Find a suitable ARMA model

Question

I know that the ACF and PACF shown below is either of a MA(2) or AR(2) process. How can I decide which one it is by just looking at the plots?

Answer 1

With the prior knowledge that the plots come from either an $\operatorname{AR}(2)$ or an $\operatorname{MA}(2)$ process, we should be able to recognize which one it is based on the following observations:

AR:

• The ACF plot of an $\operatorname{AR}(p)$ will be decaying exponentially if the parameters are all positive.
• For an $\operatorname{AR}(p)$ process, we can roughly read off the parameters from a PACF plot.
• The PACF plot will be insignificant for all lag terms $> p$.

MA:

• For an $\operatorname{MA}(q)$ process, we can read off the parameters from an ACF plot.
• The ACF plot will be insignificant for all lag terms $> q$.

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My comments above you can either take for granted or, much better: you can convince yourself that they are good guidelines by simulating processes and comparing their plots. Here is an example that you can run with R. Note that the parameters chosen here for the two processes are (most likely) not the parameters that the process in the question.

n <- 1e5
parm <- c(0.5,-0.3)

myAR <- arima.sim(list(ar = parm),n)
myMA <- arima.sim(list(ma = parm),n)

## AR
par(mfrow = c(1,2))
acf(myar)
pacf(myar)

## MA
par(mfrow = c(1,2))
acf(myma)
pacf(myma)