Welfare 3rd order

Welfare calculations are typically done using 2nd order approximations following Schmitt Grohe and Uribe.
But why not consider 3rd order approximations for welfare? Does it make sense to do this with dynare?

Here is what I learned to do for 2nd order approximation:
One needs to define welfare recursively in the model block e.g.
Welf = ©^(1-sigma)/(1-sigma)+betaWelf(+1);
Having solved the model, conditional welfare is given by:
Uncoditional welfare is given by:

Now if i choose 3rd order approximation I guess unconditional welfare can be obtained in the same way. Yet what would conditional welfare be?
It is not clear to me what is the corresponding term in 3rd order to the term 0.5*oo_.dr.ghs2 in 2nd order. Is it oo_.dr.g_0?

Many thanks!

Second order approximations are the lowest order required for correct welfare computations. To my knowledge, people have not come up yet with applications where higher order approximations are required.
You are talking about welfare conditional on being in steady state. That is why all other terms of the state space solution for welfare drop out at second order (your formula is only correct for welfare conditional on being in steady state). If you go to third order, there will be additional terms, but they all involve products with deviations of state variables from steady state or shock terms. As these terms are all 0 in steady state, conditional welfare at second and third order should be identical