Quasi Instrumental Variable (QIV)

• QIV is an instrument-like variable that relaxes classical IV criteria by allowing structured violations of exogeneity or exclusion.
• Different QIV frameworks redistribute traditional IV requirements through complementary quasi-IVs, MQIV with stable direct effects, and prognostic QIVs for binary outcomes.
• QIV methods enhance causal inference in settings with weak or questionable instruments by offering robust alternatives to standard IV approaches.

A quasi instrumental variable (QIV) is an instrument-like variable that does not satisfy all standard instrumental-variable conditions, although the exact meaning depends on the framework in which it is defined. In "Identification with possibly invalid IVs," a quasi-IV is “a relevant but possibly invalid IV because it is not exogenous or not excluded” (Bruneel-Zupanc et al., 2024). In "The Multiplicative Quasi-Instrumental Variable Model," a valid QIV retains relevance and independence but may violate exclusion through a stable direct effect (Liu et al., 5 May 2026). In "Quasi Instrumental Variable Methods for Stable Hidden Confounding and Binary Outcome," a QIV is instead “a variable that is only assumed to be predictive of the outcome,” more precisely predictive of $Y$ among untreated individuals (Liu et al., 22 Aug 2025). The surveyed literature therefore treats QIV less as a single canonical object than as a family of instrument-relaxation strategies, all motivated by the difficulty of finding variables that are simultaneously relevant, excluded, and exogenous.

1. Conceptual scope and relation to standard IV

A standard instrumental variable combines two core properties beyond relevance: exogeneity and exclusion. In the 2024 QIV framework, those two properties are explicitly split across two complementary variables rather than imposed on a single one. In the 2026 MQIV framework, exogeneity is retained while exclusion is relaxed. In the 2025 binary-outcome QIV framework, neither classical exclusion nor classical independence is required; instead, identification is shifted to stability restrictions on confounding and treatment effects. This suggests that the term QIV is best understood as denoting a structured departure from the classical IV template rather than a single universal definition.

Framework	Meaning of QIV	Core identifying structure
Complementary quasi-IVs (Bruneel-Zupanc et al., 2024)	A relevant but possibly invalid IV because it is not exogenous or not excluded	$Z$: excluded but possibly endogenous; $W$: exogenous conditional on $Z$, but possibly included
MQIV (Liu et al., 5 May 2026)	Instrument may violate exclusion through a stable direct effect	Relevance, independence, latent exchangeability, stable direct effect, multiplicative treatment model
Stable hidden confounding QIV (Liu et al., 22 Aug 2025)	Variable only assumed predictive of outcome among the untreated	Relevance to untreated outcome, multiplicative parallel trends, no current treatment value interaction

This heterogeneity matters substantively. A common misconception is to treat QIV as a synonym for “approximately valid IV.” The named QIV papers do not support that simplification. One formulation redistributes exclusion and exogeneity across two variables; one permits a single excluded violation of a specific form; one replaces classical IV logic by prognostic relevance plus hidden-confounding stability. The term is therefore descriptive only when tied to a particular structural model.

2. Complementary quasi-IVs and identification with two imperfect variables

The 2024 paper proposes a general identification strategy based on two complementary quasi-IVs: $Z$, an excluded but possibly endogenous quasi-IV, and $W$, an exogenous conditional on $Z$ but possibly included quasi-IV (Bruneel-Zupanc et al., 2024). The baseline structure is a nonseparable triangular model

$Y = h(D,W,U),$

with the conditional exogeneity restriction

$U \perp W \mid Z.$

At the same time, $Z$ is excluded from the outcome equation. Neither variable is a valid IV alone. $Z$0 fails because it may be endogenous; $Z$1 fails because it may directly affect $Z$2. Identification comes from their joint variation.

The paper’s central mechanism is “local irrelevance.” If for some $Z$3,

$Z$4

then changing $Z$5 from $Z$6 to $Z$7 does not change treatment selection at $Z$8. Any observed change in outcomes can therefore be attributed to the direct effect of $Z$9, not to treatment-selection effects. Once that direct effect is identified, reduced-form contrasts can be purged and structural treatment effects recovered. This logic is developed for several model classes, including quantile models with rank invariance, additive models with homogeneous treatment effects, local average treatment effect models, marginal treatment effect models, and models with discrete or continuous endogenous treatment.

For LATE-type analysis, the generalized local average treatment effect is

$W$0

and the MTE is

$W$1

For discrete treatment, identification is obtained from a nonlinear system whose Jacobian has the form

$W$2

with full column rank of $W$3 as the relevance condition. For continuous treatment, identification again proceeds through a conditional moment system generated jointly by $W$4.

