State Space Representation

[Lorenzo]

Hi,
I am trying to retrieve the following state-space representation from Dynare:

s_t = A*s_t-1 + B*epsilon_t
x_t = C*s_t

where s_t is a mx1 vector of states and x_t is a nx1 vector of controls. A is a mxm matrix, epsilon_t is wx1 and B is mxw. C is the policy funciton and is nxm.

However, to my knowledge, Dynare combines the two equations as one:

x_t = C*A*s_t-1 + C*B*epsilon_t

My question: How can I retrieve the coefficients of A, B, and C using Dynare (up to third-order)?

Thanks so much

[jpfeifer]

At first order, this is straightforward. See https://github.com/JohannesPfeifer/DSGE_mod/tree/master/FV_et_al_2007 and the content of the ABCD_test.m
At higher order, the state-space needs cannot be easily represented linearly (unless you use pruning). This is not yet supported.

[Lorenzo]

Dear Johannes,
Thanks for your reply. Thanks for the function ABCD_test.m. However, this function simply uses the output from Dynare and reshuffles it a bit. Basically, I am trying to estimate a model with Bayesian estimation up to the third order. The estimation itself should be fine. However, I would need to be able to retrieve the state space representation from somewhere - I was thinking about Dynare. As a first step, I tried to replicate the examples (2nd order) from Schmitt-Grohe and Uribe (2004) "Solving dynamic general equilibrium models using a second-order approximation to the policy function" - basically get the same coefficients of the 2nd order approximation. However, this seems not to be possible with Dynare... Do you have any idea how this could be accomplished?
Thanks so much,
Lorenzo

[jpfeifer]

As I said, it is not clear which state space representation you mean. At higher order, you have a nonlinear statespace so that you won't get

Code: Select all

s_t = A*s_t-1 + B*epsilon_t
x_t = C*s_t

but rather a more general

Code: Select all

s_t = f(s_t-1,epsilon_t)
x_t = g(s_t)

As there is no standard form, you would need to do this yourself. The only help you can find is the unpacking of the g_* matrices conducted in
https://github.com/DynareTeam/dynare/blob/master/matlab/k_order_pert.m
and the simulations in
[url]https://github.com/DynareTeam/dynare/blob/master/matlab/simult_.m
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