Efficient nonparametric estimation with difference-in-differences in the presence of network dependence and interference

Didong Li; Michael G. Hudgens; Michael Jetsupphasuk

arxiv: 2502.03414 · v4 · submitted 2025-02-05 · 📊 stat.ME

Efficient nonparametric estimation with difference-in-differences in the presence of network dependence and interference

Michael Jetsupphasuk , Didong Li , Michael G. Hudgens This is my paper

Pith reviewed 2026-05-23 03:43 UTC · model grok-4.3

classification 📊 stat.ME

keywords difference-in-differencesnetwork interferencecausal inferencedoubly robust estimationnonparametric estimationlongitudinal dataexposure effects

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The pith

A doubly robust estimator extends difference-in-differences to networks with interference and latent dependence while remaining consistent and efficient.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends difference-in-differences to settings where treatment effects and exposure probabilities can vary across units, one unit's treatment can affect outcomes in neighboring units, and outcomes, treatments, and covariates can be correlated through latent factors. It defines the target as the network-averaged expected exposure effect when all units receive a fixed exposure level. A doubly robust estimator is proposed that allows flexible, data-adaptive estimation of nuisance functions. Under a conditional parallel trends assumption plus network dependency and heterogeneity conditions, this estimator is shown to be consistent, asymptotically normal, and efficient. Readers would care because many observational studies involve connected units such as counties, firms, or individuals where standard methods break down.

Core claim

Under a conditional parallel trends assumption and suitable network dependency and heterogeneity conditions, a doubly robust estimator allowing for data-adaptive nuisance function estimation is proposed and shown to be consistent, asymptotically normal, and efficient for the network-averaged expected exposure effect if units received a specific exposure level.

What carries the argument

The doubly robust estimator combining outcome regression and exposure probability models, adjusted for network features, to estimate network-averaged exposure effects.

If this is right

The estimator applies directly to longitudinal data with non-identically distributed units and heterogeneous treatment effects.
It remains valid when treatment of one unit spills over to affect outcomes of its neighbors.
Data-adaptive methods such as machine learning can be plugged in for the nuisance functions without losing efficiency.
The approach was evaluated in simulations and applied to county-level effects of power plant emission controls on cardiovascular mortality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same doubly robust structure could be adapted to other longitudinal causal estimands such as total network effects.
Researchers studying spatial policies or social networks could use this to relax the no-interference assumption common in standard difference-in-differences.
Sensitivity checks that vary the definition of network neighbors would test how robust the estimates are to network misspecification.
Extensions to time-varying exposures or staggered adoption could follow by redefining the exposure mapping accordingly.

Load-bearing premise

Expected potential outcome trajectories are parallel between treatment groups under the counterfactual where all units receive one specific treatment, after conditioning on network features.

What would settle it

A simulation or real dataset with known network structure where the estimator converges to an incorrect value or its asymptotic variance fails to match the efficiency bound despite the conditional parallel trends and dependence conditions holding.

Figures

Figures reproduced from arXiv: 2502.03414 by Didong Li, Michael G. Hudgens, Michael Jetsupphasuk.

**Figure 1.** Figure 1: Quantile-quantile plots comparing σˆ −1 n √ n(ˆτ − τ ) with the N(0, 1) distribution where nuisance functions were estimated using the Superlearner under scenarios: (a) ring network, ind. errors, bandwidth 0; (b) ring network, dep. errors, bandwidth 15; (c) bipartite network, ind. errors, bandwidth 0; (d) bipartite network, dep. errors, bandwidth 1.1 [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗

**Figure 2.** Figure 2: Counties by interference burden group and power plants by scrubber status in 2007. [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

**Figure 3.** Figure 3: Estimated effects and corresponding 95% confidence intervals of coal power plant scrub [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

read the original abstract

Differences-in-differences (DiD) is a causal inference method for observational longitudinal data that assumes parallel expected potential outcome trajectories between treatment groups under the counterfactual scenario where all units receive a specific treatment. In this paper DiD is extended to allow for: (i) non-identically distributed treatment effects and exposure probabilities; (ii) interference, where treatment of one unit can affect outcomes in neighboring units; and (iii) latent variable dependence, where outcomes, treatments, and covariates may exhibit between-unit correlation. The causal estimand of interest is the network-averaged expected exposure effect if units received a specific exposure level, where a unit's exposure is a function of its own treatment and its neighbors' treatments. Under a conditional parallel trends assumption and suitable network dependency and heterogeneity conditions, a doubly robust estimator allowing for data-adaptive nuisance function estimation is proposed and shown to be consistent, asymptotically normal, and efficient. The proposed methods are evaluated in simulations and applied to study the effects of adopting emission control technologies in coal power plants on county-level mortality due to cardiovascular disease.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper derives a doubly robust DiD estimator for network-averaged exposure effects under interference and latent dependence.

