Autocorrelation

Autocorrelation is a measure of the correlation between the lagged values of a time series. For example, r₁ is the autocorrelation between y_t and y_t-1; similarly, r₂ is the autocorrelation between y_t and y_t-2. This can be summarized in the following formula:

In the preceding formula, T is the length of the time series.

For example, say that we have the correlation coefficients, as shown in the following diagram:

To plot it, we get the following:

The following are some observations from this autocorrelation function plot:

• r₄ is higher than other lags, which is mostly because of a seasonal pattern
• The blue lines are the indicators of whether correlations are significantly different from zero
• Autocorrelation at lag 0 is always 1