*4.3. Diagnostic Test and Robustness Check*

To further consider the reliability and validity of the model estimate, diagnostic tests are considered. There are two critical diagnostic tests for the panel PMG/ARDL method in Eview: coefficient diagnosis and residual diagnostic. However, according to Wooldridge (2015), based on the asymptotic theory, when there is a sufficient number of observations, it is not necessary to test the normal distribution of the residuals. With 275 observations, this study omits the residual diagnostic and only performs the coefficient diagnostic by coefficient confidence intervals and the Wald test, with the Null Hypothesis that the coefficients are all equal to 0. The results of the diagnostic coefficients are presented in Table 6 below.


**Table 6.** Coefficient diagnostics.

Note: \*\*\* for statistical significance at the 0.01 levels, respectively. Source: Results of Wald test.

Table 6 provides the values of the coefficients at the 95% and 99% confidence intervals. Accordingly, the maximum and minimum values of lnTC, lnHR and ln RF are all greater than 0. In contrast, the values of Dummy are all less than 0. The Wald test gives significance at 0.01 level for both F and Chi-squared statistics. Therefore, the null hypothesis is rejected and the alternative hypothesis is accepted, meaning that the estimated coefficients in the model are all non-zero, and they are all necessary for the model. This evidence lends support to the reliability and validity of the estimated model.

Next, the robustness check is performed by comparing the estimated results among PMG/ARDL, cointegration regression and OLS for panel data (assuming the cointegration series from the Pedroni test result). In the OLS method, Random Effects Model (REM) is selected from the Pooled OLS model, Fixed Effect Models (FEM) and REM. In the cointegration regression, the FMOLS estimator is chosen because there is a quite large difference in the long-term coefficient of variance in lnVA (Table 2). The estimated results by FMOLS and OLS methods are detailed in Table A2 in Appendix A.2. The coefficients estimated by PMG/ARDL, FMOLS and OLS methods are compared in Table 7.


**Table 7.** Differences in coefficients estimated by PMG/ARDL, FMOLS and OLS.

Note: \*\* and \*\*\* for statistical significance at the 0.05 and 0.01 levels, respectively. Source: Estimation results from PMG/ARDL, FMOLS and OLS.

According to Table 7, although the methods produce different estimation results, the signs of the coefficients are similar. To be more detailed, lnTC has quite similar results (bias of no more than 10%), lnHR has a maximum bias of 27.4% and Dummy variable has a bias of no more than 40%. Particularly, lnRF estimated by FMOLS and REM do not reach significance at the 0.05 level. Despite certain differences, it is believed that the results from the PMG/ARDL are more appropriate because of the advantage of PMG/ARDL discussed above, and the cointegration series is still in doubt.
