# Linear decoupling field matrix

Hello,

I have a non-linear deterministic model implemented in Dynare.

Is there a way to get the sensitivities of the forward-looking variables with regard to the predetermined ones in the linearized model around the steady state?

In other words, the decoupling field (the forward-looking variables as a function of the predetermined ones) of the linearized model is a linear application, and I would like to access its matrix.

Best,
Thibaut

@stepan-a One should be able to get this matrix from the perfect foresight simulations. Is there an easy way to access it?

Sorry but I think I do not understand what is asked here. I am not sure that this is about perfect foresight models, even if the model is deterministic. Why do not use the `ghx` matrix from the first order perturbation (or a composition of `ghx` with itself if a mapping for the variables in t+1 is required)?

Best,
Stéphane.

Hello both of you,

Matrix ghx is, if I understand well, the transition matrix of the linearized model.
What I am looking for is the matrix of the decoupling field of this linearized model.
To precise my question, here is the original article for the Blanchard and Kahn condition:
http://people.bu.edu/rking/SGZ2007WK1/blanchardkahnem80.pdf
We can see on page 1308 (page 4 of the pdf), equation (3), that what I am looking for is the matrix - C_22^(-1) * C_21
This involves the Jordan decomposition of the transition matrix ghx, plus some submatrix manipulations. I guess Dynare must do that work when we call the command “check”.

Thibaut

Dear Thibaut,

The C matrix is an output of the the Jordan decomposition, but we do not use the same strategy as in Blanchard and Kahn paper. We even do not work with the same representation of the linearised model. Blanchard and kahn work with:

z_{t+1} = A z_{t}

where z_t is the stacked vector in the first equation (1.a), while we consider models of the form:

Bz_{t+1} = A z_{t}

where B is not necessarily full rank, meaning that we cannot simply rewrite the model as in Blanchard and Kahn by inverting B. So I am not sure it is possible to extract the C_{22} and C_{21} matrices from Dynare’s routines. I think the simplest way to go is to start from the Jacobian matrices computed by Dynare (the dynamic matlab routine), build A as in Blachard and Kahn, and compute the Jordan decomposition.

Best,
Stéphane.

Dear Stephan,

If I’m not mistaken, Dynare’s strategy uses indeed the Schur decomposition of A and B instead of a Jordan decomposition.
It is still possible, through some simple submatrix manipulations, to obtain the matrix I am looking for from this Schur decomposition.
So do you know if I can access this decomposition?

Thanks a lot!

Best,
Thibaut

I do not think we save what you want, but you can probably easily change the Dynare routines to fit your needs. The generalized Schur decomposition with the partitions of the matrices between the stable and unstable directions is done in `dyn_first_order_solver.m` after line 182 (this routine is called by `stochastic_solvers.m`). The algebra is described here.

Best,
Stéphane.

That’s perfect, thanks a lot for your help!

Best,
Thibaut