# PACF and ACF for AR and MA

I once heard the following statement:

The PACF (partial autocorrelation function) for MA processes behaves much like the ACF for AR processes; the PACF for AR processes behaves much like the PACF for AR processes.

How to understand the logic underlying this statement?

In addition to the strict mathematical proof, are there any approaches to understand this statement, the inherent relationship of AR, MA along with their PACF/ACF, from time series properties, statistics, or any other high-level thoughts?

• @Mumbo.Jumbo by getting the expression of the autocovariances in an AR process, it can be checked the decaying ACF of an AR(1) model, see for example this. For the MA model, you can get its AR representation (assuming invertibility) and see that it has coefficients of the form $\theta^i$ ($i=1,2,\dots$ and $\theta$ is the MA coefficient), this suggests a decaying FACP of the MA model. – javlacalle Jun 6 '17 at 19:48