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I'm looking for an R package (or a combination of packages) that would allow me to perform MCMC estimation of a GMM model, with a user-specified moments function.

I've looked at the CRAN Bayesian task-view, but I can't seem to find what I'm looking for, packages being either too general (eg. mcmc) or too specific (MCMCpack)... Ideally I would like to be able to implement Stochastic GMM or to use package R-INLA to do iterative nested Laplace approximation, again with a user-specified moments function.

Any help greatly appreciated!

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I've implemented Bayesian GMM estimators in STAN before and it is pretty straight-forward. The way STAN works is that one simply defines how to calculate he log-likelihood and then a very efficient sampler (NUTS) based on Hybrid Monte Carlo is created automatically, and for the GMM likelihood one just creates the $U_i$ and $\Sigma$ by hand (which will depend on $\beta$) and increment the log-likelihood by doing something like

lp__ <- lp__ - 0.5 * U[i]' * Sigma * U[i]

(make sure to double check that this is correct though). It may even be possible to trick JAGS/BUGS into doing it with the "zeros trick" but STAN seems like a fantastic option for this (although I suppose it may also depend on the amount of data you have). I'm taking for granted that your pseudo-likelihood defines a valid posterior, of course!

EDIT: should definitely be a Sigma inverse rather than Sigma in the formula above!

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In Bayesian generalized method of moment we estimate the pseudo-likelihood and then multiply the specified priors of model parameters and then use the usual MCMC procedure to estimate the model parameters.

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