The reduced-rank vector autoregressive (VAR) model can be interpreted as a supervised factor model, where two factor modelings are simultaneously applied to response and predictor spaces. This article introduces a new model, called vector autoregression with common response and predictor factors, to explore further the common structure between the response and predictors in the VAR framework. The new model can provide better physical interpretations and improve estimation efficiency. In conjunction with the tensor operation, the model can easily be extended to any finite-order VAR model. A regularization-based method is considered for the high-dimensional estimation with the gradient descent algorithm, and its computational and statistical convergence guarantees are established. For data with pervasive cross-sectional dependence, a transformation for responses is developed to alleviate the diverging eigenvalue effect. Moreover, we consider additional sparsity structure in factor loading for the case of ultra-high dimension. Simulation experiments confirm our theoretical findings and a macroeconomic application showcases the appealing properties of the proposed model in structural analysis and forecasting.

翻译：降秩向量自回归（VAR）模型可被解释为一种有监督因子模型，其中对响应空间和预测空间同时应用了两种因子建模方法。本文提出一种名为"具有共同响应与预测因子的向量自回归"的新模型，以进一步探究VAR框架中响应变量与预测变量之间的共同结构。该模型能提供更清晰的物理解释并提升估计效率。结合张量运算，该模型可轻松扩展至任意有限阶VAR模型。针对高维估计问题，我们采用基于梯度下降算法的正则化方法，并建立了其计算与统计收敛性保证。针对具有普遍横截面依赖性的数据，我们开发了一种响应变量变换方法以缓解特征值发散效应。此外，针对超高维情形，我们还在因子载荷中考虑了额外的稀疏结构。仿真实验验证了理论发现，宏观经济学应用则展示了该模型在结构分析与预测中的优异特性。