The goal of ADF is to test whether we have unit root or not.
DF (Dickey-Fuller) test equation (regression equation) may include autocorrelation, and its result is not so reliable.
In ADF, additional lags of the differenced variable are added to the right of the ADF regression equation. These added lags of the differenced variable reduces autocorrelation.
Now, assume that we are performing ADF test when max lag length (say 14) is given (or assume max lag length (say 14) is the obtained from Schwetz's formula; as in Eviews).
If ADF test results in optimal lag length of 5 (via AIC or ABC criterion) in ADF test regression equation, then do we really know that the ADF equation with 5 lag length includes no autocorrelation?
OR, do we have to check that the ADF test regression equation with optimal lag length of 5 includes no autocorrelation by the help of some autocorrelation tests?
Any answer with related sources will be greatly appreciated.