## 4 Empirical Analysis
### Results of the Cointegration Test  
A cointegration test is conducted to detect the number of cointegrations. If an analysis is conducted using differenced time series despite the existence of a cointegrated relationship, it can lead to excessive differencing, resulting in the loss of important information regarding the long-term relationships between variables. The Johansen maximum likelihood method and the Engle-Granger (EG) two-step method are currently the most widely used cointegration test methods [79]. The former accurately determines the number of cointegration vectors in addition to testing for the existence of a cointegration relationship between variables, while the latter is a cointegration test that uses only one equation. This paper thus employs the Johansen maximum likelihood method.  
If cointegration exists, the vector error correction model (VECM) is applied; otherwise, the VARM is applied. In this study, the presence of cointegration was determined using [PERSON]'s cointegration test by analyzing the Trace statistic.  
Before performing the cointegration test, the optimal lag length was determined using the VARM. To decide how far back to include in the time series, the AIC was calculated for the most recent eight periods of the raw data. As shown in Table 4, the optimal lag length for 14 of the global logistics companies, excluding COSCO Shipping, was determined to be 2. For COSCO Shipping, the optimal lag length was 7, but considering the uniformity across variables, the second-lowest AIC value corresponding to Lag 2 was chosen. Therefore, the optimal lag length for the first-differenced data is 1 (i.e., lag length of raw data--1).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Global Logistics Company** & **Original Data (t-Value, \(p\)-Value)** & **First Difference (t-Value, \(p\)-Value)** \\ \hline Delta Air Lines & \(-2.538975\) [0.1096] & \(-9.956550\) [0.0000] \\ American Airlines Group & \(-1.149828\) [0.6933] & \(-10.99714\) [0.0000] \\ United Airlines Holdings & \(-1.797645\) [0.3797] & \(-9.442409\) [0.0000] \\ Lufthans Group & \(-1.091319\) [0.7168] & \(-5.270122\) [0.0000] \\ United Parcel Service & \(-1.065696\) [0.4758] & \(-10.07340\) [0.0000] \\ FedEx & \(-2.197930\) [0.2084] & \(-10.10082\) [0.0000] \\ DSV & \(-1.149230\) [0.6935] & \(-9.911817\) [0.0000] \\ Energy Transfer & \(-2.367533\) [0.1536] & \(-9.565179\) [0.0000] \\ Enterprise Products Partners & \(-2.613802\) [0.0937] & \(-10.98261\) [0.0000] \\ Plains GP Holdings & \(-2.085411\) [0.2510] & \(-10.23675\) [0.0000] \\ Enbridge & \(-2.865735\) [0.0532] & \(-10.94215\) [0.0000] \\ Cheniere Energy & \(0.326018\) [0.9786] & \(-9.463355\) [0.0000] \\ COSCO Shipping & \(-1.182857\) [0.6794] & \(-11.99130\) [0.0001] \\ Maersk Group & \(-1.677142\) [0.4396] & \(-11.45171\) [0.0001] \\ Hapag-Lloyd & \(-1.593584\) [0.4820] & \(-10.27446\) [0.0000] \\ GSCPI & \(-2.190238\) [0.2112] & \(-7.442573\) [0.0000] \\ \hline \hline \end{tabular} Source: Authors’ own compilation.  
\end{table}
Table 3: Results of the ADF test.  
Based on the analysis of the Johansen cointegration test's Trace statistic, using the optimal lag length for each global logistics company, no cointegration was identified between any of the global logistics companies and the GSCPI. Therefore, we assumed that no cointegration exists between the variables and proceeded with the VARM analysis. The results of the Johansen cointegration test's Trace statistic are shown in Table 5.