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Identifying Causal Efgects in Experiments with Social Interactions and Non-compliance Francis J. DiTraglia 1 Camilo Garca-Jimeno 2 Rossa OKeefge-ODonovan 1 Alejandro Sanchez 3 1 University of Oxford 2 Federal Reserve Bank of Chicago 3

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SLIDE 1

Identifying Causal Efgects in Experiments with Social Interactions and Non-compliance

Francis J. DiTraglia1 Camilo García-Jimeno2 Rossa O’Keefge-O’Donovan1 Alejandro Sanchez3

1University of Oxford 2Federal Reserve Bank of Chicago 3University 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.

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SLIDE 2

Empirical Example with Potential for Indirect Treatment Efgects

Crepon et al. (2013; QJE)

◮ Large-scale job-seeker assistance program in France. ◮ Randomized ofgers of intensive job placement services.

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.”

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SLIDE 3

Studying Social Interactions Without Network Data

Partial Interference

Spillovers within but not between groups.

Randomized Saturation

Two-stage experimental design.

0% 25% 50% 75% 100% 1 1

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SLIDE 4

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)

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SLIDE 5

Notation

Sample Size and Indexing

◮ Groups: g = 1, . . . , G ◮ Individuals in g: i = 1, . . . , Ng

Observables

◮ Zig = binary treatment ofger to (i, g) ◮ Dig = binary treatment take-up of (i, g) ◮ Yig = outcome of (i, g) ◮ Sg = saturation of group g ◮ ¯ Dig = take-up fraction in g excluding (i, g)

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SLIDE 6

Overview of Assumptions

(i) Experimental Design: Randomized Saturation (ii) Potential Outcomes: Correlated Random Coeffjcients Model (iii) Treatment Take-up: 1-sided Noncompliance & “Individualized Ofger Response” (iv) Exclusion Restriction for (Zig, Sg) (v) Rank Condition

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SLIDE 7

Assumption (ii) – Correlated Random Coeffjcients Model

Yig(D) = Yig(Dg) = Yig(Dig, ¯ Dig) = f(¯ Dig)′ (1 − Dig)θig + Digψig

  • ◮ f ≡ vector of known functions, Lipschitz continuous on [0, 1]

◮ (θig, ψig) ≡ RVs, possibly dependent on (Dig, ¯ Dig).

This Talk

Focus on linear potential outcomes model.

7/ 19

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SLIDE 8

Yig(Dig, ¯ Dig) = αig + βigDig + γig ¯ Dig + δigDig ¯ Dig

¯ Dig αig + βig Yig(1, ¯ Dig) γig + δig αig Yig(0, ¯ Dig) γig

Indirect Efgects

Treated: γig + δig Untreated: γig

Direct Efgects

βig + δig ¯ Dig

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SLIDE 9

Assumption (iii) – Treatment Take-up

1-sided Non-compliance

Only those ofgered treatment can take it up.

Individualistic Ofger Response (IOR)

Dig(Z) = Dig(Zg) = Dig(Zig, ¯ Zig) = Dig(Zig)

Notation

Cig = 1 ifg (i, g) is a complier; ¯ Cig ≡ share of compliers among (i, g)’s neighbors. (IOR) + (1-Sided) ⇒ Dig = Cig × Zig

9/ 19

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SLIDE 10

No Evidence Against IOR in Our Example

Data from Crepon et al. (2013; QJE)

(IOR) + (1-Sided)

Take-up only depends on own ofger: Dig = Cig × Zig

Testable Implication

❊[Dig|Zig = 1, Sg] = ❊[Dig|Zig = 1]

Figure at right

Take-up among ofgered doesn’t vary with saturation (p = 0.62)

0.25 0.50 0.75 1.00 0.0 0.2 0.4 0.6 0.8 1.0

Saturation Treatment Take-up among Ofgered

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SLIDE 11

Assumption (iv) – Exclusion Restriction

Notation

◮ Bg = random coeffjcients for everyone in group g. ◮ Cg = complier indicators for everyone in group g ◮ Zg = treatment ofgers for everyone in group g

Exclusion Restriction

(i) Sg | = (Cg, Bg, Ng) (ii) Zg | = (Cg, Bg)|(Sg, Ng)

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SLIDE 12

Näive IV Does Not Identify the Spillover Efgect

Unofgered Individuals

Yig = αig + ✘✘✘

✘

βigDig + γig ¯ Dig + ✘✘✘✘

✘

δigDig ¯ Dig = E[αig] + E[γig]¯ Dig + (αig − E[αig]) + (γig − E[γig]) ¯ Dig = α + γ ¯ Dig + εig

IV Estimand

γIV = Cov(Yig, Sg) Cov(¯ Dig, Sg) = γ + Cov(εig, Sg) Cov(¯ Dig, Sg) = . . . = γ + Cov(γig, ¯ Cig) ❊(¯ Cig)

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SLIDE 13

Identifjcation – Average Spillover Efgect when Untreated

One-sided Noncompliance

(1 − Zig)Yig = (1 − Zig)(αig + ✘✘ ✘ βigDig + γig ¯ Dig + ✘✘✘✘ ✘ δigDig ¯ Dig ) = (1 − Zig)

  • 1

¯ Dig ′ αig γig

  • Theorem

(Zig, ¯ Dig) | = (αig, γig)|(¯ Cig, Ng).

