Estimate unobservable $X_t$ with observable estimation {$Y_t$}, where $Y_t= X_t+\delta e_t$ Suppose there are measurement values {$Y_t, Y_{t-1},..., Y_0$} which come from the relationship $Y_t= X_t+\delta e_t$, where $\delta$ is a known constant, $e_t\sim N(0,\sigma^2_e)$ is a Gaussian distribution with known moments while $X_t\sim N(\mu_X, \sigma^2_x)$ is unobservable whose mean needs to be estimated ($\sigma^2_X$ is known).
Empirically, how does one estimate $\mu_X$ with the measurement value {$Y_t,...$}? Can the estimation be performed if both $\mu_X$ and $\sigma_X^2$ are unknown?
Btw please also tell me if there's any literature I can refer to (or start with). I've come across bunches of papers about Bayesian estimation in macro but can't figure out where to start.
Thanks a lot in advance!
It seems all you need to do is calculate the mean of Y (take the average of observations), which gives you the mean of X, since the expected value of epsilon is zero.
I would approach this problem as follows:
$ X_t = Y_t - \delta e_t$
$\mu_X   = E(X_t)= E(Y_t - \delta e_t) = \mu_y - \delta E(e_t) = \mu_y$