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This paper proposes a new approach to identifying the effective cointegrationrank in high-dimensional unit-root (HDUR) time series from a predictionperspective using reduced-rank regression. For a HDUR process $\mathbf{x}_t\in\mathbb{R}^N$ and a stationary series $\mathbf{y}_t\in \mathbb{R}^p$ ofinterest, our goal is to predict future values of $\mathbf{y}_t$ using$\mathbf{x}_t$ and lagged values of $\mathbf{y}_t$. The proposed frameworkconsists of a two-step estimation procedure. First, the Principal ComponentAnalysis is used to identify all cointegrating vectors of $\mathbf{x}_t$.Second, the co-integrated stationary series are used as regressors, togetherwith some lagged variables of $\mathbf{y}_t$, to predict $\mathbf{y}_t$. Theestimated reduced rank is then defined as the effective cointegration rank of$\mathbf{x}_t$. Under the scenario that the autoregressive coefficient matricesare sparse (or of low-rank), we apply the Least Absolute Shrinkage andSelection Operator (or the reduced-rank techniques) to estimate theautoregressive coefficients when the dimension involved is high. Theoreticalproperties of the estimators are established under the assumptions that thedimensions $p$ and $N$ and the sample size $T \to \infty$. Both simulated andreal examples are used to illustrate the proposed framework, and the empiricalapplication suggests that the proposed procedure fares well in predicting stockreturns.

Determination of the effective cointegration rank in high-dimensional time-series predictive regressions