Existing models for high-dimensional time series are overwhelmingly developed within the finite-order vector autoregressive (VAR) framework, whereas the more flexible vector autoregressive moving averages (VARMA) have been much less considered. This paper introduces a high-dimensional model for capturing VARMA dynamics, namely the Scalable ARMA (SARMA) model, by combining novel reparameterization and tensor decomposition techniques. To ensure identifiability and computational tractability, we first consider a reparameterization of the VARMA model and discover that this interestingly amounts to a Tucker-low-rank structure for the AR coefficient tensor along the temporal dimension. Motivated by this finding, we further consider Tucker decomposition across the response and predictor dimensions of the AR coefficient tensor, enabling factor extraction across variables and time lags. Additionally, we consider sparsity assumptions on the factor loadings to accomplish automatic variable selection and greater estimation efficiency. For the proposed model, we develop both rank-constrained and sparsity-inducing estimators. Algorithms and model selection methods are also provided. Simulation studies and empirical examples confirm the validity of our theory and advantages of our approaches over existing competitors.

翻译：现有高维时间序列模型主要受限于有限阶向量自回归（VAR）框架，而更具灵活性的向量自回归移动平均（VARMA）模型尚未得到充分研究。本文通过结合新颖的重参数化与张量分解技术，提出了一种捕捉VARMA动态的高维模型——可扩展ARMA（SARMA）模型。为确保可识别性与计算可行性，我们首先对VARMA模型进行重参数化，并发现该过程在时间维度上等价于对AR系数张量施加Tucker低秩结构。受此发现启发，我们进一步对AR系数张量的响应变量与预测变量维度进行Tucker分解，从而提取跨变量与时滞的潜在因子。此外，我们引入因子载荷的稀疏性假设以实现自动变量选择与更高估计效率。针对所提模型，我们分别开发了秩约束估计量与稀疏诱导估计量，并提供了相应的算法与模型选择方法。仿真实验与实证分析验证了理论的有效性，并表明所提方法相较于现有竞争模型具有显著优势。