read the letter

The core contribution is an extension of difference-in-differences that drops the usual identical distribution assumption, incorporates interference through neighbors' treatments, and allows latent dependence across units. They target the network-averaged expected exposure effect under a fixed exposure level and build a doubly robust estimator that stays consistent and efficient when nuisance functions are estimated adaptively. The identification rests on a conditional parallel trends assumption plus explicit conditions on network dependence and heterogeneity, and the asymptotics follow the standard influence-function path adapted to this setting. Simulations and the coal-plant emissions application to county mortality give some practical grounding. The conditional parallel trends assumption is the main practical limitation here, as it must hold after conditioning on network features and for the specific counterfactual where every unit receives the same treatment; that will often require strong substantive justification in real data. The network dependence conditions are stated up front rather than hidden, which helps, but their restrictiveness will matter for how widely the results apply. This is aimed at causal inference people working with spatial or networked observational data in policy or environmental settings. A reader already familiar with doubly robust methods will see the technical steps clearly enough to assess the extension. It deserves peer review because the claims are concrete and the problem is relevant, even if the proofs need close checking.

Referee Report

1 major / 1 minor

Summary. The paper extends difference-in-differences to observational longitudinal data with non-iid treatment effects, interference (treatment of one unit affecting neighbors), and latent variable dependence. The target is the network-averaged expected exposure effect under a specific exposure level. Under a conditional parallel trends assumption plus network dependency and heterogeneity conditions, it proposes a doubly robust estimator that permits data-adaptive nuisance estimation and claims to establish consistency, asymptotic normality, and efficiency. The approach is assessed via simulations and an empirical application to emission-control adoption by coal plants and county-level cardiovascular mortality.

Significance. If the asymptotic claims hold under the stated conditions, the work would supply a practical, doubly robust tool for causal inference in networked settings where standard DiD fails due to interference and dependence. The explicit allowance for data-adaptive nuisances and the real-data application are strengths that could increase adoption in applied work.

major comments (1)

[Abstract] The manuscript states that consistency, asymptotic normality, and efficiency are shown, yet the actual proof steps, the precise network-dependence conditions, the form of the influence function, and the handling of data-adaptive nuisance estimators are not verifiable from the text provided. These derivations are load-bearing for the central claim.

minor comments (1)

The abstract refers to 'suitable network dependency and heterogeneity conditions' without enumerating them; these should be stated explicitly early in the paper so readers can assess their plausibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for clearer verifiability of the asymptotic results. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] The manuscript states that consistency, asymptotic normality, and efficiency are shown, yet the actual proof steps, the precise network-dependence conditions, the form of the influence function, and the handling of data-adaptive nuisance estimators are not verifiable from the text provided. These derivations are load-bearing for the central claim.

Authors: We agree that explicit pointers improve readability. The consistency, asymptotic normality, and semiparametric efficiency of the doubly robust estimator are stated in Theorem 3.1 (Section 3). The proof appears in full in Appendix A, which derives the influence function (Equation A.12) under the network dependence conditions of Assumption 2.3 (Section 2.3) and the conditional parallel trends assumption. Data-adaptive nuisance estimation is handled via cross-fitting in Section 3.3, with the efficiency result following from the Neyman orthogonality of the influence function. In the revision we will insert forward references from the abstract and introduction to these specific locations so that the derivations are immediately locatable without searching the supplement. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a conditional parallel trends assumption, derives an identification result for the network-averaged exposure effect, and constructs a doubly robust estimator whose consistency, asymptotic normality, and efficiency follow from standard influence-function arguments under the listed network dependence and heterogeneity conditions. No equation reduces a claimed result to a fitted quantity by construction, no uniqueness theorem is imported from the authors' prior work, and no self-citation is load-bearing for the central claims. The derivation is self-contained given the explicit assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the conditional parallel trends assumption and network dependency/heterogeneity conditions stated in the abstract. No free parameters or invented entities are mentioned.

