PaIII Period of the Study
The empirical work is based on daily closing prices from April 1 2000 to March 31, 2015.To better analyze the relations between stock prices before and after Global Financial Crisis, the entire period is sub divided into two sub-periods and call the first sub-period which covered from 2000 April to 2007 July as pre-crisis period and the sub-period which covered from 2007 July to 2012 March as post-crisis period. The sample period choice was motivated by the fact that it’s during the summer of 2007 the U.S. “subprime” crisis was spread to a number of other advanced economies through a combination of direct exposures to subprime assets, the gradual loss of confidence in a number of asset classes and the drying-up of wholesale financial markets. Hence, the year 2007 was taken for a standing point for analysis
IV Data and Objectives of the Study
The empirical work is based on daily closing prices of the stock indices. Only secondary data were used. The data consisted of daily series of the S&P CNX Nifty (India), NASDAQ (USA), Nikkei (Japan), Hang–Seng (Hong Kong), TW11 (Taiwan) and STI (Singapore). Shanghai Composite (China) Jakarta Composite (Indonesia ) KLSE Composite (Malaysia ) Seoul Composite (South Korea ) Australia (AORD), FTSE (UK), Dax( France) and CAC 40 (Germany). The data on daily stock markets indices were downloaded from the Yahoo Finance website (http://finance.yahoo.com).
Objectives of the study
To test the weak form of Market efficiency of the stock markets of Indian and International Stock Markets.
To determine the volatility levels in the stock markets of India and International Stock Markets.
To draw a comparison between the Stock Markets of India and International Stock Markets important parameters like market capitalization, GDP, return and Volatility.
To examine cointegrationa and causality among Indian and International Markets.
V HYPOTHESES
The Following null hypotheses have been formulated for testing.
• All the international stock markets are weak form efficient.
• Indian stock market is not cointegrated with other international markets.
VI METHODOLOGY
A. Market Efficiency
To test the market efficiency, the study uses both traditional tests (such as, Kolmogorov Smirnov Goodness of Fit, run test, autocorrelation test and Ljung Box test ) and dynamic time series model Dickey–Fuller Test (Unit Root) and Augmented Dickey–Fuller (ADF) test. The following formulas are used for that.
Kolmogorov Smirnov Goodness of Fit Test
Kolmogorov Smirnov Goodness of Fit Test (KS) is a non –parametric test and is used to determine how well a random sample of data fits a particular distribution (normal). It is based on comparison of the sample’s cumulative distribution against the standard cumulative function for each distribution and test whether the distribution are homogeneous. We use normal parameters to test distribution.
The Kolmogorov-Smirnov test statistic is defined as11
where F is the theoretical cumulative distribution of the distribution being tested which must be a continuous distribution (i.e., no discrete distributions such as the binomial or Poisson), and it must be fully specified (i.e., the location, scale, and shape parameters cannot be estimated from the data). The calculated value of Z is compared with a critical value of a Z in the One –Sample Kolmogorov Goodness –of –Fit Test table for a given sample size 12.
Our hypothesis for this test is shown as the following;
H0: The data follow a normal distribution
Ha: The data do not follow the normal distribution
The Runs Analysis
The run test is another approach to detect the statistical independencies which means randomness. A run is a sequence of consecutive positive or negative returns. By comparing the total number of runs in the data with the expected number of runs under random walk hypothesis, the test of the random walk hypothesis may be constructed. To perform the test, the sampling distribution of the total number of runs in a sample is required. The r –statistic follows a normal distribution with a mean and standard deviation given by the following formulae. 13
This test is performed by examining a time series of returns for a security and testing whether the number of consecutive price gains or drops shows a pattern. A price gain is represented by a “ +”, a price drop is represented by a “- ” and “0” shows that return is zero. The run test converts the total number of runs into a Z statistic. For large samples the Z statistics gives the probability of difference between actual and expected number of runs. The Z values is greater than or equal to 1.96, then the null hypothesis that stock returns follow random walk is rejected. The null hypothesis is that stock returns depict a random walk through time. If the absolute value of Z is greater than Z critical value 1.96 for =0.05)
Our hypothesis for run test shown as the following;
H0: Price changes are random.
H1: Price changes are not random.
Autocorrelation Test ACF(k)
The autocorrelation coefficient provides a measure of the relationship between the value of a random variable in time t ( t) and its value at k periods earlier ( t-k). This indicates whether price changes at time t are influenced by price changes occurring at k periods earlier. The autocorrelation function ACF(k) for the time series Yt and the k–lagged series Yt-k is defined as:14
Where is the overall mean of the series with n observations.
The standard error of ACF (k) is given by:
ACF (k) = 1 /
When n is sufficiently large (n > 50), the approximate value of the standard error of ACF(k) is given by15 :
Where
ri = Autocorrelation at lag i
k = The time lag
n = Number of observations in the data series
To test whether ACF (k) are significantly different from zero, the following distribution of t is used:
t = ACF(k) / Se ACF(k)
Ljung–Box Statistic
Ljung–Box (LB) statistic is used to test the joint hypothesis that all the autocorrelations for lag one through m are simultaneously equal to zero. The value of the LB statistic is compared with the chi-square table to assess if it is significant.
