Asset Pricing Implications of Log-Linearized NK Model

Hi there,

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.

Is there a suggestion how to proceed?

Many thanks!

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

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?

Many thanks again

I would simply use the nonlinear model and go to second order. See on how to write the NKPC recursively. There, you can also see the nonlinear version of the NK model.

Thank you for the link, this helps me a lot