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

Abstract

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.