I do not know how wald estimator is derived I found an expla

I do not know how wald estimator is derived. I found an explanation, but I could not understand the last two steps.

Could anyone explain the how the derivation proceeds from the last three to the last two step?

2.1 Wald Estimator for Homogeneous (Constant) Effect Recall that regressingy on 1 and d produces a slope estimate (associated with d) equal to the group (indicated by d) miean difference. Although the consistency of an IV estimator for can be shown easily, there is an illuminating interpretation of an IV estimator when the instrument z is binary, linking the IV estimator to group mean differences E (yle-1)-E (yle = 0) and E (4:-1)-E (4: 0) Recall the previous In model: Define zi (1,2i)\', di (1, d). (1, 2)\', and (1, 2). [B (zd))-1 For an IV estimator, the slope 2 in E (zy) can be written as Cov(y,z)EEGE( Cov (d,z) E(dz) - E (d) E(z) Because E(dz) = E (dz = 1) Pr (z-1) and d = d [z + (1-z)], we can rewrite the denominator as E(dz)-E(d)E(e) -E (d|z = 1) Pr (z = 1)-E(d|z + (1-z)) Pr(z-1) -E (4:-1) Pr (z = 1) _ E(dz) Pr(z = 1) _ E [d(1-z)] Pr(z = 1) -E (d|z = 1) Pr (z = 1) Pr(z = 0)-Eld|c-0] Pr(z-0) Pr(z-1) [B (d|z = 1) _ E(d|c-0)] Pr (z = 1 ) Pr(z-0)

Solution

It is onn the basis of late ie the average effect of treatement for the subpopulation of compilers.