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

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\$)?

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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.)

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