6. Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.4. Univariate Time Series Models
6.4.4.6. Box-Jenkins Model Identification

6.4.4.6.3.

Partial Autocorrelation Plot

Purpose: Model Identification for Box-Jenkins Models

Partial autocorrelation plots (Box and Jenkins, pp. 64-65, 1970) are a commonly used tool for model identification in Box-Jenkins models.

The partial autocorrelation at lag \(k\) is the autocorrelation between \(X_t\) and \(X_{t-k}\) that is not accounted for by lags 1 through \(k-1\).

There are algorithms, not discussed here, for computing the partial autocorrelation based on the sample autocorrelations. See (Box, Jenkins, and Reinsel 1970) or (Brockwell, 1991) for the mathematical details.

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. We look for the point on the plot where the partial autocorrelations essentially become zero. Placing a 95 % confidence interval for statistical significance is helpful for this purpose.

The approximate 95 % confidence interval for the partial autocorrelations are at \(\pm 2/\sqrt{N}\).

Sample Plot

This partial autocorrelation plot, for the southern oscillations data set, shows clear statistical significance for lags 1 and 2 (lag 0 is always 1). The next few lags are at the borderline of statistical significance. If the autocorrelation plot indicates that an AR model is appropriate, we could start our modeling with an AR(2) model. We might compare this with an AR(3) model.

Definition

Partial autocorrelation plots are formed by

Vertical axis:	Partial autocorrelation coefficient at lag \(h\).
Horizontal axis:	Time lag \(h \, (h=0, \, 1, \, 2, \, 3, \, \ldots)\).

In addition, 95 % confidence interval bands are typically included on the plot.

Questions

The partial autocorrelation plot can help provide answers to the following questions:

1. Is an AR model appropriate for the data?
2. If an AR model is appropriate, what order should we use?

Related Techniques

Autocorrelation Plot
Run Sequence Plot
Spectral Plot

Case Study

The partial autocorrelation plot is demonstrated in the Negiz data case study.

Software

Partial autocorrelation plots are available in many general purpose statistical software programs.