## 4 Experimental Results
### Analyzing the Interrelationships  
LVM is adopted to analyze interactions between Walmart, Costco, and Kroger. In Table 6, it is found that the relationship known as _mutualism_ exists between all pairs. This means that each firm can benefit from the existence of the other retailers. This result may imply the whole retail market is still a growing pie. Since the three firms position themselves in different geographical locations, product categories, and consumer groups, they do not intensively compete with one another. Table 7 further estimates stable sales equilibriums, considering the interactive dynamics continue. Compared to the sales in 2021/Q4, Costco (+5.9%) and Kroger (+7.8%) significantly increased sales at market equilibriums, while Walmart (\(-2.8\%\)), slightly decreased. In Table 7, the MAPEs for the three retailers are around 10% and these results justify the validity of using LVM in interactive regression. More importantly, they provide a quantitative basis to estimate the degree of change in sales revenues. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Outcome** & **Predictor** & \multicolumn{2}{c}{**Parameters (\(a\),\(b\),\(c\))**} & **Relationship** \\ \hline \multirow{3}{*}{Walmart} & \multirow{3}{*}{Costco} & 1.466 ***, 5.287 \(\times\)\(10^{-6}\) *, & \multirow{3}{*}{Mutualism} \\  & & & \(-\)5.697 \(\times\)\(10^{-6}\) * & & \\  & & & & \(-\) & \(-\) & \\ \multirow{3}{*}{Costco} & \multirow{3}{*}{Walmart} & 0.767 ***, 8.976 \(\times\)\(10^{-6}\) ***, & \multirow{3}{*}{} & \multirow{3}{*}{} & \multirow{3}{*}{} \\  & & & \(-\)4.405 \(\times\)\(10^{-6}\) ** & & \\ \multirow{3}{*}{Costco} & \multirow{3}{*}{Kroger} & 0.768 *, 11.88 \(\times\)\(10^{-5}\) ***, & \multirow{3}{*}{} & \multirow{3}{*}{} & \multirow{3}{*}{} \\  & & & \(-\)2.266 \(\times\)\(10^{-5}\) ***, & \multirow{3}{*}{} & \multirow{3}{*}{} & \multirow{3}{*}{} \\ Kroger & \multirow{3}{*}{Costco} & 1.553 ***, 3.481 \(\times\)\(10^{-5}\) **, & & & \\  & & & \(-\)1.234 \(\times\)\(10^{-5}\) ** & & & \\ Kroger & \multirow{3}{*}{Walmart} & 0.612 ***, 2.450 \(\times\)\(10^{-5}\) **, & \multirow{3}{*}{} & \multirow{3}{*}{} & \multirow{3}{*}{} \\  & & & \(-\)8.671 \(\times\)\(10^{-6}\) ** & & & \\ \multirow{3}{*}{Walmart} & Kroger & 1.294 ***, 4.169 \(\times\)\(10^{-6}\) **, & & & & \\  & & & \(-\)8.028 \(\times\)\(10^{-6}\) ** & & & \\ \hline \hline \end{tabular} Significance level used: *** \(<\) 0.001, ** \(<\) 0.01, * \(<\) 0.05.  
\end{table} 
Table 6: Pairwise analyses for the three retail firms. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Training** & \multicolumn{2}{c}{**Walmart**} & \multicolumn{3}{c}{**Costco**} & \multicolumn{3}{c}{**Kroger**} \\ \hline  & **MARS** & **SVR** & **DNN** & **MARS** & **SVR** & **DNN** & **MARS** & **SVR** & **DNN** \\ \hline RMSE & 6250 & 6583 & 6616 & 3557 & 3889 & 3841 & 3541 & 3606 & 3414 \\ MAE & 295,647 & 289,486 & 5383 & 171,969 & 141,518 & 2526 & 141,155 & 136,233 & 2387 \\ MAPE & 4.5\% & 4.36\% & 4.86\% & 11.09\% & 8.53\% & 9.43\% & 12.27\% & 8.48\% & 9.41\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Forecasting errors for the training set (2005/Q1\(\sim\)2019/Q4). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Testing** & \multicolumn{2}{c}{**Walmart**} & \multicolumn{3}{c}{**Costco**} & \multicolumn{3}{c}{**Kroger**} \\ \hline  & **MARS** & **SVR** & **DNN** & **MARS** & **SVR** & **DNN** & **MARS** & **SVR** & **DNN** \\ \hline RMSE & 7190 & 8986 & 9022 & 8080 & 10,210 & 9355 & 5699 & 5353 & 4978 \\ MAE & 48,830 & 61,676 & 7762 & 43,001 & 63,978 & 6982 & 29,132 & 37,298 & 3835 \\ MAPE & 4.27\% & 5.37\% & 5.43\% & 10.22\% & 15.12\% & 13.1\% & 9.4\% & 13.25\% & 10.49\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Forecasting errors for the testing set (2020/Q1\(\sim\)2021/Q4). In reality, sales revenues are affected by many factors, including manufacturing cost, pricing policies, promotion plans, channel competition, product positioning, geographical location, customer defection, etc.