I found this interesting explanation for how one can use Dynare to deal with ill-formed Hessian matrices that seems like it could help out my current project greatly.
My question is: How does dynare draw the thetas? An MHRW seems like an obvious choice but without knowing the mode of the posterior or the hessian at the mode, it’s not obvious how to calibrate the proposal density.
As in a standard metropolis we use draws from a Gaussian, centered on the previous state of \theta. For the covariance, we first use a diagonal matrix (I don’t remember if it’s an identity matrix or a diagonal matrix with the prior variances) and update the covariance matrix of the proposal with the output of the recursive definition of the posterior covariance, \Sigma, every 20 000 simulations (or the value given to options_.gmhmaxlik.number). We do not use the mode, even if we track an estimate of it.
Best,
Stéphane