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\begin{slide}
\center \red Bayesian Estimation of a DSGE model \black \flushleft
Why bother?
Practical problem: as the number of parameters to be estimated
increases the likelihood might flatten out.
Bayesian approach brings in prior information to the problem, which
can give us local identification where we had none before.
\end{slide}
\begin{slide}
\center \red How does it work \black \flushleft Bayes' rule:
Let $A$ and $B$ be two events defined on a sample space, then
\[ P(A/B) = \frac{P(B/A)P(A)}{P(B)} \]
Now let $X$ and $Y$ be random variables
\[ P_{x/y}(X/Y) = \frac{P_{y/x}(Y/X)P_{x}(X)}{P_y(Y)}
\]
If $Y$ is observed data and $X$ is a collection of parameters, the
$P_{y/x}(Y/X)$ is the likelihood. $P_x(X)$ is the prior and
$P_{x/y}(X/Y)$ is the posterior.
\end{slide}
\begin{slide}
\center \red Analytical Integration \black \flushleft
If we could compute
\[
\int_{x} P_{y/x}(Y/X) P_x(X) dx
\]
Then we could also construct $P_y (Y)$, at that point we would know
$P_{x/y}(X/Y)$.
This problem can be solved analytically for ``judicious'' choices of
priors and models.
Analytical procedure generally not available for the estimation of
DSGE models, since the parameters of interest are embedded in the
reduced form of the model.
\end{slide}
\begin{slide}
\center \red Numerical Integration \black \flushleft
Can we save the day with numerical methods? We most certainly can.
The most widely used numerical method when estimating DSGE models is
the Metropolis-Hastings algorithm.
It is one implementation of importance sampling:
- sample at random, but stick around a little longer where the
posterior has a lot of mass
- still allow for escapes from areas of high mass
\end{slide}
\begin{slide}
\center \red The MH Algorithm \black \flushleft
Choose $\theta_0$
Draw $\theta^*$ from $q(./\theta_i)$
Calculate ratio $r =
\frac{P(Y/\theta^*)P(\theta*)}{P(Y/\theta_i)P(\theta_i)}$
If $r \geq 1$ then $\theta_{i+1} = \theta^*$ otherwise
\begin{eqnarray}
&&\theta_{i+1} = \theta^* \hbox{with probability r} \nonumber \\
&&\theta_{i+1} = \theta_i \hbox{with probability 1-r} \nonumber
\end{eqnarray}
Draws from $q$ constructed as above will converge to draws from the
posterior if some regularity conditions are met. These conditions
include that $q(./\theta)$ needs to cover the posterior
distribution.
\end{slide}
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