Bayes’ rule in "The sources of capital misallocation"

Question

I am reading a paper titled "The sources of capital misallocation". 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

Answer 1

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 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.