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Statistical Inference in Causal Partial Identification with Smooth Densities

Sirui Lin, Zijun Gao, Jose Blanchet, Peter Glynn

Abstract

Many causal quantities are only partially identifiable due to the inherent missingness of potential outcomes, and the associated partial identification (PI) sets can be obtained by solving an optimal transport (OT) problem. Covariates often provide additional information about the potential outcomes and thus yield tighter PI sets, which can be obtained via conditional optimal transport (COT). However, COT-based PI set estimators are susceptible to the curse of dimensionality in the covariates and outcomes, which precludes the asymptotic normality and hinders statistical inference. In this paper, we exploit smoothness in the marginal densities of covariates and potential outcomes and develop a wavelet-based primal method for COT with multivariate outcomes and covariates. Moreover, for quadratic cost functions, we establish a stability result for COT and prove asymptotic normality of the proposed estimator. This characterization of the asymptotic distribution enables valid statistical inference for the partial identification set. Empirically, we validate the estimation and inference performance of our approach through numerical experiments in comparison with existing benchmarks.

Statistical Inference in Causal Partial Identification with Smooth Densities

Abstract

Many causal quantities are only partially identifiable due to the inherent missingness of potential outcomes, and the associated partial identification (PI) sets can be obtained by solving an optimal transport (OT) problem. Covariates often provide additional information about the potential outcomes and thus yield tighter PI sets, which can be obtained via conditional optimal transport (COT). However, COT-based PI set estimators are susceptible to the curse of dimensionality in the covariates and outcomes, which precludes the asymptotic normality and hinders statistical inference. In this paper, we exploit smoothness in the marginal densities of covariates and potential outcomes and develop a wavelet-based primal method for COT with multivariate outcomes and covariates. Moreover, for quadratic cost functions, we establish a stability result for COT and prove asymptotic normality of the proposed estimator. This characterization of the asymptotic distribution enables valid statistical inference for the partial identification set. Empirically, we validate the estimation and inference performance of our approach through numerical experiments in comparison with existing benchmarks.
Paper Structure (53 sections, 19 theorems, 134 equations, 21 figures, 2 tables, 2 algorithms)

This paper contains 53 sections, 19 theorems, 134 equations, 21 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Organization.
  3. Notation.
  4. Preliminary
  5. Potential Outcome Model and Conditional Optimal Transport
  6. Curse of Dimensionality and Wavelet Density Estimator
  7. Assumption
  8. Main Result
  9. Wavelet-based COT Estimator
  10. Fast Convergence Rate with General Objective
  11. Statistical Inference with Quadratic Objective
  12. Numerical Experiment
  13. Estimation
  14. Data generation mechanism.
  15. Implementation.
  16. ...and 38 more sections

Key Result

Proposition 2.1

Suppose that $\widehat{\mu}, \mu \in \mathcal{P}_{\textup{ac}}([0,1]^d)$ with densities $\hat{f}, f \in L^2([0,1]^d)$. If $f(x) \geq \gamma^{-1}\,\forall x\in[0,1]^d$ for some $\gamma > 0$, then there is a universal constant $C_{0}>0$ such that

Figures (21)

  • Figure 1: Location model
  • Figure 2: Quadratic model
  • Figure 3: Scale model
  • Figure 5: Location model
  • Figure 6: Quadratic model
  • ...and 16 more figures

Theorems & Definitions (43)

  • Definition 2.1: Conditional optimal transport
  • Definition 2.2: Besov space
  • Proposition 2.1: niles2022minimax
  • Definition 2.3: Wavelet density estimator
  • Definition 2.4: Hölder space
  • Proposition 2.2: c.f.,manole2024plugin
  • Definition 3.1: Wavelet-based COT estimator
  • Proposition 3.1: Stability
  • Proposition 3.2: Key estimation bound
  • Theorem 3.1: Fast convergence of COT under smooth densities
  • ...and 33 more