This paper first establishes some new asymptotic properties for the first k largest eigenvalues of the sample covariance matrices of the time series model. It shows that near unit root part is the leading term under weak conditions. We then propose a new estimator for near unit roots. The paper further develops a new near unit root test for the high–dimensional case. The proposed test naturally extends the well–known Phillips and Perron (PP) type of test to the high–dimensional near unit root setting. We also evaluate the finite–sample performance of the proposed test and then demonstrates that it outperforms its existing competitors.

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