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Causal Effect Identification in lvLiNGAM from Higher-Order Cumulants
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Proves that causal effects in lvLiNGAM are identifiable from higher-order cumulants with a single proxy variable (even when the proxy causes the treatment) or a single instrument shared by multiple treatments, and gives estimators.
Problem
Causal effect identification and estimation in latent variable LiNGAM when a single proxy variable may causally affect the treatment, or when instrumental variables are underspecified (fewer instruments than treatments), settings where covariance-based and existing higher-order methods fail.
Approach
Uses higher-order cumulant polynomial equations (roots of a degree l+1 polynomial plus a linear cumulant system) applied to pairs of variables, combined with subtracting proxy/instrument contributions to remove edges, to identify treatment effects with one proxy or one instrument.
Key results
- Proxy with edge from proxy to treatment: the proposed Cumulant method is consistent where Cross-Moment and GRICA fail; relative error decays with sample size.
- Underspecified IV (two treatments, one instrument): the Cumulant method is the only one that consistently estimates both effects among compared methods.
- Real data: estimates of 2.68 and 2.71 under G1 and G2 versus 8.26 under G3.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Cumulant instead of Cross-Moment | Proxy-variable setting, G1 · G1 | relative absolute error | Hurt |
| Use Cumulant instead of competing methods | Proxy-variable setting with multiple latent variables or an edge from Z to T · G2 and G3 | relative absolute error | Helped |
| Add the minimization technique to Cumulant (ablation) | Cumulant on the proxy-variable graph G3 · G3 | relative absolute error | Helped |
| Use Cross-Moment or GRICA instead of a consistent estimator | Proxy-variable settings with multiple latent variables or an edge from Z to T · G2 and G3 | relative absolute error | Hurt |
| Use ReLVLiNGAM instead of Cumulant | Proxy-variable setting · G1 and G3 | relative absolute error | Hurt |
| Use Cumulant instead of Min Norm, GRICA, or ReLVLiNGAM | Underspecified instrumental-variable setting with two treatments and one instrument · Underspecified instrumental-variable graph | average relative absolute error | Helped |
| Use GRICA instead of Cumulant at small sample sizes | Underspecified instrumental-variable setting · Underspecified instrumental-variable graph, small samples | average relative absolute error | Helped |
Surprising findings
- Second- and third-order cumulants alone do not suffice to identify the effect in the single-latent proxy graph with a proxy-to-treatment edge (Gröbner basis argument).
- GRICA outperforms the cumulant method at small sample sizes, attributed to high variance of k-statistics.
Limitations
- Cumulant estimators based on fourth-order (or higher) k-statistics have high variance at small sample sizes.
- Real-data estimate under the G3 graph (8.26) deviates substantially from literature estimates, with uncertainty not assessed.
Experimental setup
Datasets: synthetic lvLiNGAM simulations New Jersey fast-food minimum wage dataset (Card and Krueger)
Benchmarks: synthetic lvLiNGAM graphs G1-G3 underspecified IV graph New Jersey minimum wage data
Metrics: relative absolute error of causal effect median over 100 simulations interquartile range
Baselines: Cross-Moment GRICA ReLVLiNGAM Min Norm TSLS projection
Builds on: LiNGAM Cross-Moment method GRICA two-stage least squares ReLVLiNGAM
Closest papers by meaning
- Causal Effect Identification with a Single Agnostic Proxy (NeurIPS 2026 · similarity 0.93)
- Causal Effect Identification in LiNGAM Models with Latent Confounders (ICML 2024 · similarity 0.92)
- Identification of Causal Structure with Latent Variables Based on Higher Order Cumulants (AAAI 2024 · similarity 0.90)
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- On the Parameter Identifiability of Partially Observed Linear Causal Models (NeurIPS 2024 · similarity 0.88)
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