Identifying Causal Efgects in Experiments with Social Interactions and Non-compliance Francis J. DiTraglia 1 Camilo García-Jimeno 2 Rossa O’Keefge-O’Donovan 1 Alejandro Sanchez 3 1 University of Oxford 2 Federal Reserve Bank of Chicago 3 University of Pennsylvania October 30, 2020 The views expressed in this talk are those of the authors and do not necessarily refmect the position of the Federal Reserve Bank of Chicago or the Federal Reserve System.
Empirical Example with Potential for Indirect Treatment Efgects Crepon et al. (2013; QJE) Displacement Efgects of Labor Market Policies “Job seekers who benefjt from counseling may be more likely to get a job, but at the expense of other unemployed workers with whom they compete in the labor market. This may be particularly true in the short run, during which vacancies do not adjust: the unemployed who do not benefjt from the program could be partially crowded out.” 2/ 19 ◮ Large-scale job-seeker assistance program in France. ◮ Randomized ofgers of intensive job placement services.
Studying Social Interactions Without Network Data 0 1 0 0 0 0 1 0 3/ 19 Partial Interference Two-stage experimental design. Randomized Saturation groups. Spillovers within but not between 0 % 25 % 50 % 75 % 100 %
This Paper: Non-compliance in Randomized Saturation Experiments Identifjcation Beyond Intent-to-Treat: Direct & indirect causal efgects under 1-sided non-compliance. Estimation Simple, asymptotically normal estimator under large/many-group asymptotics. Application French labor market experiment: Crepon et al. (2013; QJE) 4/ 19
Notation Sample Size and Indexing Observables 5/ 19 ◮ Z ig = binary treatment ofger to ( i , g ) ◮ Groups: g = 1 , . . . , G ◮ D ig = binary treatment take-up of ( i , g ) ◮ Individuals in g : i = 1 , . . . , N g ◮ Y ig = outcome of ( i , g ) ◮ S g = saturation of group g ◮ ¯ D ig = take-up fraction in g excluding ( i , g )
Overview of Assumptions (ii) Potential Outcomes: Correlated Random Coeffjcients Model (iii) Treatment Take-up: 1-sided Noncompliance & “Individualized Ofger Response” (v) Rank Condition 6/ 19 (i) Experimental Design: Randomized Saturation � (iv) Exclusion Restriction for ( Z ig , S g )
Assumption (ii) – Correlated Random Coeffjcients Model This Talk Focus on linear potential outcomes model. 7/ 19 D ig ) ′ � � Y ig ( D ) = Y ig ( D g ) = Y ig ( D ig , ¯ D ig ) = f (¯ ( 1 − D ig ) θ ig + D ig ψ ig ◮ f ≡ vector of known functions, Lipschitz continuous on [ 0 , 1 ] ◮ ( θ ig , ψ ig ) ≡ RVs, possibly dependent on ( D ig , ¯ D ig ) .
8/ 19 0 D ig D ig Direct Efgects D ig Untreated: Indirect Efgects Y ig ( D ig , ¯ D ig ) = α ig + β ig D ig + γ ig ¯ D ig + δ ig D ig ¯ Y ig ( 1 , ¯ α ig + β ig D ig ) Treated: γ ig + δ ig γ ig + δ ig γ ig α ig β ig + δ ig ¯ Y ig ( 0 , ¯ D ig ) γ ig ¯
Assumption (iii) – Treatment Take-up 1-sided Non-compliance Only those ofgered treatment can take it up. Individualistic Ofger Response (IOR) Notation 9/ 19 D ig ( Z ) = D ig ( Z g ) = D ig ( Z ig , ¯ Z ig ) = D ig ( Z ig ) C ig = 1 ifg ( i , g ) is a complier; ¯ C ig ≡ share of compliers among ( i , g ) ’s neighbors. (IOR) + (1-Sided) ⇒ D ig = C ig × Z ig
No Evidence Against IOR in Our Example 0.75 Treatment Take-up among Ofgered Saturation 1.0 0.8 0.6 0.4 0.2 0.0 1.00 0.50 Data from Crepon et al. (2013; QJE) 0.25 with saturation (p = 0.62) Take-up among ofgered doesn’t vary Figure at right Testable Implication Take-up only depends on own ofger: (IOR) + (1-Sided) 10/ 19 D ig = C ig × Z ig ❊ [ D ig | Z ig = 1 , S g ] = ❊ [ D ig | Z ig = 1 ]
