows a random walk.

H1: Data series does NOT follow a random walk.

Notes on the Interpretations of Variance Ratio Tests - Heteroskedasticity assumption:

✓ Watch the p-value.

✓ If the P-value for the joint test is below alpha (0.05) and therefore the test is statistically significant at 5% , which suggests the rejection of the null hypothesis of the random walk in daily return series.

✓ If the p-value of less than 0.05, reject the null hypothesis and concluded that daily return series do not follow random walk.

✓ If the p-values for a share index is greater than the significant level of 5%, accept the null hypothesis to conclude that the stock exchange index is a random walk across all sample periods.

✓ If variance ratio is not equal to one then study has to reject the null hypothesis of random walk behaviour.

Jarque-Bera Normality Test:

✓ Efficiency in the stock market requires that the return series are normally distributed; hence, the normality test for normal distribution of the data.

✓ Jarque-Bera normality test statistic has been used to examine whether the stock returns follow a normal distribution.

✓ JB test is a statistic for testing whether or not a series is normally distributed. It measures the difference of the skewness and kurtosis of a series with those from a normal distribution.

✓ JB test measures the degree of deviation in the kurtosis and skewness of the distribution of daily returns with the kurtosis and skewness of a normal distribution.

✓ For a normally distributed series, skewness = 0 and kurtosis = 3.

✓ Therefore, the JB test of normality is a test joint hypothesis that skewness and kurtosis are 0 and

3 respectively. If JB > χ2 (2) where 2 is the degree of freedom, then the null hypothesis is rejected.

✓ This is when the p-value is lower than the level of significance, 1% in this case.

✓ Under the null hypothesis of normality in distribution, the JB is equal to 0. The Jarque-Bera test hypothesis are stated thus:

H0: Data is normally distributed. H1: Data is NOT normal distributed.

✓ If Jarque-Bera test firmly rejects normality, this implies that the stock exchange daily returns series is not normally distributed.

Notes on Interpretation of JB Normality Test:

ü  If the p-value of JB test is less than the significance level of 0.01 (1%), JB test rejected the null hypothesis (that stock returns are not originated randomly), hence, the stock market index of daily return series did not follow a normal distribution; thus suggesting that the returns of the stock exchange do not follow the theory of random walk.

ü  If Jarque-Bera test statistics is less than 0.05 (5%) significant level, it indicates the non-normality in the distributions.

ü  The Jarque-Bera with their corresponding probabilities revealed that daily is not normally distributed with a probability of 0.001901 but weekly and monthly are normally distributed with 0.707594 and 0.120037 respectively at 5 percent.

ü  Interpretations: A tiny p-value and a large test statistics value from this test means that you can reject the null hypothesis that the data is normally distributed.

ü  This indicates that the test statistic is 2209.871, with a p-value of 0.000. We would reject the null hypothesis that the data is normally distributed in this circumstance. There is enough evidence to conclude that the data in this scenario is not normally distributed.

ü  This indicates that the test statistic is 0.057628, with a p-value of 0.971597. We would not be able to reject the null hypothesis that the data is normally distributed in this scenario. It is evident here that the data in this scenario is normally distributed.

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