Estimation technique 53

3.3. Methodology 50

3.3.2. Estimation technique 53

The estimation technique adopted for this study is the panel data error correction approach proposed by Westerlund(2007)⁴. Westerlund (2007) developed a new panel cointegration test that makes use of the structure dynamics rather than residual dynamics (see Demetriades and Fielding, (2010). Therefore since the new panel cointegration test makes use of structural dynamics it does not impose the restriction of common factors like the one that makes use of residual dynamics. The procedure is divided into three. The first stage is to test for stationarity,i.e panel unit root test.

The second stage is the error correction based panel cointegrationtest and the third stage is the estimation of both the long run and the short-run equations. Another rationale for this study’s choice of this approach lies in the fact that Westerlund’s (2007) panel cointegration is based on error correction. Therefore, short run relationships/dynamics are given more attention. This is necessary because of the near consensus that monetary policy often exhibits weak or no long-run relationship with real variables (see Tobins, 1965; Mundel, 1963). Again, the panel cointegration test is an array of tests that are normally distributed and can accommodate generally unit-specific short-run dynamics, unit-specific trends and slope parameters as well as cross-sectional dependence.

As earlier stated, Westerlund developed four panel cointegration tests which are divided into two separate groups. The first two tests developed by Westerlund(2007) test the alternative hypothesis that at least one cross-sectional unit is cointegrated, while the other two test the alternative hypothesis that the whole panel is cointegrated. The process of estimating our model starts with the panel unit root test.

4 The Westerlund cointegration implements a relatively new command in stata. The idea is to test for the absence of cointegration, that is to determine if any of the panel test is error correcting or not (Westerlund and Persyn, 2007).

3.3.2.1 Panel unit root test

The panel unit root test explores the data characteristics of the panel before proceeding to the panel cointegration test. The idea here is to test for stationarity of each variable used in the study. According to Engel and Granger (1987) a variable may not be stationary but a linear combination of the non-stationary variables maybe stationary. So, we test for cointegration just to verify this. The method of panel unit root test adopted for this study is the Im, Pesaran and Shin (IPS) test. The test has been proven to be suitable in verifying stationarity of variables in panel data (see Im, Pesaran and Shin, 2003; Maddala and Wu, 1999).

Again, the choice of IPS has to do with the problem of intercept heterogeneity associated with many estimates. IPS allows for heterogeneity of the intercept. The IPS unit root tests are superior to the homogenous test when N (cross sectional unit) is relatively smaller than T (series). Therefore, in this study where T>N, the IPS is more suitable for the unit root test (See Chris Brooks, 2013). The basic IPS specification is given by:

∆𝑦_𝑖,𝑡 = 𝛼𝑦_{𝑖,𝑡−1}+ ∑^𝑝_𝑗=1^𝑖 𝛾_𝑖𝑗∆𝑦_{𝑖,𝑡−𝑗}+ 𝛽₀+ 𝛽₁𝑡 + 𝛽₁𝑥_𝑖,𝑡+ 𝜀_𝑖,𝑡… (3.11) where, β0 is the constant, xi,t represents the explanatory variables, ∆yi,t is the explained variable, β1t is a time trend and pi is the required lag length. The null hypothesis to be tested for the IPS is H0:αi=0, for all ‘i’s while the alternative hypothesis is H1:αi<0, for at least one i. The lag lengths are selected using the Akaike Information Criterion which is used mostly in panel estimation. According to Westerlund (2007) all series must be largely non-stationary series i.e. I(1) before a panel cointegration test can be carried out.

3.3.2.2 Error-correction based panel cointegration test

After the stationarity test, we proceed to the cointegration testor the error-correction based panel cointegration test developed by Westerlund (2007).The panel cointegration test model is given by:

∆𝐺_𝑖,𝑡 = 𝛼_𝑖 + 𝜃_𝑖𝐸𝐶𝑇_{𝑖,𝑡−1}+ ∑^𝑝_𝑗=1^𝑖 𝛽_𝑖𝑗∆𝐺_{𝑖,𝑡−𝑗}+ ∑^𝑝_𝑗=0^𝑖 𝜋_𝑖𝑗∆𝑋_{𝑖,𝑡−𝑗}+ 𝜀_𝑖,𝑡 (3.12) Where X and 𝐺𝑖,𝑡 are explanatory variables and manufacturing growth rate respectively, αi, βi and πi are parameter estimates.ECT is the error correction term while parameter θi

determines the speed of adjustment at which the system corrects back to equilibrium when there is a sudden shock. If θi< 0, it means that there is error correction, which simply implies that 𝐺𝑖,𝑡it and mpit are cointegrated. But if θi= 0, then there is no error correction hence there is no cointegration. Therefore the null hypothesis is stated as H0:θi = 0 for all i while the alternative hypothesis depends on the assumption we made about the homogeneity of θiand this varies in the four panel cointegration tests (see Demetriades and Fielding, 2010; Westerlund, 2007). The four tests are divided into two, namely: the group-mean test and the panel test.