axioms (2)

domain assumption conditional parallel trends assumption
Invoked as the identifying assumption that allows the DiD extension to identify the network-averaged exposure effect.
domain assumption suitable network dependency and heterogeneity conditions
Required for the consistency and asymptotic normality of the proposed estimator.

pith-pipeline@v0.9.0 · 5723 in / 1015 out tokens · 30225 ms · 2026-05-23T03:43:22.212474+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Under a conditional parallel trends assumption and suitable network dependency and heterogeneity conditions, a doubly robust estimator allowing for data-adaptive nuisance function estimation is proposed and shown to be consistent, asymptotically normal, and efficient.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The causal estimand of interest is the network-averaged expected exposure effect if units received a specific exposure level

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[3]

n−1 nX i=1 1( ¯Git =¯ gt) P( ¯Git =¯ gt) E[Yit(¯ g′ t)−Y i,t−δ(¯ g′ t)|Xi, ¯Git =¯ gt] # =n −1 nX i=1 E[∆δYi| ¯Git =¯ gt]−E

*Corresponding author. Email: jetsupphasuk@unc.edu 37 Assumption S2 provides regularity conditions of the parametric model forP( ¯Git =¯ gt)≡ pi1(¯ gt)in the case when there is network effect and exposure heterogeneity. These assumptions are standard assumptions for parametric models and estimators. Assumption S2(Regularity assumptions of parametric model...

work page 2021
[4]

S1.4 Proof of Proposition 2 Proof of Proposition 2.Letϕ ∗(O;P)as stated in Proposition 2 be the conjectured efficient influ- ence function

In particular, Theorem 1 in Sant’Anna and Zhao (2020) can be applied, replacing treatmentDin Sant’Anna and Zhao (2020) with1( ¯Gt =¯ gt), 1−Dwith1( ¯Gt =¯ g′ t), and generalizing the time periods from{0,1}to{t−δ, t}. S1.4 Proof of Proposition 2 Proof of Proposition 2.Letϕ ∗(O;P)as stated in Proposition 2 be the conjectured efficient influ- ence function. ...

work page 2020
[5]

40 S1.5 Proof of Theorem 1 Proof ofTheorem 1.By Theorem 3.1 in Kojevnikov, Marmer, and Song (2021) and Assumptions 7 – 9, the following result holds asn→ ∞: n−1 nX i=1 (ˆh1( ¯Git)− ˆh0( ¯Git,X i; ˆπi))(∆δYi −ˆµi,¯ g′ t,δ(Xi))− E[(ˆh1( ¯Git)− ˆh0( ¯Git,X i; ˆπi))(∆δYi −ˆµi,¯ g′ t,δ(Xi))] →p

work page 2021
[6]

The first term, 1 , in (S3) can be shown to be equal to: E n−1 nX i=1 ˆh1( ¯Git)∆δYi −h 1( ¯Git)∆δYi 41 = E n−1 nX i=1 ∆δYi1( ¯Git =¯ gt) pi1(¯ gt)−ˆpi1(¯ gt) pi1(¯ gt)ˆpi1(¯ gt)

The expression above can be decomposed into the following: (∗) := E n−1 nX i=1 ˆh1( ¯Git)∆δYi −h 1( ¯Git)∆δYi | {z } 1 (S3) − ˆh1( ¯Git)ˆµi,¯ g′ t,δ(Xi)−h 1( ¯Git)ˆµi,¯ g′ t,δ(Xi) | {z } 2 − ˆh0( ¯Git,X i; ˆπi)∆δYi −h 0( ¯Git,X i;π i)∆δYi | {z } 3 + ˆh0( ¯Git,X i; ˆπi)ˆµi,¯ g′ t,δ(Xi)−h 0( ¯Git,X i;π i)µi,¯ g′ t,δ(Xi) | {z } 4 . The first term, 1 , in (S3...

work page 2021
[7]

=n −1 nX i=1 E[(h1( ¯Git)−h 0( ¯Gt,X i;π 0 i ))(∆δYi −µ i,¯ g′ t,δ(Xi))], ˆτ(π0) =n −1 nX i=1 (ˆh1( ¯Git)− ˆh0( ¯Gt,X i; ˆπ0 i ))(∆δYi −ˆµi,¯ g′ t,δ(Xi)) Theorem 1 in Sant’Anna and Zhao (2020) shows thatτ(π

work page 2020
[8]

