0
$\begingroup$

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!

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ 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 $\endgroup$
    – 1muflon1 ♦
    Commented Oct 27, 2020 at 10:37
  • 3
    $\begingroup$ 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 $\mu_X$. $\endgroup$
    – Dayne
    Commented Oct 27, 2020 at 11:34

1 Answer 1

Reset to default
2
$\begingroup$

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$

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

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.