Home / Curated / Algorithms / Some Invalid Some Valid Instrumental Variables Estimator (sisVIVE) and 50% Majority-Rule Identifiability

Some Invalid Some Valid Instrumental Variables Estimator (sisVIVE) and 50% Majority-Rule Identifiability

Algorithms AIMech:SomeInvalidSomeValidInstrumentalVariablesEstimatorAndMajorityRuleIdentifiability DRAFT

Aliases: sisVIVE, Some Invalid Some Valid Instrumental Variables Estimator, Kang-Zhang-Cai-Small Invalid Instrument Lasso Estimator, Majority-Rule Mendelian Randomization Instrumental Variables Estimator

An $\ell_1$-penalized instrumental variables methodology and structural identifiability theory (introduced by Hyunseung Kang, Anru Zhang, T. Tony Cai, and Dylan S. Small in 2016) that estimates the causal effect $\beta^*$ of an endogenous exposure $D_i$ on an outcome $Y_i$ in the presence of potentially invalid instruments $\mathbf{Z}_{i.} \in \mathbb{R}^L$ without prior knowledge of which instruments are valid. Under the Additive Linear, Constant Effects (ALICE) potential outcomes model, the observed outcome satisfies $Y_i = \mathbf{Z}_{i.}^T\boldsymbol{\alpha}^* + D_i\beta^* + \epsilon_i$ with $\mathbb{E}(\epsilon_i | \mathbf{Z}_{i.}) = 0$ (Equation 3), where $\boldsymbol{\alpha}^* = \boldsymbol{\phi}^* + \boldsymbol{\psi}^*$ aggregates direct pleiotropic effects ($\boldsymbol{\phi}^*$) and unmeasured confounder associations ($\boldsymbol{\psi}^*$), so instrument $j \in \{1, \dots, L\}$ is valid if and only if $\alpha_j^* = 0$ (Definition 1, p. 10). Given the reduced-form moment equations $\boldsymbol{\Gamma}^* = \boldsymbol{\alpha}^* + \boldsymbol{\gamma}^*\beta^*$ (Equation 7) and a sparsity bound $s = \|\boldsymbol{\alpha}^*\|_0 < U$, Theorem 1 (p. 13) and Corollary 1 (p. 14) establish that $(\boldsymbol{\alpha}^*, \beta^*)$ is uniquely identified without knowing the valid subset whenever $U \le L/2$ (the 50% majority rule: strictly more than half of the instruments are valid). The sisVIVE estimator (Equation 10, p. 15) solves the convex program $(\hat{\boldsymbol{\alpha}}_\lambda, \hat{\beta}_\lambda) \in \arg\min_{\boldsymbol{\alpha}, \beta} \frac{1}{2}\|\mathbf{P}_{\mathbf{Z}}(\mathbf{Y} - \mathbf{Z}\boldsymbol{\alpha} - \mathbf{D}\beta)\|_2^2 + \lambda\|\boldsymbol{\alpha}\|_1$ where $\mathbf{P}_{\mathbf{Z}} = \mathbf{Z}(\mathbf{Z}^T\mathbf{Z})^{-1}\mathbf{Z}^T$, which reduces algebraically to a two-step Lasso on $\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z}$ followed by linear projection onto $\hat{\mathbf{D}} = \mathbf{P}_{\mathbf{Z}}\mathbf{D}$ (Theorem 3, Equation 17, p. 21) and is solvable along the entire regularization path in $O(nL^2)$ via LARS.

