---
tag: T1VJ281P1V
kind: definition
name: Partial autocorrelation function
msc: [62M10]
defines: 11YRFWD951
uses: [YCZHYY8VE5, DT0QXJE705, 0B8KZRT28M, 0DNN9B6Q2F, 2JC96FMGA1]
sources:
  - url: https://en.wikipedia.org/wiki/Partial_autocorrelation_function
---

## Definition

Let $(X_t)_{t \in \Z}$ be a real weakly [[stationary process|DT0QXJE705]] with mean $\mu$ and autocovariance $\gamma$, with $\gamma(0) > 0$.
Let $L^2$ be the space of square-integrable random variables.
For $t \in \Z$ and $h \ge 2$, let $P_{t,h} Y$ be the [[orthogonal projection|0B8KZRT28M]] in $L^2$ of $Y$ onto the span of $1, X_{t+1}, \dots, X_{t+h-1}$.
So $P_{t,h} Y$ is the best linear predictor of $Y$ from these variables.
The *partial autocorrelation function* $\alpha$ is defined for $h \ge 1$ by:

1. $\alpha(1) = \rho(1)$, where $\rho$ is the [[autocorrelation function|YCZHYY8VE5]].
2. For $h \ge 2$, $\alpha(h)$ is the [[correlation coefficient|0DNN9B6Q2F]] of $X_{t+h} - P_{t,h} X_{t+h}$ and $X_t - P_{t,h} X_t$.

By stationarity, $\alpha(h)$ does not depend on $t$.

## Remarks

Equivalently, $\alpha(h) = \phi_{hh}$, the last coefficient of the best linear predictor $\mu + \sum_{j=1}^{h} \phi_{hj} (X_{t+h-j} - \mu)$ of $X_{t+h}$ from $X_{t+h-1}, \dots, X_t$, provided the $h \times h$ matrix $[\gamma(i - j)]$ is nonsingular.
The sample partial autocorrelation is obtained by putting the sample autocovariances into these formulas, for instance by the Durbin–Levinson recursion.

Example: let $X_t = \phi X_{t-1} + \varepsilon_t$ be a stationary [[autoregressive model|2JC96FMGA1]] of order $1$ with $\abs{\phi} < 1$ and uncorrelated noise $\varepsilon_t$.
Then $\alpha(1) = \rho(1) = \phi$.
For $h \ge 2$, the best linear predictor of $X_{t+h}$ from $X_{t+1}, \dots, X_{t+h-1}$ is $\phi X_{t+h-1}$, and the residual $\varepsilon_{t+h}$ is uncorrelated with $X_t - P_{t,h} X_t$, so $\alpha(h) = 0$.

Non-example: a random walk is not weakly stationary, so clause 1 and clause 2 do not define a function of the lag alone.