# Models with non-normal shocks (Non-linear transformation of shocks)

Hi,

I wonder how to implement gamma shocks (or other non-normal shocks) in dynare? I guess for these non-normal shocks one would need to do the non-linear transformation to tranform normal shocks. And if the curvature of this non-linear transformation is high, the accuracy of dynare (perturbation) is low.

I wonder if there is a general approach to address this issue. One thing I notice is the perturbation AIM method developed here

Did anyone use it before? In the abstract it says it replies on non-stochastic steady state, so I guess it won’t fit for calculating equity premium with time-varying volatility, as in asset pricing literature with epstein-zin preference?

1. Perturbation AIM is essentially standard perturbation as in Dynare, with the underlying computation algorithm working somewhat differently.
2. Have a look at Stochastic Simulation based on Non-Normal Shocks
3. Computing the equity premium is not really about the approximation point, but the approach to capture it. Jermann (1998) for example uses a first order approximation around the deterministic steady state but still is able to compute an equity premium.

Thank you for your reply. In that reference, you said “A limitation of Dynare currently is that it does not allow for skewed distributions, i.e. even at order 3 the skewness of the shock distribution is assumed to be 0”

Is it still the case? The skewness is what matters in my model, is there any way to capture the effect of skewness in dynare?

Also, according to 1, perturbation AIM would not solve this problem either, given that its underlying is standard dynare?

perhaps put my question differently, suppose I do the non-linear transformation to normal distribution , such that the resulting distribution has some skewness, and perhaps kurtosis as well. Dynare would be able to capture those higher order moments, right?

Then, I think the problem is the curvature associated with this transformation. That’s why I wonder if perturbation AIM would help

1. This is a complicated issue. In Dynare we do not yet allow for non-normal shocks - although the used techniques can handle them. From the PDF you linked above, this is different with AIM, where they explicitly allow for that.
2. The important distinction is endogenous propagation vs. exogenous shocks. Dynare is able to approximate any differentiable endogenous dynamics, but it does not allow for arbitrary exogenous shock distributions yet.
1. “although the used techniques can handle them” – what are the used techniques that can handle non-normal shocks, would you please elaborate a bit?

2. endogenous propagation vs. exogenous shocks. I guess you are saying that by non-linear transformation, I am changing the endogenous dynamics, rather than the shock distribution? But if a normal variable always enter the model after a non-linear transformation, that would effectively become a different distribution for exogenous shock, right?

Thanks!

1. I mean perturbation. Both AIM and Dynare’s implementation rely on the moments of the exogenous processes for computing the solution. But Dynare only allows specifying the first two moments, therefore working with a normal distribution. But as shown in https://www.dynare.org/assets/dynare++/k_order.pdf, any moments \Sigma could theoretically be allowed for (as long as they are finite)
2. No. Dynare relies on the shocks u in the linked document being normal. But there are no restrictions on the function f transforming these normal shocks.