ACF vs PACF in ARIMA Given a time series problem, 


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

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

*To know that the system is non-stationary and that differencing may help, you calculate ACF and PACF. Then, after differencing, you calculate ACF and PACF again to see if they are decaying relatively rapidly and the time series is stationary. If yes, you use them for identification of the modeling framework for the mean process (AR vs MA vs ARMA). You use them on squared residuals for identification of the modeling framework for the variance (constant vs ARCH vs GARCH)... Note, ACF & PACF should not be used solely. They are just a part of the model selection toolkit. Other methods to look at: Box-Ljung test on residuals, AIC, BIC, the test on whether Implied_CDF_of_Model_Forecast(model forecast at time t) is uniformly distributed on [0,1].


\2. ACF and PACF are not in competition. They should be used jointly in decision making. Check out the Box-Jenkins approach.
A: Answer 1.
Remember that ACF and PACF are useful instruments in specifying the model and are not in competition. They are rather complementary. Therefore,


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*they are commonly used before differencing the ARIMA to understand what process you have and potentially detect non-stationarity (that however must be confirmed via appropriate testing! This is just a first-instance visual inspection);

*then they are used again after identifying a non-stationary process in order to specify the first difference of the process.
Let me clarify as follows.
A random walk process for example has a very high persistency, which implies a very strong ACF with almost no decay over time. Instead a stationary Arma process has a decay. A MA(q) process has a decay to 0 of the ACF after q lags. And so on.... So, as you may notice, the ACF allows you to grab a visual idea of the persistency of the process, and identify what kind process we are facing, including stationarity.
The PACF instead is used to further define the model once you have contemplated the ACF: indeed, given for clarity a AR(q) process whose ACF shows a gradual/exponential decay over time, the PACF helps you formulate an hypothesis about q. Indeed if the process is an actual AR(q) then you should see the PACF converging to 0 after q lags. In that sense they should be used complementary. 
Therefore, after identifying that a process is an ARIMA with the help of the ACF and PACF (as described above), then you can repeat this exercise and use ACF and PACF on the first difference of the process to further specify the form of this first difference.
Answer 2
Very often ACF and PACF show different results, which do not contradict themself and are both useful and valuable pieces of informations. Their use has been explained above, so I will just repeat that given a generic process, the ACF is used to have a visual idea of the persistency/memory of the process (is it a ARIMA, is it a AR, is it a MA?). While the PACF is used to better specify the process and define the process order (is it a AR(q) or AR(q-1) or AR(q-3)). These pieces of info must be used together. So, once you have visually inspected an ARIMA process via ACF and appropriately tested for non-stationarity via other tools, then the ACF on the first difference will let you understand whether the first difference is a MA/AR/ARMA and the PACF will help you understand the order of such MA/AR/ARMA (is the first difference of the ARIMA a AR(q) or AR(q-1)...).
See also this question and this article and this for clarity
A: All of what @stans says is true IFF there is no deterministic structure in the series i.e pulses, level/step shifts , seasonal pulses and local time trends AND the parameters of the underlying model are invariant over time. This is why we follow the more general paradigm https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf suggesting an iterative self-checking process.
QUICK ANSWERS TO BOTH QUESTIONS:


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*not necessarily , the data will dictate the best approach

*not necessarily , the data will dictate the best approach
