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I have yearly data. When I do a Dickey-Fuller test it gives me insignificant results, indicating that the series are non-stationary. After differencing them the DFT tells me they are now significant and stationary.

When using an ACF/PACF plot on the original series it is obvious that an AR(1) process should be used.

When plotting ACF/PACF for the first difference I can't make that same inference again because there is no significant spike in the ACF.

ACF

PACF

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There is nothing conflicting here. Differencing a unit root process removes the unit root. If the original process is AR(1) with a unit root, $y_t=y_{t-1}+\varepsilon_t$, the differenced process is just white noise, $\Delta y_t=\varepsilon_t$. This is what you see in the ACF and PACF for the differenced data.

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  • $\begingroup$ Based on that you would conclude that you would use an AR(1) process? $\endgroup$
    – Narog
    May 14 at 20:03
  • $\begingroup$ @Narog, why the question? You write: When using an ACF/PACF plot on the original series it is obvious that an AR(1) process should be used. If it is obvious, then it is obvious, or?.. $\endgroup$
    – Richard Hardy
    May 14 at 20:06
  • $\begingroup$ shouldn't you look at the ACF/PACF of the stationary data? The non-stationary looks like an AR1 but the stationary more like a MA1 or am I incorrect assuming this. $\endgroup$
    – Narog
    May 15 at 12:06
  • $\begingroup$ @Narog, all the bars are inside the confidence bounds. That is a signature of white noise. Where do you see any signs of MA(1)? (Note: ACF(0) is always 1, regardless of the data generating process.) $\endgroup$
    – Richard Hardy
    May 15 at 12:10

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