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# Theoretical reminder
Consider the order $`p`$ VAR representation for the $`1\times m`$ vector of observed variables $`y_t`$:
```math
y_{t}=\mathbf C+\sum_{k=1}^{p} y_{t-k} \mathbf{A}_{k} + u_t
```
where $`u_t\sim \mathcal N\left( 0,\Sigma_u\right)`$. Let $`z_t`$ be the $`(mp+1)\times 1`$ vector $`\left[1,y_{t-1}',...,y_{t-p}'\right]'`$ and define $`\mathbf{A}=\left[\mathbf C', \mathbf A_1',...,\mathbf A_p'\right]'`$, the VAR representation can then be written in matrix form as:
```math
Y=Z\mathbf A +\mathcal U
```
where $`Y = (y_1',\ldots,y_T')'`$, $`Z = (z_1',\ldots,z_T')'`$ and $`\mathcal U = (u_1',\ldots,u_T')'`$.
Dummy observations prior for the VAR can be constructed using the VAR likelihood function for $`\mathcal T = [\lambda T]`$ artificial data simulated with the DSGE $`\left( Y^{\ast },Z^{\ast}\right)`$, combined with diffuse priors (see Tiao and Zellner (1962) or Zellner (1971)). The prior is then given by:
```math
p_{0}\left( \mathbf A, \Sigma \mid Y^*,Z^* \right)
\propto
\left\vert \Sigma \right\vert ^{-\frac{\lambda T+m+1}{2}}e^{-\frac{1}{2}tr\left[ \Sigma^{-1}\left(
{Y^*}'Y^*-\mathbf{A}'{Z^*}'Y^*-{Y^*}'Z^*\mathbf A+ \mathbf A'{Z^*}'Z^*\mathbf A \right) \right] }
```
implying that $`\Sigma`$ follows an inverted Wishart distribution and $`\mathbf A`$ conditional on $`\Sigma`$ is Gaussian. Assuming that observables are covariance stationary, Del Negro and Schorfheide use the DSGE theoretical autocovariance matrices for a given $`n\times 1`$ vector of model parameters $`\theta`$, denoted $`\Gamma_{YY}\left( \theta \right)`$, $`\Gamma_{ZY}\left( \theta \right)`$,
$`\Gamma_{YZ}\left( \theta\right)`$, $`\Gamma_{ZZ}\left( \theta \right)`$ instead of the (artificial) sample moments $`{Y^*}'Y^*`$, $`{Z^*}'Y^*`$, $`{Y^*}'Z^*`$, $`{Z^*}'Z^*`$. In addition, the $`p`$-th order VAR approximation of the DSGE provides the first moment of the prior distributions through the population least-square regression:
```math
\begin{array}{ll}
\mathrm A^*( \theta ) = \Gamma_{ZZ}\left( \theta\right)^{-1}\Gamma_{ZY}\left( \theta \right) & \qquad\qquad\textrm{(P1a)}\\
\Sigma^*(\theta) = \Gamma_{YY}(\theta) -\Gamma_{YZ}(\theta)\Gamma_{ZZ}\left( \theta \right)^{-1}\Gamma_{ZY}\left( \theta \right) & \qquad\qquad\textrm{(P1b)}
\end{array}
```
Conditional on the deep parameters of the DSGE $`\theta`$ and $`\lambda`$, the priors for the VAR parameters are given by:
```math
\mathrm{vec} \mathbf A \mid \Sigma ,\theta, \lambda \sim \mathcal N \left(\mathrm{vec}\mathbf A^*(\theta),\Sigma
\otimes \left[\lambda T \Gamma_{ZZ}(\theta)\right]^{-1}\right)
\Sigma \mid \theta,\lambda \sim \mathcal{IW}\left(\lambda T \Sigma^*(\theta),\lambda
T-mp-m\right)\qquad \textrm{(P2)}
```
where $`\Gamma_{ZZ}(\theta)`$ is assumed to be non singular and $`\lambda \geq \frac{mp+m}{T}`$ for the priors to be proper. The *a priori* density of $`\mathbf A`$ is defined by $`n+1`$ parameters ($`\theta`$ and $`\lambda`$), which is likely to be less than $`m^2p`$ (the VAR number of parameters). If we have a one-to-one relationship (no identification issues) between $`(\theta,\lambda)`$ and $`\mathbf A`$ it will be a good idea to estimate $`(\theta,\lambda)`$ instead of $`\mathbf A`$, *ie* to estimate fewer free parameters. To do so, Del Negro and Schorfheide complete the prior by specifying a prior distribution over the structural model's deep parameters:
$`p_0(\theta)`$. We still have to set the weight of the structural prior, $`\lambda`$. Del Negro and Schorfheide choose the value of $`\lambda`$ that maximizes the marginal density. They estimate a limited number of DSGE-VAR models, say $`N`$ models, with different values of $`\lambda`$. For each model they also estimate the marginal density and select the model (*ie* the value of $`\lambda`$) with highest marginal density. Implicitly, they compare different models considering a (discrete) uniform prior distribution over the collection of $`N`$ models. With Dynare it is also possible to estimate $`\lambda`$ as another parameter, instead of doing a loop over the values of this parameter. So we define a prior on $`\lambda`$, which is assumed to be independent from $`\theta`$.
