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A Causal Inference Framework for Data Rich Environments

Alberto Abadie, Anish Agarwal, Devavrat Shah

TL;DR

This paper addresses unobserved confounding in data-rich settings where both the number of units $N$ and the number of measurements per unit $T$ are large. It develops a latent-factor framework that links structural causal models with latent factor models, showing that potential outcomes $Y^{(a)}_{n,t}$ can be approximated by a low-rank linear factor representation under Hölder-smooth latent structure. The authors derive identification results for $ATE$, $ATT$, and $ATU$ under a linear-span condition and propose a PCR-based estimator that achieves finite-sample consistency and asymptotic normality under tractable conditions. The approach nests classical econometric models (e.g., two-way FE, interactive FE) and provides practical tools for counterfactual inference in high-dimensional data environments, with explicit rates and assumptions on smoothness, spectrum, and low-rank approximation.

Abstract

We propose a formal model for counterfactual estimation with unobserved confounding in "data-rich" settings, i.e., where there are a large number of units and a large number of measurements per unit. Our model provides a bridge between the structural causal model view of causal inference common in the graphical models literature with that of the latent factor model view common in the potential outcomes literature. We show how classic models for potential outcomes and treatment assignments fit within our framework. We provide an identification argument for the average treatment effect, the average treatment effect on the treated, and the average treatment effect on the untreated. For any estimator that has a fast enough estimation error rate for a certain nuisance parameter, we establish it is consistent for these various causal parameters. We then show principal component regression is one such estimator that leads to consistent estimation, and we analyze the minimal smoothness required of the potential outcomes function for consistency.

A Causal Inference Framework for Data Rich Environments

TL;DR

This paper addresses unobserved confounding in data-rich settings where both the number of units and the number of measurements per unit are large. It develops a latent-factor framework that links structural causal models with latent factor models, showing that potential outcomes can be approximated by a low-rank linear factor representation under Hölder-smooth latent structure. The authors derive identification results for , , and under a linear-span condition and propose a PCR-based estimator that achieves finite-sample consistency and asymptotic normality under tractable conditions. The approach nests classical econometric models (e.g., two-way FE, interactive FE) and provides practical tools for counterfactual inference in high-dimensional data environments, with explicit rates and assumptions on smoothness, spectrum, and low-rank approximation.

Abstract

We propose a formal model for counterfactual estimation with unobserved confounding in "data-rich" settings, i.e., where there are a large number of units and a large number of measurements per unit. Our model provides a bridge between the structural causal model view of causal inference common in the graphical models literature with that of the latent factor model view common in the potential outcomes literature. We show how classic models for potential outcomes and treatment assignments fit within our framework. We provide an identification argument for the average treatment effect, the average treatment effect on the treated, and the average treatment effect on the untreated. For any estimator that has a fast enough estimation error rate for a certain nuisance parameter, we establish it is consistent for these various causal parameters. We then show principal component regression is one such estimator that leads to consistent estimation, and we analyze the minimal smoothness required of the potential outcomes function for consistency.

Paper Structure

This paper contains 35 sections, 13 theorems, 137 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Notation
  4. Data Generating Process
  5. Latent Factorization of Confounders
  6. Encoding Latent Structure Across Potential Outcomes
  7. Nesting Classical Econometric Models
  8. Intervention Assignment Function
  9. Nesting Classical Intervention Assignment Mechanisms
  10. Identification and Estimation of Treatment Effects with Factor Structure in Potential Outcomes
  11. Identification
  12. Estimator
  13. Finite-Sample Consistency
  14. Asymptotic Normality
  15. Estimation results using Principal Component Regression (PCR)
  16. ...and 20 more sections

Key Result

Proposition 1

Suppose Assumption assumption:holder.mech holds. Then, for any $\delta>0$, there exist latent variables $\lambda_n, \rho_{t, a} \in \mathbb R^r$ such that: where for $C$ that is allowed to depend on $(q,S)$,

Figures (1)

  • Figure 1: A DAG representation consistent with the data generating process.

Theorems & Definitions (28)

  • Definition 1: Hölder continuity, e.g., jiaming_spectral_convergence, jiaming_spectral_convergence
  • Remark 1
  • Proposition 1: Hölder low rank matrix approximation, agarwal2021robustness, agarwal2021causal
  • Example 1: Two-way fixed effects model
  • Example 2: Interactive fixed effects model, bai2009interactive, bai2009interactive
  • Example 3: Binary choice
  • Example 4: Dictionary basis expansion
  • Example 5: Randomized trial
  • Example 6: Selection on observables
  • Example 7: Regression Discontinuity
  • ...and 18 more