DeepVARwT: Deep Learning for a VAR Model with Trend

TL;DR

DeepVARwT integrates an LSTM-based trend generator with a VAR($p$) model for the residuals in a joint, likelihood-based framework to address nonstationary multivariate time series. By learning $mu_t$ and the VAR parameters simultaneously and enforcing causality via the Ansley-Kohn transform, it provides stable, end-to-end maximum likelihood estimation under a Gaussian error structure. Across simulations and three real-data sets (US macro series and global temperatures), DeepVARwT yields improved forecast accuracy and tighter prediction intervals compared with VAR with trend and several deep-learning baselines. The approach offers a principled, trend-aware forecasting tool with potential extensions to high-dimensional settings and non-Gaussian data.

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.

Figures (16)

• Figure 1: US macroeconomic series spanning 1955Q1 to 2003Q1.
• Figure 2: US macroeconomic series and estimated polynomial trends (red lines).
• Figure 3: The computation unit for hidden state $\bm{h}_{t}$ in an LSTM.
• Figure 4: Simulated series from VAR(2) model with trend.
• Figure 5: True (solid, blue) and estimated trends from DeepVARwT (left pane) and VARwT (right pane) with $\mathrm{MAD}$ at first quartile (short dashed, black), third quartile (long dashed, black), and median (dotted, red).