Reasonable acceptance rate coexist with posterior spikes


I’m confused about the presence of 20%-30% acceptance rate and very tight credibility intervals of the parameters i.e. spikes in the posterior check plots. As I understand it, a sound acceptance rate should have guranteed a comprehensive search in the parameter space, whereas the RWMH draws seem to get stuck around the posterior mode and fail to jump around in the parameter space.

Thank you very much if anyone can point out the source of the problem.


Please post the check plots to show what you mean with

I mean the posteriors are just spikes that implies the posterior sampler only explores around the initial values.

Please see the attached file. Thanks.
CheckPlots.pdf (15.1 KB)

As we are talking about prior-posterior plots, I would need to see the mod-file or the prior.
See also here [Evaluation (robustness) of the Bayesian estimation results)

Please see the priors here (original shocks are rescaled to 100*shocks). Btw, since all of the mode-finders will result in non-positive-definite covariance matrix i.e. they can’t find the posterior mode, I initialize the RWMH draws with mode_compute=6.

And output of estimation as below (as one can see, the credibility intervals are strangely tight):

Many thanks.

Look at your AR-coefficient estimates. They are basically 1. If a proposed draw that hits the 1, this draw is rejected. Thus, you need a small jumping matrix just to get a decent acceptance rate. Those parameters hitting the bound should also explain the mode finding problems. Given the large persistence implied by those estimates, I would guess your data were not correctly treated/stationarized.

Thanks for pointing this out. I think you are right about the jumping scale. But it seems to me normal that some shocks are extremely persistent, say Gali, Smets&Wouters (2011) and Gertler, Sala&Trigari (2008) also find high level (.99) of persistence for price markup or gov spending shocks. What may worry me is the potential underestimation of standard deviations of shocks, which will also push AR coefficients to unity. However, I do want smaller standard deviations of the shocks to account for plausible theoretical moments of the model variables and more reasonable magnitude of IRFs. Actually, that is the reason why I reject the first pass of estimation and have the ongoing attempt.

As for the measurement eqs., I think they should be fine since except for interest rate and inflation rate, all of the observables are filtered and demeaned when necessary.

In fact, when I run more draws, spikes in posteriors of some parameters disappear though others still remain. But no one hits the boundary of priors. So I guess they may hit the boundary of determinacy parameter region where is a high probability region as well. In this case, it may take much longer time (1 million draws in my experiment) to jump away from the boundary of determinacy region.

A few quick notes. There is a large difference between 0.99 in Gali, Smets, Wouters, and what you find. With 0.99, the halflife of your shock is 69 quarters, with 0.9988 as the lower bound of the HPDI for rho_xi, the halflife is 577 quarters.
Essentially, your distribution is degenerate given that many of your shock processes basically have a unit root. That’s why I conjectured there is something wrong.

I hope you use a correct filter for your data (see e.g.

Any positive definite jumping matrix in the Metropolis-Hastings will yield draws from the posterior. You just may need a lot of draws. What you describe is exactly this. I would recommend trying to run the MCMC with the prior_variance option for mcmc_jumping_covariance and see whether your distribution looks better. This will also be inefficient but should at least assure you of a better feeling what the posterior coverage is. The current estimates seem to be stuck due to a too narrow proposal density.

In short, you have to find out if the results you got indicate that there is a problem with the model as the posterior is really at the upper bound or whether your MCMC sampler just got stuck and you have not enough draws.

Many thanks for your suggestions. I’ll try and see.