appendix E Technical lemma for the alternative instrumental variable estimator

In chapter 9, we provided an alternative formulation to the instrumental variable (IV) estimator in equation 9.2, which relies on the following technical result: if R is a binary variable, then for any variable Q,

cov(Q, R ) = (E[Q|R = 1] – E[Q|R = 0])P (R = 1)P (R = 0)

That is,

cov(Q, R ) = E[Q R ] – E[Q ]E [R] =

E[Q|R = 1] P(R = 1) – (E[Q|R = 1)P (R = 1) + E[Q|R = 0]P (R = 0))P (R = 1) =

(E[Q|R = 1] – E[Q|R = 0])P (R = 1)P (R = 0)