Methodology

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3.2 Methodology

Generally speaking, most of the time series variables possess the characteristic of nonstationarity. A time series variable is stationary under the condition that this variable’s mean, variance and function of autocorrelation will not vary in accordance with the change of the time. Such condition, however, will be violated by the fluctuations over time. If we employ the traditional econometric models and methods in dealing with time series data having the characteristic of nonstationarity, the residuals we get may be non-stationary due to the high autocorrelation within the residuals. Then, we may end up producing one spurious regression which will not help us understand the true relationship among the variables. Stock prices and exchange rates employed in this study are typically non-stationary. In order to cope with non-stationary economic and financial time series analysis, Granger (1986), Engle & Granger (1987) and Johansen (1988, 1991) put forward the cointegration test.

This model provides a solution to the problem of spurious regression.

In the beginning, this research employs unit-root tests including Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test, and Johansen cointegration test as the primary methodologies. In addition to these two tests, the correlation of coefficients between stock prices and exchange rates will be calculated annually. By employing these methodologies, whether the long-term relationship exists between stock prices and exchange markets in the countries of interest will be tested and observed. The process of this study begins with the unit-root tests on each country’s samples. After determining the time series order Ι (*), we proceed to Johansen cointegration test to determine the optimal cointegration vector. The methodologies are explained as follows:

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3.2.1 Unit-root tests: The ADF test and PP test

Granger and Newbold (1974) firstly point out that using non-stationary macroeconomic variables in time series analysis causes superiority problems in regression. The issue of unit-root of such variables is empirically demonstrated in Nelson and Plosser (1982) and since then this important property of macroeconomic and financial data series has been generally accepted. Many studies have lately shown that majority of time series variables are non-stationary or integrated of order 1. Thus, a unit-root test should precede any empirical study employing such variables. There have been a variety of proposed methods for implementing stationarity test and principally Augmented Dickey-Fuller test and Phillips-Perron test have been widely used in econometric data. Also this study, as a first step, executes both unit-root tests to investigate whether the time series of stock prices and exchange rates are stationary or not. ADF unit-root test (Dickey and Fuller 1979) is often used to determine the time series order Ι (*). Its unit-root test regression is:

(2)

where ∆ is the first difference of variable X, is the white noise.

ADF test uses to establish the null and alternative hypothesis:

: 0

: 0

If is significant different from zero, null hypothesis will be rejected, which means the variable does not have a unit-root.

Although ADF test takes the problem of autocorrelation into consideration, the characteristic of homoscedasticity in time series exists. Phillips and Perron (1988) made a more relaxed assumption on the basis of ADF test and this leads to the

∆ ∆  

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development of PP test. The test is robust with respect to unspecified autocorrelation and heteroscedasticity in the disturbance process of the test equation.

3.2.2. Coefficient of Correlation

Correlation coefficient is a measure of the degree of association between two variables. Its definition is:

∑

∑ ∑

∑ ∑ ∑

∑ ∑ ∑ ∑

(3)

where , . This is also known as sample correlation

coefficient.

3.2.3. Johansen cointegration test

According to the concept of cointegration, two or more non-stationary time series share a common trend, then they are said to be cointegrated. The theoretical framework is expressed as follow: the component of the vector

, _,… , are considered to be cointegrated of order d, b, denoted

~ , if first, all the component are stationary after n difference, or integrated of order d and noted as ~ ; second, presence of a vector

, , … , in such that linear combination

where the vector β in named the cointegrating vector. Some noteworthy characteristics of this model are that the cointegration relationship obtained indicates a linear combination of non-stationary variables, in which all variables must be integrated of the same order and moreover if there are n series of variables, there may be as many as n-1 linearly independent cointegrating vectors.

Johansen’s (1991) cointegration test is adopted to determine whether the linear

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combination of the series possesses a long-run equilibrium relationship. The numbers of significant cointegrating vectors in non-stationary time series are tested by using the maximum likelihood based on λ and λ introduced by Johansen and Juselius (1990). The advantage of this test is that it utilizes test statistic that can be used to evaluate cointegration relationship among a group of two or more variables.

Therefore, it is a superior test as it can deal with two or more variables that may be more than one cointegrating vector in the system.

In the Johansen procedure, it involves the identification of rank of the n x n matrix Π in the specification given by:

∆ ∆ (4)

where Yt is a column vector of the n variables, Δ is the difference operator, Γ and Π are the coefficient matrices, k denotes the lag length and δ is a constant. In the absence of cointegrating vector, Π is a singular matrix, which means that the cointegrating vector rank is equal to zero. On the other hand, in a cointegrated scenario, the rank of Π could be anywhere between zero.

The Johansen Maximum likelihood test provides a test for the rank of Π, namely the trace test (λ ), on which this study focuses, and the maximum eigenvalue test (λ ). The statistic tests the null hypothesis that the number of distinct cointegrating vectors is less than or equal to r against alternative hypothesis. The test statistic is given as follow:

1 (5)

where p is the number of separate series to be analyzed, T is the number of usable observations and is the estimated eigenvalues.

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