<p>It follows immediately from Bayesian Updating using Bayes Rule for normal random variables. You can find derivations in e.g. <a href="https://econ-papers.upf.edu/papers/1797.pdf" rel="nofollow noreferrer">Baley/Veldkamp (2021): Bayesian Learning</a>
See also the related answer at
<a href="https://stats.stackexchange.com/questions/237037/bayesian-updating-with-new-data">https://stats.stackexchange.com/questions/237037/bayesian-updating-with-new-data</a></p>
<p>The variance shows up because you are combining the expected movement based on the AR process and the signal information into one value. The relative weight of the two pieces of information depends on the precision, i.e. the inverse of the variances.</p>

CC BY-SA 4.0
2022-03-07T12:48:12.617Z
Thank you very much. Really helpful.
Seyed Mahdi
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2022-03-07T10:19:35.730Z
2022-03-07T10:19:35.730Z
jpfeifer
<p>I am reading a paper titled &quot;<a href="https://www.nber.org/papers/w23129" rel="nofollow noreferrer">The sources of capital misallocation</a>&quot;. In the model, firms are facing incomplete information about their future productivities. In particular, the productivity process is an AR(1):</p>
<p><span class="math-container">$$ a_{it} = \rho \ a_{it-1} + \mu_{it} \hspace{2cm} \mu_{it} \sim  N(0,\sigma_\mu^2) $$</span></p>
<p>At time t, firms observe the next period's innovation <span class="math-container">$\mu_{it+1} $</span> with a noise, in fact they observe <span class="math-container">$s_{it+1}$</span>:</p>
<p><span class="math-container">$$s_{it+1} = \mu_{it+1} + e_{it+1}\hspace{2cm} e_{it+1} \sim N(0,\sigma_e^2) $$</span></p>
<p>My question is here: The authors derive the expected value of next period's productivity, conditional on firm's information at time t, simply by saying &quot;A direct application of
Bayes’ rule yields:&quot;</p>
<p><span class="math-container">$$E_{it}\ [a_{it+1}]= \rho a_{it} + \frac{V}{\sigma^2_e}s_{it+1} $$</span></p>
<p>Where V is the posterior variance and is equal to:</p>
<p><span class="math-container">$$V=\left(\frac{1}{\sigma^2_\mu}+\frac{1}{\sigma^2_e}\right)^{-1}$$</span></p>
<p><strong>What do the authors mean exactly by Bayes’ rule? The famous formula in conditional probability? Then what's the relation between that formula and this expected value calculation?
How is the variance showing up in the expected value? I really have no idea.</strong></p>
<p>I searched and read a lot but it wasn't helpful. Any explanation is greatly appreciated. Thanks</p>

<probability><information><stochastic-processes>
Bayes’ rule in "The sources of capital misallocation"
Seyed Mahdi
# Bayes’ rule in "The sources of capital misallocation"

(Asked by: username_0 on 2022-03-07)

I am reading a paper titled "[The sources of capital misallocation](https://www.nber.org/papers/w23129)". In the model, firms are facing incomplete information about their future productivities. In particular, the productivity process is an AR(1):

$$ a\_{it} = \rho \ a\_{it-1} + \mu\_{it} \hspace{2cm} \mu\_{it} \sim N(0,\sigma\_\mu^2) $$

At time t, firms observe the next period's innovation $\mu\_{it+1} $ with a noise, in fact they observe $s\_{it+1}$:

$$s\_{it+1} = \mu\_{it+1} + e\_{it+1}\hspace{2cm} e\_{it+1} \sim N(0,\sigma\_e^2) $$

My question is here: The authors derive the expected value of next period's productivity, conditional on firm's information at time t, simply by saying "A direct application of
Bayes’ rule yields:"

$$E\_{it}\ [a\_{it+1}]= \rho a\_{it} + \frac{V}{\sigma^2\_e}s\_{it+1} $$

Where V is the posterior variance and is equal to:

$$V=\left(\frac{1}{\sigma^2\_\mu}+\frac{1}{\sigma^2\_e}\right)^{-1}$$

**What do the authors mean exactly by Bayes’ rule? The famous formula in conditional probability? Then what's the relation between that formula and this expected value calculation?
How is the variance showing up in the expected value? I really have no idea.**

I searched and read a lot but it wasn't helpful. Any explanation is greatly appreciated. Thanks

**Tags:** probability, information, stochastic-processes


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## 1 answers

### Answer by username_1 on 2022-03-07. Score: 3 | Accepted answer

It follows immediately from Bayesian Updating using Bayes Rule for normal random variables. You can find derivations in e.g. [Baley/Veldkamp (2021): Bayesian Learning](https://econ-papers.upf.edu/papers/1797.pdf)
See also the related answer at
<https://stats.stackexchange.com/questions/237037/bayesian-updating-with-new-data>

The variance shows up because you are combining the expected movement based on the AR process and the signal information into one value. The relative weight of the two pieces of information depends on the precision, i.e. the inverse of the variances.

> **Comments:**
> **username_0**: Thank you very much. Really helpful.  

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