<p>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.</p>
<p>I would approach this problem as follows:</p>
<p><span class="math-container">$ X_t = Y_t - \delta e_t$</span></p>
<p><span class="math-container">$\mu_X   = E(X_t)= E(Y_t - \delta e_t) = \mu_y - \delta E(e_t) = \mu_y$</span></p>

CC BY-SA 4.0
2020-10-27T11:28:12.670Z
2020-10-27T11:28:12.670Z
BB King
<p>Suppose there are measurement values {<span class="math-container">$Y_t, Y_{t-1},..., Y_0$</span>} which come from the relationship <span class="math-container">$Y_t= X_t+\delta e_t$</span>, where <span class="math-container">$\delta$</span> is a known constant, <span class="math-container">$e_t\sim N(0,\sigma^2_e)$</span> is a Gaussian distribution with known moments while <span class="math-container">$X_t\sim N(\mu_X, \sigma^2_x)$</span> is unobservable whose mean needs to be estimated (<span class="math-container">$\sigma^2_X$</span> is known).</p>
<p>Empirically, how does one estimate <span class="math-container">$\mu_X$</span> with the measurement value {<span class="math-container">$Y_t,...$</span>}? Can the estimation be performed if both <span class="math-container">$\mu_X$</span> and <span class="math-container">$\sigma_X^2$</span> are unknown?</p>
<p>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.</p>
<p>Thanks a lot in advance!</p>

CC BY-SA 4.0
2020-10-27T10:37:29.230Z
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
CC BY-SA 4.0
2020-10-27T11:34:54.300Z
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 $\mu_X$.
Dayne
<macroeconomics><econometrics><statistics>
Estimate unobservable $X_t$ with observable estimation {$Y_t$}, where $Y_t= X_t+\delta e_t$
Hao
# Estimate unobservable $X_t$ with observable estimation {$Y_t$}, where $Y_t= X_t+\delta e_t$

(Asked by: username_0 on 2020-10-27)

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!

**Tags:** macroeconomics, econometrics, statistics

> **Comments:**
> **username_1**: 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  
> **username_2**: 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 $\mu_X$.  

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## 1 answers

### Answer by username_3 on 2020-10-27. Score: 2

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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