# Forecasting Seasonal Sales with Many Drivers: Shrinkage or Dimensionality Reduction?
## 4 Empirical Study
### Discussion  
Overall, we see that the models with covariates outperform their counterparts without covariates across all the forecast horizons, according to both accuracy measures used, in line with the findings of [17]. These findings confirm our initial claim that the inclusion of the main drivers which affect demand at the store level always improves accuracy. In terms of how to best include covariates we find that in agreement with the literature [11; 1, 37] both work well to incorporate the rich information available. When promotions are dominant then PCA encoding is beneficial. However, to make the PCA-based models transparent to the users, the principal components need to be remapped to  
Figure 5: MCB test results at a 5% significance level based on MASE (**left**) and on RMSSE (**right**) for promotional (**top**) and non-promotional (**bottom**) periods (forecast horizon 1–13). \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Promotion**} & \multicolumn{2}{c}{**No Promotion**} \\ \cline{2-5}  & **MASE** & **RMSSE** & **MASE** & **RMSSE** \\ \hline RidgeX & 3.148 & 2.096 & **0.742** & **0.632** \\ PCRegARIMAF & 2.947 & 2.031 & 0.754 & 0.652 \\ PCRegARIMA & **2.865** & **1.937** & 0.762 & 0.658 \\ RidgeFX & 3.011 & 2.006 & 0.763 & 0.638 \\ ESXPC & 2.924 & 1.982 & 0.772 & 0.666 \\ \hline Ridge & 4.181 & 2.667 & 0.762 & 0.640 \\ ARIMA & 4.191 & 2.683 & 0.774 & 0.661 \\ RegARIMAF & 4.173 & 2.665 & 0.776 & 0.651 \\ TBATS & 4.174 & 2.667 & 0.782 & 0.659 \\ RidgeF & 4.091 & 2.606 & 0.797 & 0.657 \\ ES & 4.257 & 2.726 & 0.798 & 0.677 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model’s forecasting performance on promotional and non-promotional periods, for the horizon 1–13. the raw explanatory variances so that the respective coefficients can be inferred. This is not necessary with shrinkage estimators, that perform well overall, and can be a desirable solution when the promotional intensity is not very high. In terms of the use of trigonometric seasonality, the results are mixed, but again in some agreement with past literature [33; 63; 64]. Shrinkage estimators are able to introduce sparsity as needed, through the specification of the \(\lambda\) hyper-parameter. Therefore, the trigonometric representation is not beneficial. In fact, we find that the trigonometric representation performs relatively poorly for longer-term forecasts. Considering that all models are optimised on one-step-ahead errors, then we can argue that this indicates that RidgeF and RidgeFX in fact have overfitted to the data more than their counterparts Ridge and RidgeX, with better performance around for short-term forecasts, and vice versa for the longer term. On the other hand, for ARIMA that contains no shrinkage, the trigonometric seasonality is beneficial, with increasing relative performance in the long term. Therefore, we conclude that the performance of the trigonometric encoding is estimator dependent. Furthermore, it is interesting to note that all models used here are fairly transparent, in that one can infer the effect of specific covariates on the sales of each SKU, and potentially use that to optimise the pricing and promotional strategy [39]. As our models were estimated after a logarithmic transformation of the data, any coefficients can be interpreted as elasticities and inform marketing activities, beyond the benefit of having accurate forecasts for operations, such as for inventory planning. In this study, we have not considered ML and Artificial Intelligence (AI) methods, as this was not compatible with the objectives of our evaluation. Similarly, we excluded using more complex combination schemes of models, for example as in [11].