# Forecasting Seasonal Sales with Many Drivers: Shrinkage or Dimensionality Reduction?
## 5 Conclusions  
Demand forecasting at Store \(\times\) SKU level is a complex problem mainly due to the requirements imposed by large retailers. A forecasting system should include a great variety of drivers that affect SKU demand at store level, such as calendar events and promotional information. Additionally, the forecasts are needed for a large number of products and consequently, the forecasting process must be automated, reliable and computationally efficient. In this study, we design novel approaches to forecast retailer product sales taking into account the main drivers that affect SKU demand at store level. We propose a feasible solution to include all relevant effects, including promotions, into the statistical ARIMA and ES models based on principal components selected automatically, which prevents overfitting. We also propose an automatic approach to model the short-term dynamics and the seasonality of the demand with Ridge regression. Both, stochastic seasonality represented by seasonal lags, and deterministic seasonality included as Fourier terms, are implemented for comparison. Using a diverse retail dataset, we compare the forecast performance of our proposed models over several forecast horizons based on two error measures. The forecasting performance results enable us to conclude that the models with covariates outperform their counterparts without covariates across all the forecast horizons, according to both metrics used. These findings confirm that the inclusion of the main drivers which affect demand at store level can significantly impact the performance of forecasting methods, but also that the proposed modelling approach can take advantage of them. RidgeX is generally the most accurate out of all competing models. This suggests that shrinkage is relatively more accurate in estimating effects from covariates. Nonetheless, when we focus on the forecast performance solely for promotional periods the PCA based models perform best. This finding helps to synthesise different results from the literature that advocates for both approaches. Furthermore, the shrinkage based models enable inference directly, if this is needed. We found a distinct difference in the behaviour of trigonometric seasonality compared to more standard approaches such as stochastic seasonality. It was not beneficial when a shrinkage estimator was used but provided significant accuracy gains otherwise. This research helps to clarify some of the modelling preferences for retail forecasting, as most of the past contributions in the literature have focused on demonstrating the performance of a single, often novel, algorithm. Our work consolidates some of this research. However, as we were motivated by computational efficiency we did not consider ML and AI approaches. The investigated forecasting approaches have the advantage of being relatively transparent against ML and AI methods. The latter do not inform the users on estimated promotional effects that can be useful inputs for promotional and pricing plans. Moreover, for these approaches to become useful in such a setting, they have substantial computational and data requirements. This study has several implications for practice. Retailers, and in particular supermarkets like the one in our case study, have to face the challenge of incorporating extensive promotional information into their models, along with other covariates. This is often done inefficiently resulting in miscalibrated models, the forecasts of which often require substantial manual adjustments by demand planners [42]. Our recommended models can be automatically calibrated for each item, including relevant effects. On the one hand, this provides gains in forecast accuracy. On the other hand, the models are sufficiently transparent to support promotional planning activities and can increase the trust of users towards the models [31]. For these benefits to be realisable the recommended models need to be relatively easy to implement.