Finally, the DSGE-VAR model has the following prior structure:
```math
p_0\left( \mathbf A,\Sigma, \theta, \lambda \right) = p_0\left(
\mathbf A, \Sigma \mid \theta ,\lambda \right) \times
p_0\left( \theta \right) \times p_0\left( \lambda \right)\qquad\textrm{(P3)}
```
where $`p_0\left(\mathbf A, \Sigma \mid \theta ,\lambda \right)`$ is defined by (P1a,P1b) and (P2).
The posterior distribution, may be factorized in the following way:
```math
p\left( \mathbf A, \Sigma , \theta , \lambda \mid \mathcal Y_T\right) = p\left(\mathbf A, \Sigma
\mid \mathcal Y_T, \theta, \lambda\right) \times
p\left( \theta ,\lambda \mid \mathcal Y_T\right)\qquad\textrm{(Q3)}
```
where $`\mathcal Y_T`$ stands for the sample. A closed form expression for the first density function on the right hand side of (Q3) is available. Conditional on $`\theta`$ and $`\lambda`$, (P1a,P1b) and (P2) define a conjugate prior for the VAR model, so its posterior density belongs to the same family: the distribution of $`\mathbf A`$ conditional on $`\Sigma`$, $`\theta`$, $`\lambda`$ and the sample is matric-variate normal, and the distribution of $`\Sigma`$ conditional on $`\theta`$, $`\lambda`$ and the sample is inverted Wishart. More formally, we have:
```math
\begin{array}{ll}
\mathrm{vec} \mathbf A \mid \Sigma, \theta , \lambda, \mathcal Y_T
\sim \mathcal N \left(\mathrm{vec}\widetilde{\mathbf A}(\theta,\lambda),\Sigma \otimes
V(\theta,\lambda)^{-1}\right) & \qquad\qquad\textrm{(Q2a)}\\
\Sigma \mid \theta, \lambda, \mathcal Y_T \sim
\mathcal{IW} \left( (\lambda+1) T~\widetilde{\Sigma}(\theta,\lambda),(\lambda+1)T-mp-m\right) & \qquad\qquad\textrm{(Q2b)}
\end{array}
```
where:
```math
\begin{array}{ll}
\widetilde{\mathbf A}(\theta,\lambda) = V(\theta,\lambda)^{-1}\left( \lambda T~\Gamma_{ZY}(\theta)+Z'Y\right)
& \qquad\qquad\textrm{(Q1a)}\\
\widetilde{\Sigma}(\theta,\lambda) = \frac{1}{(1+\lambda)T}
\left[ \lambda T~\Gamma_{YY}(\theta) + Y'Y - \left(\lambda T~\Gamma _{YZ}(\theta)
+Y'Z\right) V(\theta,\lambda)^{-1}\left(
\lambda T~\Gamma_{ZY}(\theta)+Z'Y\right)\right] & \qquad\qquad\textrm{(Q1b)}
\end{array}
```
with:
```math
V(\theta,\lambda) = \lambda T~\Gamma_{ZZ}(\theta) +Z'Z
```
The posterior mean of $`\mathbf A`$ is a convex combination of $`A^*(\theta)`$, the prior mean, and of the OLS estimate of $`\mathbf A`$. When $`\lambda`$ goes to infinity the posterior mean shrinks towards the prior mean, *ie* the projection of the DSGE model onto the VAR($`p`$). We do not have a closed form expression for the joint posterior density of $`\theta`$ and $`\lambda`$ (the second term on the right hand side of (Q3)). So the posterior distribution of $`(\theta,\lambda)`$ is recovered from an MCMC algorithm, as described in Del Negro and Schorfheide, except that we do estimate $`\lambda`$ jointly with the deep parameters $`\theta`$.
