I have written a NK 2-country model which is now log-linearized. The simulation works and the model itself looks absolutely correct.
Now: As the model is log-linearized, I observe only steady state values of zero. Theoretical means are also zero but standard deviations are not.
I’m interessted in my model’s asset pricing implications. As in Jermann’s (1998) paper, I would like to simulate my model in a first step to capture standard macroeconomic variables behavior, and then in a second step, observe prices and returns of assets.
I’d say for asset price implications a log-linearized model won’t do. You need a second or even third Taylor approximation in order to account a time-varying risk premium.
At first order, certainty equivalence holds so that risk in the economy will not generate any risk premia in the logged variables. That is why @pepito_bm suggested you need to go to higher order. That is generally true. But @Cybueh referred to Jermann (1998). What that paper does is so-called log-normal asset pricing. It solves model in log-linear form with normally distributed shocks. Thus, certainty equivalence holds. But it then uses the property of the log-normal distribution that if x is log-normal with mean 0 and variance \sigma^2, then e^x is normally distributed with mean e^{0+0.5\sigma^2}. As you can see, the undoing of the log-transformation results in a Jensen’s Inequality term that gives you a risk-premium. That will give you results virtually indistinguishable from the second-order approximation to the nonlinear model.
However, I find the nonlinear model easier to use. See https://github.com/JohannesPfeifer/DSGE_mod/blob/master/Jermann_1998/Jermann_1998.mod
Many thanks for your helpful answers. I appreciate!
I would really like to examine asset pricing implications of a multiple country NK model. Therefore I have built a 2-country NK model with especially Calvo Pricing, habit persistence, and different types of adjustment costs (at the end I have a working model with 36 equations).
The only reason why I did the log-linearization was the NK-Phillips curve. I simply do not know how to enter this equation (infinite sum) in the nonlinear form.
@jpfeifer Generally, is it a good idea to do such an extension to Jermann 1998?
I’m not sure if I understand your comment correctly: When I simulate the log-linearized model, as I did so far, I can in fact undo the log transformation (for example by defining asset pricing variables powered with e^[] ) and simulate risk premia similar to a 2nd order approximated non-linear model?