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| Consider the order {{{#latex $p$}}} VAR representation for the $1\times m$ vector of observed variables $y_t$: |
Consider the order [[latex($p$)]] VAR representation for the [[latex($1\times m$)]] vector of observed variables [[latex($y_t$)]]: {{{#!latex |
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| where $u_t\sim \mathcal N\left( 0,\Sigma_u\right)$. Let $z_t$ be the $mp\times 1$ vector $\left[ y_{t-1}',...,y_{t-p}'\right]'$ and define $\mathbf{A}=\left[\mathbf A_1',...,\mathbf A_p'\right]'$, the VAR representation can then be written in matrix form as: |
}}} 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: {{{#!latex |
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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: {{{#!latex \[ p_{0}\left( \mathbf A, \Sigma \mid Y^*,Z^* \right) \propto \left\vert \Sigma \right\vert ^{-\frac{\lambda T+m+1}{2}}e^{-\frac{1}{2}tr\left[ \Sigma^{-1}\left( {Y^*}'Y^*-\mathbf{A}'{Z^*}'Y^*-{Y^*}'Z^*\mathbf A+ \mathbf A'{Z^*}'Z^*\mathbf A \right) \right] } \] }}} implying that [[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: {{{#!latex \begin{subequations} \begin{equation}\tag{P1a}%\label{prior1a} \mathrm A^*( \theta ) = \Gamma_{ZZ}\left( \theta \right)^{-1}\Gamma_{ZY}\left( \theta \right) \end{equation} \begin{equation}\tag{P1b}%\label{prior1b} \Sigma^*(\theta) = \Gamma_{YY}(\theta) -\Gamma_{YZ}(\theta) \Gamma_{ZZ}\left( \theta \right)^{-1}\Gamma_{ZY}\left( \theta \right) \end{equation} \end{subequations} }}} Conditional on the deep parameters of the DSGE [[latex($\theta $)]] and [[latex($\lambda$)]], the priors for the VAR parameters are given by: {{{#!latex \begin{equation}\tag{P2}%\label{prior2} \begin{split} \mathrm{vec} \mathbf A \mid \Sigma ,\theta, \lambda &\sim \mathcal N \left(\mathrm{vec}\mathbf A^*(\theta),\Sigma \otimes \left[\lambda T \Gamma_{ZZ}(\theta)\right]^{-1}\right)\\ \Sigma \mid \theta,\lambda &\sim \mathcal{IW}\left(\lambda T \Sigma^*(\theta),\lambda T-mp-m\right) \end{split} \end{equation} }}} where [[latex($\Gamma_{ZZ}(\theta)$)]] is assumed to be non singular and [[latex($\lambda \geq \frac{mp+m}{T}$)]] for the priors to be proper (Note that it would not be possible to estimate the VAR model by OLS (or maximum likelihood) if we had [[latex($\mathcal T < m(p+1)$)]]. In this case we would not have more observations than parameters to estimate). The ''a priori'' density of [[latex($\mathbf A$)]] is defined by [[latex($n+1$)]] parameters ([[latex($\theta$)]] and [[latex($\lambda$))]], which is likely to be less than [[latex($mp$)]] (the VAR number of parameters). If we have a one-to-one relationship (no identification issues) between [[latex($(\theta,\lambda)$)]] and [[latex($\mathbf A$)]] it will be a good idea to estimate [[latex($(\theta,\lambda)$)]] instead of [[latex($\mathbf A$)]], ''ie'' to estimate fewer free parameters. To do so, Del Negro and Schorfheide complete the prior by specifying a prior distribution over the structural model's deep parameters: [[latex($p_0(\theta)$)]]. We still have to set the weight of the structural prior, [[latex($\lambda$)]]. Del Negro and Schorfheide choose the value of [[latex($\lambda$)]] that maximizes the marginal density. They estimate a limited number of DSGE-VAR models with different values of [[latex($\lambda$)]]. For each model they also estimate the marginal density and select the model (''ie'' the value of [[latex($\lambda$)]]) with highest marginal density. Dynare can also estimate directly [[latex($\lambda$)]] as another parameter, instead of doing a loop over the values of this parameter. So we define a prior on [[latex($\lambda$)]], which is assumed to be independent from [[latex($\theta$)]]. Finally, the DSGE-VAR model has the following prior structure: {{{#!latex \begin{equation}\tag{P3} p_0\left( \mathbf A,\Sigma, \theta, \lambda \right) = p_0\left( \mathbf A, \Sigma \mid \theta ,\lambda \right) \times p_0\left( \theta \right) \times p_0\left( \lambda \right) \end{equation} }}} where [[latex($p_0\left(\mathbf A, \Sigma \mid \theta ,\lambda \right)$)]] is defined by [P1a,P1b] and [P2]. The posterior distribution, may be factorized in the following way: {{{#!latex \begin{equation}\tag{Q3} p\left( \mathbf A, \Sigma , \theta , \lambda \mid \mathcal Y_T\right) = p\left(\mathbf A, \Sigma \mid \mathcal Y_T, \theta, \lambda\right) \times p\left( \theta ,\lambda \mid \mathcal Y_T\right) \end{equation} }}} where [[latex($\mathcal Y_T$)]] stands for the sample. A closed form expression for the first density function on the right hand side of [Q3] is available. Conditional on [[latex($\theta$)]] and [[latex($\lambda$)]], [P1a,P1b] and [P2] define a conjugate prior for the VAR model, so its posterior density belongs to the same family: the distribution of [[latex($\mathbf A$)]] conditional on [[latex($\Sigma$)]], [[latex($\theta$)]], [[latex($\lambda$)]] and the sample is matric-variate normal, and the