Simulating chaotic dynamics or endogenous cycles

[robjump1]

I am attempting to use the Dynare deterministic simulator to simulate endogenous cycles. For backward looking models this does not seem to be a problem, but for forward looking models it appears to be very difficult.

Attached are two mod files to illustrate. "logmap.mod" is just the standard backwards looking logistic map, which can easily be simulated in Dynare with the deterministic simulator (works fine for any parameterisation 0 < r < 4). "logmapf.mod" is the standard logistic map with the addition of a forward looking term. Confusingly, even if this forward looking term is made irrelevant (by setting the parameter s = 0), so the model should behave in exactly the same way as the backwards looking logistic map, Dynare cannot simulate it.

Can anyone advise why this is the case? Any general advice on how to simulate forward looking models that are not saddle path stable (i.e. that generate endogenous cycles) would also be greatly appreciated!!

All best,

Robert Jump

[jpfeifer]

We are investigating the issue: https://github.com/DynareTeam/dynare/issues/1124
Conceptually, I am not sure what to expect when s is different from 0. You currently impose that x in the last period is 0. Does this make sense?

[robjump1]

Thanks!

No perhaps it does not make sense - I wonder what a sensible final value to impose would be? Ideally, with this type of model, one would like to avoid specifying a final value for the simulation, but I don't think this is possible in Dynare?

All best, and thanks for your help,

Rob Jump

[jpfeifer]

Without a terminal condition, the model would be underdetermined. Thus, there must be something missing. Maybe you could look into the

Code: Select all

differentiate_forward_variables

option, see http://www.dynare.org/manual/index_18.html#Model-declaration and http://ecares.org/ecaresdocuments/seminars1314/nbb-juillard-slides.pdf

[robjump1]

Thanks again - I think having the difference in jump variables equal to zero as a terminal condition is probably not that useful in a model designed to produce permanent endogenous fluctuations - but I will give it a go.

Is there a way to set arbitrary terminal conditions in Dynare? In this case one could write a program to search over a space of terminal conditions until convergence.

Many thanks,

Rob

[jpfeifer]

For a model where you do not know the terminal condition, having a 0 change in the last period as opposed to a small endogenous change might be a better approximation than jumping from an endogenously determined point in the second-to-last period to an arbitrary terminal value that might be far away.

But more fundamentally, you have to ask yourself what determines the terminal condition. If there is a unique terminal condition as you seem to suggest, there must be an economic mechanism to determine it endogenously. In your current setup, something like this is missing.