Dimension Reduction for High‐Dimensional Vector Autoregressive Models*
将高维VAR模型分解为一个小规模VAR和一个白噪声，从而用少量共同成分捕捉整个系统的动态，并提供了检测和估计方法，可用于识别商业周期频率的主要冲击。
Abstract This article aims to decompose a large dimensional vector autoregressive (VAR) model into two components, the first one being generated by a small‐scale VAR and the second one being a white noise. Hence, a reduced number of common components generates the entire dynamics of the large system through a VAR structure. This modelling, which we label as the dimension‐reducible VAR, extends the common feature approach to high‐dimensional systems, and it differs from the dynamic factor model in which the idiosyncratic component can also embed a dynamic pattern. We show the conditions under which this decomposition exists. We provide statistical tools to detect its presence in the data and to estimate the parameters of the underlying small‐scale VAR model. Based on our methodology, we propose a novel approach to identify the shock that is responsible for most of the common variability at the business cycle frequencies. We evaluate the practical value of the proposed methods by simulations as well as by an empirical application to a large set of US economic variables.