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Estimation: First-difference (FD) estimator
- Exceptionally, we start with two equations this time
- First-difference (FD) estimator (e.g. Wikipedia)
- Δyit=yit−yit−1=θΔdit+βΔxit+Δεit
- i is the index for individuals, t is the index for the time points (t0, t1, t2 etc.)
- Δyit is the first-differenced outcome variable (difference between t0 and t1)
- θ is the treatment variable coefficient (estimate of the “causal” effect)
- Δdit the difference in the treatment variable d between t0 and t1
- βΔxit one first-differenced covariate Δxit and its effect β
- …with more covariates we would have a matrix ΔXit of first-differenced covariates and a vector of βs
- Δεit is the first-differenced error term
- Constant covariates drop out through first differencing (see top two equations)