Dynamic Stochastic General Equilibrium

Lecture Notes and Assignments

The main notes for this section are found in DSGE2013.pdf.

There is also a supplementary file in DSGE2013extra.pdf.

The homework associated with this section can be found in Econ_Week2.pdf.

You will need the python version of Uhlig’s toolkit. It can be found in uhlig.py.

Below we will show the documentation for it.

Documentation for Python Uhlig

Python module for using the method outlined by Uhlig (1997) to solve a log-linearized RBC model for policy functions.

Original adaptation of MATLAB code done by Spencer Lyon in May 2012

Additional work has been done by Chase Coleman

byumcl.uhlig.uhlig.solvePQRS(AA=None, BB=None, CC=None, DD=None, FF=None, GG=None, HH=None, JJ=None, KK=None, LL=None, MM=None, NN=None)

This function mimics the behavior of Harald Uhlig’s solve.m and calc_qrs.m files in Uhlig’s toolkit.

In order to use this function, the user must have log-linearized the model they are dealing with to be in the following form (assume that y corresponds to the model’s “jump variables”, z represents the exogenous state variables and x is for endogenous state variables. nx, ny, nz correspond to the number of variables in each category.) The inputs to this function are the matrices found in the following equations.

\[Ax_t + Bx_t-1 + Cy_t + Dz_t = 0\]\[E\{Fx_{t+1} + Gx_t + Hx_{t-1} + Jy_{t+1} + Ky_t + Lz_{t+1} Mz_t \} = 0\]

The purpose of this function is to find the recursive equilibrium law of motion defined by the following equations.

\[X_t = PX_{t-1} + Qz_t\]\[Y_t = RY_{t-1} + Sz_t\]

Following outline given in Uhhlig (1997), we solve for \(P\) and \(Q\) using the following set of equations:

\[FP^2 + Gg + H =0\]\[FQN + (FP+G)Q + (LN + M)=0\]

Once \(P\) and \(Q\) are known, one ca solve for \(R\) and \(S\) using the following equations:

\[R = -C^{-1}(AP + B)\]\[S = - C^{-1}(AQ + D)\]

Parameters :

AA : array_like, dtype=float, shape=(ny, nx)

The matrix represented above by \(A\). It is the matrix of derivatives of the Y equations with repsect to \(X_t\)

BB : array_like, dtype=float, shape=(ny, nx)

The matrix represented above by \(B\). It is the matrix of derivatives of the Y equations with repsect to \(X_{t-1}\).

CC : array_like, dtype=float, shape=(ny, ny)

The matrix represented above by \(C\). It is the matrix of derivatives of the Y equations with repsect to \(Y_t\)

DD : array_like, dtype=float, shape=(ny, nz)

The matrix represented above by \(C\). It is the matrix of derivatives of the Y equations with repsect to \(Z_t\)

FF : array_like, dtype=float, shape=(nx, nx)

The matrix represetned above by \(F\). It the matrix of derivatives of the model’s characterizing equations with respect to \(X_{t+1}\)

GG : array_like, dtype=float, shape=(nx, nx)

The matrix represetned above by \(G\). It the matrix of derivatives of the model’s characterizing equations with respect to \(X_t\)

HH : array_like, dtype=float, shape=(nx, nx)

The matrix represetned above by \(H\). It the matrix of derivatives of the model’s characterizing equations with respect to \(X_{t-1}\)

JJ : array_like, dtype=float, shape=(nx, ny)

The matrix represetned above by \(J\). It the matrix of derivatives of the model’s characterizing equations with respect to \(Y_{t+1}\)

KK : array_like, dtype=float, shape=(nx, ny)

The matrix represetned above by \(K\). It the matrix of derivatives of the model’s characterizing equations with respect to \(Y_t\)

LL : array_like, dtype=float, shape=(nx, nz)

The matrix represetned above by \(L\). It the matrix of derivatives of the model’s characterizing equations with respect to \(Z_{t+1}\)

MM : array_like, dtype=float, shape=(nx, nz)

The matrix represetned above by \(M\). It the matrix of derivatives of the model’s characterizing equations with respect to \(Z_t\)

NN : array_like, dtype=float, shape=(nz, nz)

The autocorrelation matrix for the exogenous state vector z.

Returns :

P : array_like, dtype=float, shape=(nx, nx)

The matrix \(P\) in the law of motion for endogenous state variables described above.

Q : array_like, dtype=float, shape=(nx, nz)

The matrix \(P\) in the law of motion for endogenous state variables described above.

R : array_like, dtype=float, shape=(ny, nx)

The matrix \(P\) in the law of motion for endogenous state variables described above.

S : array_like, dtype=float, shape=(ny, nz)

The matrix \(P\) in the law of motion for endogenous state variables described above.

References

[R1]	Uhlig, H. (1999): “A toolkit for analyzing nonlinear dynamic stochastic models easily,” in Computational Methods for the Study of Dynamic Economies, ed. by R. Marimon, pp. 30-61. Oxford University Press.