Estimate unobservable $X_t$ with observable estimation {$Y_t$}, where $Y_t= X_t+delta e_t$

Economics Asked on January 5, 2021

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, $$e_tsim N(0,sigma^2_e)$$ is a Gaussian distribution with known moments while $$X_tsim N(mu_X, sigma^2_x)$$ 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.

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:

$$X_t = Y_t - delta e_t$$

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

Answered by BB King on January 5, 2021