I am confused with the purpose of unit root tests in relation to serial correlation tests. I know they test for different things, but ultimately tests if a time series is stationary or not.
From what I understand, you perform a unit root test on a time series and when you reject the null, that time series is I(0), and is stationary.
So then, what will be the purpose of the auto-correlation test on the residuals for in this case? Since the time series is already proven to be stationary by the unit root test, what then is the purpose of the auto-correlation test? Or do you only choose one test? Seems to me that if a time series has a unit root, performing an auto correlation test on the residuals will indicate non stationarity anyways, so why not just use the auto-correlation test in the first place.
Both test indicates the stationarity of the time series, and hence why should I perform both tests?