Ljung–Box statistic, is defined as 16
The LB statistic follows the chi-square distribution with m (the number of time lags to be tested) degrees of freedom. The null hypothesis is that all rk are zero. If the computed LB exceeds the critical LB value from the chi-square table at the chosen level of significance, the null hypothesis is rejected. In such a situation, the result suggests dependency of the series, which violates the assumption of market efficiency. The chi-square value for fourteen degrees of freedom at 5 per cent level of significance is 24.99.
Augmented Dickey–Fuller Test (Unit Root)
The Dickey–Fuller statistic applies only to an autoregression (1) model. For some series autoregression (1) model does not capture all the serial correlation in (yt). In Dickey–Fuller test it is assumed that the white noise error term is uncorrelated. Dickey and Fuller have developed a higher order autoregression, the Augmented Dickey–Fuller (ADF) test, in case µt are correlated.
The Augmented Dickey–Fuller (ADF) test is estimated by the following formula: 17
Where µt is a white noise error term and ∆ Yt-1 = ( Yt-1–Yt-2). The null hypothesis H0 : δ = 0, and the alternative hypothesis H1: δ < 0.The critical value of ADF test at 5 per cent level of significance is –3.41.
B. Volatility
Close to close volatility, open to open volatility, open close volatility and high low volatility models are used to calculate volatility. ARCH and GARCH model is used to capture volatility transmission.
Return is calculated using logarithmic method as follows.
rt = (log pt–log pt-1)*100
Inter–day Volatility
The variation in share price return between the two trading days is called inter–day volatility. Inter–day volatility is computed by close to close and open to open value of any index level on a daily basis. Standard deviation is used to calculate inter–day volatility
Close to close volatility
Close to close volatility (standard estimation volatility) is measured with the following formula
Open to open volatility
Open to open volatility is considered necessary for many market participants because opening prices of shares and the index value reflect any positive or negative information that arrives after the close of the market and before the start of the next day’s trading .The following formula is used to calculate open-to-open volatility:18
Inter–day volatility takes into account only close to close and open to open index value and it is measured by standard deviation of returns.
Intra–day Volatility
The variation in share price return within the trading day is called intra–day volatility. It indicates how the indices and shares behave in a particular day. Intra–day volatility is calculated with the help of Parkinson Model and Garman and Klass model.
Parkinson Model
High–low volatility is calculated with the following formula:19
Garman and Klass Model
The Garman and Klass model is used to calculate the open–close volatility. The formula for Garman and Klass model (1980) takes the following form.20
Engle Granger Dickey–Fuller test for cointegration EG–DF (Engle–Granger 1987), The Engle Granger Augmented Dickey–Fuller test and Granger Causality test is used to examine the short run between the stock markets.
C. Cointegration
Before conducting cointegration test it is of interesting to determine if there are any common forces driving the long-run movement of the data series or if each individual stock index is driven solely by its own fundamentals. Cointegration analysis is used to investigate long term relationship between Indian and other International stock markets and it is estimated by ordinary least squares under the following formula:21
X t = β 1 + δ Yt + µt
The Engle Granger Augmented Dickey–Fuller test is applied on the ‘cointegrtating residuals’ µt obtained from the equation (1). The formula for EG–ADF test is as follows22
∆ µt represents the first differences of the residuals The specific hypotheses are :
H0 : δ = 0
H1 : δ ≠ 0
Null hypothesis is that there is no co integration among the stock indices. The value of a calculated absolute tau (τ) value is greater than the tabulated critical (τ) value; the null hypothesis of no cointegration is rejected. Engle and Granger have provided the critical values of ADF statistics.
D. Granger Causality
Short run integration is examined using Granger’s (1969) causality test. Formally, a time series xt Granger – causes another time series yt if series yt can be predicted with better accuracy by using past values of xt rather than by not doing so, other information being identical. In other words, variable xt fails Granger –cause yt if
Pr ( yt+m| Ωt) = Pr (yt+m | Ψ t),
Where Pr ( yt+m| Ωt) denotes conditional probability of yt , Ωt is the set of all information available at time t, and Pr( yt+m| Ψ t) denotes conditional probability of yt obtained by excluding all information on xt from yt this set of information is depicted as Ψ t . To examine the causality, if a cointegration relationship is found, a Vector Error Correction Model (VECM) is estimated.23
= + x + y +
= + x + y +
where represents the deviation from long – run equilibrium in period t-1 obtained from the cointegration regression. Where k is a suitably chosen positive integer, and , j = 0,1 ……k are parameters and ’ are constants and ’s are disturbance terms with zero means and finite variances. The null hypothesis that does not Granger – cause is not accepted if the ’s, j>0 in equation (4) are jointly significantly different from zero using a standard joint test (e.g., and F test). Similarly, Granger – causes , if the ’s j >0 coefficients in equation (5) are jointly different from zero. For non -cointegrating series, Granger causality is examined by the Vector Autoregressive (VAR) model. The form of the VAR model is obtained by deleting the terms in (4) and (5).
The stationary series are also cross – correlated. The cross – correlation between the time series are tested by using the following formula:24
( – ) ( – )
(k) =
Where k is greater than, equal to, or less than zero.
The significance of estimated cross – correlation is assessed by using approximate standard error, T-½, (Bartlet, 1966), of the sample of cross – correlation. This helps to identify the causality patterns associated with, γ xt yt (k). This method was used, among others, by Ahmad (1989) in estimating the cross – correlations between two interest rate series.
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