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Unit Root Tests

The unit root test is used to determine whether a time series variable is stationary or non-stationary (Griffiths, Hill, & Lim, 2008, p. 336). As Granger and Newbold (1974) note that a significant regression relationship can be spurious if the variables involved are nonstationary. Thus, the non-stationarity of variables need to be addressed before a sensible modelling exercise can be carried out. Dickey and Fuller (1979, 1981) developed a formal procedure to examine the presence of a unit root. The Dickey- Fuller test is commonly applied to check whether a time series variable contain a unit root. In the Dickey-Fuller test, the null hypothesis is that the time series has a unit root against the alternative hypothesis that it is stationary (Enders, 2015; Maddala, 2001). In terms of Equation (7) and (8) in this current study, the stationarity of each variable needs to be determined by using the unit root test.

Cointegration

Engle and Granger (1987) were the first scholars to introduce the concept of cointegration. Enders (1995, p. 358) notes “cointegration refers to a linear combination of nonstationary variables”. Wooldbridge (2013, p. 646) points out that when a linear combination in the nonstationary variables is stationary, these variables are cointegrated. A cointegration regression estimates the long-run equilibrium properties of time series variables in economic models (Enders, 1995; Gujarati & Porter, 2009). Theoretically, the number of cointegrating vectors is infinite. Therefore, the cointegrating

cannot be cointegrated. Thus, cointegration implies that the dependent variable and independent variables share similar stochastic trends (Baltagi, 2008; Stock & Watson, 1988).

The Engle-Granger test is one of the methods used to examine the stationarity of the residuals and identify cointegration (Granger, 1986; Kennedy, 2008). Hill, Griffiths and Lim (2011) report that if the residuals are stationary, then the dependent variable and independent variables are cointegrated. Wooldridge (2013, pp. 647-648) notes the benchmark of critical values for the cointegration test as shown in Table 4.2:

Table 3-2 Critical values for the cointegration test

Regression Model 1% 2.5% 5% 10%

1. y = β1 + β2x+ e -3.90 -3.59 -3.34 -3.04

2. yt = β1+ δt + β2xt + et -4.32 -4.03 -3.78 -3.50

Multiple Regression Analysis

Multiple regression analysis reveals the relationships between the variables of interest: the dependent variable and the explanatory variable(s) (Hill et al., 2011). Nieuwenhuis (2009) notes that the values of explanatory variable(s) are assumed to have positive or negative effects on the value of the single dependent variable.

In the Equation (7) and (8), the dependent variables are Service Exports and Service Imports. The explanatory variables are the Real Effective Exchange Rate, World Income, Domestic Income, FDI,

Service Exports and Service Imports in the two equations, respectively. The signs of α1 ... α4 and β1 ...

β 4 indicate positive or negative effects of the explanatory variables on Service Exports and Service

Imports. In addition, the magnitude of α1 ... α4 and β1 ... β4 measure the amount of change in each

explanatory variable. Further, the error term µ represents residuals between observed values and predicted values of Service Exports and Service Imports.

Coefficient of Determination

The coefficient of determination (R2_{) measures the degree of usefulness of the estimated model (Hill}

et al., 2011). The R2 _{is the proportion of the reduction of the variation of the dependent variable that}

is explained by all independent variables (Gujarati & Porter, 2009). The calculation of R2 _{is as follows}

(Nieuwenhuis, 2009): R2 ₌ 𝑆𝑆𝑅 𝑆𝑆𝑇 = 𝑆𝑆𝑇−𝑆𝑆𝐸 𝑆𝑆𝑇 = 1 - 𝑆𝑆𝐸 𝑆𝑆𝑇

Where: SSR stands for the sum of squares regression; SST stands for the total sum of squares; SSE stands for sum of the squared errors. Nieuwenhuis (2009) lists the rules of R2_:

0 ≤ R2 _{≤ 1}

If the value of the R2 _{is close to 0, then the reduction in variation is only a small proportion. This result}

implies that the regression model has a weak predictive ability. If the R2 _{is close to 1, then the reduction}

in variation is a large proportion, which means the regression line perfectly fits the sample data. Therefore, the R2 _{is considered as a measure of “goodness–of–fit”.}

Based on these definitions, the measurements of SST, SSE and SSR for service exports and imports for the model in this study are as follows:

Where 𝐸𝑋̅̅̅̅ and 𝐼𝑀̅̅̅̅are the means of service exports and imports.

2. SSEex =∑(𝐸𝑋̂ − 𝐸𝑋̅̅̅̅)2 and

SSEim = ∑(𝐼𝑀̂ − 𝐼𝑀̅̅̅̅)2

Where 𝐸𝑋̂ and 𝐼𝑀̂ are the predicted values for service exports and imports.

3. SSRex = ∑(EX − 𝐸𝑋̂ )2 and (EX − 𝐸𝑋̅̅̅̅)2 and (IM − 𝐼𝑀̅̅̅̅)2 are the total variations of the actual values of

services exports and imports about their sample mean, respectively. (𝐸𝑋̂ − 𝐸𝑋̅̅̅̅)2and (𝐼𝑀̂ − 𝐼𝑀̅̅̅̅)2 are the variation of the estimated service exports and imports, which are explained by the regression,

respectively. The smaller (𝐸𝑋̂ − 𝐸𝑋̅̅̅̅)2 and (𝐼𝑀̂ − 𝐼𝑀̅̅̅̅)2 the more reliable the predictions obtained from the regression model. (EX − 𝐸𝑋̂ )2 and (IM − 𝐼𝑀̂ )2are defined as a residual, which describes the difference between what actually occurred and what the model predicted would occur (Halcoussis,

2005).

In terms of the Equation (7) and (8), the R2 _{represents the percentage of variation in the dependent}

variables (Service Exports and Service Imports) explained by the variation in the conducted explanatory variables. Higher values of a R2 _{indicate that the model fits the data well (Hill et al., 2011). Therefore,}

the adjusting coefficient of determination can improve the model fit.

Test of Significance of the Regression Model

The F-test is the test statistic estimating the overall significance of the regression model. The value of the F-test statistic is calculated as follows (Hill et al., 2011):

N is the number of observations.

Regarding to Equation (7) and (8), SSTex =∑(EX − 𝐸𝑋̅̅̅̅)2and SSTim =∑(IM − 𝐼𝑀̅̅̅̅)2 , and SSEex

=∑(𝐸𝑋̂ − 𝐸𝑋̅̅̅̅)2 and SSEim = ∑(𝐼𝑀̂ − 𝐼𝑀̅̅̅̅)2. K is 8 independent variables and N is 31 observations in

each equation.

The null hypothesis for the F-test is that all the regression coefficients are equal to 0. Hill et al. (2011) conclude that when a regression model has no significant predictability for the dependent variable, the null hypothesis is accepted. Conversely, when at least one of the independent variables has significant predictability of the dependent variable, the null hypothesis is rejected.

Gujarati (2003) notes that the coefficient of determination R2 _{and the F test have an intimate}

relationship when used in the analysis of variance, which is presented as follows:

1. When R2 _{=0, F = zero;}

2. The larger the R2_{, the greater the F value;}

3. When R2 _{= 1, F is infinite.}

Therefore, the F test is both a measurement of overall significance of the estimated regression and a test of significance of the R2_{. The F-test can test the joint null hypotheses for the three hypotheses}

formulated for this study. If the null hypothesis is true, there is no significant independent variables in the model. Thus, the independent variables cannot explain the dependent variables (Service Exports and Service Imports). Therefore, the constructed variables need to be modified and developed. Whereas, if the null hypothesis is rejected, the constructed variables are valuable in explaining the two dependent variables.