---
tag: 11YRFWD951
kind: concept
name: Partial autocorrelation function
msc: [62M10]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_autocorrelation_function
same_as:
  wikidata: Q1258253
  wikipedia: Partial autocorrelation function
---

## Summary

The partial autocorrelation function of a weakly stationary process gives, for each lag $h$, the correlation between $X_{t+h}$ and $X_t$.
It first removes the linear effect of the values in between.
It complements the [[autocorrelation function|YCZHYY8VE5]].
For an autoregressive model of order $p$ it vanishes beyond lag $p$.
This makes it the main tool for choosing the order of such a model.

## Explanations

### L1 human

Today's value may be correlated with the value two days ago only because both are linked to yesterday's value.
The partial autocorrelation at lag $2$ asks what correlation is left once yesterday's value has been taken into account.
In a process where each value depends only on the previous one plus fresh noise, nothing is left: the partial autocorrelation at lag $2$ is $0$.
The ordinary autocorrelation at lag $2$ of the same process is not $0$, because the influence passes through the middle day.