One of the paper’s main interpretive contributions is its treatment of difference-in-differences as a QIV design. Time plays the role of an exogenous but included quasi-IV; group assignment plays the role of an excluded but possibly endogenous quasi-IV. The framework therefore does not merely weaken IV assumptions; it redistributes them. Identification no longer requires one variable that is both excluded and exogenous. It requires a pair whose roles are complementary.

3. The MQIV model: a single QIV with stable direct effect

The MQIV framework studies observed data $W$5, where $W$6 is treatment, $W$7 is outcome, $W$8 is the candidate quasi-instrument, $W$9 are measured covariates, and $Z$0 is an unmeasured confounder (Liu et al., 5 May 2026). A valid QIV satisfies relevance,

$Z$1

independence,

$Z$2

latent exchangeability,

$Z$3

and stable direct effect,

$Z$4

The QIV is therefore not required to satisfy the standard exclusion restriction

$Z$5

A direct effect of $Z$6 on $Z$7 is allowed, provided it may depend on $Z$8 but not on $Z$9 or $Z$0.

The defining treatment-selection restriction is multiplicative: $Z$1 with $Z$2. The target estimand is the average treatment effect on the treated,

$Z$3

Under Assumptions 1–5, Theorem 1 shows that the conditional ATT is identified by a modified Wald ratio,

$Z$4

where

$Z$5

Equivalently,

$Z$6

The correction term $Z$7 subtracts the direct $Z$8 effect from the reduced form.

The paper derives an efficient influence function and a semiparametric efficiency bound, and proposes a cross-fitted EIF estimator using DDML. Its moment equation is multiply robust under the union of three nuisance-model collections: $Z$9

$W$0

$W$1

Under boundedness, consistency of nuisances, and cross-product remainder conditions,

$W$2

with $W$3.

Empirically, the framework is designed for settings in which exclusion is doubtful but multiplicative treatment selection is substantively plausible. In simulations with exclusion restriction violation and $W$4, the proposed IF-based estimator $W$5 had bias $W$6 and coverage $W$7, whereas standard Wald and single-arm Wald had large non-diminishing bias and zero coverage. In the fertility–labor supply application, the MQIV estimator gave

$W$8

with estimated average direct effect among treated

$W$9

4. Binary outcomes, stable hidden confounding, and prognostic QIVs

The 2025 binary-outcome QIV paper adopts a more radical departure from classical IV logic (Liu et al., 22 Aug 2025). Here $Z$0 with $Z$1, and the QIV is defined operationally by relevance to the untreated outcome: $Z$2 This is not relevance to treatment. The QIV need not satisfy exclusion restriction, need not be independent of unmeasured confounders, and need not even have classical IV relevance.

The target is the marginal ATT

$Z$3

with conditional versions

$Z$4

and multiplicative confounding bias

$Z$5

The paper’s key restrictions are multiplicative parallel trends,

$Z$6

and no current treatment value interaction,

$Z$7

Under these conditions,

$Z$8

The main identification theorem states that if relevance, multiplicative parallel trends, and no current treatment value interaction hold, then

$Z$9

and

$Y = h(D,W,U),$0

The marginal ATT is then

$Y = h(D,W,U),$1

Under the causal null $Y = h(D,W,U),$2 for all $Y = h(D,W,U),$3, only QIV relevance and multiplicative parallel trends are needed for a valid null test, since no-current-treatment-value interaction is automatic under the null.

Because $Y = h(D,W,U),$4 is binary, the paper introduces a generalized odds product nuisance parameter,

$Y = h(D,W,U),$5

and proves that

$Y = h(D,W,U),$6

is a diffeomorphism. This supports both a likelihood-based estimator and a triply robust semiparametric locally efficient estimator. The triply robust estimator is CAN in the union model

$Y = h(D,W,U),$7

where $Y = h(D,W,U),$8 specifies $Y = h(D,W,U),$9, $U \perp W \mid Z.$0, and $U \perp W \mid Z.$1; $U \perp W \mid Z.$2 specifies $U \perp W \mid Z.$3 and $U \perp W \mid Z.$4; and $U \perp W \mid Z.$5 specifies $U \perp W \mid Z.$6, $U \perp W \mid Z.$7, and $U \perp W \mid Z.$8.

In simulation, with true marginal ATT $U \perp W \mid Z.$9, the triply robust estimator was consistent across all four reported scenarios: all models correct, only $Z$0 correct, only $Z$1 correct, and only $Z$2 correct. In the UK Biobank application on overweight and hypertension, QIV methods applied to three SNPs yielded ATT estimates below the fully adjusted g-formula estimate, and estimated marginal $Z$3 values were close to $Z$4, interpreted as modest upward confounding.

