# RWMH Behavior when Jacobian of dynamic model contains Inf

Question: How does Dynare treat parameter draws which result in the Jacobian of the dynamic model containing Inf? Do these draws get discarded and the likelihood set to a penalty as in the cases listed in the cross-post https://archives.dynare.org/phpBB3/viewtopic.php%3Ff=1&t=5171.html?

Background: I am working on a replication .mod for Schmitt-Grohe and Uribe (2012) which incorporates the JR preference structure used in Jaimovich and Rebelo (2009) to govern the wealth effect of labor supply using a parameter γ∈ (0,1] which is to be estimated. Very low values of this parameter imply the steady state value of the (stationarized) stock of current and past-habit adjusted consumption levels approaches zero, and the inverse of this near-zero steady-state value enters one of the FONCs. As a result of this division by (near) zero, the solver is returning Inf for elements of the Jacobian associated to this FONC.

Yes, anything that does not lead to a unique determinate solution that you can compute will be rejected during the MCMC. This amounts to implicitly truncating the prior. I find it a bit unusual that this happens in their model, because as far as I remember they bounded that parameter away from 0 by introducing a lower bound unequal to 0.

Thank you for the quick response. In the paper they say “We adopt a uniform prior distrbution for γ, with a support spanning the interval (0,1]” but they do not comment in the paper nor in their replication files on how the lower bound is achieved.

I know there is an implicit lower bound which can be derived below which the labor disutility parameter will imply the argument of the utility function will be negative. I wonder if this is what they used. For now I will try a parameter transformation and add a small constant to pull it away from zero and see how that works.

Did you encounter any similar troubles with this parameter from your 2013 JEDC? I know you used a gamma prior instead of a uniform for what you called σs, which should still have technically included zero in the support.

You can verify in their replication files that the mode for \gamma>0.002. Typically, people impose a lower prior bound of 0.001 or so. That is small enough to not matter much in terms of truncation, but large enough to prevent numerical problems.

In our JEDC paper, we used a beta distribution that pulls the parameter away from 0

To clarify, are you referring to something contained in the package of files on the Econometrica website? I see their transformations on a few of the parameters in their steady state file (gx_hx_inputs.m), and the values of the mean/median from their estimations (parameter_vector.mat), but nothing about restricting gamma nor about the estimated mode. Perhaps I am not seeing something…?

They do not provide replication files for the estimation itself. But they provide mode-files. And there gamma is away from 0. My guess is that they introduced a lower bound different from 0. If I were to repeat the estimation, I would put the lower bound at 0.001.

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