Identity Notes & Concept Scope Resolved against all 442 pinned AIO concepts and 666 maintained AIMech concepts. Distinct from `AIMech:SelfTuningPivotalHighDimensionalInstrumentalVariablesAndSensitivityConfidenceSets` (Gautier & Rose 2011/2021, which solves a self-normalized conic program for high-dimensional endogenous structural parameters $\beta \in \mathbb{R}^{d_X}$ and scale $\sigma$ rather than proving the Theorem 1 / Corollary 1 50% majority-rule combinatorial identifiability criterion on $\boldsymbol{\Gamma}^* = \boldsymbol{\alpha}^* + \boldsymbol{\gamma}^*\beta^*$ and the Theorem 3 exact two-step orthogonal-projection Lasso reduction $\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z}$ for a scalar exposure with invalid instruments), distinct from `AIMech:SelfNormalizedLassoOptimalInstrumentEstimationAndSupNormWeakIdentificationTesting` (Belloni, Chen, Chernozhukov, & Hansen 2012, which assumes all instruments satisfy the exclusion restriction $\mathbb{E}[\epsilon_i | Z_i] = 0$ and uses first-stage Lasso to select strong instruments from a large valid set rather than penalizing structural exclusion-restriction violations $\boldsymbol{\alpha}^*$), and distinct from `AIMech:PenalizedFocusedGeneralizedMethodOfMomentsForHighDimensionalEndogenousRegression` (Fan & Liao 2014, which estimates high-dimensional sparse structural regressors under valid over-identifying moments via nonconcave penalized Focused GMM).

Claims & Grounded Evidence

3 Claims

Mechanism #combinatorial-consistency-criterion-and-fifty-percent-majority-rule

Given reduced-form moments $\boldsymbol{\Gamma}^* = E(\mathbf{Z}^T\mathbf{Z})^{-1}E(\mathbf{Z}^T\mathbf{Y}) = \boldsymbol{\alpha}^* + \boldsymbol{\gamma}^*\beta^*$ and $\boldsymbol{\gamma}^* = E(\mathbf{Z}^T\mathbf{Z})^{-1}E(\mathbf{Z}^T\mathbf{D})$ with $\gamma_j^* \ne 0$ ($j = 1, \dots, L$) and at most $s = \|\boldsymbol{\alpha}^*\|_0 < U$ invalid instruments (Equation 7), Theorem 1 (Section 3.1, p. 13) proves that $(\boldsymbol{\alpha}^*, \beta^*)$ has a unique solution in $\mathcal{C}_U = \{(\boldsymbol{\alpha}, \beta) : \|\boldsymbol{\alpha}\|_0 < U\}$ if and only if there do not exist two subsets $C_m, C_{m'} \subseteq \{1, \dots, L\}$ of size $|C_m| = |C_{m'}| = L - U + 1$ with constants $q_m \ne q_{m'}$ such that $\gamma_j^* q_m = \Gamma_j^*$ for $j \in C_m$ and $\gamma_j^* q_{m'} = \Gamma_j^*$ for $j \in C_{m'}$. By pigeonhole overlap $|C_m \cap C_{m'}| \ge 2(L - U + 1) - L = L - 2U + 2 \ge 2$ whenever $U \le L/2$, Corollary 1 (p. 14) establishes that $s < U \le L/2$ is a universal sufficient condition for unique identifiability of $(\boldsymbol{\alpha}^*, \beta^*)$ without knowing which instruments are valid.

supports Source: Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization · Locator: Section 3.1, pp. 10-15, Equations (7)-(9), Theorem 1, Corollary 1; Appendix E.1-E.2, pp. 61-63

Mechanism #two-step-orthogonal-projection-lasso-reduction-and-lars-homotopy

Theorem 3 (Section 3.4, p. 21, Equation 17) proves that minimizing the joint sisVIVE criterion $\frac{1}{2}\|\mathbf{P}_{\mathbf{Z}}(\mathbf{Y} - \mathbf{Z}\boldsymbol{\alpha} - \mathbf{D}\beta)\|_2^2 + \lambda\|\boldsymbol{\alpha}\|_1$ (Equation 10) is algebraically equivalent to the two-step estimator: Step 1 solves the standard Lasso problem $\hat{\boldsymbol{\alpha}}_\lambda \in \arg\min_{\boldsymbol{\alpha} \in \mathbb{R}^L} \frac{1}{2}\|\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{P}_{\mathbf{Z}}\mathbf{Y} - \mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z}\boldsymbol{\alpha}\|_2^2 + \lambda\|\boldsymbol{\alpha}\|_1$, and Step 2 computes $\hat{\beta}_\lambda = \frac{\hat{\mathbf{D}}^T(\mathbf{Y} - \mathbf{Z}\hat{\boldsymbol{\alpha}}_\lambda)}{\|\hat{\mathbf{D}}\|_2^2}$ where $\hat{\mathbf{D}} = \mathbf{P}_{\mathbf{Z}}\mathbf{D}$ and $\mathbf{P}_{\hat{\mathbf{D}}^\perp} = \mathbf{I} - \hat{\mathbf{D}}\hat{\mathbf{D}}^T / \|\hat{\mathbf{D}}\|_2^2$, enabling the entire regularization path $(\hat{\boldsymbol{\alpha}}_\lambda, \hat{\beta}_\lambda)$ for all $\lambda > 0$ to be computed in $O(nL^2)$ via LARS.

