Define the two moving average processes $$u(t)=a_0e(t)+a_1e(t-1)+\dots a_Ne(t-N)\qquad e(t)\sim WN(0,1)$$ $$v(t)=a_0\eta(t)+a_1\eta(t-1)+\dots a_N\eta(t-N)\qquad \eta(t)\sim WN(0,1)$$ where $e$ and $\eta$ are independet white noises. Is it true that $y(t)=u(t)+v(t)$ is still a moving average process?
Note that if $y(t)$ is a moving average process, exist $\{c_n\}_n$ such that $$y(t)=c_0\nu(t)+c_1\nu (t-1)+\dots c_N\nu(t-N)\qquad \nu(t)\sim WN(0,1)$$ and, by imposing that the covariance function of $y$ is equal in the two cases, we have $$\forall 0\le\tau\le N \qquad \sum_{n=0}^N c_nc_{n+\tau}=\sum_{n=0}^N a_na_{n+\tau}+\sum_{n=0}^N b_nb_{n+\tau}$$ but solving this system of equations with respect to $c_n$ is not simple