Autocorrelation Function and Partial Autocorrelation [/ˈautokorrelaʃən ˈfəŋkʃən ənd ˈpɑrʃəl ˈautokorrelaʃən/] n - The autocorrelation function measures how nearly a series keeps company with its own past. For a return series, the ACF commonly hovers near zero with occasional spikes of noise:

Autocorrelation function of returns

At lag $k$, it is:

$${\rho }_k=\frac{\text{Cov}(y_t,y_{t-k})}{\text{Var}(y_t)}$$

A slow, linear decay in the ACF usually means the series wants differencing. A lone spike at lag one, then a cutoff, suggests a first-order moving-average process; an ACF that decays after differencing suggests an autoregressive component.

The partial autocorrelation function strips away the indirect effects of the shorter lags. It measures the correlation between $y_t$ and $y_{t-k}$ once $y_{t-1},\dots ,y_{t-k+1}$ have been accounted for. For an AR(p) process the PACF cuts off sharply after lag $p$; for an MA(q) process the ACF cuts off sharply after lag $q$.

These rules of thumb are the Box-Jenkins recipe for choosing orders:

• AR(p): PACF cuts off after lag p, ACF decays.
• MA(q): ACF cuts off after lag q, PACF decays.
• ARMA(p,q): both tail off, and estimation must disentangle them.

In finance the signals are seldom so clean. Returns carry small, noisy autocorrelations; volatility carries persistent autocorrelation; spreads may show plain mean reversion. The ACF tells you where to look, but it does not relieve you of model diagnostics.

Confidence bands for the ACF and PACF are commonly drawn on the null that the true autocorrelation is zero. A spike beyond the band is statistically significant, yet statistical significance is not economic significance: a lag-one autocorrelation of 0.01 may be significant with data enough, and still untradeable after costs.

Compute the ACF and PACF of the residuals always, not of the raw series alone. If the residuals still show structure, the model is incomplete. See ARIMA and Conditional Mean Models for what is to be done once the structure has been named.