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PCM Selector: Penalized Covariate-Mediator Selection Operator for Evaluating Linear Causal Effects
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Proposes PCM Selector, a two-stage L1/L2-penalized regression that estimates linear total causal effects using covariates and intermediate variables (front-door-like criterion), with penalties that select intermediate variables and covariates for lower-variance estimates.
Problem
Estimating total causal effects in linear structural causal models when back-door covariates are unobserved or when covariates/intermediate variables cause multicollinearity or high-dimensional problems for OLS.
Approach
A two-stage penalized regression that does not penalize prior-knowledge covariates and mediators, penalizes the remaining candidate covariates and intermediate variables with adaptive weights, and then debiases with OLS/ridge re-estimation on active sets, generalizing PAL_pMA to include intermediate variables.
Key results
- In Setting (a), PCM Selector had bias -0.010 for true total effect 0.045, versus -0.033 for LASSO and 0.009 for PAL_1MA (which had larger SD 0.792 vs 0.718 for PCM Selector).
- In Setting (b), PCM Selector had bias 0.046 (SD 0.549) and sign-coincidence 0.808, compared with 0.066 bias for minimal front-door estimator.
- LASSO, adaptive LASSO, and Elastic Net had low sign-coincidence rates (0.117-0.156) in Setting (a).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PCM Selector instead of PAL1MA | Total-effect estimation · Setting (a) | SD: 0.792 -> 0.718 | Helped |
| Use PCM Selector instead of back-door estimation | Total-effect estimation · Setting (a) | SD: 1.591 -> 0.718 | Helped |
| Use PCM Selector instead of front-door-like estimation including X | Total-effect estimation · Setting (a) | SD: 1.577 -> 0.718 | Helped |
| Use PCM Selector instead of front-door-like estimation without X | Total-effect estimation · Setting (a) | SD: 1.051 -> 0.718 | Helped |
| Use PCM Selector instead of LASSO | Total-effect estimation · Setting (a) | Sign: 0.117 -> 0.526 | Helped |
| Use PCM Selector instead of adaptive LASSO | Total-effect estimation · Setting (a) | Sign: 0.138 -> 0.526 | Helped |
| Use PCM Selector instead of Elastic Net | Total-effect estimation · Setting (a) | Sign: 0.156 -> 0.526 | Helped |
| Use PCM Selector instead of the minimally sufficient front-door set | Total-effect estimation · Setting (b) | SD: 0.552 -> 0.549 | Helped |
| Use PCM Selector instead of the whole front-door set | Total-effect estimation · Setting (b) | SD: 0.692 -> 0.549 | Helped |
Surprising findings
- Standard penalized regressions (LASSO, adaptive LASSO, Elastic Net) yield sign-coincidence rates as low as 0.117 for the total effect, suggesting misleading conclusions about treatment effect direction.
Limitations
- Formulated only for linear (single/joint) regression models; extension to GLMs left to future work.
- Evaluation uses simulated linear SCM data with a small set of settings; the case study is only in the technical appendix.
- Estimator is biased, and the theoretical variance guarantees hold only approximately under normality and optimal tuning.
Experimental setup
Datasets: simulated linear SCM data
Benchmarks: simulated linear SCM settings (a) and (b)
Metrics: mean standard deviation bias sign coincidence rate
Baselines: LASSO adaptive LASSO Elastic Net PAL_1MA OLS TSLS front-door adjustment
Builds on: PAL_pMA adaptive LASSO front-door criterion back-door criterion elastic net
Closest papers by meaning
- Local Learning for Covariate Selection in Nonparametric Causal Effect Estimation with Latent Variables (NeurIPS 2025 · similarity 0.87)
- Data-Driven Covariate Selection for Nonparametric and Cycle-Agnostic Causal Effect Estimation (NeurIPS 2026 · similarity 0.86)
- Latent Variable Causal Discovery under Selection Bias (ICML 2025 · similarity 0.85)
- Data-Driven Selection of Instrumental Variables for Additive Nonlinear, Constant Effects Models (ICML 2025 · similarity 0.85)
- Automating the Selection of Proxy Variables of Unmeasured Confounders (ICML 2024 · similarity 0.85)
- Score-Based Causal Discovery of Latent Variable Causal Models (ICML 2024 · similarity 0.84)
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