I want to ask if it is possible to do the estimation in Dynare with the projection method as the solution part.

For example, can I implement my Chebyshev polynomials code into the source code, and let Dynare do the Kalman/particle filter and MCMC? But I imagine it can be very difficult.

You could do this, but in this case, it might be easier to do that from scratch. The Kalman filter relies on linear Gaussian state-space representations. Combining it with projection might therefore not make much sense as you use this technique for nonlinear solutions. Programming the Metropolis-Hastings algorithm is just a few lines so it is not worth the effort taking it from Dynare. Regarding the particle filtering, relying on these routines would be a lot of effort. In this case, I would rather go for a Julia, C or Fortran implementation of particle filters.

Thanks @jpfeifer for the answer! You are right, Kalman filter is not useful under this context, because the reason to use the projection method is to obtain a more accurate nonlinear state space representation. Regarding the particle filter, I’m trying to program it in Matlab because I don’t really know C or Fortran.