0.1 Partial Autocorrelation Function (PACF)
The partial autocorrelation at lag \(n\), \(\alpha(n) = \phi_{nn}\), is the correlation between \(X_t\) and \(X_{t+n}\) after removing the linear effect of the intervening observations \(X_{t+1}, \dots, X_{t+n-1}\). Equivalently, \(\phi_{nn}\) is the coefficient on \(X_{t-n}\) in the best linear predictor (see Time Series Best Linear Predictors) of \(X_t\) from \(X_{t-1}, \dots, X_{t-n}\). By convention \(\alpha(0)=1\).
For a causal \(AR(p)\) process, \(\alpha(n) = \phi_{nn} = 0\) for all \(n > p\) — the PACF cuts off at lag \(p\), mirroring how the ACF cuts off at lag \(q\) for \(MA(q)\) (see Moving Average Processes). This complementary behavior is the basis of the classical Box-Jenkins identification strategy for AR vs MA models.
Example (PACF of MA(1)). For \(\theta(B)=1+\theta_1B\), \[ \alpha(1) = \rho_X(1) = \frac{\theta_1}{1+\theta_1^2}, \qquad \alpha(k) = \frac{(-1)^{k+1}\theta_1^k}{1+\theta_1^2+\dots+\theta_1^{2k}}\ (k>0), \] which decays geometrically rather than cutting off — consistent with \(MA\) having infinite AR order.
0.2 Estimation
The sample PACF \(\hat\alpha(h) = \hat\phi_{hh}\) is obtained either by solving the sample Yule-Walker Equations at each order \(h\), or recursively via the Durbin-Levinson Algorithm.
0.3 References
- Time Series Analysis
- [[pstat274-time-series-summary-notes]]