Is SARIMA(0,0,1)$_\text{s}$ actually MA(s)?

Question

Since for SARIMA(0,0,1)$_\text{s}$ the model equation is $x_t =e_t+a e_{t-s}$, can we say this is a kind of a MA(s) model?

Answer 1

I have some doubts about the notation, i.e. whether SARIMA(0,0,1)$_\text{S}$ should be interpreted as SARIMA with all the nonseasonal lag and differentiation orders set to zero. It would be less less ambiguous to write something like SARIMA(0,0,0)$\times$(0,0,1)$_\text{S}$. But notation aside, it is indeed a restricted MA(S) model where all but the last (S$^{\text{th}}$) lag are set to zero.