I’m interested in doing a very simplied variation on FernandezVillaverde and RubioRamírez recent NBER paper  i.e. something on parameter drift.
Two questions arise

Can this meaningfully/correctly be done in a linearized version of the model, i.e. estimated with Dynare? They use a second order approximation and a particle filter for the estimation.

I need to bound the drifting parameter below 1 for my model to have a solution. I let the parameter follow an AR1. My prior is on the process (persistence, variance of innovation) of the parameter, not it’s value, so I can’t bound the parameter by using the prior.
BUT
I did notice that Dynare let me specify the a nonlinear process for the param:
log(M_t)=min( rho_M*log(M_t1) + (1  rho_M)*MBAR + eps_t , 0 )
How much trust should I put in the Dynare solution and estimation of this type of setup? Clearly it’s important that the bound (almost) never binds b/c we are using a linear solution algorithm.
I’d be grateful for a response on any of the two questions.