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Long-Term Causal Inference with Many Noisy Proxies

Apoorva Lal, Guido Imbens, Peter Hull

TL;DR

This work demonstrates that regularized regression methods substantially outperform naive proxy selection and shows in particular that the bias of Ridge regression decreases as more proxies are added, with closed-form expressions for the bias-variance tradeoff.

Abstract

We propose a method for estimating long-term treatment effects with many short-term proxy outcomes: a central challenge when experimenting on digital platforms. We formalize this challenge as a latent variable problem where observed proxies are noisy measures of a low-dimensional set of unobserved surrogates that mediate treatment effects. Through theoretical analysis and simulations, we demonstrate that regularized regression methods substantially outperform naive proxy selection. We show in particular that the bias of Ridge regression decreases as more proxies are added, with closed-form expressions for the bias-variance tradeoff. We illustrate our method with an empirical application to the California GAIN experiment.

Long-Term Causal Inference with Many Noisy Proxies

TL;DR

This work demonstrates that regularized regression methods substantially outperform naive proxy selection and shows in particular that the bias of Ridge regression decreases as more proxies are added, with closed-form expressions for the bias-variance tradeoff.

Abstract

We propose a method for estimating long-term treatment effects with many short-term proxy outcomes: a central challenge when experimenting on digital platforms. We formalize this challenge as a latent variable problem where observed proxies are noisy measures of a low-dimensional set of unobserved surrogates that mediate treatment effects. Through theoretical analysis and simulations, we demonstrate that regularized regression methods substantially outperform naive proxy selection. We show in particular that the bias of Ridge regression decreases as more proxies are added, with closed-form expressions for the bias-variance tradeoff. We illustrate our method with an empirical application to the California GAIN experiment.
Paper Structure (30 sections, 45 equations, 10 figures)

This paper contains 30 sections, 45 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. The Causal Model
  4. Economic Foundations.
  5. The Identification Challenge.
  6. Data Structure and the Feasible Estimator
  7. Theoretical Analysis
  8. Motivating Cases
  9. High-Dimensional Selection with the Ridge Surrogate Index
  10. Model and Estimator
  11. Asymptotic Bias
  12. Asymptotic Mean Squared Error (AMSE)
  13. Variance estimation and inference
  14. Delta-method decomposition.
  15. First-stage variance (experimental sample).
  16. ...and 15 more sections

Figures (10)

  • Figure 1: Graphical representation of the causal model with unobserved surrogate, and two partially overlapping data sources. The gray node denotes that $S$ is unobserved.
  • Figure 2: Estimation Performance when increasing number of proxies for 1 scalar surrogate $S$
  • Figure 3: Estimation Performance when increasing number of proxies for a 5-vector-valued surrogate $S$
  • Figure 4: Estimation Performance when increasing number of surrogates while fixing the number of proxies for each surrogate at 6
  • Figure 5: Raw differences (top panel), and naive and surrogate estimates (bottom panel)
  • ...and 5 more figures

Theorems & Definitions (1)

  • Remark 1