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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?

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  • $\begingroup$ As far as I'm concerned, the unit root test can in no way "prove" that a series is stationary. See my question here: stats.stackexchange.com/questions/450468/… - I didn't get an answer but I'm pretty sure I'm right on that matter. $\endgroup$ May 16 '20 at 10:23
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I know they test for different things, but ultimately tests if a time series is stationary or not.

The mistake in your argument is that tests of autocorrelation assess stationarity. Actually, they assess presence of autocorrelation (unsurprisingly). Now, an autocorrelated time series can be stationary or nonstationary. (Note that a stationary time series can have zero or nonzero autocorrelation; nonzero autocorrelation by itself does not make a time series nonstationary.) Therefore, learning that a time series is autocorrelated does not indicate the time series is nonstationary. If you want to assess nonstationarity of the unit root kind (there are other kinds, too, e.g. nonstationarity due to a structural break or changing variance), you need a unit root test.

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  • $\begingroup$ Ok, I have done some research and would like to see if I am in the right direction. 1. Auto-correlation of a time series does not necessary access the stationarity of a time series. But for a stationary time series, ACF should converge towards 0. 2. Serial correlation in the residual of a regression model picks up the composite effect of everything not accounted for by the explanatory variables, e.g. misspecfication, etc. Hence, even if a the residuals have no auto-correlation, it does not mean that the regression is stationary. 3. To test stationarity of a time series, one can use ADF. $\endgroup$
    – OGARCH
    May 17 '20 at 4:32
  • $\begingroup$ I have asked the above in another post with more clarity - stats.stackexchange.com/questions/466932/… $\endgroup$
    – OGARCH
    May 17 '20 at 6:52
  • $\begingroup$ @OGARCH, you are correct, if by ACF should converge towards 0 you mean convergence as the time lag goes to infinity (as opposed to the sample size going to infinity for a fixed lag). $\endgroup$ May 17 '20 at 12:48

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