  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.
