11.3 Estimation: Fixed-effects (FE) estimator
- Fixed-effects (FE) estimator (see e.g. Gangl 2010, 33–35, Wikipedia)
- yit−¯yi=θ(dit−¯di)+β(xit−¯xi)+(εit−¯εi)
- i is the index for individuals, t is the index for the time points (t0, t1, t2 etc.)
- yit−¯yi is the de-meaned outcome variable (¯yi = individual mean across time)
- θ(dit−¯di) is the de-meaned treatment variable (dit−¯di) and its coefficient θ (estimated “causal effect”)
- β(xit−¯xi) is one first-differenced covariate (xit−¯xi) and its effect β
- …with more covariates we would have a matrix of de-meaned covariates and a vector of βs
- (εit−¯εi) is a de-meaned error term
- …sometimes error terms for unobserved constant variables are added to the equation to show that they drop out (see Wikipedia).
- yit−¯yi=θ(dit−¯di)+β(xit−¯xi)+(εit−¯εi)
References
Gangl, Markus. 2010. “Causal Inference in Sociological Research.” Annual Review of Sociology 36 (1): 21–47.