We have the following situation: \begin{aligned} y_t &= f(x_t)+u_t, \\ u_t &= au_{t-1}+\epsilon_t, \\ \epsilon_t &\overset{iid}{\sim} N(0,\sigma^2). \end{aligned} To make it simple, let's assume $f$ is deterministic or known.
I think we can see this as a random-effect model, since $y_t\mid u_{t-1}\sim N(f(x_t)+au_{t-1},\sigma^2)$ and $u_{t-1}$ is randomly distributed (see this CV stackexchange answer).
Am I right? If not, when could we consider it to be a random effect?
