Estimation: First-difference (FD) estimator
- Exceptionally, we start with two equations this time
- First-difference (FD) estimator (e.g. Wikipedia)
- \(\Delta y_{it} = y_{it} - y_{it-1} = \theta \Delta d_{it} + \beta \Delta x_{it} + \Delta \varepsilon_{it}\)
- \(i\) is the index for individuals, \(t\) is the index for the time points (t0, t1, t2 etc.)
- \(\Delta y_{it}\) is the first-differenced outcome variable (difference between t0 and t1)
- \(\theta\) is the treatment variable coefficient (estimate of the “causal” effect)
- \(\Delta d_{it}\) the difference in the treatment variable \(d\) between t0 and t1
- \(\beta \Delta x_{it}\) one first-differenced covariate \(\Delta x_{it}\) and its effect \(\beta\)
- …with more covariates we would have a matrix \(\Delta X_{it}\) of first-differenced covariates and a vector of \(\beta\)s
- \(\Delta \varepsilon_{it}\) is the first-differenced error term
- Constant covariates drop out through first differencing (see top two equations)