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.

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$

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

Answered by BB King on January 5, 2021

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