Partial autocorrelation function

Partial autocorrelation function of Lake Huron's depth

In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorrelation function, which does not control for other lags.

This function plays an important role in data analysis aimed at identifying the extent of the lag in an autoregressive (AR) model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, whereby 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.

Definition

Given a time series $z_{t}$, the partial autocorrelation of lag $k$, denoted $\phi _{k,k}$, is the autocorrelation between $z_{t}$ and $z_{t+k}$ with the linear dependence of $z_{t}$ on $z_{t+1}$ through $z_{t+k-1}$ removed. Equivalently, it is the autocorrelation between $z_{t}$ and $z_{t+k}$ that is not accounted for by lags $1$ through $k-1$, inclusive.$$\phi _{1,1}=\operatorname {corr} (z_{t+1},z_{t}),{\text{ for }}k=1,$$$$\phi _{k,k}=\operatorname {corr} (z_{t+k}-{\hat {z}}_{t+k},\,z_{t}-{\hat {z}}_{t}),{\text{ for }}k\geq 2,$$where ${\hat {z}}_{t+k}$ and ${\hat {z}}_{t}$ are linear combinations of $\{z_{t+1},z_{t+2},...,z_{t+k-1}\}$ that minimize the mean squared error of $z_{t+k}$ and $z_{t}$ respectively. For stationary processes, ${\hat {z}}_{t+k}$ and ${\hat {z}}_{t}$ are the same.

Calculation

The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm:$$\phi _{n,n}={\frac {\rho (n)-\sum _{k=1}^{n-1}\phi _{n-1,k}\rho (n-k)}{1-\sum _{k=1}^{n-1}\phi _{n-1,k}\rho (k)}}$$where $\phi _{n,k}=\phi _{n-1,k}-\phi _{n,n}\phi _{n-1,n-k}$ for $1\leq k\leq n-1$ and $\rho (n)$ is the autocorrelation function.

The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series.

Examples

The partial autocorrelation of white noise is zero for all lags.

AR models have nonzero partial autocorrelations for lags less than or equal to its order. In other words, the partial autocorrelation of an AR(p) process is zero at lags greater than p.

For moving average (MA) models, their partial autocorrelation exponentially decays to 0. For MA models that have $\phi _{1,1}>0$, the decay is oscillating and the other models with $\phi _{1,1}<0$ have geometric decay.

The partial autocorrelation function of an ARMA(p, q) model also exponentially decays but only after lags greater than p.

Autoregressive Model Identification

Sample partial autocorrelation function of a simulated AR(3) time series

Partial autocorrelation plots 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 at least moderately large (say $n>30$) and that the underlying process has finite second moment.

References

 This article incorporates public domain material from http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4463.htm. National Institute of Standards and Technology. {{citation}}: External link in |title= (help)