# Forecasting Seasonal Sales with Many Drivers: Shrinkage or Dimensionality Reduction?
## 1 Introduction  
Retailers depend strongly on accurate sales forecasts to manage their supply chains and make decisions concerning purchasing, logistics, marketing, finance, human resources, etc. [1, 2, 3]. Forecasts are principally needed at the Store \(\times\) SKU (Stock-Keeping Unit) level, i.e., all combinations of SKUs and store locations as argued, for example, by [4]. Inaccurate forecasts of product sales in-store can lead to stock-outs which negatively impact the business [5]. If the product is not available on shelf, its potential sales are lost and there is the chance of customers looking to competitors, making loyalty difficult to maintain [6]. Ordering excess inventory, to reduce the risk of stock-outs and to improve customer's satisfaction, increases costs significantly (e.g., labor and storage) reducing the profit margin [7]. Additionally, there is an increasing awareness that food waste should be reduced [8, 9] with the European Parliament calling for urgent measures to halve food waste by 2025 [10]. Efficient inventory management relies on accurate forecasts of SKU sales at the store level, which enable the retailer to replenish in time and meet the customersexpectations. As a consequence, there has been increasing interest in identifying more accurate forecasting methods, both by researchers but also by retailers and their software suppliers [3]. The latest research on demand forecasting at Store \(\times\) SKU level has considered two major questions with immediate impact on retail stores: (i) the first considers the effects of various factors including promotions e.g., [11; 12; 13; 14] and weather effects e.g., [15]. The inclusion of many demand drivers raises the challenging question of model specification, where standard regression and model selection techniques have serious limitations; (ii) the second question is whether more advanced techniques deliver improved value. [PERSON] et al. [3] summarized the evidence as moot, despite Machine Learning (ML) methods having a long history in retail starting with [16]. Recently, the M5 competition [17] and the latest two Kaggle sales forecasting competitions [18] explored this issue in some depth and found that ML methods, specifically LightGBM [19], can improve on standard benchmarks, leading to a more positive reappraisal in [PERSON] et al. [3] update of retail forecasting research. However, the major improvements were mainly on the top and middle levels of the retail data hierarchies, while at the more disaggregated levels, i.e., at the SKU, SKU-state and SKU-store levels, where demand patterns are more difficult to capture due to high volatility, easy to implement and computationally cheap methods such as exponential smoothing (ES) were competitive. [PERSON] and [PERSON] [20] showed that advanced ML methods for meta-learning tasks, to select the best forecast, can be effective in a heavily promoted environment. Nonetheless, the best forecasting method that was found to be most often the most accurate was exponential smoothing, the workhorse of many forecasting systems in practice, the performance of which can be augmented substantially when promotions and other indicators are included [11]. The usage of ML methods by retailers remains an open question, as it needs to ensure that any forecast value added is meaningful given the extra costs [4; 21; 22; 23]: skilled data scientists, significant amount of time for training the models, sufficient computational and data infrastructures, among other issues [24; 25; 26; 27; 28; 29]. [PERSON] et al. [30] used the M5 data to evaluate the forecasting and inventory performance of both established statistical approaches and advanced ML methods and concluded that simple methods may result in similar if not lower monetary costs than more sophisticated approaches. Moreover, to facilitate the adoption and continued use of a forecasting method in practice, the expertise within the organization [17] as well as model transparency and intelligibility are important attributes to gain user trust [31].