# Second-order approximation Schmitt-Grohe and Uribe vs Dynare

Hi all,

I would like to compare SGU (2004) JME second-order approximate solution with the one produced by Dynare.
SGU (2004) solution is given by :

For the state variables:
x(t+1)=x + A * ( x(t) – x) + 1/2 * B* kron ( ( x(t) – x), x(t) – x) ) + ½C sigma^2

For the control variables:
y(t)= y +D* ( x(t) – x) +1/2 * F* kron ( ( x(t) – x), x(t) – x) ) + ½G sigma^2

where y(t) is the vector of endogenous control variables
x(t) the vector of endogenous and exogenous state variables
x, y their long-run solution,
sigma is a scalar which measures uncertainty.
And A,B,C,D,F,G are solution matrices.

While the Dynare solution is given by:

y(t)=ys +0.5D^2 +A x(t-1) +B* u(t) +0.5C kron(x(t-1),x(t-1) )+0.5D kron(u(t),u(t)) +E* kron(x(t-1),u(t))

I suppose that if the vector u(t) is equal to zero the two solutions are identical?

Do they differ because SGU (2004) assume that the expected value of u(t) is zero, while dynare formula is for the general case where u(t) can be different from zero?

P.

SGU use a typical state space representation with the first equation describing the law of motion for the states (x_t as a function of x_t-1). Here, innovations are basically treated as states (no u_t in the equation) if your notation is correct. The second equation then maps the time t states into the time t observables (y_t as a function of x_t).

In contrast, Dynare essentially plugs in the state LOM into the observation equation in order to provide the variable vector z_t=[x_t y_t] as a function of past states x_t-1 and current shocks u_t (instead of lumping them together into x_t).

You can find a lot more details in Andreasen, Fernandez-Villaverde, Rubio-Ramirez (2013).

Dear jpfeifer,