# Partial autocorrelation function

In time series analysis, the partial autocorrelation function (PACF) plays an important role in data analyses aimed at identifying the extent of the lag in an autoregressive model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, where by plotting the partial autocorrelative functions one could determine the appropriate lags p in an AR (p) model or in an extended ARIMA (p,d,q) model.

## Description

Given a time series $z_t$, the partial autocorrelation of lag k, denoted $\alpha(k)$, is the autocorrelation between $z_t$ and $z_{t+k}$ with the linear dependence of $z_{t+1}$ through to $z_{t+k-1}$ removed; equivalently, it is the autocorrelation between $z_t$ and $z_{t+k}$ that is not accounted for by lags 1 to k − 1, inclusive.

$\alpha(1) = \operatorname{Cor}(z_t,z_{t+1})$
$\alpha(k) = \operatorname{Cor}(z_{t+k} - P_{t,k}(z_{t+k}),\, z_t - P_{t,k}(z_t)),\text{ for }k\geq 2,$

where $P_{t,k}(x)$ denotes the projection of $x$ onto the space spanned by $z_{t+1}, \dots, z_{t+k-1}$.

There are algorithms, not discussed here, for estimating the partial autocorrelation based on the sample autocorrelations. See (Box, Jenkins, and Reinsel 2008) or (Brockwell and Davis, 2009) for the mathematical details. These algorithms derive from the exact theoretical relation between the partial autocorrelation function and the autocorrelation function.

Partial autocorrelation plots (Box and Jenkins, Chapter 3.2, 2008) are a commonly used tool for identifying the order of an autoregressive model. The partial autocorrelation of an AR(p) process is zero at lag p + 1 and greater. If the sample autocorrelation plot indicates that an AR model may be appropriate, then the sample partial autocorrelation plot is examined to help identify the order. One looks for the point on the plot where the partial autocorrelations for all higher lags are essentially zero. Placing on the plot an indication of the sampling uncertainty of the sample PACF is helpful for this purpose: this is usually constructed on the basis that the true value of the PACF, at any given positive lag, is zero. This can be formalised as described below.

An approximate test that a given partial correlation is zero (at a 5% significance level) is given by comparing the sample partial autocorrelations against the critical region with upper and lower limits given by $\pm 1.96/\sqrt{n}$, where n is the record length (number of points) of the time-series being analysed. This approximation relies on the assumption that the record length is moderately large (say n>30) and that the underlying process has finite second moment.