Conditional Quasi Likelihood Ratio Test

• The paper demonstrates that conditional quasi likelihood ratio tests use projection-based QLR metrics tailored for moment inequality models and weak-IV setups.
• It shows that conditioning on active constraints or sufficient statistics yields chi-squared decision rules with sample-dependent degrees of freedom.
• The methods achieve exact and asymptotically valid inference without simulation or tuning, offering advantages over bootstrap and subsampling techniques.

Searching arXiv for recent and foundational papers on conditional quasi likelihood ratio tests and related CLR/CQLR methods. arxiv_search(query="conditional quasi likelihood ratio test moment inequality models conditional likelihood ratio weak instruments", max_results=10, sort_by="relevance") Conditional quasi likelihood ratio testing denotes a family of hypothesis-testing procedures that retain a likelihood-ratio form while adapting to nuisance structure through conditioning, projection, or both. In moment inequality models, it is a projection-based quasi-likelihood ratio statistic that measures the squared distance from the sample moment vector to a feasible polyhedron and compares that distance to a chi-squared critical value whose degrees of freedom are determined by the active inequalities (Cox et al., 2019). In weak-instrument instrumental-variables regression, the closely related label conditional quasi-likelihood ratio (CQLR) is used in work by Andrews, Moreira, and Stock, while the same statistic is commonly called the conditional likelihood-ratio (CLR) test in Moreira’s terminology; under the Gaussian reduced-form maintained in Moreira (2003), LR and QLR coincide, so CLR and CQLR are the same object in that setting (Londschien, 4 Sep 2025).

1. Terminology and scope

The expression “conditional quasi likelihood ratio test” is not attached to a single inferential environment. It has two technically distinct but conceptually related uses in the literature covered here. In one use, the statistic is defined by projecting sample moments onto a set determined by linear inequalities. In the other, the statistic is an LR-type test in IV models whose conditional distribution is used to eliminate nuisance dependence, especially under weak identification (Cox et al., 2019).

Usage of the term	Core statistic	Adaptation mechanism
Moment inequality models	Projection-based $Q(\theta)$	Chi-squared critical value with df equal to the rank of active inequalities
Weak-IV regression	CLR/CQLR statistic	Conditioning on nuisance-sufficient statistics; modified critical values when variance is estimated

This dual usage matters because “conditional” refers to different operations across settings. In conditional moment inequalities, conditioning appears in the moment restrictions themselves and in the treatment of nuisance parameters entering linearly. In weak-IV regression, conditioning refers to the use of sufficient or asymptotically sufficient statistics so that the null distribution is free of nuisance parameters. A common misconception is therefore to treat CQLR as a single standardized test; the sources instead describe a family of closely related procedures whose common feature is nuisance adaptation without abandoning an LR or QLR structure (Ayyar et al., 2022).

2. Projection-based QLR in moment inequality models

In the moment-inequality framework, a canonical unconditional model is

$A\, E_F[\overline{m}_n(\theta)] \le b,$

where

$$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$

with $A \in \mathbb{R}^{d_A \times d_m}$ and $b \in \mathbb{R}^{d_A}$. This formulation covers upper and lower bounds, equalities, and more general polyhedral restrictions through appropriate choices of $A$ and $b$ (Cox et al., 2019).

The scaled sample moment vector and covariance are

$$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$

The quasi-likelihood ratio statistic is

$$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$

This is the squared distance from $Z(\theta)$ to the feasible polyhedron $A\, E_F[\overline{m}_n(\theta)] \le b,$0 under the $A\, E_F[\overline{m}_n(\theta)] \le b,$1-norm. Geometrically, it is a cone or polyhedron projection problem. If $A\, E_F[\overline{m}_n(\theta)] \le b,$2 denotes the projection solution, the active inequalities are

$A\, E_F[\overline{m}_n(\theta)] \le b,$3

The QLR equals the squared $A\, E_F[\overline{m}_n(\theta)] \le b,$4-distance from $A\, E_F[\overline{m}_n(\theta)] \le b,$5 to the face defined by the active set (Cox et al., 2019).

The decision rule is correspondingly simple: $A\, E_F[\overline{m}_n(\theta)] \le b,$6 where $A\, E_F[\overline{m}_n(\theta)] \le b,$7 is the rank of the active constraints. The critical-value degrees of freedom are therefore sample dependent. This is the central adaptation device: inequalities that are slack do not contribute to $A\, E_F[\overline{m}_n(\theta)] \le b,$8, so the critical value is not inflated by the full ambient dimension of the inequality system (Cox et al., 2019).

