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 ($m$)?
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.)