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A linear regression is performed as part of some unit root tests such as ADF, ADF-GLS (ERS), and PP.

Should I pay attention to the correlation coefficients (ordinary, adjusted) of the regression?

For example, do poor correlations invalidate any results of these tests?

Or, can I trust the test statistic to incorporate sufficient information, so I can safely ignore the correlation coefficients?

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When your goal is to learn about stationarity, I do not see added value in the correlation coefficient.

For example, the ADF coefficient statistic $T(\hat\rho-1)$, with $$\hat\rho=\frac{\sum_ty_ty_{t-1}}{\sum_t y_{t-1}^2}$$ the coefficient of the AR(1) regression, already contains more or less the information contained in the correlation coefficient.

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