Extended Autocorrelation Function

Definition

The Extended Autocorrelation Function (EACF) is a method for identifying the orders $(p,q)$ of an ARMA model by filtering out the AR component and examining the resulting MA structure.

Procedure

For ARMA(p,q), neither ACF nor PACF cuts off cleanly. EACF provides a systematic way to identify both $p$ and $q$.

1. Start with AR(0): Compute sample ACF of $Y_t$ (original series)
2. For each candidate AR order $k$: Fit AR($k$) and compute residuals ${\hat{a}}_t$
3. For each candidate MA order $j$: Examine ACF of filtered series

$W_{t,k,j}=Y_t-{\hat{\varphi }}_1{Y}_{t-1}-\cdots -{\hat{\varphi }}_k{Y}_{t-k}$

4. Create EACF table: Mark whether ACF at lag $j+1$ is significantly different from zero
5. Identify p,q: For ARMA(p,q), the EACF table shows a triangle of zeros starting at row $p$, column $q$:
   • If AR order $k=p$ and MA order $j\ge q$: $W_{t,k,j}$ is MA($q$)
   • ACF at lag $j+1$ should be zero for $j\ge q$

EACF Table Pattern

For ARMA(1,1):

	MA(0)	MA(1)	MA(2)	MA(3)
AR(0)	X	X	X	X
AR(1)	X	O	O	O
AR(2)	X	X	O	O
AR(3)	X	X	X	O

Triangle of zeros starts at (AR=1, MA=1) → ARMA(1,1)

Practical Limitation

Sample EACF rarely shows a perfect triangle pattern. Look for approximate patterns.

Decision Rule

EACF at lag $j+1$ is significant if:

$|\text{ACF}|>\frac{1.96}{\sqrt{n-k-j}}$

Related

• Sample Autocorrelation (ACF)
• Partial Autocorrelation (PACF)
• ARMA(p,q) Process Model
• Box-Jenkins Strategy