Partial Autocorrelation Function

Study guide: Exam MAS-II

The partial autocorrelation function (PACF) gives the correlation between $Y_t$ and $Y_{t-k}$ after removing the effect of the intervening lags $Y_{t-1}, \dots, Y_{t-k+1}$ . It is what separates a direct dependence at lag $k$ from one inherited through the lags in between.

$\phi_{kk} = \mathrm{Corr}\!\left(Y_t, Y_{t-k} \mid Y_{t-1}, \dots, Y_{t-k+1}\right)$

$\phi_{kk} = \text{the } k\text{-th coefficient in an AR}(k) \text{ fit}$

The problem it solves: in an AR(1) with $\phi = 0.8$ , the ACF at lag 2 is $0.64$ — not because $Y_{t-2}$ matters directly, but because it acted through $Y_{t-1}$ . The PACF at lag 2 is 0, correctly reporting no direct link
The identification rule, the mirror of the ACF's: an AR( $p$ ) process has PACF cutting off after lag $p$ ; an MA( $q$ ) has a PACF that tails off
Same significance bands, $\pm 1.96/\sqrt{n}$
$\phi_{11} = \rho_1$ always; the two functions agree at lag 1 and diverge after
Read together with the ACF, the pair identifies the model:

	ACF	PACF
AR( $p$ )	tails off	cuts off after $p$
MA( $q$ )	cuts off after $q$	tails off
ARMA( $p,q$ )	tails off	tails off

ACF and PACF Together A stationary series of 100 observations has ACF 0.71, 0.48, 0.35, 0.24, 0.17 (a smooth geometric decay) and PACF 0.71, −0.05, 0.08, 0.02, −0.03. Identify the model.

Answer Bands: $\pm 0.196$ . The ACF decays geometrically and stays significant for several lags — tailing off. The PACF has one significant spike at lag 1 and nothing after — cutting off at 1. AR(1), with $\hat\phi_1 \approx 0.71$ . Consistent: the ACF of an AR(1) is $\rho_k = \phi^k$ , and $0.71^2 = 0.50 \approx 0.48$ , $0.71^3 = 0.36 \approx 0.35$ ✓