In a nonlinear model which includes Calvo style price and wage markups, could the estimation include shocks to elasticity of substitution of labor and goods? I know as the elasticities appear in the exponent and are state dependent, they cannot be dealt with as time varying in a recursive form, but does first order simulation (the bayesian estimation is apparently also first order) some how not have this problem? Because I read in one of the posts that you said it cannot be done in second order and I got suspicious of its validity in first order apporximation. Christiano et al. (2014) have used price markup shock in their bayesian estimation of their model in “Risk Shocks” but it doesnt seem to me that they have had log-linearized their model. I have Calvo style sticky price and wage in my model and when I add their respective shocks for the estimation i get better estimations (bigger absolute log likelihood number) vs for when I remove those shocks and add two other shocks. Its like -5380 vs -5150 in log likelihood. So their shocks result in better estimation results theoretically but if you say it is not right to add their shocks to a nonlinear model then I have to compromise I guess.

I don’t think their implementation is correct. It looks to me as if they simply derived a recursive representation without markup shocks then added a time-varying markup after that.

Thank you for your response Prof. Pfeifer. So for Calvo style markup shocks you cannot use recursive form even in first order approximation. And currently their is no other possible way other than linearizing the model. Am I correct?

I have received a response from Lawrence Christiano. I will put the related part as follows:

"We do analyze the linearized version of the model. The risk shock evidently has a first order impact on the variables in our model. This reflects the nature of the loan contract.

You are right that there does not exist a recursive representation of the (nonlinear) optimality conditions for sticky prices when markups are measured the way they usually are. However, if you consider the equilibrium conditions in sequence form (i.e., as infinite sums) you can linearize those around steady state. Those linearized equations do have a recursive representation. I know, this is weird and maybe there is a problem here at some deep level. However, this is how the literature deals with markup shocks. (Actually, I have another paper where we model shocks to the markup in a slightly different way so that the nonlinear equations have a recursive form.)"

I would appreciate if anyone can help me understanding this. First it says they analyze the linearized version of the model then it explains about how the linearization of the recursive form takes place. I do not understand why you would write it in recursive form initially if you want to linearize it and then rewrite it as recursive form later (if I understand correctly). Also, it is said that they have another paper that has modeled the markup shocks differently such that it is possible to have a nonlinear recursive form. Does anyone recognize the paper he is referring to?

In principle, you can just linearize the pricing first order condition (which is not yet recursive but features an infinite sum), express it recursively after linearization, and link it to the rest of the nonlinear model. That’s what we did in earlier work and it allows you to not linearize everything.

What puzzles me is that the replication files of CMR (2014) do not seem to pursue this usual road. Instead, there is a recursive nonlinear equation entered in the mod-file that Dynare then approximates. I haven’t checked whether the resulting linearized approximation would be identical to first linearizing then expressing it recursively, but I have my doubts.

Thank you for your response. Is it possible for me to just linearize the price FOC and leave ALL of the rest of the equations, including the price dispersion equation, aggregate price equation and linkage between total output and firm output via price dispersion equation, as nonlinear? In that case the variables in the price FOC will become linearized but the same variables that are linearized in this equation also appear in other equations as nonlinear in the exp(variable) form. Is this okay? Is there any similair setting in a piece of code or a pdf file available so I can look in to it?