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Dynamic treatment effects: high-dimensional inference under model misspecification

Yuqian Zhang, Weijie Ji, Jelena Bradic

TL;DR

The paper tackles dynamic treatment effects under time-varying confounding with high-dimensional covariates and potential misspecification. It introduces the sequential model doubly robust (SMDR) estimator, built on moment-targeted nuisance estimates and a suite of losses that enforce orthogonality conditions even when nuisance models are misspecified. The authors prove root-$N$ inference for the dynamic treatment effect parameter $ heta_{1,1}$ under minimal assumptions, and derive nuisance-estimator rates under misspecification and correct specification, along with sparsity conditions. Through simulations and semi-synthetic NJCS analysis, SMDR consistently outperforms competing methods in bias, RMSE, and coverage, demonstrating robustness to misspecification in high dimensions. The work advances high-dimensional causal inference in longitudinal settings and paves the way for extensions to more complex dynamic regimes and dense models.

Abstract

Estimating dynamic treatment effects is crucial across various disciplines, providing insights into the time-dependent causal impact of interventions. However, this estimation poses challenges due to time-varying confounding, leading to potentially biased estimates. Furthermore, accurately specifying the growing number of treatment assignments and outcome models with multiple exposures appears increasingly challenging to accomplish. Double robustness, which permits model misspecification, holds great value in addressing these challenges. This paper introduces a novel "sequential model doubly robust" estimator. We develop novel moment-targeting estimates to account for confounding effects and establish that root-$N$ inference can be achieved as long as at least one nuisance model is correctly specified at each exposure time, despite the presence of high-dimensional covariates. Although the nuisance estimates themselves do not achieve root-$N$ rates, the carefully designed loss functions in our framework ensure final root-$N$ inference for the causal parameter of interest. Unlike off-the-shelf high-dimensional methods, which fail to deliver robust inference under model misspecification even within the doubly robust framework, our newly developed loss functions address this limitation effectively.

Dynamic treatment effects: high-dimensional inference under model misspecification

TL;DR

The paper tackles dynamic treatment effects under time-varying confounding with high-dimensional covariates and potential misspecification. It introduces the sequential model doubly robust (SMDR) estimator, built on moment-targeted nuisance estimates and a suite of losses that enforce orthogonality conditions even when nuisance models are misspecified. The authors prove root- inference for the dynamic treatment effect parameter under minimal assumptions, and derive nuisance-estimator rates under misspecification and correct specification, along with sparsity conditions. Through simulations and semi-synthetic NJCS analysis, SMDR consistently outperforms competing methods in bias, RMSE, and coverage, demonstrating robustness to misspecification in high dimensions. The work advances high-dimensional causal inference in longitudinal settings and paves the way for extensions to more complex dynamic regimes and dense models.

Abstract

Estimating dynamic treatment effects is crucial across various disciplines, providing insights into the time-dependent causal impact of interventions. However, this estimation poses challenges due to time-varying confounding, leading to potentially biased estimates. Furthermore, accurately specifying the growing number of treatment assignments and outcome models with multiple exposures appears increasingly challenging to accomplish. Double robustness, which permits model misspecification, holds great value in addressing these challenges. This paper introduces a novel "sequential model doubly robust" estimator. We develop novel moment-targeting estimates to account for confounding effects and establish that root- inference can be achieved as long as at least one nuisance model is correctly specified at each exposure time, despite the presence of high-dimensional covariates. Although the nuisance estimates themselves do not achieve root- rates, the carefully designed loss functions in our framework ensure final root- inference for the causal parameter of interest. Unlike off-the-shelf high-dimensional methods, which fail to deliver robust inference under model misspecification even within the doubly robust framework, our newly developed loss functions address this limitation effectively.

Paper Structure

This paper contains 22 sections, 23 theorems, 343 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Estimation of causal effects under dynamic setups
  3. Existing results
  4. Addressing misspecification: direct bias control
  5. The sequential model doubly robust estimator
  6. Sequential model doubly robust estimation and inference
  7. Theoretical results for the nuisance estimators
  8. Results for misspecified models
  9. Results for correctly specified models
  10. Numerical Experiments
  11. Simulation studies
  12. A semi-synthetic analysis based on the National Job Corps Study (NJCS)
  13. Discussion
  14. Notation
  15. Uniqueness of moment-targeted parameters
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let Assumptions cond:basic-cond:subG hold. Define Then, as $N,d_1,d_2\to\infty$, $\sigma^2:=\mathbb{E}\{\psi(\mathbf{W};\boldsymbol{\eta}^*)-\theta_{1,1}\}^2\asymp\|\boldsymbol{\beta}^*\|_2+1$ and

Figures (2)

  • Figure 1: Causal diagrams for dynamic settings with two exposure occasions.
  • Figure 2: The p-values for the null $H_0: \theta = 0$ as $\hat{\theta}_O$ varies. Since the true $\theta$ is unknown, the x-axis denotes the oracle difference-in-mean estimate of $\theta$.

Theorems & Definitions (43)

  • Remark 1: Correctness of nuisance models
  • Theorem 3.1: Convergence rates
  • Theorem 3.2: Inference under model misspecification
  • Remark 2: Sequential model double robustness
  • Remark 3: Required sparsity conditions under model misspecification
  • Theorem 3.3: Inference under correctly specified models
  • Theorem 4.1
  • Theorem 4.2
  • Lemma S.1: Lemmas D.1 and D.2 of zhang2023double
  • Lemma S.2: Lemma S.4 of bradic2024high
  • ...and 33 more