5. Related constructions: latent proxy instruments, semi-instruments, screening, and bounded invalidity

Several adjacent literatures develop objects that are QIV-like in function even when they are not labeled QIV. "Ivy: Instrumental Variable Synthesis for Causal Inference" does not use the term “quasi instrumental variable,” but it explicitly synthesizes a summary IV as a latent variable from many candidate instruments $Z$5 (Kuang et al., 2020). The latent-variable model is

$Z$6

and the synthesized instrument is

$Z$7

The paper assumes, among other conditions, that a majority of IV candidates are valid and that invalid candidates satisfy $Z$8. Its output is not merely a weighted score but a probabilistically synthesized latent IV that can be plugged into a downstream estimator such as the Wald ratio. The paper itself frames this as latent valid-IV recovery rather than quasi-IV inference, but the interpretive connection is strong because $Z$9 is a constructed proxy instrument from weak, correlated, or invalid candidates. On three UK Biobank tasks known to be noncausal, Ivy produced median effect sizes $Z$00, whereas allele scores gave median effect sizes $Z$01.

The paper "Semi-Instrumental Variables: A Test for Instrument Admissibility" formalizes another neighboring concept (Chu et al., 2013). A semi-instrument is a variable $Z$02 that satisfies the additive-model analog of IV relevance and exogeneity, but may directly affect $Z$03 provided that direct effect is a linear function of the direct effect on $Z$04: $Z$05 Every instrument is a semi-instrument with linear coefficient $Z$06. Theorem 1 characterizes semi-instrumentality through additivity of $Z$07 and the variance restriction

$Z$08

The paper then shows that, under additional prior assumptions, if two independent semi-instruments have the same linear coefficient, then with probability $Z$09 they are both instruments. This is not a QIV formulation in name, but it is a formally defined relaxation of exclusion and therefore QIV-like in spirit.

"Necessary and Probably Sufficient Test for Finding Valid Instrumental Variables" addresses a different but related problem: how to rank or screen candidate instruments when validity is uncertain (Sharma, 2018). It does not define QIV, but it proposes the NPS test, combining a necessary test with a Bayesian comparison between valid-IV and invalid-IV model classes. Its central score is the Validity Ratio,

$Z$10

The method can reject candidate instruments that fail necessary constraints and can compare candidates by a marginal-likelihood-based validity score. In a QIV reading, this functions as a validation layer for variables whose validity is uncertain rather than assumed.

"Possibilistic Instrumental Variable Regression" moves further toward bounded-invalidity sensitivity analysis (Steiner et al., 20 Nov 2025). In the linear structural model

$Z$11

validity is represented by $Z$12, and approximate validity is encoded by a user-specified set

$Z$13

for the direct-effect vector $Z$14. The sample-implied invalidity for a candidate $Z$15 is

$Z$16

and the partial-identification region is

$Z$17

The paper’s validified contour

$Z$18

yields confidence sets

$Z$19

with type-I error control when $Z$20 contains the true $Z$21. This is not a named QIV method, but it operationalizes the idea of an approximately valid or bounded-invalidity instrument, including the difficult case of a single potentially invalid instrument.

6. Quasi-Bayesian IV, local projections, and the older post-selection meaning of QIV

A separate use of “quasi” in the IV literature refers not to possibly invalid instruments but to quasi-likelihood or quasi-posterior inference. "Bayesian variable selection in linear regression models with instrumental variables" develops a working quasi-likelihood from IV moment restrictions,

$Z$22

and a corresponding quasi-posterior for sparse high-dimensional linear IV regression (Sabnis et al., 2019). "Quasi-Bayesian Dual Instrumental Variable Regression" defines a Gibbs/quasi-posterior

$Z$23

for nonparametric IV regression under the conditional moment restriction $Z$24 (Wang et al., 2021). "Quasi-Bayesian Local Projections: Simultaneous Inference and Extension to the Instrumental Variable Method" constructs a GMM-based quasi-likelihood for LP-IV,

$Z$25

with stacked IV moments

$Z$26

In these papers, “quasi” refers to the inferential device, not to quasi-valid instruments.

An older and terminologically distinct use appears in "Inference for biased models: a quasi-instrumental variable approach" (Lin et al., 2014). There the QIV is not a causal instrument with relaxed exogeneity or exclusion; it is a constructed variable used to debias post-selection linear regression. Starting from the biased working model

$Z$27

the paper introduces

$Z$28

and reconstructs the model as

$Z$29

with $Z$30. Under its linearity and regularity conditions, the resulting estimator is root-$Z$31 consistent and asymptotically normal. This QIV is therefore a bias-correction device for post-selection inference rather than a relaxed-IV object for causal identification.

Taken together, these strands show that “quasi” has at least three distinct meanings in the IV literature: quasi-valid instruments, quasi-likelihood or quasi-posterior procedures, and quasi-instrumental variables for post-selection debiasing. The surveyed literature suggests that any use of the term QIV requires immediate specification of the underlying structural assumptions, because the inferential target, the role of $Z$32, and even the meaning of “quasi” vary substantially across frameworks.