Theoretical reminder

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 (see Tiao and Zellner (1962) or Zellner (1971)). 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($m^2p$) (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, say latex($N$) 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. Implicitly, they compare different models considering a (discrete) uniform prior distribution over the collection of latex($N$) models. With Dynare it is also possible to estimate 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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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 [Q3]). 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($\theta$).

Estimation of a DSGE-VAR model with Dynare

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 latex($\lambda$). 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:

options_.varlag = 12;

before the estimation command. The weight of the DSGE prior can be :

1. Calibrated to a positive real number (greater than latex($\frac{mp+m}{T}$)) or to Inf (latex($\lambda=\infty$) 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.

Example

First run the file attachment:simul_hybrid.mod to create a data set from a new keynesian model with an hybrid Phillips curve. Second run the file attachment:bvar_forward.mod to estimate the DSGE-VAR model (where the DSGE prior is a new keynesian model with an hybrid Phillips curve).

References

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.