This paper proposes a new approach to identifying the effective cointegration rank in high-dimensional unit-root (HDUR) time series from a prediction perspective using reduced-rank regression. For a HDUR process $\mathbf{x}_t\in \mathbb{R}^N$ and a stationary series $\mathbf{y}_t\in \mathbb{R}^p$ of interest, our goal is to predict future values of $\mathbf{y}_t$ using $\mathbf{x}_t$ and lagged values of $\mathbf{y}_t$. The proposed framework consists of a two-step estimation procedure. First, the Principal Component Analysis is used to identify all cointegrating vectors of $\mathbf{x}_t$. Second, the co-integrated stationary series are used as regressors, together with some lagged variables of $\mathbf{y}_t$, to predict $\mathbf{y}_t$. The estimated reduced rank is then defined as the effective cointegration rank of $\mathbf{x}_t$. Under the scenario that the autoregressive coefficient matrices are sparse (or of low-rank), we apply the Least Absolute Shrinkage and Selection Operator (or the reduced-rank techniques) to estimate the autoregressive coefficients when the dimension involved is high. Theoretical properties of the estimators are established under the assumptions that the dimensions $p$ and $N$ and the sample size $T \to \infty$. Both simulated and real examples are used to illustrate the proposed framework, and the empirical application suggests that the proposed procedure fares well in predicting stock returns.

翻译：本文从预测视角出发，提出了一种基于降秩回归识别高维单位根时间序列有效协整秩的新方法。针对高维单位根过程 $\mathbf{x}_t\in \mathbb{R}^N$ 和关注的平稳序列 $\mathbf{y}_t\in \mathbb{R}^p$，我们的目标是利用 $\mathbf{x}_t$ 及 $\mathbf{y}_t$ 的滞后值预测 $\mathbf{y}_t$ 的未来取值。所提框架包含两步估计流程：首先，采用主成分分析法识别 $\mathbf{x}_t$ 的所有协整向量；其次，将协整后的平稳序列作为回归变量，并结合 $\mathbf{y}_t$ 的部分滞后变量对 $\mathbf{y}_t$ 进行预测。此时，估计出的降秩即定义为 $\mathbf{x}_t$ 的有效协整秩。在自回归系数矩阵为稀疏（或低秩）的假设下，我们应用最小绝对收缩与选择算子（或降秩技术）对高维情形下的自回归系数进行估计。当维度 $p$、$N$ 及样本量 $T$ 均趋于无穷时，我们建立了估计量的理论性质。通过模拟与真实数据案例验证了所提框架的有效性，实证应用表明，该方法在预测股票收益方面表现良好。