Projection method and nonstationarity

Hi, I’m trying to use the Chebyshev polynomial method to solve a RBC model.

I know in Dynare where the perturbation method is used, the model needs to be stationary so that the policy functions are approximated around the steady state.

However, for Chebyshev projection, only a projection range of state variables are needed. As a result, if I assume there is a stochastic trend in technology, can I solve the nonstationary model with projection method directly (there is an optimal solution), or do I still need to first rescale the model, and then solve the stationary model?

My understanding is that, the projection method is purely makes the residuals as close to zero as possible. So even if the model is nonstationary, the approximated policy function should still be correct. But I’m not sure because naturally we detrend the model.

This is tricky. In principle, you could use projection to solve for the non-stationary policy functions. As this is a global solution technique, it should work over the whole state space, which has unbounded support in this case. The problem this does not work with many polynomials you use for projection. Either they are only locally accurate due to a small convergence radius or they are defined only on a bounded domain. The latter applies to Chebyshev polynomials, which are defined between [-1,1] and then scaled up. The scaling does not work if there is no sensible upper bound as would be the case without detrending.
So in a nutshell, you want to work with the detrended model.