Identifying model via ACF/PACF

Model	ACF	PACF
AR(1)	${\rho }_k={\varphi }^k$ (exponential decay)	${\varphi }_{11}=\varphi$, ${\varphi }_{kk}=0$ for $k>1$ (cuts off)
AR($p$)	Decays exponentially/oscillates	Cuts off after lag $p$
MA(1)	${\rho }_1\ne 0$, ${\rho }_k=0$ for $k>1$ (cuts off)	Decays exponentially
MA($q$)	Cuts off after lag $q$	Decays exponentially
ARMA($p,q$)	Decays exponentially	Decays exponentially

Procedure

Use ACF and PACF patterns to identify the order of AR, MA, and ARMA models.

Step 1: Examine ACF

1. Plot sample ACF $r_k$ vs lag $k$
2. Check for significance: $|r_k|>\frac{1.96}{\sqrt{n}}$
3. Identify pattern:
   • Cuts off sharply after lag $q$ → suggests MA($q$)
   • Decays exponentially → suggests AR or ARMA

Step 2: Examine PACF

1. Plot sample PACF ${\varphi }_{kk}$ vs lag $k$
2. Check for significance: $|{\varphi }_{kk}|>\frac{1.96}{\sqrt{n}}$
3. Identify pattern:
   • Cuts off sharply after lag $p$ → suggests AR($p$)
   • Decays exponentially → suggests MA or ARMA

Step 3: Combine Information

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

Step 4: Use EACF or Information Criteria

For ARMA models where ACF/PACF both decay:

• Use EACF to find approximate $(p,q)$
• Compare AIC/BIC across candidate models

Sample Size

Need at least $n=50$ observations for reliable ACF/PACF estimates. Calculate up to lag $K\approx n/4$.

Related

• ACF
• PACF
• EACF
• AIC
• Box-Jenkins Strategy