I'm following Tsay's "Analysis of Financial Time Series", chapter 11.
The topic is derive the Kalman filter in the simple case of the local linear model $$ y_t = \mu_t + e_t \\ \mu_{t+1} = \mu_t + \eta_t $$ with $\{e_t\}$, $\{\eta_t\}$ independent zero-mean gaussian noise with variances, respectively, $\sigma^2_e$ and $\sigma^2_\eta$.
Further notation:
- $F_t = \{y_1, \ldots , y_t \}$ is the set of observations from time $1$ to time $t$;
- $y_{t \mid t-1} = E \left( y_t \mid F_{t-1} \right)$ is the prediction of the observation;
- $v_t = y_t - y_{t \mid t-1}$, is the 1-step forecast error.
I find "the forecast error $v_t$ is independent of $F_{t-1}$" so $$ Var \left( v_t \mid F_{t-1} \right) = Var \left( v_t \right), $$ and also "the information set $F_t$ can be written as $F_t = \left\{ F_{t-1}, y_t \right\} = \left\{ F_{t-1}, v_t \right\}$".
I'm not seeing this independence, or "equivalence" in knowing $y_t$ or $v_t$.
Someone could help? Thank you