Partial autocorrelation function

Partial autocorrelation function

In time series analysis, the partial autocorrelation function (PACF) or PARtial autoCORrelation (PARCOR) 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 zt, the partial autocorrelation of lag k, denoted α(k), is the autocorrelation between zt and zt + k with the linear dependence of zt + 1 through to zt + k − 1 removed; equivalently, it is the autocorrelation between zt and zt − 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 Pt,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 1970) or (Brockwell, 1991) 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, pp. 64–65, 1970) are a commonly used tool for model identification in Box-Jenkins models. Specifically, partial autocorrelations are useful in 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 a multivariate normal distribution.

See also

References

  • Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis, Forecasting and Control, 3rd ed. Prentice Hall, Englewood Cliffs, NJ.
  • Brockwell, Peter and Davis, Richard (2006) Time Series: Theory and Methods, 2nd ed. Springer

 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".

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