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Consider the following regression model:$$y_{it}=x_{it}'\beta+\alpha_{i}+\epsilon_{it}$$ Here, $\alpha_{i}$ is an individual specific intercept (fixed effect). Now, by estimating the model by fixed effects, we are effectively demeaning both the response variable the regressors. As such, we are using only within individual variation and are unable to identify coefficients on those variables that are time invariant. However, I wish to use between individual variation as well for one of the variables. Can I demean a subset of $x_{it}$ such that the estimator uses only within individual variation for $x_{it}$ by between variation for the other regressors?

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I don't think that you can treat different variables differently with a fixed effects model specification like yours. You will be demeaning all variables in $X_{it}$ and utilizing the withing individual variation. Also note, however, that if you instead used a random effects model with following assumptions

  1. $E(\varepsilon_{it} | X_{it}, \alpha_{i}) = 0$
  2. $E(\alpha_{i} | X_{it})$ = 0,

then you will be using both within and between variation in your regressors (all the variables in $X_{it}$ however will be used).

Check Colin and Trivedi, Chapter 21 and 22, Microeconometrics: Methods and Applications.

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