Partial Autocorrelation

Definition

The partial autocorrelation ${\varphi }_{kk}$ measures the correlation between $Y_t$ and $Y_{t-k}$ after removing the linear effects of the intermediate variables $Y_{t-1},Y_{t-2},\dots ,Y_{t-k+1}$.

${\varphi }_{kk}=\mathrm{Corr}(Y_t-{\hat{Y}}_t,Y_{t-k}-{\hat{Y}}_{t-k})$

where ${\hat{Y}}_t$ and ${\hat{Y}}_{t-k}$ are linear predictors based on intermediate values.

Convention

${\varphi }_{11}={\rho }_1\text{(first lag PACF equals ACF)}$

Formula for Lag 2

${\varphi }_{22}=\frac{{\rho }_2-{\rho }_1^2}{1-{\rho }_1^2}$

Key Distinction

• ACF measures total correlation (direct + indirect)
• PACF measures only direct correlation between $Y_t$ and $Y_{t-k}$

Properties by Model

AR(1)

${\varphi }_{11}=\varphi ,{\varphi }_{kk}=0\text{ for }k>1$

PACF cuts off after lag 1.

MA(1)

${\varphi }_{22}=\frac{-{\theta }^2}{1+{\theta }^2+{\theta }^4}$

PACF never exactly zero but decays exponentially.

Model Identification

Model	ACF Behavior	PACF Behavior
AR(p)	Decays exponentially	Cuts off after lag p
MA(q)	Cuts off after lag q	Decays exponentially
ARMA(p,q)	Decays exponentially	Decays exponentially

Related

• Sample Autocorrelation (ACF)
• Autocovariance and Autocorrelation
• AR(p) Process Model
• MA(1) Process Model