Visualizing the autocorrelation function

Estimating the autocorrelation function (ACF) at many lags allows us to assess how a time series x relates to its past. The numeric estimates are important for detailed calculations, but it is also useful to visualize the ACF as a function of the lag.

In fact, the acf() command produces a figure by default. It also makes a default choice for lag.max, the maximum number of lags to be displayed.

Three time series x, y, and z have been loaded into your R environment and are plotted on the right. The time series x shows strong persistence, meaning the current value is closely relatively to those that proceed it. The time series y shows a periodic pattern with a cycle length of approximately four observations, meaning the current value is relatively close to the observation four before it. The time series z does not exhibit any clear pattern.

In this exercise, you'll plot an estimated autocorrelation function for each time series. In the plots produced by acf(), the lag for each autocorrelation estimate is denoted on the horizontal axis and each autocorrelation estimate is indicated by the height of the vertical bars. Recall that the ACF at lag-0 is always 1.

Finally, each ACF figure includes a pair of blue, horizontal, dashed lines representing lag-wise 95% confidence intervals centered at zero. These are used for determining the statistical significance of an individual autocorrelation estimate at a given lag versus a null value of zero, i.e., no autocorrelation at that lag.