Independent or i.i.d. innovations is an essential assumption in the literature for analyzing a vector time series. However, this assumption is either too restrictive for a real-life time series to satisfy or is hard to verify through a hypothesis test. This paper performs statistical inference on a sparse high-dimensional vector autoregressive time series, allowing its white noise innovations to be dependent, even non-stationary. To achieve this goal, it adopts a post-selection estimator to fit the vector autoregressive model and derives the asymptotic distribution of the post-selection estimator. The innovations in the autoregressive time series are not assumed to be independent, thus making the covariance matrices of the autoregressive coefficient estimators complex and difficult to estimate. Our work develops a bootstrap algorithm to facilitate practitioners in performing statistical inference without having to engage in sophisticated calculations. Simulations and real-life data experiments reveal the validity of the proposed methods and theoretical results. Real-life data is rarely considered to exactly satisfy an autoregressive model with independent or i.i.d. innovations, so our work should better reflect the reality compared to the literature that assumes i.i.d. innovations.

翻译：独立或独立同分布新息是文献中分析向量时间序列的基本假设，然而，该假设对于现实时间序列而言要么过于严格难以满足，要么难以通过假设检验进行验证。本文对稀疏高维向量自回归时间序列进行统计推断，允许其白噪声新息存在相依性，甚至非平稳性。为实现这一目标，采用后选择估计量拟合向量自回归模型，并推导该估计量的渐近分布。自回归时间序列中的新息不假定独立，因此自回归系数估计量的协方差矩阵复杂且难以估计。本研究开发了一种自助法算法，使从业者无需进行复杂计算即可进行统计推断。模拟实验和真实数据实验验证了所提方法与理论结果的有效性。真实数据极少能精确满足具有独立或独立同分布新息的自回归模型，因此相比假设独立同分布新息的文献，本研究应能更准确地反映现实情况。