supports Source: Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization · Locator: Section 3.2, p. 15, Equations (10)-(11); Section 3.4, p. 21, Theorem 3, Equation (17)

Scope #nonasymptotic-causal-effect-bounds-under-restricted-isometry-and-mutual-incoherence

Let $\delta_{2s}^-(\mathbf{Z}), \delta_{2s}^+(\mathbf{Z})$ and $\delta_{2s}^+(\mathbf{P}_{\hat{\mathbf{D}}}\mathbf{Z})$ denote the restricted isometry constants of $\mathbf{Z}$ and $\mathbf{P}_{\hat{\mathbf{D}}}\mathbf{Z}$ (Equation 12). Under the condition $2\delta_{2s}^-(\mathbf{Z}) > \delta_{2s}^+(\mathbf{Z}) + 2\delta_{2s}^+(\mathbf{P}_{\hat{\mathbf{D}}}\mathbf{Z})$ and regularization $\lambda \ge 3\|\mathbf{Z}^T\mathbf{P}_{\hat{\mathbf{D}}^\perp}\boldsymbol{\epsilon}\|_\infty$, Theorem 2 (Section 3.3, pp. 18–19, Equation 13) establishes the non-asymptotic causal-effect error bound $|\hat{\beta}_\lambda - \beta^*| \le \frac{|\hat{\mathbf{D}}^T\boldsymbol{\epsilon}|}{\|\hat{\mathbf{D}}\|_2^2} + \frac{1}{\|\hat{\mathbf{D}}\|_2}\frac{(4/3\sqrt{5})\lambda\sqrt{s\delta_{2s}^+(\mathbf{P}_{\hat{\mathbf{D}}}\mathbf{Z})}}{2\delta_{2s}^-(\mathbf{Z}) - \delta_{2s}^+(\mathbf{Z}) - 2\delta_{2s}^+(\mathbf{P}_{\hat{\mathbf{D}}}\mathbf{Z})}$. Under pairwise instrument coherence $\mu = \max_{i \ne j}|\mathbf{Z}_{.i}^T\mathbf{Z}_{.j}| / (\|\mathbf{Z}_{.i}\|_2\|\mathbf{Z}_{.j}\|_2)$ and instrument-strength balancing $\rho = \max_j \|\hat{\mathbf{D}}^T\mathbf{Z}_{.j}\|_2 / (\|\hat{\mathbf{D}}\|_2\|\mathbf{Z}_{.j}\|_2)$ (Equation 14) with $s < \min(\frac{1}{12\mu}, \frac{1}{10\rho^2})$ (Equation 15), Corollary 2 (pp. 19–20, Equation 16) proves $|\hat{\beta}_\lambda - \beta^*| \le \frac{|\hat{\mathbf{D}}^T\boldsymbol{\epsilon}|}{\|\hat{\mathbf{D}}\|_2^2} + \frac{1}{\|\hat{\mathbf{D}}\|_2}\frac{(4\sqrt{105}/9)\lambda s\rho}{1 - s(5\rho^2 + 6\mu)}$.

supports Source: Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization · Locator: Section 3.3, pp. 16-20, Equations (12)-(16), Theorem 2, Corollary 2; Appendix E.3-E.4, pp. 64-72

