The standard vector autoregressive (VAR) models suffer from overparameterization which is a serious issue for high-dimensional time series data as it restricts the number of variables and lags that can be incorporated into the model. Several statistical methods, such as the reduced-rank model for multivariate (multiple) time series (Velu et al., 1986; Reinsel and Velu, 1998; Reinsel et al., 2022) and the Envelope VAR model (Wang and Ding, 2018), provide solutions for achieving dimension reduction of the parameter space of the VAR model. However, these methods can be inefficient in extracting relevant information from complex data, as they fail to distinguish between relevant and irrelevant information, or they are inefficient in addressing the rank deficiency problem. We put together the idea of envelope models into the reduced-rank VAR model to simultaneously tackle these challenges, and propose a new parsimonious version of the classical VAR model called the reduced-rank envelope VAR (REVAR) model. Our proposed REVAR model incorporates the strengths of both reduced-rank VAR and envelope VAR models and leads to significant gains in efficiency and accuracy. The asymptotic properties of the proposed estimators are established under different error assumptions. Simulation studies and real data analysis are conducted to evaluate and illustrate the proposed method.

翻译：标准向量自回归模型面临过度参数化问题，这对高维时间序列数据尤为严重，因为它限制了模型中可包含的变量数量和滞后阶数。已有多种统计方法尝试解决VAR模型的参数空间降维问题，例如多元（多变量）时间序列的降秩模型以及包络VAR模型。然而，这些方法在提取复杂数据中的相关信息时可能效率不足——它们要么无法区分相关与不相关信息，要么不能有效处理秩亏缺问题。我们将包络模型思想融入降秩VAR模型，以同时应对这些挑战，并提出了一种经典VAR模型的简约化新版本，称为降秩包络VAR模型。本文提出的REVAR模型融合了降秩VAR和包络VAR模型的优势，在效率和精度上均取得显著提升。在不同误差假设下，我们建立了所提出估计量的渐近性质，并通过模拟研究和实际数据分析对方法进行了评估与验证。