# How to Remove Fixed Effects to Reduce Heterogeneity?

When discussing GMM estimation, Toni Whited and Luke Taylor suggest to reduce heterogeneity by ''eliminating fixed effects,'' see here on Taylor's slides (slide 36):

My question: I'm not quite sure what they mean with this statement and how to implement it.

My aim is to use portfolio returns to calculate the moment conditions. Let $$R_{i,t}$$ denote monthly returns. Following wikipedia, I can

In this case, I do not only subtract the means $$\frac{1}{T}\sum\limits_t R_{i,t}$$ and $$\frac{1}{N}\sum\limits_i R_{i,t}$$ but also the cross-term $$\frac{1}{T}\frac{1}{N}\sum_\limits{i}\sum\limits_t R_{i,t}$$?

In this case, I can consider $$\Delta R_{i,t}=R_{i,t}-R_{i,t-1}$$ but what would be the cross-sectional lag term? $$R_{i,t}-R_{i-1,t}$$ does not make sense, does it?

In the end, I compute the GMM moment conditions from either the (properly) demeaned returns or the (somehow?) first differenced returns?