⚠️ Curated Limitations & Negative Results

Without prior domain knowledge of which instruments are valid, unique identifiability of $(\boldsymbol{\alpha}^*, \beta^*)$ breaks down when $50\%$ or more of the candidate instruments are invalid ($s \ge L/2$) and the invalid instruments' direct effects $\alpha_j^*$ align proportionally with their first-stage strengths $\gamma_j^*$ to form a competing cluster of size $\ge L - U + 1$ (Theorem 1, Section 3.1, p. 13).
The finite-sample $\ell_2$ performance guarantees in Theorem 2 and Corollary 2 require the projected matrix $\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z}$ to satisfy restricted isometry ($2\delta_{2s}^-(\mathbf{Z}) > \delta_{2s}^+(\mathbf{Z}) + 2\delta_{2s}^+(\mathbf{P}_{\hat{\mathbf{D}}}\mathbf{Z})$) or mutual incoherence $s < \min(1/(12\mu), 1/(10\rho^2))$, which constrains both pairwise instrument correlation $\mu$ and the dominance $\rho$ of any single instrument in $\hat{\mathbf{D}}$.
Primary-source typographical audit of `arXiv:1401.5755v3`: (1) Section 3.1 line 1 (p. 12) writes '$\boldsymbol{\Gamma}^* = E(\mathbf{Z}^T\mathbf{Y})E(\mathbf{Z}^T\mathbf{Z})^{-1}$' and '$\boldsymbol{\gamma}^* = E(\mathbf{Z}^T\mathbf{D})E(\mathbf{Z}^T\mathbf{Z})^{-1}$' placing the $L \times 1$ column vector on the left of the $L \times L$ inverse matrix instead of '$E(\mathbf{Z}^T\mathbf{Z})^{-1}E(\mathbf{Z}^T\mathbf{Y})$'; (2) Appendix B.4 lines 10 and 13 (p. 48) write '$k \in \operatorname{supp}(\boldsymbol{\alpha}^*)^C$ (i.e. set of invalid instruments)' instead of '(i.e. set of valid instruments)' and '$j$ now belongs to $\operatorname{supp}(\boldsymbol{\alpha}^*)^C$ and $k$ belongs to $\operatorname{supp}(\boldsymbol{\alpha}^*)^C$' (repeating '$^C$'); (3) Appendix B.4 paragraph 2 line 3 (p. 51) spells 'sisSIVE\'s performance' instead of 'sisVIVE\'s performance'; (4) Appendix E.1 line 5 (p. 62) writes 'let $A^{(1)}$ and $^{(2)}$ be any pair' omitting '$A$' before '$^{(2)}$'; (5) Appendix E.2 Proof of Corollary 1 line 6 (p. 63) writes 'whenever $U \ge L/2$' with reversed inequality instead of 'whenever $U \le L/2$'; (6) Appendix E.3 line 13 (p. 67) writes '$2\delta_{2s}^-(\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z}) > \delta_{2s}^-(\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z})$' with a minus superscript on the right-hand side instead of '$\delta_{2s}^+(\mathbf{P}_{\hat{\mathbf{D}}^\perp}\mathbf{Z})$'; and (7) Appendix Figure 28 caption line 2 (p. 98) writes 'There are ten ($L = 100$) instruments'.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:LassoRegression` Lasso Regression	related_match	Penalizes the l1 norm of direct instrument-outcome effects alpha in a two-step orthogonal-projection Lasso program.
`AIO:LinearRegression` Linear Regression	related_match	Estimates linear structural causal effect beta* of an endogenous exposure D on outcome Y using projected linear regression.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`kzcs2016-sisvive-pdf`	Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization Hyunseung Kang, Anru Zhang, T. Tony Cai, Dylan S. Small	primary paper full_text	ID: https://doi.org/10.1080/01621459.2014.994705 sha256:74945ff92ecb…
`kzcs2016-sisvive-abs`	Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization Hyunseung Kang, Anru Zhang, T. Tony Cai, Dylan S. Small	primary paper abstract_only	ID: https://arxiv.org/abs/1401.5755 sha256:a1e29bfe75fc…
`kzcs2016-sisvive-crossref`	Instrumental Variables Estimation with Some Invalid Instruments and its Application to Mendelian Randomization Hyunseung Kang, Anru Zhang, T. Tony Cai, Dylan S. Small	primary paper abstract_only	ID: https://doi.org/10.1080/01621459.2014.994705 sha256:20df538012f7…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini

Created: 2026-10-19T04:57:00Z

Updated: 2026-10-19T04:57:00Z

Source File: data/curated/SomeInvalidSomeValidInstrumentalVariablesEstimatorAndMajorityRuleIdentifiability.yaml · Schema: curated_concept.yaml

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