Solution

Differentiating the residualized objective gives

β_{4} = \frac{E [ A Y ]}{E [ A ^{2} ]} = \frac{E [( A - e ( X )) ( Y - μ ( X ))]}{E [( A - e ( X ) ) ^{2} ]} = β_{3} .

e (x) = E [A ∣ X = x]

and

μ (x) = E [Y ∣ X = x]

flexibly. With cross-fitting, obtain held-out predictions

e_{i}, μ_{i}

and regress the residualized outcome on the residualized treatment through the origin:

β_{3} = \frac{\sum _{i = 1}^{n} ( A _{i} - e _{i} ) ( Y _{i} - μ _{i} )}{\sum _{i = 1}^{n} ( A _{i} - e _{i} ) ^{2}} .

Cross-fitting limits overfitting bias and permits flexible nuisance estimators under the usual convergence and overlap conditions.

 Ancestors (11)

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