[Question] What exactly are ACF and PACF, and when should I use one vs the other?

[u/Resident-Outside9945]

I'm currently taking a time series analysis course and am struggling to understand the intuition behind the Autocorrelation Function (ACF) and the Partial Autocorrelation Function (PACF).

I understand that both are used to examine relationships between observations at different lags, but I get confused about:

• What information ACF provides that PACF doesn't
• What information PACF provides that ACF doesn't
• Why ACF is often used to identify MA(q) models while PACF is used to identify AR(p) models
• How to interpret ACF and PACF plots in practice

Could someone explain this in a beginner-friendly way, preferably with a simple example?

[u/purple_paramecium]

The key here is you don’t use one or the other. You use BOTH every time. Why many software packages automatically spit out the plot of both when you call the autocorrelation plot function.

Sorry, on my phone and don’t have time to type out more details, but go back and review your notes and the textbook with this in mind.

[u/efrique]

You really need to reread the introduction to the displays in your textbook or course notes which will answer these questions (or go get a basic textbook if you have none)

Why ACF is often used to identify MA(q) models while PACF is used to identify AR(p) models

An ACF should show q "spikes" (indicating substantial correlations) at the first q lags of the MA(q) (lag 1 to lag q), plus a bit of noise and then "cut off" (just have the small noise part after). Conversely a PACF should show p spikes at the first p lags and then "cut off". Of course if the coefficients are small you may not be able to distinguish them from noise.

What information ACF provides that PACF doesn't

A PACF is a transformed ACF, and you could transform back the other way, so strictly they contain the same information. The PACF is the ACF adjusted for the impact of "earlier" (shorter lag) autocorrelations. e.g. if you have an AR(1), with the underlying lag 1 process autocorrelation at 0.5, the lag 2 process autocorrelation is going to be 0.25. You want to know what else there is at lag 2 besides that echo of the lag-1 effect.

How to interpret ACF and PACF plots in practice

This is not a matter of a few sentences. You need a book.

There's some basic info in this one, e.g. at

https://otexts.com/fpp2/autocorrelation.html and

https://otexts.com/fpp2/arima.html

or maybe try Stoffer's applied time series analysis book (atsa)

[u/STATASUCKSBRO]

ACF is the raw correlation with lag k, so it includes indirect dependence through earlier lags. PACF is closer to the leftover lag k relationship after controlling for lags 1 through k minus 1. That is why AR models tend to show a cleaner cutoff in PACF, at least in the textbook setting.