We propose a vector auto-regressive (VAR) model with a low-rank constraint on the transition matrix. This new model is well suited to predict high-dimensional series that are highly correlated, or that are driven by a small number of hidden factors. We study estimation, prediction, and rank selection for this model in a very general setting. Our method shows excellent performances on a wide variety of simulated datasets. On macro-economic data from Giannone et al. (2015), our method is competitive with state-of-the-art methods in small dimension, and even improves on them in high dimension.(1995); Francq and Zakoian (2019). In this paper, we propose a vector auto-regressive (VAR) model that is suitable to predict high-dimensional series that are strongly correlated, or that are driven by a reasonable number of hidden factors. These features are captured by imposing a low-rank constraint on the transition matrix. The coefficients can be efficiently computed by convex optimization techniques.Let us briefly describe the motivation for this model. Assume we deal with an R M -valued process (X t ) t≥0 withfor a very large M . For examples, think of daily sales of items on Amazon, where an item might be iPhone 7 128Go black, Lord of the Rings -Harper Collins Box Set, 1991 or Estimation of Dependences Based on Empirical Data: Second Edition, by Vladimir Vapnik, Springer.Then it is obvious that even with a few years of observations, we observe at most X t for a few thousands of days t while M is probably of the order or 10 5 or 10 6 . Thus the estimation of the M 2 coefficients of the matrix A is impossible. Some constraints are necessary to reduce the dimension of the problem. We believe that the sparsity of A, studied by Davis et al. (2016) in another context, does not make sense here.On the other hand, it is clear that a few factors, like the current economic conditions, the period of the year, have a strong influence on the series. Assuming these factors H t are linear functions of X t , we can write them asThen, assuming that X t+1 can be linearly predicted by H t , we can predict X t+1 by U H t = U V X t for some M × r matrix U . At the end of the day, we indeed predict X t+1 by (U V )X t = AX t , but the rank of A is r M . Note that the assumption that the coefficient matrix A is low-rank in a multivariate regression model Y i = AX i +ξ i where Y i ∈ R s and X i ∈ R t was studied in Econometric theory as early as in the 50's Anderson (1951);Izenman (1975). It was referred to as RRR (reduced-rank regression). We refer the reader to Koltchinskii et al. (2011); Suzuki (2015); Alquier et al. (2017); Klopp et al. (2017a,b); Moridomi et al. (2018) for state-of-the-art results. Low-rank matrices were actually used to model high-dimensional time series by De Castro et al. (2017) and Alquier and Marie (2019), however, the models described in these papers cannot be straightforwardly used for prediction purposes. Here, we study estimation and prediction for the model (1).The paper is designed as follows. In the end of the in...

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We revisit the interest of classical statistical techniques for sales forecasting like exponential smoothing and extensions thereof (as Holt's linear trend method). We do so by considering ensemble forecasts, given by several instances of these classical techniques tuned with different (sets of) parameters, and by forming convex combinations of the elements of ensemble forecasts over time, in a robust and sequential manner. The machine-learning theory behind this is called "robust online aggregation", or "prediction with expert advice", or "prediction of individual sequences" (see Cesa-Bianchi and Lugosi, 2006). We apply this methodology to a hierarchical data set of sales provided by the e-commerce company Cdiscount and output forecasts at the levels of subsubfamilies, subfamilies and families of items sold, for various forecasting horizons (up to 6-week-ahead). The performance achieved is better than what would be obtained by optimally tuning the classical techniques on a train set and using their forecasts on the test set. The performance is also good from an intrinsic point of view (in terms of mean absolute percentage of error). While getting these better forecasts of sales at the levels of subsubfamilies, subfamilies and families is interesting per se, we also suggest to use them as additional features when forecasting demand at the item level.