How to enter Jensen's term correctly

General DSGE Modeling
1

Hello,

I am working with Epstein-Zin utilities, and I would like to have a term of the form

(E_t(X_{t+1}^alpha))^(beta)

Is the correct way to define Y_t = E_t(X_{t+1}^alpha) and then use Y_t^beta?

Thanks for all the help.

2

That’s one way to do it. Rudebusch and Swanson in the papers on bond pricing use this method. To be honest, I don’t really trust this setup. If you use Dynare’s approximation to try to get the SDF, it gives pretty poor results. The problem is that you’re approximating around the nonstochastic steady state, but for the risk-adjusted continuation value, you really need to be at a different point. I usually use other methods, e.g. projection. I’m currently getting nice results using the generalized stochastic simulation algorithm proposed in judd, maliar, and maliar (2010).