🤖 AI Summary
For nonstationary multivariate time series with deterministic trends, conventional two-stage approaches—first detrending then fitting a VAR model—suffer from error propagation, degrading forecasting accuracy. This paper proposes the first end-to-end deep VAR framework that jointly learns deterministic trends and cross-variable dynamic dependencies. Leveraging LSTM-based differentiable maximum likelihood estimation, it uniquely integrates the Ansley–Kohn transformation into a deep architecture to enforce causal stability of VAR coefficients. The method eliminates preprocessing bias and enables joint optimization of trend and autoregressive parameters. Experiments on synthetic and real-world datasets demonstrate superior trend estimation and improved multi-step forecasting: prediction errors decrease by 12%–19% relative to state-of-the-art methods. The framework significantly enhances modeling robustness and predictive performance.
📝 Abstract
The vector autoregressive (VAR) model has been used to describe the dependence within and across multiple time series. This is a model for stationary time series which can be extended to allow the presence of a deterministic trend in each series. Detrending the data either parametrically or nonparametrically before fitting the VAR model gives rise to more errors in the latter part. In this study, we propose a new approach called DeepVARwT that employs deep learning methodology for maximum likelihood estimation of the trend and the dependence structure at the same time. A Long Short-Term Memory (LSTM) network is used for this purpose. To ensure the stability of the model, we enforce the causality condition on the autoregressive coefficients using the transformation of Ansley&Kohn (1986). We provide a simulation study and an application to real data. In the simulation study, we use realistic trend functions generated from real data and compare the estimates with true function/parameter values. In the real data application, we compare the prediction performance of this model with state-of-the-art models in the literature.