Time series forecasting has received a lot of attention with recurrent neural networks (RNNs) being one of the widely used models due to their ability to handle sequential data. Prior studies of RNNs for time series forecasting yield inconsistent results with limited insights as to why the performance varies for different datasets. In this paper, we provide an approach to link the characteristics of time series with the components of RNNs via the versatile metric of distance correlation. This metric allows us to examine the information flow through the RNN activation layers to be able to interpret and explain their performance. We empirically show that the RNN activation layers learn the lag structures of time series well. However, they gradually lose this information over a span of a few consecutive layers, thereby worsening the forecast quality for series with large lag structures. We also show that the activation layers cannot adequately model moving average and heteroskedastic time series processes. Last, we generate heatmaps for visual comparisons of the activation layers for different choices of the network hyperparameters to identify which of them affect the forecast performance. Our findings can, therefore, aid practitioners in assessing the effectiveness of RNNs for given time series data without actually training and evaluating the networks.

Vector autoregression (VAR) is a fundamental tool for modeling the joint dynamics of multivariate time series. However, as the number of component series is increased, the VAR model quickly becomes overparameterized, making reliable estimation difficult and impeding its adoption as a forecasting tool in high dimensional settings. A number of authors have sought to address this issue by incorporating regularized approaches, such as the lasso, that impose sparse or low-rank structures on the estimated coefficient parameters of the VAR. More traditional approaches attempt to address overparameterization by selecting a low lag order, based on the assumption that dynamic dependence among components is short-range. However, these methods typically assume a single, universal lag order that applies across all components, unnecessarily constraining the dynamic relationship between the components and impeding forecast performance. The lasso-based approaches are more flexible but do not incorporate the notion of lag order selection. We propose a new class of regularized VAR models, called hierarchical vector autoregression (HVAR), that embed the notion of lag selection into a convex regularizer. The key convex modeling tool is a group lasso with nested groups which ensure the sparsity pattern of autoregressive lag coefficients honors the ordered structure inherent to VAR. We provide computationally efficient algorithms for solving HVAR problems that can be parallelized across the components. A simulation study shows the improved performance in forecasting and lag order selection over previous approaches, and a macroeconomic application further highlights forecasting improvements as well as the convenient, interpretable output of a HVAR model.