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: 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

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_gzε_(t+1) such that the conditional covariance, or equivalently, risk premium, is determined by the coefficients on shocks, e.g., risk premium_t=risk aversionη_rzη_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?

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.

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):

Assume the productivity z is AR(1):

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

or equivalently by expanding the last term:

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:

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.

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.