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!

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    $\begingroup$ 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
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    $\begingroup$ 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


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$

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


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