BK conditions satisfied when model is in one form but not when written in equivalent form

Stochastic simulations
1

Hi

I want to solve the following simple model in Dynare:

x_t = E_t x_{t+1} - \frac{1}{\sigma} (i_{t-1} - r_t^n) \\ \mu_t^x = \frac{1}{\beta} \mu_{t-1}^x - x_t \\ E_t \mu_{t+1}^x = 0.

simple_impinf.mod (488 Bytes)
When I try to do so, I get the following error:

There are 2 eigenvalue(s) larger than 1 in modulus for 2 forward-looking variable(s)
The rank condition ISN’T verified!

If I eliminate \mu_t^x from the equations and write the model in the equivalent form

x_t = E_t x_{t+1} - \frac{1}{\sigma} (i_{t-1} - r_t^n) \\ E_t x_{t+1} = \frac{1}{\beta} (E_{t-1} x_t - x_t).

then Dynare solves the model without a problem: simple_impinf_nomu.mod (472 Bytes)

What’s going wrong in the former case? Can it be fixed without eliminating the \mu_t^x (because I want to expand the model in such a way that eliminating the Lagrange multipler by hand may be difficult).

2

I don’t really see why the two are equivalent. Could you provide additional steps?

3

Hi Johannes

Starting with the first system, I iterate the second equation forward and take the expectation conditional on period t information to get

E_t \mu_{t+1}^x = \frac{1}{\beta} \mu_t^x - E_t x_{t+1}.

Using the third equation - i.e. E_t \mu_{t+1}^x = 0 - I get \frac{1}{\beta} \mu_t^x = E_t x_{t+1}. Substituting this expression in for \mu_t^x and \mu_{t-1}^x in the second equation then gives

E_t x_{t+1} = \frac{1}{\beta} (E_{t-1} x_t - x_t)

which is the second equation in the second system.

Conversely, to go from the second system to the first system, I define \mu_t^x = \beta E_t x_{t+1}. Substituting into the second equation of the second system for E_t x_{t+1} and E_{t-1} x_t gives the second equation of the first system. Further

E_t \mu_{t+1}^x = \frac{1}{\beta} \mu_t^x - E_t x_{t+1} = 0

so the third equation of the first system also holds.

The two systems are therefore equivalent (if my reasoning is correct).

1 Like
4

I see. You uncovered a bug in Dynare 4.6, see Preprocessor simplification incorrectly treats foward-looking equations. (#83) · Issues · Dynare / preprocessor · GitLab. Both files will still run in 4.5.7 and will work again in Dynare 4.7.

5

Thanks Johannes! I’ll try it in the older version.

6

You can also stay with Dynare 4.6.4 for the moment.
You can prevent the preprocessor bug by tagging the third equation for \mu_x.
E.g. this model block should solve the problem

model;
x = x(+1) - 1/sigma*(i(-1) - rn);
mu_x = 1/beta*mu_x(-1) - x;
[static]
mu_x = 0;
[dynamic]
mu_x(+1) = 0;
end;
7

Yes, that is a workaround to disable preprocessor simplifications.