# Second order approximation - Simulations

Dear Professor Juillard,
while simulating a (large) nonlinear model with Dynare I encontered a strange problem. Namely, the Dynare computes the theoretical moments (let say properly), but If I try to use the solution for the second order approximation to simulate the moments (using the same mod-file but with a command “period=…”) the state vector explodes after 5-6 iterations. I think there must be something wrong with the model solution but what could be the reason for this. And more important, how to tackle the problem.
I will appreciate any feedback from you.
Ernest

The theoretical moments calculated in Dynare Matlab are the second order accurate. So this is only a gross approximation to true ergodic distribution. For the second order approximation the first moment at t depends on the second moments at t-1. By induction, for the ergodic distribution, the first moment will depend on all other moments (not only on the second). As far as I know it is impossible to calculate this theoretically.

Your problem with the exploding simulations is that the second order decision rule is not stationary, although the first order is (Blanchard Kahn linear stability).

The possible solutions are:

1. decrease the volatility of the shocks

2. decrease the degree of non-linearity in the model

3. perhaps your policy is not enough agressive?

4. It is still possible that your model yields a stationary solution but the second order approximation is wrong. So try a higher order approximation using Dynare++, maybe higher orders will compensate error of the second order approximation? If so, let me know.

Regards,

Ondra K.

Yes, you are right, the reason for the non-stationarity is in this case the high volatility of structural shocks. I will definitely try the higher order approximations.

The another question I have is whether it is possible to calculate the second order accurate theoretical moments in the case where some eigenvalues are equal to unity. I am talking here about a two-country model with definitions of relative prices which are responsible for introducing the eigenvalues equal to unity. Is there any way to calculate theoretical moments (more precisely the mean of consumption) then? What can I use instead of Sylvester formula? Do you know any usefull tricks?
I will appreciate any feedback from you,
E.

I don’t know about any trick.

However, if there is some algorithm for solving your problem having units in the spectra, I would be very very careful to prove that the algorithm is numerically stable and would definitely try to theoretically or experimentally calculate its forward error bounds. This is not encouraging, I know, but I am a purist. I always put in doubt algorithms which blindly rely on float(1)-float(1)=exact(0).

regards,

Ondra K.