# Determine the number of parameters of ARIMA models for AIC calculation

It is my understanding of the AIC formula that: $$AIC = -2\log(L) + 2m$$ 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 ($m$)?

• I think it's mostly right, although perhaps +1 should be added everywhere (except for SARIMA) because of the fact that also $\sigma^2_{\varepsilon}$ is being estimated extra to the AR and MA coefficients. But I am not sure. Also, what is your logic behind the +1 for SARIMA but not in the other cases? – Richard Hardy Apr 21 '17 at 6:54
• @RIchard The +1 is correct but nothing is lost by ignoring it with AIC (since they all have it). For BIC and AICc though, you can't just leave it out. Also you usually have a mean but for differenced models you often don't – Glen_b -Reinstate Monica Apr 21 '17 at 7:00
• @Glen, thanks. I presume you mean that nothing is lost in comparisons where such constants cancel out. But if one takes AIC of a given (as opposed to chosen) model and interprets it as the expected likelihood, then it starts to matter, right? – Richard Hardy Apr 21 '17 at 7:02
• Yes, nothing is lost in comparisons when it's common to both. – Glen_b -Reinstate Monica Apr 21 '17 at 7:27
• I actually thought the +1 was for the seasonal component of the seasonal ARIMA model. My interest is finding the exact number of parameters, so I do not want to leave out any additional parameters just for the ease of comparison. – niallStudent Apr 21 '17 at 11:51

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.)