As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and difficulty of interpretation, especially for high-dimensional time series. This paper proposes a novel sparse infinite-order VAR model for high-dimensional time series, which avoids all above drawbacks while inheriting essential temporal patterns of the VARMA model. As another attractive feature, the temporal and cross-sectional structures of the VARMA-type dynamics captured by this model can be interpreted separately, since they are characterized by different sets of parameters. This separation naturally motivates the sparsity assumption on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved with little loss of temporal information. We introduce two $\ell_1$-regularized estimation methods for the proposed model, which can be efficiently implemented via block coordinate descent algorithms, and derive the corresponding nonasymptotic error bounds. A consistent model order selection method based on the Bayesian information criteria is also developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis.

翻译：作为一种特殊的无限阶向量自回归（VAR）模型，向量自回归滑动平均（VARMA）模型能够捕捉比广泛使用的有限阶VAR模型丰富得多的时序模式。然而，其不可识别性、计算难解性以及解释困难等问题长期制约着其实用性，尤其是在处理高维时间序列时。本文针对高维时间序列提出了一种新颖的稀疏无限阶VAR模型，该模型在继承VARMA模型关键时序模式的同时，避免了上述所有缺陷。作为另一项吸引人的特性，该模型捕获的VARMA型动态的时序结构与截面结构可通过不同参数集分别刻画，从而实现独立解释。这种分离自然引出了对决定截面依赖关系的参数施加稀疏性假设，从而在几乎不损失时序信息的前提下，显著提升统计效率与可解释性。我们为所提模型引入了两种基于ℓ1正则化的估计方法，可通过块坐标下降算法高效实现，并推导了相应的非渐近误差界。同时还开发了一种基于贝叶斯信息准则的一致模型阶数选择方法。仿真实验与真实宏观经济数据分析验证了所提方法的优越性。