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).

References

Gangl, Markus. 2010. “Causal Inference in Sociological Research.” Annual Review of Sociology 36 (1): 21–47.