Texts: Causal Inference
Prove (6.7) and (6.9) below. If you do not have enough knowledge about this, please do not make it a poop by pretending to know how to solve it. Try it only if you surely know the solution. Only a complete solution will be accepted. If you are not confident about it, do not ruin it and just leave it for other people to try. I am sure that you will be reported if you ruin it.
6.2.2.2 Understanding Lin (2013)'s estimator via predicting the potential outcomes
We can view Lin (2013)'s estimator as a predictive estimator based on OLS fits of the potential outcomes on the covariates. We build a prediction model for Y(1) based on X using the data from the treatment group:
Y(1) = β0 + β1X. (6.4)
Similarly, we build a prediction model for Y0 based on X using the data from the control group:
Y0 = γ0 + γ1X. (6.5)
TABLE 6.1: Predicting the potential outcomes
X Z Y1 Y0 Y(1) Y(0)
X1 1 Y1(1) ? Y1(X1) Y0(X1)
Xn1 1 Yn(1) ? Yn(Xni) Y0(Xni)
Xn+1 0 ? Yn+1(0) Y1(Xn1+1) Y0(Xn+1)
Xn 0 ? Yn(0) Y1(Xn) Y0(Xn)
Table 6.1 illustrates the prediction of the potential outcomes. If we predict the missing potential outcomes, then we have the following predictive estimator:
Ŷ = Σ(Z=1)Y(1) + Σ(Z=0)Y(0). (6.6)
We can verify that with (6.4) and (6.5), the predictive estimator equals Lin (2013)'s estimator:
T̂pred = β1L. (6.7)
If we predict all potential outcomes, even if they are observed, we have the following projective estimator:
T̂proj = Σ(Z=1)(Y1(Xi) - Y0(Xi)). (6.8)
We can verify that with (6.4) and (6.5), the projective estimator equals Lin (2013)'s estimator:
TÌ‚proj = L. (6.9)
