# 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?

Regarding the meaning, on the same slide 36, they claim an assumption:

"Common identifying assumption: Parameter values are constant across all firms and years within the sample."

Because of this assumption, theoretically, the fixed effects of firm and year don't vary in your model (if they are present, you're "controlling" them to get parameter estimates). When you take them out, you can see heterogeneity that is associated with the model outcome (in this case, it sounds like you're modelling moments).

As for the implementation, that might depend on the question at hand. It sounds like an exploratory strategy.

• Thanks a lot for your answer! I see the motivation for their suggestion but I'm not still not sure about the meaning (and implementation) of ''removing fixed effects.'' Is it just applying this within-transformation of the fixed effects estimator?
– Alex
Mar 26, 2021 at 18:36
• I think it's actually simpler than you're imagining; you should be able to literally leave the effects out of the model (or set the parameter estimates to 0 and do not allow them to vary, if you prefer). It might not super intuitive, because you're purposely making the model fit less well. If those fixed effects are the only predictors of the outcome, I believe you could call it an "empty model", where you're just looking at the variability in the outcome.
– pep
Mar 30, 2021 at 22:06