distribution of [[latex($\Sigma$)]] conditional on [[latex($\theta$)]], [[latex($\lambda$)]] and the sample is inverted Wishart. More formally, we have: {{{#!latex \begin{equation}\tag{Q2} \begin{split} \mathrm{vec} \mathbf A \mid \Sigma, \theta , \lambda, \mathcal Y_T & \sim \mathcal N \mathrm{vec} \left(\widetilde{\mathbf A}(\theta,\lambda),\Sigma \otimes V(\theta,\lambda)^{-1}\right)\\ \Sigma \mid \theta, \lambda, \mathcal Y_T &\sim \mathcal{IW} \left( (\lambda+1) T~\widetilde{\Sigma}(\theta,\lambda),(\lambda+1)T-mp-m\right) \end{split} \end{equation} }}} where: {{{#!latex \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) \end{equation} \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] \end{equation} \end{subequations} }}} with: {{{#!latex \[ V(\theta,\lambda) = \lambda T~\Gamma_{ZZ}(\theta) +Z'Z \] }}} The posterior mean of [[latex($\mathbf A$)]] is a convex combination of [[latex($A^*(\theta)$)]], the prior mean, and of the OLS estimate of [[latex($\mathbf A$)]]. When [[latex($\lambda$)]] goes to infinity the posterior mean shrinks towards the prior mean, ''ie'' the projection of the DSGE model onto the VAR([[latex($p$)]]). We do not have a closed form expression for the joint posterior density of [[latex($\theta$)]] and [[latex($\lambda$)]] (the second term on the right hand side of \equaref{posterior1}). So the posterior distribution of [[latex($(\theta,\lambda)$)]] is recovered from an MCMC algorithm, as described in Del Negro and Schorfheide, except that we do estimate [[latex($\lambda$)]] as the deep parameters [[latex($\theta$)]]. ''' Example ''' |
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}$) for the priors to be proper (Note that it would not be possible to estimate the VAR model by OLS (or maximum likelihood) if we had latex($\mathcal T < m(p+1)$). In this case we would not have more observations than parameters to estimate). The a priori density of latex($\mathbf A$) is defined by latex($n+1$) parameters (latex($\theta$) and latex($\lambda$)), which is likely to be less than latex($mp$) (the VAR number of parameters). If we have a one-to-one relationship (no identification issues) between latex($(\theta,\lambda)$) and latex($\mathbf A$) it will be a good idea to estimate latex($(\theta,\lambda)$) instead of latex($\mathbf A$), ie to estimate fewer free parameters. To do so, Del Negro and Schorfheide complete the prior by specifying a prior distribution over the structural model's deep parameters: latex($p_0(\theta)$). We still have to set the weight of the structural prior, latex($\lambda$). Del Negro and Schorfheide choose the value of latex($\lambda$) that maximizes the marginal density. They estimate a limited number of DSGE-VAR models with different values of latex($\lambda$). For each model they also estimate the marginal density and select the model (ie the value of latex($\lambda$)) with highest marginal density. Dynare can also estimate directly latex($\lambda$) as another parameter, instead of doing a loop over the values of this parameter. So we define a prior on latex($\lambda$), which is assumed to be independent from latex($\theta$). Finally, the DSGE-VAR model has the following prior structure:
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where latex($p_0\left(\mathbf A, \Sigma \mid \theta ,\lambda \right)$) is defined by [P1a,P1b] and [P2].
The posterior distribution, may be factorized in the following way:
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where latex($\mathcal Y_T$) stands for the sample. A closed form expression for the first density function on the right hand side of [Q3] is available. Conditional on latex($\theta$) and latex($\lambda$), [P1a,P1b] and [P2] define a conjugate prior for the VAR model, so its posterior density belongs to the same family: the distribution of latex($\mathbf A$) conditional on latex($\Sigma$), latex($\theta$), latex($\lambda$) and the sample is matric-variate normal, and the distribution of latex($\Sigma$) conditional on latex($\theta$), latex($\lambda$) and the sample is inverted Wishart. More formally, we have:
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where:
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with:
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The posterior mean of latex($\mathbf A$) is a convex combination of latex($A^*(\theta)$), the prior mean, and of the OLS estimate of latex($\mathbf A$). When latex($\lambda$) goes to infinity the posterior mean shrinks towards the prior mean, ie the projection of the DSGE model onto the VAR(latex($p$)). We do not have a closed form expression for the joint posterior density of latex($\theta$) and latex($\lambda$) (the second term on the right hand side of \equaref{posterior1}). So the posterior distribution of latex($(\theta,\lambda)$) is recovered from an MCMC algorithm, as described in Del Negro and Schorfheide, except that we do estimate latex($\lambda$) as the deep parameters latex($\theta$).
Example