It is my understanding of the AIC formula that:

$$

AIC = -2log(L) + 2m

$$

where $m =$ number of parameters.

I am wondering how to determine the value of $m$ in AIC for the following models:

- AR(1)
- MA(1)
- ARMA(1,1)
- ARIMA(1,1,1)
- Seasonal ARIMA(1,1,1)(1,1,1)12

I believe for an AR(p) model $m=p$,

MA(q) model $m=q$,

ARMA(p,q) model $m = p+q$,

ARIMA(p,d,q) model $m=p+q$

and for a seasonal ARIMA(p,d,q)(P,D,Q)s model $m = p+q+P+Q+1$.

Could you please let me know if this is correct?

And if not, how would I determine the correct number of parameters ($m$)?

**Contents**hide

#### Best Answer

I think you are *almost* right. For a general SARIMA model, it would be $$ m=p+q+P+Q+1 $$ where $+1$ comes from the fact that also $sigma^2_{varepsilon}$ is being estimated extra to the AR and MA coefficients.

For submodels such as AR, MA or ARIMA, just set the appropriate coefficients to zero. Thus $m=p+1$ for AR; $m=q+1$ for MA; and $m=p+q+1$ for ARIMA.

(As @Glen_b notes, the $+1$ is not so important when we compare AICs across models; when we take a difference between two AICs, the $+1$ in each cancels out.)