Determination of the effective cointegration rank in high-dimensional time-series predictive regressions
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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 𝐱_t∈ℝ^N and a stationary series 𝐲_t∈ℝ^p of interest, our goal is to predict future values of 𝐲_t using 𝐱_t and lagged values of 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 𝐱_t. Second, the co-integrated stationary series are used as regressors, together with some lagged variables of 𝐲_t, to predict 𝐲_t. The estimated reduced rank is then defined as the effective cointegration rank of 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 →∞. 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.
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