4.4.1. Recuritment to a Program

• 내용 요약
  • 정부에서 job training program에 투자함.
  • pilot randomized experiments를 통해 해당 program이 effective하다는 것을 보여줌. (program을 통과한 사람들이 그렇지 못한 사람들보다 높은 확률로 직업을 구함.)
  • 결과로, 해당 program은 통과되었고 정식 program으로서 운영됨. 심지어 randomized pilot study 때보다 더 높은 취업률을 보여줌.
• Critics claim that the program is a waste of taxpayers' money.
  • Those who self-enroll, the critics say, are more intelligent, more resourceful, and more socially connected than the eligible who did not enroll, and are more likely to have found a job regradless of the training.
  • what we need to estimate is the differential benefit of the program on those enrolled: the extent to which hiring rate has increased among the enrolled, compared to what it would have been had they not been trained. → 프로그램의 효과를 측정하고 싶음!
• notations
  • $X=1$ represent training
  • $Y=1$ represent hiring
  • $ETT$ (effect of treatment on the treated) represent the effect of training on the trained.

$$ ETT=E[Y_1-Y_0|X=1] $$

• $Y_1-Y_0$ represents the causal effect of training $(X)$ on hiring $(Y)$ for a randomly chosen indivisual.
• $X=1$ limits the choice to those anctually choosing the training program on their own initiative.
• section 4.1의 고속도로 예와는 다르게 "I should have taken the freeway"라는 반사실적 후회(?)는 가시적인 결과가 없지만, 해당 예시는 경제적 영향이 있을 것이다. (the consequences have serious economic implications, such as terminating a training program, or possibly restructuring the recruitment strategy to attract people who would benefit more from the program offered.)
• (Counterfactual expectation) $E[Y_0|X=1]$ stands for the expectation that a trained person ($X=1$) would find a job had he/she not been trained. (defy empirical measurement because we can never rerun history and deny training to those who recevied it.) → 이미 지난 일들은 다시 되돌릴 수 없고, 진행할 수도 없다! estimable expressions in many, though not all, situations로 표현을 압축시키자!

$$ P(Y_x=y|x\prime)=\sum_zP(Y_x=y|z,x\prime)P(z|x\prime) $$

$$ P(Y_x=y|X=x\prime)=\sum_zP(Y=y|X=x,Z=z)P(Z=z|X=x\prime) $$

• Standard adjustment formula (3.5)

$$ P(Y=y|do(X=x))=\sum P(Y=y|X=x, Z=z)P(Z=z) $$

• Both formulas call for conditioning on $Z=z$ and averaging over $z$, except that eq calls for a different weighted average, with $P(Z=z|X=x\prime)$ replacing $P(Z=z)$.
• Noncounterfactual expression for $ETT$

$$ ETT=E[Y_1-Y_0|X=1] $$

$$ =E[Y_1|X=1]-E[Y_0|X=1] $$