Hello everyone!
I am currently working on a version of the Krusell-Smith (1998) model, but instead of a fixed transition matrix, I need to work with two continuous autoregressive processes and discretize them into the Markov chain. However, to get structure similar to KS, I need to impose a correlation between micro and macro-level shocks, which makes the stochastic system VAR, not just two independent ARs.
I would be grateful for any suggestion, how to perform this discretization. Is there some nice approach similar to Tauchen/Rouwenhorst for AR(1)s? Or I need to perform Monte-Carlo simulations and fit the Markov chain on it?
Best,
Honza
Hi @Honza9723. Good question. If it was me, I would perform MC simulations and fit the chain to it in the usual way.
Actually, I usually prefer to use fitted value function iteration or some other fitted solution method that avoids discretization. If you can use the endogenous grid method then you can certainly solve the household problem quickly without discretization (see, e.g., https://python.quantecon.org/ifp_advanced.html).
Discretization suffers most severely from the curse of dimensionality. It’s advantage is that it is well suited to vectorized code. But with JIT compilation, vectorization is no longer necessary.
Just my point of view…
Thank you very much!
So, your advice is simply to solve Krusell-Smith with two continuous shocks with GH quadrature?
Best,
Honza
Yes, something like that. I personally like linear interpolation because it preserves the contraction map property of the Bellman operator.
If you can use the endogenous grid method then that will help a lot. And integration can be done by quadrature or straight Monte Carlo. Just simulate expectations in the second case. (But if you do that, be sure to generate a sample of shocks and use the same one every time you update the value function / policy function. Otherwise you will introduce instability.)
Good luck :-)