# 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.

• I think you should consider posting this on cross-validated. While econometrics is definitely on topic here this is quite abstract and does not have much connection to economics, and at cross-validated there are statisticians from all sorts of fields and focusing on this sort of abstract problems
– 1muflon1
Commented Oct 27, 2020 at 10:37
• Are you sure $X_t$ is not a stochastic process? Otherwise this just looks lilke that sample mean of $Y_t$ is the best estimate of $\mu_X$. Commented Oct 27, 2020 at 11:34

$$X_t = Y_t - \delta e_t$$
$$\mu_X = E(X_t)= E(Y_t - \delta e_t) = \mu_y - \delta E(e_t) = \mu_y$$