# Deriving of stare space matrices in DSGE

Hi everybody.

We have the following state space form in a DSGE model.

S_{t}=\Psi S_{t-1}+\Omega u_{t}
P_{t}=\Gamma S_{t-1}+ \Lambda u_{t}

And in a total form we have :

Y_{t}=Y^{s} + A S_{t-1}+B u_{t}

We can derive matrix A by this command in Dynare oo_.dr.ghx

In my DSGE model I ordered the variables in dynare according to the decision rule order and the first three variables are static variables. We I derive the matrix A in dynare I can select the rows in relation to the state variables of the model. In my DSGE model this matrix is 6 \times 6 and when I calculate it’s eigenvalues in MATLAB, these eigenvalues are the same with Dynare output (matrix \Psi ). These eigenvalues are less than one. But when I derive matrix \Gamma for explosive part of the system the eigenvalues are not true.They are not more than one. In my DSGE model in the A matrix I derive matrix \Gamma by removing of relation rows of static variables and purely predetermined variables.I derive this matrix for mixed variables and purely forward variables. In my DSGE model this is a 6 \times 6 matrix. I consider the rows of mixed variables and purely forward variables for the \Gamma matrix. In my model matrix A is 12 \times 6 and three first rows are in relation to the static variables. The rows of 4 until 9 are in relation to state variables and matrix \Psi . The rows 7 until 12 are in relation to jumpers or \Gamma matrix. But eigenvalues of the \Gamma matrix are not more than one.

What is wrong in deriving of \Gamma matrix??

Eigenvalues of the \Psi matrix are the same with dynare output results.

I don’t understand your post. The eigenvalues of the matrix \Gamma are irrelevant. In fact, the matrix is generally not even square.
You need some unstable eigenvalues in the AE(x_{t+1})=Bx_t formulation to be able to compute a unique stable solution. But once you have the computed the solution to the system, everything should be stable.

Thanks professor for your comment, but in my DSGE model \Gamma matrix is square and is 6 \times 6. This matrix may not be square in some DSGE models unlike my model.

I derived \Psi matrix in Dynare.This matrix is square too and is 6 \times 6.The eigenvalues were less than one in absolute form and were the same with Dynare output results.

But according to your comment by this way we can not derive \Gamma matrix, but as I did there is not any problem in recovering of \Psi matrix.

If you have n_x states and n_y non-predetermined variables, then \Gamma is n_y\times n_x and generally not square. But even it is square, what does “explosiveness” even mean when mapping from states to controls? The objects on the right of the equation are always bounded if \Psi has all eigenvalues inside of the unit circle.