I am trying to estimate inverse of negative hessian for the posterior . However , the matrix turns out to be not positive definite. Upon investigation I find that one of the entry associated with measurement error variance is creating the trouble. But, when I estimate the inverse of negative hessian for the likelihood , the matrix is positive definite.
Will it be correct to take scale version of the Variance covariance matrix estimated just with likelihood as my step size in MH. My Priors are not heavy tailed (mostly gamma , beta and uniform) .
Any positive definite covariance matrix should theoretically work. An/Schorfheide even suggest using the inverse negative Hessian of the likelihood. Asymptotically, there is no distinction between the two.
Thanks professor , another small query I will like to clear. What is the optimal value of the "scale parameter " for taking the scale version of the covariance matrix ?Is the scale value in the range of 1e-14 acceptable , if I am obtaining a desired acceptance rate of 0.25-0.30 . In such case I understand the standard deviation of the posterior will be too tight, or can I say certainty associated with the parameter is high and tightly identified around mode. The adaptive MH is indicating the scale in such range.
No, 1e-14 typically is not OK, but rather indicates a problem in at least one parameter direction. Investigating the trace plots often helps to identify the issue.
Thanks professor. I was also doubting the same. Will it be possible to share some reference codes for regime switching DSGE . I want to check if any optimisation routines (other than Sims routine) are being used and the checks and balances especially in sub-steps involving Gibbs within Metropolis.