In the absence of network heterogeneity,ϕ ∗(O1:n;P) =ϕ i(Oi;P)and is equal to the efficient influence function (EIF) discussed in Sant’Anna and Zhao (2020)

S1.6 Proof of Theorem 2 Letϕ ∗(O1:n;P) =n −1Pn i=1 ϕi(O1:n;P)denote the influence function whose form is given in the main text. In the absence of network heterogeneity,ϕ ∗(O1:n;P) =ϕ i(Oi;P)and is equal to the efficient influence function (EIF) discussed in Sant’Anna and Zhao (2020). In the following proof, we generally considerφ(O 1:n;P) =ϕ ∗(O1:n;P)sin...

work page 2020
[9]

Then,Ψ( ˆP)−Ψ(P)can be further decomposed (whereP nϕ∗ =ϕ ∗ sinceϕ ∗ is already a sample average), Ψ(ˆP)−Ψ(P) =−P{ϕ ∗(ˆP)}+R 2(ˆP,P), = (Pn −P)ϕ ∗(ˆP)−(P n −P)ϕ ∗(P) + (Pn −P)ϕ ∗(P) +R 2(ˆP,P) = (Pn −P)ϕ ∗(P) + (Pn −P)(ϕ ∗(ˆP)−ϕ ∗(P)) +R 2(ˆP,P), where the second equality usesP nϕ∗(ˆP) = 0and adds and subtracts(P n −P)ϕ ∗(P); and the third equality re-arra...

work page 2021
[10]

Consider the following, ˜σ∗2 n = 1 n X s≥0 X i∈Nn X k∈N ∂n (i;s) E[ϕi(ˆP)ϕk(ˆP)]ω(s/bn), ˆσ2 n = 1 n X s≥0 X i∈Nn X k∈N ∂n (i;s) ϕi(ˆP)ϕk(ˆP)ω(s/bn)

The remainder of the proof shows that the variance estimator is consistent for the first term σ2,∗ n = 1 n P s≥0 P i∈Nn P k∈N ∂n (i;s) E[ϕi(P)ϕk(P)]and is therefore conservative forσ 2 n with the bias equal toV n = 1 n P s≥0 P i∈Nn P k∈N ∂n (i;s)(Ψi(P)−Ψ(P))(Ψ k(P)−Ψ(P)), or the sample covariance of the network exposure effects. Consider the following, ˜σ...

work page 2021
[11]

corrections

= 0.5 +α i, the true individual effects wereAEE i = 5 +θ i, and the true total effect wasAEE =n −1Pn i=1(5 +θ i). Table S2: Results from 1000 simulations. Data generation Estimator parameters Results Exposure prob. het. Network corr. Exposure prob. estimator Bias + variance correction Bias ESE ASE Coverage (%) No NA Sample average No 0.000 0.040 0.048 98....

work page 2024
[12]

These summary statistics were similar for other study 68 Table S3: Number (percent) of counties within each exposure cohort. Exposure cohort Low interference burden High interference burden <2007 0 (0.0%) 27 (4.2%) 2007 5 (0.8%) 98 (15.2%) 2008 70 (10.9%) 198 (30.7%) 2009 367 (57.1%) 288 (44.7%) 2010 147 (22.9%) 27 (4.2%) >2010 54 (8.4%) 6 (0.9%) years. T...

work page 2007
[13]

Semiparametric Difference-in-Differences Estimators

Covariate Mean (SD) Quartile 3 Quartile 4 County Proportion White 0.849 (0.163) 0.857 (0.162) Proportion Black 0.118 (0.161) 0.108 (0.146) Proportion Hispanic 0.026 (0.029) 0.025 (0.04) Proportion female 0.508 (0.015) 0.51 (0.016) Median age 36.8 (2.9) 37.2 (3.0) Average household size 2.5 (0.1) 2.5 (0.1) Proportion urban 0.419 (0.287) 0.468 (0.304) Propo...

work page doi:10.1002/joc.3413 2005
[14]

