# Second Order Approximation General Questions

Hi Johannes,

I have a few general questions about Second Order approximations.

1. What would be the best go-to paper/guide to learn the general procedure? (preferably with examples and not a description of the general matrix procedure only).
2. Can you solve a DSGE model that combines first and second order approximations?
3. Any matlab codes or (dynare codes) that can approximate a non-linear system into a second order approximation would be much appreciated.

Note: I know dynare can solve Second Order Approximations by just setting order=2, but I need to do this by hand for several equations as it is important for me to understand the behavior of certain dynamics. Thus, simply plugging in the non-linear model into dynare wouldn’t be of use for me (hence this thread)

Thanks

1. The usual reference is the Schmitt-Grohé/Uribe 2004 paper with codes at https://github.com/JohannesPfeifer/DSGE_mod/blob/master/SGU_2004/SGU_2004.mod. The Dejong/Dave book also contains an example.
2. You can combine linear and nonlinear equations if you know what you are doing. As the system is not fully nonlinear anymore, you are going to lose some of its properties.
3. You might want to have a look at the original replications files of the SGU 2004 paper. There you can see the approximations.

Thanks Johannes. I may be missing something here, but a first and second order linear approximations don’t have any non linearities right? What I meant in 2 is that I want to combine a first and second order approximation into the equilibrium of the system. I don’t want to exploit the original non-linear system.

A second order approximation is not linear. There will be quadratic terms.

Yes that was silly on my part.

Back to my previous question, can I mix linear and non-linear equations in dynare. With respect to my case, first and second order approximations?

Yes, of course, with the caveat mentioned above.

Are you aware of any code in dynare that does this that I could reference?

Not really. What people sometimes do is have a nonlinear mod-file with `order=2` where some linearized equations are appended.