Can you mix log linearized equation with level equation?

[hewei2004]

Hi,
I think all the New Keynesian models are written in log-linrearized form.

In addition to a New Keynesian setup, I also want to use Epstein-Zin preference in it, and all the examples I see are written in levels, ie. exp(log(At))=d+exp(log(Bt)+Et[exp(log(Ct+1))]) something like that.

Is it allowed to to mix log-linearized equation, with level equations, in dynare? I hope I can have both in the mod file, and hope it doesn't mess up the perturbation method and calculation of steady states.

Thank you.

[jpfeifer]

No, you cannot do this. The reason is that all equations must be approximated up to the same order to get correct results (up to that order). You can use linear equations together with nonlinear ones if you use a first order approximation. Dynare will then make the nonlinear equation linear. But that does not help with Epstein-Zin. Here, you must go at least to second order. But your linear equations will only be correct up to first order. For this reason, you need the nonlinear equations. The problem with New Keynesian models is that the Calvo pricing introduces an infinite sum. This infinite sum disappears when log-linearizing. However, there are other ways to get around this, notably the Schmitt-Grohe/Uribe trick that transforms the infinite sum into a recursive equation (see http://www.dynare.org/phpBB3/viewtopic.php?f=2&t=6109). An example of this is in the NK_baseline.mod in the Dynare examples folder that does exactly that.

[hewei2004]

Thank you Sir.

1.So am I correct in saying that, if I have 2nd order or 3rd order approximation for the mod file [order=2]. I still cannot do the mixing?
If I have that the NK equations are still written in log linearization in dynare mod file; But the Epstein Zin are written in levels. The equations are of different approximation order, but I can still get an approximation that is numerically more accurately than solely using linear equations in mod file, right? My logic is that some equations in 2nd order and some in 1st order, is closer to truth than all equations in 1st order.

2.If it is theoretically forbidden to mix nonlinear with linear.
Can I keep all equations in log-linear form by log-linearizing Epstein Zin preference also, such as in the Uhlig tutorial?
http://sfb649.wiwi.hu-berlin.de/fedc/ev ... SGE_v2.pdf

Does log linearizing Epstein Zin somehow take away some of the interesting dynamics of Epstein Zin preference?

[jpfeifer]

1. Your results will still only be accurate up to first order.
2. I haven't read the Uhlig paper, but the logic seems to be similiar to Jermann's (1998)-paper. Using log-linear asset pricing allows using first order instead of second order approximations. But unless you do this as well, you typically need higher order, because at first order, Epstein-Zin collapses to CRRA preferences.