How to apply second order approximation to Stigliz-Dixit equation?

Classically,the so called S-D equation is Yt=(yt(i)^(1-1/θ))^(θ/(θ-1)),the second order approximation of it in Interest and Price is Ythat=yt(i)hat+1/2·(1-1/θ)·var(yt(i)).How to calculate the approximation?I spent two days calculating it but didn’t work.
THANKS very much if anybody could help me.

Should be Ythat=yt(i)hat+1/2·(1-1/θ)·var(yt(i)hat),sorry

Are you not missing an integral in your aggregator. Usually the trick is two rewrite the integral, for the price index for instance, recursively and then do the local approximation. You will find this explained in papers deriving properly the NK model with Calvo price setting.


Thank you very much,I have calculated this equation successfully😬