I have an exercise to compute the covariance between the prediction errors, but I'm not sure if it is correct, this is the exercise;
I have an AR(1) model, $y_t = \phi y_{t-1} + \epsilon_t$, where $\epsilon \sim^{i.i.d.} N(0, \sigma^2)$, assuming $\sigma^2$ is known, what is the $Cov(e_n(1), e_n(2))$? Where $e_n(k)$ is the prediction $k$ steps ahead, for example, $$e_n(1) = y_{n+1} - y_n(1) = y_{n+1} - \phi y_n$$ and $$e_n(2) = y_{n+2} - y_n(2) = y_{n+2} - \phi y_n(1)$$
My answer is:
\begin{align} Cov(e_n(1), e_n(2) | F_n) & = Cov(Y_{T+1} - y_T(1), Y_{T+2} - y_T(2) | F_n) \\ & = Cov(\phi y_T + \epsilon_{T+1}, \phi^2 y_T + \phi \epsilon_{T+1} + \epsilon_{T+2} | F_t) \\ & = Cov(\epsilon_{T+1}, \phi \epsilon_{T+1} | F_t) \\ & = \phi \sigma^2 \end{align}
Is this correct? I'm considering that $y_T$ is not random since they are from the observed time-series.
