# Context in which an AR(1) error term can be considered a random effect?

We have the following situation: \begin{aligned} y_t &= f(x_t)+u_t, \\ u_t &= au_{t-1}+\epsilon_t, \\ \epsilon_t &\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?

• I have changed the formatting for better visibility, but feel free to undo the change if you do not like it. – Richard Hardy Sep 29 at 9:46
• @RichardHardy thanks ;) – An old man in the sea. Sep 29 at 12:06