3.3.2.3 Estimating the group-mean tests

Considering equation 3.12 which is the error correction based panel cointegration equation, we compute the group mean tests as follows (see Persyn and Westerlund, 2008):

𝑉_𝜏 =_𝑁¹∑ _{𝑆𝐸(𝜃}^𝜃̂𝑖_̂

𝑖)

𝑁𝑖=1 (3.13)

𝑉_𝛼 =_𝑁¹∑ _𝑆𝐸𝜃^𝑇𝜃_̂^̂𝑖

𝑖(1)

𝑁𝑖=1 (3.14) Consequently, the first two tests which verify existence of cross-sectional unit cointegration are estimated using equations 3.13 and 3.14. Both 𝑉𝜏 and 𝑉𝛼 test statistics test the null hypothesis that there is no cointegration across all the cross sectional units, against the alternative hypothesis that there is cointegration in at least one cross- sectional unit(i.e 𝐻₀^𝑉:θi = 0 for all i) while 𝐻₁^𝑉:θi< 0 for at least one i. Therefore the rejection of the null hypothesis means that there is cointegration in at least one of the cross-sectional units.

3.3.2.4 Estimating the panel tests

The second sets of tests are the panel tests for cointegration. The test statistics are computed as follows:

𝐵_𝜏 =_{𝑆𝐸(𝜃}^𝜃̂_̂) (3.15) 𝐵_𝛼 = 𝑇𝜃̂ (3.16) Equations 3.15 and 3.16 are panel test statistics for cointegration. The null hypothesis is that there is no cointegration for the whole panel (i.e 𝐻₀^𝐵:θi = 0 for all i)against the alternative hypothesis that there is cointegration for all the cross-sectional unitsi.e the whole panel( 𝐻₁^𝐵:θi< 0 for all i). Once the null hypothesis is rejected it means there is cointegration for the whole panel.

The P values are computed based on both asymptotic test distribution and cross sectional dependence which makes use of the bootstrap values of the parameter estimates as the robust P values.

Asymptotic test distribution

According to Westerlund (2007) the asymptotic distribution of error-correction test is purely based on sequential limit theory which simply implies that T is taken to infinity before N. The implication of this is that for the test to be justified, T must be substantially larger than N. This is peculiar in our current study which features T=41 and N=9. Asymptotic P values are obtained for all the four categories of tests highlighted in equations 3.13, 3.14, 3.15 and 3.16 (see Elberhardt, 2011; Frimpong, 2011; Basher and Elsamadisy, 2010; Demetriades and Fielding, 2010).

Cross-sectional dependence

The bootstrap approach of computing the parameter estimates is used to capture the cross-sectional dependence. The idea behind this is that there is a possibility of cross- sectional correlation existing which can affect our results. Consequently, Westerlund (2007) developed bootstrap values which take care of any expected cross-sectional dependence (Ishibashi, 2012; Demetriades and Fielding, 2010). The P values obtained here are referred to as the robust P values even in the presence of common factors in time series. In this study the lags and the leads are set using the Akaike Information Criterion (AIC) as used by previous studies (see Westerlund, 2007; Pedroni 2000;Ishibashi, 2012; Demetriades and Fielding, 2010)and the Bartlett Kernel window width are set according to 4(T/100)^2/n. The time series in this study is 41 years i.e T=41.

Putting this into the formula for selecting the Kernel window gives approximately 3 which is the Irwindow. Considering the T which is 41, the number of replications regarding the bootstrap values is 400 (see seeWesterlund, 2007; Pedroni 2000).

Finally, after the estimation of the model, we test for cross sectional dependence. The essence of the test is to find out if the presence of common factors has an effect on the panel cointegration test result. Notwithstanding, bootstrapping has been identified as a remedy for the presence of the common factor (see Persyn, andWesterlund,2008;Ishibashi, 2012; Demetriades and Fielding, 2010).This further provides justification for bootstrapping. However, the test requires that T>N and in our study T=41 and N=9.This condition is not violated. Therefore, we can conveniently test for cross sectional dependence in the residuals.