# Estimation of a DSGE-VAR model with Dynare
## Up to version 4.1.x
The estimation of a DSGE-VAR model instead of a DSGE model is triggered by using the keyword `dsge_prior_weight` in the list of declared parameters. This parameter name is the Dynare word for $`\lambda`$. If `dsge_prior_weight` is a parameter, Dynare understands that the user wants to estimate a DSGE-VAR model using the DSGE model declared in the same file as a prior for a VAR model. The number of lags is four by default and can be changed by adding, for instance, the following line of code:
```matlab
options_.varlag = 12;
```
before the `estimation` command. The weight of the DSGE prior can be:
1. Calibrated to a positive real number (greater than $`\frac{mp+m}{T}`$) or to `Inf` ($`\lambda=\infty`$ case).
2. Estimated jointly with the deep parameters. In this case, a prior density has to be defined in the `estimated_params` block with the usual syntax. An example where the estimation of $`\lambda`$ is implemented is given in Adjemian Darracq-Pariès and Moyen (2008).
## Version 4.2.x and following
The estimation of a DSGE-VAR model instead of a DSGE model is triggered by using the keyword `dsge_var` in the `estimation` command. If the weight of the DSGE prior is calibrated (as in Del Negro and Schorfheide (2004)) the calibrated value for $`\lambda`$ must be supplied to the `estimation` command as follows:
```example
estimation(..., dsge_var = 1.4, ...);
```
The DSGE prior weight has to be a real positive number or plus infinity. If the weight of the DSGE prior is estimated (as in Adjemian, Darracq-Pariès, and Moyen (2008)), the prior of $`\lambda`$ must be defined in the `estimated_params` block, as follows:
```example
estimated_params;
...
dsge_prior_weight, uniform_pdf, , , 0, 2;
...
end;
estimation(..., dsge_var, ...);
```
The number of lags of the VAR model is four by default and can be changed by using the option `dsge_varlag` in the `estimation` command, for instance:
```
estimation(..., dsge_var, dsge_varlag=12, ...);
```
## Example
### Up to version 4.1.x
First run the file [simul_hybrid.mod](https://git.dynare.org/snippets/2) to create a data set from a new keynesian model with an hybrid Phillips curve. Second run the file [bvar_forward.mod](https://git.dynare.org/snippets/1) to estimate the DSGE-VAR model (where the DSGE prior is a new keynesian model with an hybrid Phillips curve). The second file is self documented.
### Version 4.2.x and following
Examples are in the tests directory of the (git) master repository [here](https://git.dynare.org/Dynare/dynare/tree/master/tests/dsge-var).
# References
* Adjemian, S., Darracq-Pariès, M. and S. Moyen (2008). *Towards a monetary policy evaluation framework*. [ECB Working paper 942](http://www.ecb.int/pub/pdf/scpwps/ecbwp942.pdf)
* Del Negro, M., and F. Schorfheide (2004). *Priors from General Equilibrium Models for VARs*, in International Economic Review, 45(2), 643–673.
* Del Negro, M., F. Schorfheide, F. Smets, and R. Wouters (2007). *On the Fit of New Keynesian Models*, in Journal of Business and Economic Statistics, 25(2), 123-162.
* Tiao, G. C., and A. Zellner (1964). *Bayes Theorem and the Use of Prior Knowledge in Regression Analysis*, Biometrika, 51(162), 219–230.
* Zellner, A. (1971). *An Introduction to Bayesian Inference in Econometrics*. John Wiley & Sons.