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))
please see dynare.org/manual/index_23.h … roximation
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?
Thanks in advance.