Consider the order latex($p$) VAR representation for the latex($1\times m$) vector of observed variables latex($y_t$):
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where latex($u_t\sim \mathcal N\left( 0,\Sigma_u\right)$). Let latex($z_t$) be the latex($mp\times 1$) vector latex($\left[y_{t-1}',...,y_{t-p}'\right]'$) and define latex($\mathbf{A}=\left[\mathbf A_1',...,\mathbf A_p'\right]'$), the VAR representation can then be written in matrix form as:
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where latex($Y = (y_1',\dots,y_T')'$), latex($Z = (z_1',\dots,z_T')'$) and latex($\mathcal U = (u_1',\dots,u_T')'$).
Dummy observations prior for the VAR can be constructed using the VAR likelihood function for latex($\mathcal T = [\lambda T]$) artificial data simulated with the DSGE latex($\left( Y^{\ast },Z^{\ast}\right)$), combined with diffuse priors. The prior is then given by:
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implying that latex($\Sigma$) follows an inverted Wishart distribution and latex($\mathbf A$) conditional on latex($\Sigma$) is gaussian. Assuming that observables are covariance stationary, Del Negro and Schorfheide use the DSGE theoretical autocovariance matrices for a given latex($n\times 1$) vector of model parameters latex($\theta$), denoted latex($\Gamma_{YY}\left( \theta \right)$), latex($\Gamma_{ZY}\left( \theta \right)$), latex($\Gamma_{YZ}\left( \theta\right)$), latex($\Gamma_{ZZ}\left( \theta \right)$) instead of the (artificial) sample moments latex(${Y^*}'Y^*$), latex(${Z^*}'Y^*$), latex(${Y^*}'Z^*$), latex(${Z^*}'Z^*$). In addition, the latex($p$)-th order VAR approximation of the DSGE provides the first moment of the prior distributions through the population least-square regression:
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Conditional on the deep parameters of the DSGE latex($\theta $) and latex($\lambda$), the priors for the VAR parameters are given by:
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where latex($\Gamma_{ZZ}(\theta)$) is assumed to be non singular and latex($\lambda \geq \frac{mp+m}{T}$) for the priors to be proper\footnote{Note that it
- would not be possible to estimate the VAR model by OLS (or maximum
likelihood) if we had $\mathcal T < m(p+1)$. In this case we would not have more observations than parameters to estimate.}. The
\textit{a priori} density of $\mathbf A$ is defined by $n+1$ parameters ($\theta$ and $\lambda$), which is likely to be less than $mp$ (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 $\mathbf A$, \textit{ie} to estimate fewer free parameters. To do so, \cite{DS2004} 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, $\lambda$. \citeauthor{DS2004} choose the value of $\lambda$ that maximizes the marginal density. They estimate a limited number of DSGE-VAR models with different values of $\lambda$. For each model they also estimate the marginal density and select the model (\textit{ie} the value of $\lambda$) with highest marginal density. In the present paper, we estimate directly $\lambda$ as another parameter, instead of doing a loop over the values of this parameter\footnote{In this regard, the approach followed by
- \citeauthor{DS2004} is, at least computationally, inefficient. Also, contrary to us, they do not average over different possible values of $\lambda$ but pick a single value of this parameter, which is not the bayesian way.}. So we define a prior on the distribution of
$\lambda$, which is assumed to be independent from $\theta$. Finally, the DSGE-VAR model has the following prior structure: \begin{equation}\tag{P3}\label{prior3} 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) \end{equation} where $p_0\left(\mathbf A, \Sigma \mid \theta ,\lambda \right)$ is defined by [\ref{prior1a},\ref{prior1b}] and \equaref{prior2}.}\newline
