risk premium in dynare

[David]

Hi

First of all, thanks for your time to read this post.

I am using Dynare to solve an OLG model to study asset risk premium. One question confuses me all the time but I do not know where to seek answers. Hope someone can help me to clarify my misunderstanding.

It is well known the first order approximation of a rational expectation model will not give risk premium, as mentioned in Mr. Michel Juillard's notes: https://quimbaya.banrep.gov.co/document ... ena_mj.pdf

But what if I use the first order approximation to back up asset risk premium according to

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 risk premium_t=risk aversion*cov_t (asset return_(t+1),consumption growth_(t+1)).

We can use first order approximation to express asset return and consumption growth as functions of state variables, e.g., r_(t+1)=f(k_t,z_t )+η_rz *ε_(t+1) and consumption growth_(t+1)= f(k_t,z_t )+η_gz*ε_(t+1) such that the conditional covariance, or equivalently, risk premium, is determined by the coefficients on shocks, e.g., risk premium_t=risk aversion*η_rz*η_gz*sigma^2. So risk premium in the first order approximation of the model is not zeros. However, I know taking expectation of r_(t+1) resulting a risk free rate f(k_t,z_t ) so risk premium is zero. I cannot reconcile these conflicting results from different approaches. Do I have any misunderstanding in using such a formula to derive risk premium from the first order approximation of the model?

I appreciate any reply. Thank you.

[jpfeifer]

If you perform a first order approximation around the deterministic steady state, there will be no shocks. It seems you are talking about a different first order approximation than Dynare is performing.

[David]

If you perform a first order approximation around the deterministic steady state, there will be no shocks. It seems you are talking about a different first order approximation than Dynare is performing.

Hi, thanks for your reply.

I am talking about the first order approximation in Dynare.
By setting order=1, the policy function given by Dynare looks like (variables are in log):

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ly=const+coeff1*(lk(-1)-lk)+coeff2*z

Assume the productivity z is AR(1):

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z(t)=coeff3*z(t-1)+e(t)

Then we could substitute z(t) and write the above equation of ly as:

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ly=const+coeff1*(lk(-1)-lk)+coeff2*(coeff3*z(t-1)+e(t))

or equivalently by expanding the last term:

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ly=const+coeff1*(lk(-1)-lk)+coeff2*coeff3*z(t-1)+coeff2*e(t))

Therefore, the shock appears in the first order approximation of ly, with coefficient coeff2.

Similarly, I am thinking if I define asset return as r(t) and consumption growth as g(t), dynare should also give me a first order approximation of the form:

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r(t)=constr+coeffr1*(lk(-1)-lk)+coeffr2*coeffr3*z(t-1)+coeffr2*e(t))
g(t)=constg+coeffg1*(lk(-1)-lk)+coeffg2*coeffg3*z(t-1)+coeffg2*e(t))

Then couldn't we compute the risk premium as rp(t)=risk aversion*cov_t(lr(t),g(t))=risk aversion*coeffr2*coeffg2*sigma^2?

[jpfeifer]

Yes, you can do this. Dynare just does not do it automatically. This is equivalent to using the first order solution to computing second-order accurate moments.

[lor_men]

You are perfectly right, you can either use a second order approximation or use a first order approximation to compute the way you describe the risk premium. The second approach is the one employed by Jermann (1998) and is usually called loglinear-lornormal approach.

[David]

Thanks very much for all your reply. I am delving into this issue and will have update later.