The autocorrelation function (ACF) summarizes how strongly a time series is related to its own past values at different lags. It is widely used to describe persistence and to choose and diagnose time-series models.
Definition (conceptual)
For a covariance-stationary series (y_t), the ACF at lag (k) is:
[ \rho(k) = \text{Corr}(y_t, y_{t-k}) ]
Equivalently, using autocovariances (\gamma(k)=\text{Cov}(y_t,y_{t-k})):
[ \rho(k)=\frac{\gamma(k)}{\gamma(0)} ]
In applied work, people often look at the sample ACF computed from data and plotted as a correlogram.
Why students use the ACF
- Describe persistence: slow decay suggests strong persistence; quick decay suggests weak dependence.
- Detect seasonality: spikes at seasonal lags indicate periodic structure.
- Model identification: ACF and PACF patterns help choose AR/MA orders in ARMA/ARIMA models.
- Diagnostics: after fitting a model, residual ACF should show little remaining correlation.
ACF vs PACF (quick distinction)
- ACF: correlation with lagged values (includes indirect effects through intermediate lags).
- PACF: “direct” correlation at lag \(k\) controlling for lags 1 through \(k-1\).
Practical Interpretation
Two common patterns are:
- AR(p) behavior: the PACF often cuts off after (p) lags, while the ACF tapers off.
- MA(q) behavior: the ACF often cuts off after (q) lags, while the PACF tapers off.
These are rules of thumb, not mechanical laws, especially in small samples.