\ Sant'Anna, P H C

Butts, Kyle (2021).Difference-in-Differences Estimation with Spatial Spillovers.DOI:10.48550/ arXiv.2105.03737. Callaway, Brantly and Pedro H. C. Sant’Anna (2021). “Difference-in-Differences with multiple time periods”. In:Journal of Econometrics. Themed Issue: Treatment Effect 1 225.2, pp. 200– 230.ISSN: 0304-4076.DOI:10.1016/j.jeconom.2020.12.001. Card,...

work page doi:10.1016/j.jeconom.2020.12.001 2021

[1] [3]

n−1 nX i=1 1( ¯Git =¯ gt) P( ¯Git =¯ gt) E[Yit(¯ g′ t)−Y i,t−δ(¯ g′ t)|Xi, ¯Git =¯ gt] # =n −1 nX i=1 E[∆δYi| ¯Git =¯ gt]−E

*Corresponding author. Email: jetsupphasuk@unc.edu 37 Assumption S2 provides regularity conditions of the parametric model forP( ¯Git =¯ gt)≡ pi1(¯ gt)in the case when there is network effect and exposure heterogeneity. These assumptions are standard assumptions for parametric models and estimators. Assumption S2(Regularity assumptions of parametric model...

work page 2021

[2] [4]

S1.4 Proof of Proposition 2 Proof of Proposition 2.Letϕ ∗(O;P)as stated in Proposition 2 be the conjectured efficient influ- ence function

In particular, Theorem 1 in Sant’Anna and Zhao (2020) can be applied, replacing treatmentDin Sant’Anna and Zhao (2020) with1( ¯Gt =¯ gt), 1−Dwith1( ¯Gt =¯ g′ t), and generalizing the time periods from{0,1}to{t−δ, t}. S1.4 Proof of Proposition 2 Proof of Proposition 2.Letϕ ∗(O;P)as stated in Proposition 2 be the conjectured efficient influ- ence function. ...

work page 2020

[3] [5]

40 S1.5 Proof of Theorem 1 Proof ofTheorem 1.By Theorem 3.1 in Kojevnikov, Marmer, and Song (2021) and Assumptions 7 – 9, the following result holds asn→ ∞: n−1 nX i=1 (ˆh1( ¯Git)− ˆh0( ¯Git,X i; ˆπi))(∆δYi −ˆµi,¯ g′ t,δ(Xi))− E[(ˆh1( ¯Git)− ˆh0( ¯Git,X i; ˆπi))(∆δYi −ˆµi,¯ g′ t,δ(Xi))] →p

work page 2021

[4] [6]

The first term, 1 , in (S3) can be shown to be equal to: E n−1 nX i=1 ˆh1( ¯Git)∆δYi −h 1( ¯Git)∆δYi 41 = E n−1 nX i=1 ∆δYi1( ¯Git =¯ gt) pi1(¯ gt)−ˆpi1(¯ gt) pi1(¯ gt)ˆpi1(¯ gt)

The expression above can be decomposed into the following: (∗) := E n−1 nX i=1 ˆh1( ¯Git)∆δYi −h 1( ¯Git)∆δYi | {z } 1 (S3) − ˆh1( ¯Git)ˆµi,¯ g′ t,δ(Xi)−h 1( ¯Git)ˆµi,¯ g′ t,δ(Xi) | {z } 2 − ˆh0( ¯Git,X i; ˆπi)∆δYi −h 0( ¯Git,X i;π i)∆δYi | {z } 3 + ˆh0( ¯Git,X i; ˆπi)ˆµi,¯ g′ t,δ(Xi)−h 0( ¯Git,X i;π i)µi,¯ g′ t,δ(Xi) | {z } 4 . The first term, 1 , in (S3...

work page 2021

[5] [7]

=n −1 nX i=1 E[(h1( ¯Git)−h 0( ¯Gt,X i;π 0 i ))(∆δYi −µ i,¯ g′ t,δ(Xi))], ˆτ(π0) =n −1 nX i=1 (ˆh1( ¯Git)− ˆh0( ¯Gt,X i; ˆπ0 i ))(∆δYi −ˆµi,¯ g′ t,δ(Xi)) Theorem 1 in Sant’Anna and Zhao (2020) shows thatτ(π

work page 2020

[6] [8]

In the absence of network heterogeneity,ϕ ∗(O1:n;P) =ϕ i(Oi;P)and is equal to the efficient influence function (EIF) discussed in Sant’Anna and Zhao (2020)

