Sum of $I(1)$ and WN Is the process $\Delta Y$ an MA(1)? $Y_{t} = X_{t} + w_{t}$ with $X_{t} = X_{t-1} + e_{t}$ and $e_t$, $w_t$ both independent white noise a MA(1) process?
What I did is the following:
$(1-L)X_{t} = e_{t}$ and $(1-L)Y_{t} = \Delta Y = e_{t} + w_t + w_{t-1}$
Apparently this isn't necessarily an MA(1) but I don't know why

It seems to me that
$$
\begin{aligned}
Y(t)&=X(t)+w_t, ~~~w_t \sim N(0,\sigma^2_w) \\
& = X(t-1)+e_t+w_t, ~~~  e_t \sim N(0,\sigma^2_e),\\
& = (Y(t-1)-w_{t-1})+e_t+w_t
\end{aligned}
$$
is just an AR(1) process with $Y(t)=Y(t-1)+ q_t, q_t \sim N(0,\sigma^2_e+2\sigma^2_w)$.
