Second order approximation NK model

General DSGE Modeling
1

Hi all,

I am currently working on a New Keynesian model which includes the usual macro variables such as output, consumption, investment and inflation also some asset pricing related variables such as a bond, risk premium, bond price and the discount factor.

I would like to compute the second order approximation to get the precautionary savings motive. For now this motive is very small ( I can rarely/not see a difference in comparison to order 1). I was wondering if this is common and if so, is there a way to make this motive stronger (without deviating from a RANK model) ? for example including further assets or using EZW preferences.
For now I use CRRA preferences and habit formation for consumption and capital adjustment costs.

For me it would be essential to first (without proceeding with the actual analysis I want to do) show why second order approx. is important in a New Keynesian model for shock transmissions. I am wondering if there is any modeling trick I could add or if indeed in the NK model for usual technology shocks and so on (no uncertainty shocks) the second order approx. does not play a crucial role. Unfortunately, I could not find a paper writing about the explicit comparison for first and second order in a basic NK model and its implications for the irfs.

Thank you for your help!

2

Check Jermann 1998 (https://econpapers.repec.org/article/eeemoneco/v_3a41_3ay_3a1998_3ai_3a2_3ap_3a257-275.htm) on how to increase risk premium in a RBC model. It may give you ideas for a NK model.

1 Like
3

Thank you very much Prof. Juillard, I will have a look at the paper.

But just to clarify, in general second order approximations can matter in NK model for usual shocks such as technology, monetary policy etc.? I was testing code in an RBC model with EZW, equity, bonds, habits etc. and here the irfs did not change much (not really noticeable on the plot) when I changed the order of approx. and I am a little bit confused by this result or does the second order approximation only change the theoretical moments?

4

You need to be careful with such comparisons. At first order, the IRFs are state-invariant due to linearity. That’s not true anymore at higher order. Thus, you cannot easily compare a single IRF at higher order to the one at first order.

The difference at orders of course depends on the degree of nonlinearity in the model. With EZW the difference should be noticeable.