How to Remove Fixed Effects to Reduce Heterogeneity?

Question

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):

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

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

• use the fixed effects estimator (essentially demean the returns within firms/months)

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 $\frac{1}{T}\frac{1}{N}\sum_\limits{i}\sum\limits_t R_{i,t}$?

• subtract lagged values in a first-difference way

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?

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

Answer 1

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.