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Two-Stage Shadow Inclusion Estimation: An IV Approach for Causal Inference under Latent Confounding and Collider Bias
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Proposes Two-Stage Shadow Inclusion (2SSI), an IV-based method that uses the first-stage treatment residual as a shadow variable to jointly correct latent confounding and collider (sample selection) bias.
Problem
Causal effect estimation from observational data when latent confounding and collider bias from non-random sample selection (outcome missing for S=0) occur simultaneously.
Approach
Uses the residual of the treatment regressed on instrument and covariates as both a proxy for latent confounders and a shadow variable, with a learned instrument representation Φ that is associated with selection S but independent of T given Z; the second stage fits the outcome on S=1 data and generalizes to S=0 via odds-ratio weighting.
Key results
- On collider-biased Demand data (alpha=10, beta=10), 2SSI reaches out-of-sample MSE of 0.154e-3 on S=1 and 1.278e-3 on S=0, versus best baseline Kernel IV 0.400e-3 (S=1) and Heckit 6.741e-3 (S=0).
- On Fertil2, 2SSI is the only method that consistently yields positive ATE estimates close to the reference from Wooldridge (2010), while DeepGMM, SHADOW, and DFIV have negative means.
- 2SSI achieves best MSE across all tested collider bias (alpha in {5,10,15}) and latent confounding (beta in {5,10,15}) strengths.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the treatment residual and learned instrument representation in the two-stage method (2SSI) | Demand estimators · Demand, α = 10, β = 10, S = 0 | MSE: 5.233±0.289 -> 1.278±0.155 | Helped |
| Add the treatment residual to the second-stage regression (ablation) | 2SSI · Demand, α = 10, β = 10, S = 0 | MSE: 5.233±0.289 -> 1.278±0.155 | Helped |
| Learn the decomposed representation in the first stage and use it to make the residual a shadow variable (ablation) | 2SSI · Demand, α = 10, β = 10, S = 0 | MSE: 6.673±0.219 -> 1.278±0.155 | Helped |
| Increase collider bias strength α | IV baselines · Demand, S = 0 | MSE | Hurt |
| Increase latent confounding strength β | 2SSI · Demand, α = 10, S = 0 | MSE: 1.995±0.088; 1.278±0.155; 2.655±0.630 | Mixed |
Surprising findings
- Performance at weak latent confounding (beta=5) is worse than at beta=10 for 2SSI, which the authors attribute to weak conditional dependence between the residual and Y.
- CB-IV, a method designed for confounding, sometimes gave estimates near the ground truth on Fertil2 but had the worst overall results.
Limitations
- Requires well-defined instrumental variables, which need expert knowledge.
- Assumes latent confounding is present and that IV analysis is appropriate; testing for latent confounding is not addressed under collider bias.
- Assumes unmeasured confounders do not directly affect sample selection (Assumption 4.1).
Experimental setup
Datasets: Demand dataset (collider-biased semi-synthetic version) Fertil2 (Wooldridge, 2010)
Benchmarks: Demand datasets (low- and high-dimensional) Fertil2 ATE
Metrics: out-of-sample MSE ATE estimate vs reference
Baselines: 2SRI DeepIV Kernel IV DeepGMM DFIV CB-IV Heckit IPSW SHADOW
Builds on: 2SRI DeepIV shadow variable approach (Miao & Tchetgen Tchetgen, 2016) DFIV CB-IV
Closest papers by meaning
- Learning Shadow Variable Representation for Treatment Effect Estimation under Collider Bias (ICML 2024 · similarity 0.89)
- Data Fusion for Partial Identification of Causal Effects (NeurIPS 2025 · similarity 0.87)
- Detecting and Measuring Confounding Using Causal Mechanism Shifts (NeurIPS 2024 · similarity 0.86)
- Conditional Instrumental Variable Regression with Representation Learning for Causal Inference (ICLR 2024 · similarity 0.86)
- Automating the Selection of Proxy Variables of Unmeasured Confounders (ICML 2024 · similarity 0.86)
- Generalizing Causal Effects from Randomized Controlled Trials to Target Populations across Diverse Environments (ICML 2025 · similarity 0.86)
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