I keep returning to this example I’d like to try: I’m trying to estimate the likelihood of a nonlinearized DSGE model using a particle filter. While I understand particle filtering, I have little training in economics, so I have a difficult time understanding how economists define DSGE models.
In statistics, the starting point is a definition of a state-space model. You have latent dynamic variables evolving according to a Markov chain, and you have observed variables distributed conditionally (on the states) independently.
My question is: what is a source that describes how I can write down a DSGE model in state space form? How many latent states are there and what do they mean? What are the static parameters of the model? What are the data being observed?
I’ve found this paper, but it’s light on the details I’m interested in. It writes the state space model in a nonspecific generic way.
This paper in section 3.2 describes the observation equations, but not the state transition distributions. Maybe it’s clear to economists from the discussion of the model when it’s stated as an optimization problem, but I’m not following it at all.
Another answer on this site comes pretty close to describing this. There’s a link to a paper that is slanted towards some software I’m not familiar with, and in this software there is no need to specify state transition equations. It’s very mysterious.
Edit:
The closest I have come to finding an answer is probably on page 21 of this paper. They define a state vector at each time point $S_t$, and an observation vector $\mathcal{Y}_t' = \begin{bmatrix} \Delta \log \mu_t^{-1} & \Delta \log y_t& \Delta \log l_t & \log \Pi_t & \log R_t. \end{bmatrix}$
I'm not sure how to find data for this, but that's not my biggest problems. It's the way he specifies the observation and state transition equations. Here they are:
and this
I'm not really clear on the meaning of these $\Psi$ things. I keep hearing this word perturbation a lot. Also, what's with the dimensions on the observation equation?
