11.2 Estimation: First-difference (FD) estimator

  • Exceptionally, we start with two equations this time
  • First-difference (FD) estimator (e.g. Wikipedia)
    • Δyit=yityit1=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)