E[Y|A=1] | = E[Ya=1 | A=1] | by consistency |
E[Y|A=0, B=1] | = E[Ya=0 | A=0, B=1] | by consistency |
= E[Ya=0 | A=1, B=1] | by (i) | |
= E[Ya=0 | A=1, B=0] | by (ii) | |
It follows that E[Y|A=0, B=1] = E[Ya=0 | A=1] |
E[Y|A=1] | = E[Ya=1 | A=1] | by consistency |
E[Y|A=0, B=1] | = E[Ya=0 | A=0, B=1] | by consistency |
= E[Ya=0 | A=1, B=1] | by (i) | |
= E[Ya=0 | A=1, B=0] | by (ii) | |
It follows that E[Y|A=0, B=1] = E[Ya=0 | A=1] |