\par{The posterior distribution, may be factorized in the following way: \begin{equation}\tag{Q3}\label{posterior1} 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) \end{equation} where $\mathcal Y_T$ stands for the sample. A closed form expression for the first density function on the right hand side of \equaref{posterior1} is available. Conditional on $\theta $ and $\lambda$, [\ref{prior1a},\ref{prior1b}] and \equaref{prior2} define a conjugate prior for the VAR model, so its posterior density has to belong 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: \begin{equation}\tag{Q2}\label{posterior2}
- \begin{split}
- \VEC \mathbf A \mid \Sigma, \theta , \lambda, \mathcal Y_T
& \sim \normal{\VEC \widetilde{\mathbf A}(\theta,\lambda)}{\Sigma \otimes V(\theta,\lambda)^{-1}}\\
\Sigma \mid \theta, \lambda, \mathcal Y_T &\sim
- \mathcal{IW} \left( (\lambda+1) T~\widetilde{\Sigma}(\theta,\lambda),
- \VEC \mathbf A \mid \Sigma, \theta , \lambda, \mathcal Y_T
(\lambda+1)T-mp-m\right)
- \end{split}
\end{equation} where: \begin{subequations}%\tag{Q1}\label{posterior3}
- \begin{equation}\tag{Q1a}\label{posterior3a}
- \widetilde{\mathbf A}(\theta,\lambda) = V(\theta,\lambda)^{-1}\left( \lambda
- T~\Gamma_{ZY}(\theta)+Z'Y\right)
- \widetilde{\mathbf A}(\theta,\lambda) = V(\theta,\lambda)^{-1}\left( \lambda
- \begin{equation}\tag{Q1b}\label{posterior3b}
- \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]
- \left[ \lambda T~\Gamma_{YY}(\theta) + Y'Y - \left(\lambda T~\Gamma _{YZ}(\theta) +Y'Z\right) V(\theta,\lambda)^{-1}\left(
- \widetilde{\Sigma}(\theta,\lambda) = \frac{1}{(1+\lambda)T}
\end{subequations} with: \[ V(\theta,\lambda) = \lambda T~\Gamma_{ZZ}(\theta) +Z'Z \] Not surprisingly, we find that 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, \textit{ie} the projection of the DSGE model onto the VAR($p$).\newline 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 \equaref{posterior1}). So the posterior distribution of $(\theta,\lambda)$ is recovered from an MCMC algorithm, as described in \cite[appendix B]{DS2004}, except that we do estimate $\lambda$ as the deep parameters $\theta$.\footnote{This can be done with
\href{http://www.cepremap.cnrs.fr/dynare}{Dynare} 4.}}\newline
\par{All in all, this estimation procedure allows to select the
- \textquotedblleft best specification \textquotedblright from a continuum of intermediate models indexed by $\lambda$ and ranging between the VAR($p$) with diffuse priors and the VAR($p$) approximation of the DSGE model. The posterior distribution of the deep parameters, $\theta$, can then be interpreted as the best model to be used as a prior for the corresponding VAR($p$). In addition, the posterior distribution of $\lambda$ gives an indication of the reliability of the DSGE model and of the empirical relevance of the associated economic restrictions.}\newline
\par{Notice that, when $\lambda$ is closer to its lowest possible
- value, the DSGE-VAR approximates an unrestricted VAR with diffuse priors. Given the number of observed variables in our study, such VAR has obviously poor empirical performance due to many free parameters and associated sampling errors. The econometric literature has shown that Bayesian VARs can improve on unrestricted VAR by introducing \textquotedblleft Minnesota-like \textquotedblright priors for example, which favor persistence, low cross-variable interactions and smaller coefficients at distant lags. Therefore, the DSGE-VAR approach is bound to call for higher $\lambda$ to increase the tightness of priors regarding serial correlations for instance which does not necessarily mean that the economic restrictions of the DSGE are more consistent with the data. A way to circumvent this problem would be to introduce an other source of dummy observation coming from a BVAR with some version of Minnesota priors and let the procedure infer the relative weight to put to the DSGE priors and the \textquotedblleft Minnesota priors \textquotedblright. This avenue is left for further research.}\newline