# The Stock Index Prediction Based on SVR Model with Bat Optimization Algorithm
## 5 Empirical Research and Result Analysis  
### Tests for Panel Smoothness and Cointegration  
Since 18 representative stock indices in the Chinese stock market are selected as the sample for this paper, a single time series smoothness test cannot be applied. For the research problem and the selected data, a panel unit root test is chosen to test the smoothness of the adopted data. Since the historical data of the six stock index characteristics used are all strongly balanced panels, and the sample selected for this paper is a long-sided version, i.e., the cross-sectional dimension is smaller than the time dimension. Therefore, the LLC test [27] is used here. The results are shown in Table 2, and the results indicate that four of the six stock index characteristics in this paper exhibit unbalanced and two exhibit balanced characteristics. We can also see this result more intuitively from the historical trend chart of these six characteristics (see Appendix A). In order to match the qualitative requirements of the panel cointegration test, it is necessary to ensure the smoothness of all data presented, so the unit root test is performed again on the first-order differences of all variables, and the results are shown in Table 3. It can be seen that all variables are smooth after first-order difference, and all six variables are first-order single integer.  
To further investigate whether pseudo-regressions occur and to test for the existence of long-run cointegration relationships among all variables. This paper further performs cointegration tests on the six variables. Based on the results of the previous panel unit root test, all variables are first order single integer and cointegration tests are allowed. In this paper, the Westerlund test [28] is mainly used to test the cointegration of all variables. If, the \(p\)-value of the obtained results is less than 0.01, it indicates that there is a significant long-term cointegration relationship among the variables. The results of the four statistics in the Westerlund test [28; 29] are shown in Table 4 below. The results in Table 4 show that all explanatory variables are significant at the 1% level of significance. There is a cointegration relationship among the six variables, and the possibility of pseudo-regression is excluded.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Statistics** & \(p\)**-Value** & **Stability** \\ \hline closing\_price & 3.0742 & 0.9989 & non-stationary \\ \hline opening\_price & 2.3897 & 0.9916 & non-stationary \\ \hline highest\_price & 3.6348 & 0.9999 & non-stationary \\ \hline lowest\_price & 1.7406 & 0.9591 & non-stationary \\ \hline trading\_volume & \(-27.1531\) & 0.0000 & Stable \\ \hline turnover & \(-22.1696\) & 0.0000 & Stable \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of the LLC panel unit root test for the variables.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Statistics** & \(p\)**-Value** & **Stability** \\ \hline D. closing\_price & \(-2.5\times e^{2}\) & 0.0000 & Stable \\ \hline D. opening\_price & \(-2.6\times e^{2}\) & 0.0000 & Stable \\ \hline D. highest\_price & \(-2.2\times e^{2}\) & 0.0000 & Stable \\ \hline D. lowest\_price & \(-2.3\times e^{2}\) & 0.0000 & Stable \\ \hline D. trading\_volume & \(-2.1\times e^{2}\) & 0.0000 & Stable \\ \hline D. turnover & \(-2.1\times e^{2}\) & 0.0000 & Stable \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of LLC panel unit root test for first-order differences of variables.