Partial autocorrelation function

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 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.

Description

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

${\displaystyle \alpha (1)=\operatorname {corr} (z_{t+1},z_{t}),{\text{ for }}k=1,}$

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

where ${\displaystyle P_{t,k}(x)}$ is the surjective operator of orthogonal projection of ${\displaystyle x}$ onto the linear subspace of Hilbert space spanned by ${\displaystyle z_{t+1},\dots ,z_{t+k-1}}$.

There are algorithms for estimating the partial autocorrelation based on the sample autocorrelations (Box, Jenkins, and Reinsel 2008 and Brockwell and Davis, 2009). 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 ${\displaystyle \pm 1.96/{\sqrt {n}}}$, 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

• Box, G. E. P.; Jenkins, G. M.; Reinsel, G. C. (2008). Time Series Analysis, Forecasting and Control (4th ed.). Hoboken, NJ: Wiley. ISBN 9780470272848.
• Brockwell, Peter; Davis, Richard (2009). Time Series: Theory and Methods (2nd ed.). New York: Springer. ISBN 9781441903198.
• Enders, Walter (2004). Applied Econometric Time Series (Second ed.). New York: John Wiley. pp. 65–67. ISBN 0-471-23065-0.

 This article incorporates public domain material from the National Institute of Standards and Technology document: "http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4463.htm".