I have a question concerning the Bayesian estimation of DSGE and VAR models.
In the VAR model, we do not need to find the maximum a posterior density. Indeed, we just do the sampling procedure, in particular, the Gibbs sampling algorithm, based on the conditional marginal posterior distribution of each underlying parameter. So, the initial value for this sampling algorithm is typically arbitrary.
however, in the estimation of DSGE mode, before doing the sampling algorithm, especially the MH algorithm, we have to find a ‘local’ maximum level of the conditional joint probability distribution of all underlying parameters. Indeed, this procedure is well-known as maximum a posteriori estimation (MAE).
So my question is that doing the MAE is to find the initial values for the sampling algorithm such as the MH algorithm? Why in the estimation of the DSGE model, we have to do that? But in the estimation of the BVAR, we never do it?
Thank you so much and please correct my understanding
You don’t have to do that. We know from theory that the Metropolis-Hastings algorithm will sample the posterior from any starting point as long as the jumping covariance is positive definite (under some regularity conditions). However, it may take a really long time, i.e. really many draws. To improve efficiency, one can tailor the proposal density to the problem. For the BVAR, things are often quite easy, because we know a lot about the posterior distribution (like which (conditional) distribution it follows). The same cannot be said for the DSGE model, where we typically have no clue about the posterior distribution. We only know that, again under some regularity conditions, the posterior will be asymptotically Gaussian. So we tailor a proposal distribution by finding the mode and then doing a Laplace approximation at the mode.
I am not sure I understand. For tailoring the proposal density, we need a covariance matrix. We find the mode and then use the inverse Hessian at the mode as the covariance. See e.g. Koop’s textbook.