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I am trying to implement an ARIMA model for my dataset. I performed a Dickey-Fuller test on my dataset and it said that it was stationary. I would like to know if the autocorrelation function can state if my dataset is stationary and how do I determine the order of my autoregressive model and moving average model from these diagrams. Also I would like to know exactly what does this autoccorelation and partial autocorrelation function say about my dataset? The last graph shows my dataset which is hourly wind speeds for 15 years.

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  • $\begingroup$ Have you tried SARIMA(1,0,0)x(1,0,0)? How do the ACF and PACF for the residuals look like? $\endgroup$
    – Richard Hardy
    Commented Mar 8, 2018 at 15:01

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Dickey-Fuller is not always perfect or perhaps there was an error in interpreting the results because stationary data does not produce those kinds of ACF and PACF plots. Is this data in hourly intervals? it appears there is a seasonal autocorrelation at 24 lags so you should first perform a diff(24) to remove seasonality and then maybe one more first order diff to remove trend and check out correlation plots again.

Choosing your ARIMA orders is a bit of an art. I would point you to this resource for a good place to start: https://otexts.org/fpp2/arima-r.html

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  • $\begingroup$ stationary data does not produce those kinds of ACF and PACF plots -- why not? Autocorrelation is perfectly possible in stationary data. you should first perform a diff(24) to remove seasonality and then maybe one more first order diff to remove trend Differencing should only be used for removing unit roots (seasonal or nonseasonal), not for removing deterministic components. When used in absence of unit roots, differencing induces a non-invertible moving average (MA) process in the errors. (Search for "overdifferencing" to learn more.) $\endgroup$
    – Richard Hardy
    Commented Mar 8, 2018 at 10:18
  • $\begingroup$ “As well as looking at the time plot of the data, the ACF plot is also useful for identifying non-stationary time series. For a stationary time series, the ACF will drop to zero relatively quickly, while the ACF of non-stationary data decreases slowly. Also, for non-stationary data, the value of r1 is often large and positive.” Hyndman. Forecasting Principles and Practice. Hyndman creTed the forecasting package for R so I’m partial to his advice and guidance on all things time series related. $\endgroup$
    – exchez
    Commented Mar 8, 2018 at 10:33
  • $\begingroup$ Hyndman is correct, but this is not exactly what we see in these plots. Try plotting the ACF and PACF of a stationary SARIMA with relatively large autoregressive coefficients and you will get just what you are seeing here. $\endgroup$
    – Richard Hardy
    Commented Mar 8, 2018 at 10:39
  • $\begingroup$ “Thus, time series with trends, or with seasonality, are not stationary — the trend and seasonality will affect the value of the time series at different times.“ Also from Hyndman. To me, seeing an autocorrelation at the same repeating interval implies seasonality. $\endgroup$
    – exchez
    Commented Mar 8, 2018 at 10:54
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    $\begingroup$ My point is that existence of seasonality does not imply a seasonal unit root, thus seasonal differencing is unwarranted.. $\endgroup$
    – Richard Hardy
    Commented Mar 8, 2018 at 11:55

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