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

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 xt∈RN and a stationary series yt∈Rp of interest, our goal is to predict future values of yt using xt and lagged values of yt. The proposed framework consists of a two-step estimation procedure. First, the Principal Component Analysis is used to identify all cointegrating vectors of xt. Second, the co-integrated stationary series are used as regressors, together with some lagged variables of yt, to predict yt. The estimated reduced rank is then defined as the effective cointegration rank of xt. 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 <i>p</i> and <i>N</i> and the sample size <i>T</i> increase to infinity. 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 asset returns.