Rare Disasters with Perturbation Methods?

Dear all,

I was hoping to construct and solve a model with rare disasters. The rare disaster is usually represented by a binary variable that points 1 with some small probability, causing a big negative impact to the economy (capital deterioration, TFP reversals, debt-default, etc…). Due to this discontinuity, usual perturbation methods can’t handle these models.

Indeed, as illustrated by Gourio (2012) and Marlène Isoréa and Urszula Szczerbowicz (2017), one should depend on stationarizing the equilibrium system such that it only features the (small) probability of a disaster, instead of the (large) original event itself. Thereby, (small) innovations to the (small) probability of disaster can be accurately simulated with conventional high-order perturbation methods, regardless of the disaster regime. This step usually requires some smart technical tricks that remove the binary disaster variable as a state variable.

HOWEVER, Jesús Fernández-Villaverde and Oren Levintal (2018) builds a model with rare disasters and solves it with perturbation method, with the binary disaster variable as a state variable. Therefore, they can simulate a disaster using perturbation methods, differently from Urszula Szczerbowicz (2017). I am confident that they’ve used some technical trick in their implementation to smooth this discontinuity but I can’t quite figure it out. It seems that they just provided the equilibrium conditions and took the usual derivatives for perturbation, without further consideration about the binary structure of the disaster variable. I am pretty confident that it could be implemented with dynare.

If anyone would know what technical adjustment were necessary to adapt disaster models for perturbation techniques, I would appreciate any help.

Thanks in advance.

PS: I include the Technical Appendix of Jesús Fernández-Villaverde and Oren Levintal (2018) below where they give a detailed explanation of their implementation (see README.pdf and Online_Appendix.pdf).

Gourio, F. 2012. Disaster Risk and Business Cycles. American Economic Review, 102(6), 2734-2766.

Isoré, M. and Szczerbowicz, U. 2017. Disaster Risk and Preference Shifts in a New Keynesian model. Journal of Economic Dynamics & Control, 79, 97-125.

Fernández-Villaverde, J. and Levintal, O. 2018. Solution Methods for Models with Rare Disasters. Quantitative Economics, 9, 903-944.

Matlab_Codes_Rare_Disasters.rar (1.5 MB)

@wmutschl Do you have any insights?
My impression is that they essentially use perturbation off the shelve. The binary structure will of course be approximated continuously with perturbation. But the results seem to be acceptable (although other techniques perform better)

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Thanks for your insight Professor. I’m sorry for the late reply.

Let me see if I understand. It would be equivalent to put their equations in dynare (see equations in the README.pdf file in the appendix) and declare the disaster variable as:

d_{t+1} = \mu^d+ u_{t+1}

Where \mu^d is the probability of disaster and u_{t+1} = \epsilon_{t+1} - \mu^d. Theoretically, \epsilon_{t+1} could only assume 0 or 1, but in dynare we would only specify the standard deviation of u_{t+1} as:

var(u_{t+1}) = var(\epsilon_{t+1}) = E( \epsilon_{t+1} )^2 = 1^2 \mu^d + (1-\mu^d)*0^2 = \mu^d .

Hence, “simulating” a disaster means simulating the shock where u_{t+1} assumes the value 1-\mu^d.

In this situation, the “disaster” is equivalent to simulate an extreme left tail shock, and not surprising only fifth-order shock would be accurate enough.

This implementation would be in line with theirs?

At least it sounds like that. In the end, only a look into their codes can answer what exactly they did.

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You summarized the literature quite accurately. What you can do out-of-the-box in Dynare is the approach taken by Gourio (2012) and Isoré and Szczerbowicz (2017) to detrend the system such that the binary disaster variable vanishes. You then put the detrended system in the mod file and you can do stochastic simulations with say at least order=3. What you can then analyze is a shock to the fear of a rare disaster, i.e. a rare-disaster risk shock. Isore and Szczerbowicz use a small shock to the probability of the rare disaster, whereas Fernandez-Villaverde and Levintal use a small shock to the size of the rare disaster. So there is a difference what they shock, but in both cases they study deviations due to a small shock on either the probability or the size of a rare disaster. In this way, perturbation is a valid approximation as you still do the approximation in the vicinity of the steady state.

With that said, if you want to study the effect of an actual disaster, this is much more involved, and in my opinion, perturbation even at fifth or higher orders is not a good choice. Because a rare disaster shock (not just to the fear of it) implies by definition a huge deviation from the point you do the Taylor approximations with perturbation and this is bound to be imprecise. You would need to consider a different solution technique. Levintal has proposed one, which seems promosing, but is not (yet) implemented in Dynare. I might do so in the future, if I find it useful for my research :wink:

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Thanks for your reply, Professor !

I agree with all your points. I will take the approach used by Gourio (2012) and Isoré and Szczerbowicz (2017) then. I think I will use the toolbox offered by Cao, Luo and Nie (2020) - GDSGE. They offer a toolbox that uses projection methods to compute solutions.

Many thanks for all your help !

See ya.