Assumption (iv) – Exclusion Restriction Notation Exclusion Restriction (ii) Z g 11/ 19 ◮ B g = random coeffjcients for everyone in group g . ( C g , B g , N g ) = | (i) S g ◮ C g = complier indicators for everyone in group g = ( C g , B g ) | ( S g , N g ) | ◮ Z g = treatment ofgers for everyone in group g
Näive IV Does Not Identify the Spillover Efgect D ig IV Estimand Unofgered Individuals D ig Y ig 12/ 19 β ig D ig + γ ig ¯ δ ig D ig ¯ ✘ = α ig + ✘✘✘ D ig + ✘✘✘✘ ✘ E [ α ig ] + E [ γ ig ]¯ D ig + ( α ig − E [ α ig ]) + ( γ ig − E [ γ ig ]) ¯ = α + γ ¯ = D ig + ε ig D ig , S g ) = . . . = γ + Cov ( γ ig , ¯ γ IV = Cov ( Y ig , S g ) D ig , S g ) = γ + Cov ( ε ig , S g ) C ig ) Cov (¯ Cov (¯ ❊ (¯ C ig )
Identifjcation – Average Spillover Efgect when Untreated Theorem ig D 2 D ig D ig 1 One-sided Noncompliance D 2 D ig D ig 1 D ig 1 ig 13/ 19 D ig 1 � � ′ � � α ig β ig D ig + γ ig ¯ δ ig D ig ¯ ( 1 − Z ig ) Y ig = ( 1 − Z ig )( α ig + ✘✘ D ig + ✘✘✘✘ ✘ D ig ) = ( 1 − Z ig ) ✘ ¯ γ ig ( Z ig , ¯ ( α ig , γ ig ) | (¯ D ig ) = C ig , N g ) . | �� � � � � � � � �� � ¯ � � α ig ¯ ¯ � � ( 1 − Z ig ) Y ig = ( 1 − Z ig ) ❊ ❊ C ig , N g C ig , N g � � ¯ ¯ ¯ � γ ig � � � �� � �� �� � ¯ � � α ig ¯ ¯ � � = ( 1 − Z ig ) ❊ ❊ C ig , N g C ig , N g � � ¯ ¯ � � γ ig � �� � ≡ Q 0 (¯ C ig , N g )
Identifjcation – Average Spillover Efgect when Untreated Previous Slide: D ig 1 D ig 1 14/ 19 D ig 1 �� � � � �� �� � � � α ig ¯ Q 0 (¯ ¯ � � ( 1 − Z ig ) Y ig = C ig , N g ) ❊ ❊ C ig , N g C ig , N g � � ¯ � γ ig � Rearrange + Iterated ❊ � � �� �� � � � � � �� ❊ ( α ig | ¯ � ❊ ( α ig ) C ig , N g ) Q 0 (¯ ¯ � C ig , N g ) − 1 = ❊ = ❊ ❊ ( 1 − Z ig ) Y ig C ig , N g � ❊ ( γ ig | ¯ ¯ ❊ ( γ ig ) C ig , N g ) � � � � � Q 0 (¯ = C ig , N g ) − 1 ( 1 − Z ig ) Y ig ❊ ¯
15/ 19 D ig d ig D 2 D ig D ig Average Spillover, Untreated: ❊ [ Y ig ( 0 , ¯ d )] = ❊ ( α ig ) + ❊ ( γ ig )¯ ❊ ( α ig ) Q 0 (¯ = ❊ ( 1 − Z ig ) Y ig C ig , N g ) − 1 1 ¯ ❊ ( γ ig ) � ¯ � � Q 0 (¯ ¯ C ig , N g ) ≡ ❊ ( 1 − Z ig ) 1 C ig , N g � ¯ ¯ � � Q 0 is a known function Distribution of ¯ D ig | (¯ C ig , N g ) determined by experimental design.
16/ 19 Rank Condition S 2 c g S 3 g S 2 g S 2 c E.g. Linear Model S 2 D ig ) ′ � � Rank Condition: Y ig ( D ig , ¯ D ig ) = f (¯ ( 1 − D ig ) θ ig + D ig ψ ig � D ig ) ′ � � � ¯ ✶ ( Z ig = z ) f (¯ D ig ) f (¯ � Q z (¯ c , n ) ≡ ❊ C ig = ¯ c , N g = n z = 0 , 1 , c , n ) in the support of (¯ Q 0 (¯ c , n ) , Q 1 (¯ c , n ) invertible for all (¯ C ig , N g ) . � � ❊ { 1 − S g } ¯ c ❊ { S g ( 1 − S g ) } Q 0 (¯ c , n ) = c 2 ❊ � g ( 1 − S g ) � n − 1 ❊ � S g ( 1 − S g ) 2 � ¯ ¯ ¯ c ❊ { S g ( 1 − S g ) } + � � c ❊ � � ❊ { S g } ¯ Q 1 (¯ c , n ) = c ❊ � � c 2 ❊ � � n − 1 ❊ � g ( 1 − S g ) � ¯ ¯ ¯ +
Spillover Direct Efgect on the Treated d . Indirect Efgects on the Treated Indirect Efgect on the Untreated 17/ 19 (Rank Condition) + (Assumptions i–iv) ⇒ Point Identifjed Efgects ¯ D ig → Y ig for the population, holding D ig = 0. D ig → Y ig for compliers as a function of ¯ ¯ D ig → Y ig for compliers holding D ig = 0 or D ig = 1. ¯ D ig → Y ig for never-takers holding D ig = 0.
18/ 19 0.0 Probability employed D ig 0.48 0.44 0.40 0.36 0.5 0.4 0.3 0.2 0.1 (0.06) (0.01) -0.06 0.47 Naïve IV (0.07) (0.01) -0.14 0.47 Our estimator Data from Crepon et al. (2013; QJE) D ig Average Spillover to Long-term Employment: Y ig ( 0 , ¯ D ig ) = α ig + γ ig ¯ ❊ ( α ig ) ❊ ( γ ig ) ¯
Conclusion Identifjcation Go beyond ITTs to identify average direct and indirect efgects in randomized saturation experiments with 1-sided non-compliance. Estimation Simple asymptotically normal estimator under large/many-group asymptotics. Application Detect labor market spillovers in Crepon et al. (2013; QJE) experiment. 19/ 19
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