let formula_3 denotes the potential outcome of subject i, where d is the binary indicator of subject formula_4’s treatment status. formula_5 denotes the treated potential outcome for subject i, while formula_6 denotes the untreated potential outcome. the causal effect of the treatment on subject formula_4 is formula_8. however, we can never observe both formula_5and formula_6 for the same subject. at any given time, we can only observe a subject in its treated formula_5 or untreated formula_6 state.