❊

  • 1

¯ Dig

  • (1 − Zig)Yig
  • ¯

Cig, Ng

  • =

❊

  • (1 − Zig)
  • 1

¯ Dig ¯ Dig ¯ D2

ig

αig γig

  • ¯

Cig, Ng

  • =

❊

  • (1 − Zig)
  • 1

¯ Dig ¯ Dig ¯ D2

ig

  • ¯

Cig, Ng

  • ≡Q0(¯

Cig ,Ng )

❊

  • αig

γig

  • ¯

Cig, Ng

  • 13/ 19
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SLIDE 14

Identifjcation – Average Spillover Efgect when Untreated

Previous Slide:

❊

  • 1

¯ Dig

  • (1 − Zig)Yig
  • ¯

Cig, Ng

  • =

Q0(¯ Cig, Ng) ❊

  • αig

γig

  • ¯

Cig, Ng

  • Rearrange + Iterated ❊
  • ❊(αig)

❊(γig)

  • =

❊

  • ❊(αig|¯

Cig, Ng) ❊(γig|¯ Cig, Ng)

  • = ❊
  • ❊
  • Q0(¯

Cig, Ng)−1

  • 1

¯ Dig

  • (1 − Zig)Yig
  • ¯

Cig, Ng

  • =

❊

  • Q0(¯

Cig, Ng)−1

  • 1

¯ Dig

  • (1 − Zig)Yig
  • 14/ 19
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SLIDE 15

Average Spillover, Untreated: ❊[Yig(0, ¯ d)] = ❊(αig) + ❊(γig)¯ d

 ❊(αig)

❊(γig)

  = ❊  Q0(¯

Cig, Ng)−1

  1

¯ Dig

  (1 − Zig)Yig  

Q0(¯ Cig, Ng) ≡ ❊

 (1 − Zig)   1

¯ Dig ¯ Dig ¯ D2

ig

 

  • ¯

Cig, Ng

 

Q0 is a known function

Distribution of ¯ Dig|(¯ Cig, Ng) determined by experimental design.

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SLIDE 16

Rank Condition: Yig(Dig, ¯ Dig) = f(¯ Dig)′ (1 − Dig) θig + Digψig

  • Qz(¯

c, n) ≡ ❊

  • ✶(Zig = z)f(¯

Dig)f(¯ Dig)′

  • ¯

Cig = ¯ c, Ng = n

  • ,

z = 0, 1

Rank Condition

Q0(¯ c, n), Q1(¯ c, n) invertible for all (¯ c, n) in the support of (¯ Cig, Ng).

E.g. Linear Model

Q0(¯ c, n) =

  • ❊ {1 − Sg}

¯ c ❊ {Sg(1 − Sg)} ¯ c ❊ {Sg(1 − Sg)} ¯ c2 ❊ S2

g(1 − Sg)

+

¯ c n−1❊

Sg(1 − Sg)2

  • Q1(¯

c, n) =

  • ❊ {Sg}

¯ c ❊ S2

g

  • ¯

c ❊ S2

g

  • ¯

c2 ❊ S3

g

  • +

¯ c n−1❊

S2

g(1 − Sg)

  • 16/ 19
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SLIDE 17

(Rank Condition) + (Assumptions i–iv) ⇒ Point Identifjed Efgects

Spillover

¯ Dig → Yig for the population, holding Dig = 0.

Direct Efgect on the Treated

Dig → Yig for compliers as a function of ¯ d.

Indirect Efgects on the Treated

¯ Dig → Yig for compliers holding Dig = 0 or Dig = 1.

Indirect Efgect on the Untreated

¯ Dig → Yig for never-takers holding Dig = 0.

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SLIDE 18

Average Spillover to Long-term Employment: Yig(0, ¯ Dig) = αig + γig ¯ Dig

Data from Crepon et al. (2013; QJE)

❊(αig) ❊(γig) Our estimator 0.47

  • 0.14

(0.01) (0.07)

Naïve IV 0.47

  • 0.06

(0.01) (0.06)

0.0 0.1 0.2 0.3 0.4 0.5 0.36 0.40 0.44 0.48

¯ Dig Probability employed

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SLIDE 19

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.

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