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Introduction to heter_endo_gmm in order to estimate heterogeneous peer effect when identity equals eligibility

Heterogeneous Peer Effects

The package “Heterogeneous Peer Effects” aims to estimate individual responses within groups. Our contribution to the standard linear-in-means model is that we allow individuals to respond differently to the outcomes of their peers depending on both their identity and eligibility for treatment.

Identity refers to an observable characteristic of an individual (e.g., race, gender).
Eligibility indicates whether an individual is allowed to receive the treatment.

In the function heter_endo_gmm, we assume that identity and eligibility coincide, meaning that eligibility is determined by identity (e.g., being Black or White). Specifically, we distinguish between: - Within-group peer effects (theta within): interactions between individuals sharing the same identity. - Between-group peer effects (theta between): interactions between individuals of different identities.

We propose a simple methodology to identify and estimate the model using partial population experiments, where only a subset of individuals within a group is eligible for treatment, and the proportion of eligible individuals varies across groups. The estimation procedure relies on the Generalized Method of Moments (GMM).

Assumptions

Our method is based on the following key assumptions:

Linear-in-Means Model with Panel Data
- The outcome depends linearly on both the average outcome and the treatment status.
Treatment Distribution Assumption
- For each observed share of eligible individuals in the population, there exist both treated and control groups.
Conditional Common Trends
- In the absence of treatment, the average change in aggregate outcomes among eligible individuals in treated groups would have been the same as in control groups.
Stable Share of Eligibles
- The proportion of eligible individuals within a group remains constant over time.

Data Requirements

To apply this methodology, the data must meet the following criteria:

Each group (e.g., districts) must contain both Eligible and Non-Eligible individuals.
Groups must be categorized as either Treated or Non-Treated.
Example: In analyzing the impact of Rosenwald schools on school attendance:
- Eligible individuals = Black residents
- Non-Eligible individuals = White residents
- Treated groups = Districts that received Rosenwald schools

The analysis is conducted at an aggregate level, considering average outcomes within each group.

Estimation Procedure

To estimate the model, the following variables are required:

s : Proportion of eligible individuals in the group (vector)
YN : Average outcome for non-eligible individuals in the group (vector)
YE : Average outcome for eligible individuals in the group (vector)
D : Binary treatment indicator for the group (vector)
n_param : Number of parameters to estimate (=5 by default or =3 for a restricted model)

Usage Examples

Estimating `delta`, `theta_within`, and `theta_between` with 5 parameters:

# Run estimation with default parameters (5 parameters)
test_nocov <- heter_endo_gmm(YE, YN, D, s)

# View results
print(test_nocov)

Estimating with a 3-parameter model:

# Run estimation with 3 parameters
test_bis <- heter_endo_gmm(YE, YN, D, s, n_param = 3)

Outcomes

This package return the estimated coefficients (and their p_values of ): - direct effect of treatment (delta) - intra group effect of treatment (theta_within) - inter group effect of treatment (theta_between) —

This package provides a flexible and robust framework for estimating heterogeneous peer effects in grouped data, making it applicable to various empirical settings, such as education, labor markets, and social networks.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.