11.3 Estimation: Fixed-effects (FE) estimator
- Fixed-effects (FE) estimator (see e.g. Gangl 2010, 33–35, Wikipedia)
- \(y_{it}-{\overline {y_{i}}} = \theta \left(d_{it} - {\overline {d_{i}}}\right) + \beta \left(x_{it} - {\overline {x_{i}}}\right) + \left(\varepsilon_{it} - {\overline {\varepsilon_{i}}}\right)\)
- \(i\) is the index for individuals, \(t\) is the index for the time points (t0, t1, t2 etc.)
- \(y_{it}-{\overline {y_{i}}}\) is the de-meaned outcome variable (\(\overline {y_{i}}\) = individual mean across time)
- \(\theta \left(d_{it} - {\overline {d_{i}}}\right)\) is the de-meaned treatment variable \(\left(d_{it} - {\overline {d_{i}}}\right)\) and its coefficient \(\theta\) (estimated “causal effect”)
- \(\beta \left(x_{it} - {\overline {x_{i}}}\right)\) is one first-differenced covariate \(\left(x_{it} - {\overline {x_{i}}}\right)\) and its effect \(\beta\)
- …with more covariates we would have a matrix of de-meaned covariates and a vector of \(\beta\)s
- \(\left(\varepsilon_{it} - {\overline {\varepsilon_{i}}}\right)\) 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).
- \(y_{it}-{\overline {y_{i}}} = \theta \left(d_{it} - {\overline {d_{i}}}\right) + \beta \left(x_{it} - {\overline {x_{i}}}\right) + \left(\varepsilon_{it} - {\overline {\varepsilon_{i}}}\right)\)
References
Gangl, Markus. 2010. “Causal Inference in Sociological Research.” Annual Review of Sociology 36 (1): 21–47.