dd-ie: Double Demeaning for Fixed Effects Interactions

A Python implementation of the double demeaning technique for unbiased estimation of interactions in fixed effects regression models.

The Problem

Standard fixed effects interactions (X x Z) may be biased when both variables vary within units and correlate with unobserved unit-specific moderators. The standard FE interaction estimator (FE-IE) uses between-unit differences in the effects of the interacted variables, making it susceptible to omitted variable bias from time-constant unobserved moderators.

The Solution

Double demeaning (dd-IE) provides an unbiased within-unit interaction estimator:

1. Demean each variable within units: X* = X - mean_i(X), Z* = Z - mean_i(Z)
2. Create the interaction from demeaned variables: X* x Z*
3. Use this in a standard FE regression (which demeans it again)

This eliminates all between-unit heterogeneity from the interaction term.

Installation

pip install dd-ie

For development:

git clone https://github.com/nkoutoun/dd-ie.git
cd dd-ie
pip install -e ".[dev]"

Quick Start

import pandas as pd
from dd_ie import DoubleDemeanAnalysis

# Load panel data
df = pd.read_csv("your_data.csv")

# Run analysis
analysis = DoubleDemeanAnalysis(
    data=df,
    unit_var="unit_id",      # Unit identifier
    time_var="time_id",      # Time identifier
    y_var="outcome",         # Dependent variable
    x_var="treatment",       # First interacting variable
    z_var="moderator",       # Second interacting variable
    w_vars=["control1"],     # Control variables (optional)
)

results = analysis.run_analysis()

# Access structured results
print(results.comparison.table)          # Coefficient comparison
print(results.hausman.statistic)         # Hausman test statistic
print(results.hausman.p_value)           # Hausman p-value
print(results.hausman.conclusion)        # "SYSTEMATIC_BIAS" or "NO_SYSTEMATIC_BIAS"

Model Comparison

Standard FE (potentially biased)

Y = b_x * X + b_z * Z + b_xz * (X * Z) + g * W + a_i + e

Double Demeaned FE (unbiased)

Y = b_x * X + b_z * Z + b_xz * (X* x Z*) + g * W + a_i + e
where X* = X - mean_i(X), Z* = Z - mean_i(Z)

Result Types

The package returns typed dataclasses for all results:

• AnalysisResult -- complete pipeline output with .standard_results, .dd_results, .comparison, .hausman, .processed_data
• ComparisonResult -- coefficient comparison .table (DataFrame) and .interaction_difference
• HausmanResult -- test .statistic, .p_value, .degrees_of_freedom, .conclusion, .positive_definite

All result types have a .to_dict() method for backward compatibility.

Advanced Usage

from dd_ie import (
    create_double_demeaned_interaction,
    estimate_fe_models,
    perform_hausman_test,
)

# Step-by-step analysis
df_processed = create_double_demeaned_interaction(df, "x", "z", "unit_id")
standard_results, dd_results, comparison = estimate_fe_models(
    df_processed, "y", "x", "z", ["control1"]
)
hausman = perform_hausman_test(standard_results, dd_results, "x", "z")

Data Requirements

• Panel structure: unit and time identifiers
• Minimum periods: T > 2 per unit for identification
• Within-unit variation: both X and Z must vary within units
• Format: long format with one row per unit-time observation

Interpreting Results

• Hausman test p >= 0.05: no systematic bias detected; standard FE is more efficient
• Hausman test p < 0.05: systematic bias detected; prefer the double-demeaned estimator

Citation

If you use this package, please cite the original methodology:

@article{giesselmann2022interactions,
  title={Interactions in Fixed Effects Regression Models},
  author={Giesselmann, Marco and Schmidt-Catran, Alexander W},
  journal={Sociological Methods \& Research},
  volume={51},
  number={3},
  pages={1100--1127},
  year={2022},
  publisher={SAGE Publications},
  doi={10.1177/0049124120914934}
}

License

MIT License