S1.6 Proof of Theorem 2 Letϕ ∗(O1:n;P) =n −1Pn i=1 ϕi(O1:n;P)denote the influence function whose form is given in the main text. In the absence of network heterogeneity,ϕ ∗(O1:n;P) =ϕ i(Oi;P)and is equal to the efficient influence function (EIF) discussed in Sant’Anna and Zhao (2020). In the following proof, we generally considerφ(O 1:n;P) =ϕ ∗(O1:n;P)sin...

work page 2020

[7] [9]

Then,Ψ( ˆP)−Ψ(P)can be further decomposed (whereP nϕ∗ =ϕ ∗ sinceϕ ∗ is already a sample average), Ψ(ˆP)−Ψ(P) =−P{ϕ ∗(ˆP)}+R 2(ˆP,P), = (Pn −P)ϕ ∗(ˆP)−(P n −P)ϕ ∗(P) + (Pn −P)ϕ ∗(P) +R 2(ˆP,P) = (Pn −P)ϕ ∗(P) + (Pn −P)(ϕ ∗(ˆP)−ϕ ∗(P)) +R 2(ˆP,P), where the second equality usesP nϕ∗(ˆP) = 0and adds and subtracts(P n −P)ϕ ∗(P); and the third equality re-arra...

work page 2021

[8] [10]

Consider the following, ˜σ∗2 n = 1 n X s≥0 X i∈Nn X k∈N ∂n (i;s) E[ϕi(ˆP)ϕk(ˆP)]ω(s/bn), ˆσ2 n = 1 n X s≥0 X i∈Nn X k∈N ∂n (i;s) ϕi(ˆP)ϕk(ˆP)ω(s/bn)

The remainder of the proof shows that the variance estimator is consistent for the first term σ2,∗ n = 1 n P s≥0 P i∈Nn P k∈N ∂n (i;s) E[ϕi(P)ϕk(P)]and is therefore conservative forσ 2 n with the bias equal toV n = 1 n P s≥0 P i∈Nn P k∈N ∂n (i;s)(Ψi(P)−Ψ(P))(Ψ k(P)−Ψ(P)), or the sample covariance of the network exposure effects. Consider the following, ˜σ...

work page 2021

[9] [11]

corrections

= 0.5 +α i, the true individual effects wereAEE i = 5 +θ i, and the true total effect wasAEE =n −1Pn i=1(5 +θ i). Table S2: Results from 1000 simulations. Data generation Estimator parameters Results Exposure prob. het. Network corr. Exposure prob. estimator Bias + variance correction Bias ESE ASE Coverage (%) No NA Sample average No 0.000 0.040 0.048 98....

work page 2024

[10] [12]

These summary statistics were similar for other study 68 Table S3: Number (percent) of counties within each exposure cohort. Exposure cohort Low interference burden High interference burden <2007 0 (0.0%) 27 (4.2%) 2007 5 (0.8%) 98 (15.2%) 2008 70 (10.9%) 198 (30.7%) 2009 367 (57.1%) 288 (44.7%) 2010 147 (22.9%) 27 (4.2%) >2010 54 (8.4%) 6 (0.9%) years. T...

work page 2007

[11] [13]

Semiparametric Difference-in-Differences Estimators

Covariate Mean (SD) Quartile 3 Quartile 4 County Proportion White 0.849 (0.163) 0.857 (0.162) Proportion Black 0.118 (0.161) 0.108 (0.146) Proportion Hispanic 0.026 (0.029) 0.025 (0.04) Proportion female 0.508 (0.015) 0.51 (0.016) Median age 36.8 (2.9) 37.2 (3.0) Average household size 2.5 (0.1) 2.5 (0.1) Proportion urban 0.419 (0.287) 0.468 (0.304) Propo...

work page doi:10.1002/joc.3413 2005

[12] [14]

\ Sant'Anna, P H C

Butts, Kyle (2021).Difference-in-Differences Estimation with Spatial Spillovers.DOI:10.48550/ arXiv.2105.03737. Callaway, Brantly and Pedro H. C. Sant’Anna (2021). “Difference-in-Differences with multiple time periods”. In:Journal of Econometrics. Themed Issue: Treatment Effect 1 225.2, pp. 200– 230.ISSN: 0304-4076.DOI:10.1016/j.jeconom.2020.12.001. Card,...

work page doi:10.1016/j.jeconom.2020.12.001 2021