I'm studying linear time series and I'm thinking whether is possible to characterize an invertible moving average of order q (i.e.: MA(q) ) by means of filtrations generated by the two processes. More formally let
$$Y_t = \epsilon_t + \theta_1 \epsilon_{t-1}+\theta_2 \epsilon_{t-2} + ...+\theta_q \epsilon_{t-q}$$
where $(\epsilon_t)_{t \in N}$ and $(Y_t)_{t \in N}$ are both discrete time stochastic processes.
Can we state that:
$(Y_t)_{t \in N}$ invertible if and only if the filtration generated by the two processes at a fixed time $t$ are equal? More formally:
$$(Y_t)_{t \in N} \ invertible \iff F^Y_t=F^{\epsilon}_t \quad \forall t \in N$$
For $F_t^Y$ natural filtration of the process $(Y_t)_{t \in N}$ up to time $t$ (and same for $\epsilon$).
Sufficiency ($\Rightarrow$) seems to me pretty trivial consequence of basic measure theory. But I really have no clue about how to prove or disprove the necessity part.