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Given a time series problem,

  1. Should ACF and PACF be done before or after differencing that make the time series stationary?

  2. If ACF and PACF has shown different results, should the number of orders of AR/MA follows ACF or PACF?

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  1. After differencing! A stationary time series is a key (as in, crucial) assumption of all ARMA models. You are presumably trying to determine the orders of your AR and MA parts, using the Box-Jenkins methodology (looking at graphs of acf/pacf). In this case, if the true process is indeed AR/MA/ARMA then the series must be stationary by assumption (which means, differencing the time series if needed to make it stationary, i.e. this is the I part in the ARIMA model).

  2. Look at both of the plots. For the AR(p) part of your model, look at the PACF plot and look for consecutive significant lags and then a sharp cutoff to zero (insignificant lags). For the MA(q) part, look for the same thing but using the ACF plot. If you see that one plot shows something as described above and the other not, then perhaps your model is only AR(p) or MA(q) rather than ARMA(p,q). If none of the plots seem to suggest anything, then likely you do not have a stationary time series and/or an ARMA model is inappropriate.

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