It seems that the auto.arima function in the "forecast" package in R only considers full ARIMA models. By "full" I mean that if an AR lag $k$ is included, AR lag $j$ will also be included for $k>1$, $0<j<k$ (and the same with MA in place of AR). This was claimed here by the author of the auto.arima function himself.
I am interested in non-full (restricted) ARIMA models, e.g. an AR(2) model where the first AR lag is restricted to zero: $x_t=\varphi_2 x_{t-2}+\varepsilon_t$.
Question 1: Is there a good theoretical reason for not considering the non-full ARIMA models?
Question 2: Is there a good practical reason for not considering the non-full ARIMA models? (Besides the argument of high computational burden if all sub-models within given maximum AR and MA orders are to be estimated.)
fixedparameter. $\endgroup$auto.arima(). Perhaps there are good reasons? $\endgroup$