3. Exact size, asymptotic validity, and refinement

The moment-inequality construction is designed to be size exact under a Gaussian benchmark and uniformly asymptotically exact more generally. Under the normal model with

$A\, E_F[\overline{m}_n(\theta)] \le b,$9

and known covariance, the refined conditional chi-squared (RCC) test has exact finite-sample size; when all tested inequalities bind and $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$0, the rejection probability equals $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$1. The underlying intuition is a decomposition of the Gaussian vector into components parallel and perpendicular to the cone spanned by the active constraints; conditional on the active set, the squared distance to the face is chi-squared with degrees of freedom equal to the active-set rank (Cox et al., 2019).

Outside the known-variance Gaussian case, asymptotic validity requires a CLT for the scaled moments and a consistent estimator $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$2. Sufficient conditions listed in the source include i.i.d. sampling, finite moments, and invertibility of the average conditional variance. Under these conditions the RCC test is uniformly asymptotically valid. When inequalities are binding or local-to-binding, rejection probability converges to $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$3. When some inequalities become “very slack” at the $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$4 rate, the test has the irrelevance-of-distant-inequalities (IDI) property: it reduces to the test based only on the not-very-slack subset (Cox et al., 2019).

A notable finite-sample refinement arises when exactly one inequality is active, so $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$5. In that case the plain conditional chi-squared test can be conservative because $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$6 may concentrate at zero when no inequality is active. The RCC modification replaces the one-degree-of-freedom critical value by a data-driven $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$7, where $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$8 measures how close the next inequality is to being active. As $$\overline{m}_n(\theta) = n^{-1}\sum_{i=1}^n m(W_i,\theta) \in \mathbb{R}^{d_m},$$9, $A \in \mathbb{R}^{d_A \times d_m}$0; as $A \in \mathbb{R}^{d_A \times d_m}$1, $A \in \mathbb{R}^{d_A \times d_m}$2, restoring exact size (Cox et al., 2019).

These properties motivate the paper’s comparison with alternative methods. Bootstrap, subsampling, and generalized moment selection procedures such as adjusted QLR with bootstrap and two-step methods require simulation and tuning and are computationally demanding per parameter value. Classical cone tests based on least-favorable mixtures of chi-squared distributions do not have the IDI property and tend to be conservative when many inequalities are slack. The RCC test uses no tuning, no simulation, and is computationally attractive for test inversion and confidence-set construction (Cox et al., 2019).

4. Conditional moment inequalities, nuisance elimination, and subvector inference

The same paper extends the projection-based QLR logic to conditional moment inequalities with nuisance parameters entering linearly. Let $A \in \mathbb{R}^{d_A \times d_m}$3 denote instrumental or conditioning variables, and consider

$A \in \mathbb{R}^{d_A \times d_m}$4

where $A \in \mathbb{R}^{d_A \times d_m}$5, $A \in \mathbb{R}^{d_A \times d_m}$6, and $A \in \mathbb{R}^{d_A \times d_m}$7 may depend on $A \in \mathbb{R}^{d_A \times d_m}$8 and $A \in \mathbb{R}^{d_A \times d_m}$9, and $b \in \mathbb{R}^{d_A}$0 is nuisance. If $b \in \mathbb{R}^{d_A}$1, the object of interest is the subvector $b \in \mathbb{R}^{d_A}$2, and inference proceeds by test inversion for $b \in \mathbb{R}^{d_A}$3, eliminating the nuisance linearly (Cox et al., 2019).

The key device is a duality-based nuisance elimination: $b \in \mathbb{R}^{d_A}$4 with $b \in \mathbb{R}^{d_A}$5 and $b \in \mathbb{R}^{d_A}$6, where $b \in \mathbb{R}^{d_A}$7 collects vertices of the polyhedron $b \in \mathbb{R}^{d_A}$8. Although this representation is polyhedral, the implementation described in the paper avoids full-blown Fourier–Motzkin vertex enumeration. Instead, the QLR is computed through the equivalent constrained minimization

$b \in \mathbb{R}^{d_A}$9

and the degrees of freedom are recovered from the cone

$A$0

The required computations use only quadratic programming, linear programs, and linear algebra (Cox et al., 2019).

For subvector inference, the confidence set is obtained by test inversion: $A$1 Both the active set and the effective degrees of freedom are parameter dependent, and this dependence is handled automatically by the algorithm. In discrete-$A$2 settings, the conditional variance estimator is the average of within-category variances; in continuous-$A$3 settings, the paper uses nearest-neighbor differences. The numerical notes emphasize that near-singular $A$4 may require shrinkage or regularization, and that very high-dimensional inequality systems can challenge covariance estimation and LP steps; the source suggests sparsity or dimensionality reduction, such as instrument selection, as possible remedies (Cox et al., 2019).

5. Conditional LR and quasi-LR in weak-instrument IV regression

In IV regression with many weak instruments, the conditional quasi-likelihood ratio idea appears through a modified CLR test. The structural model is

$A$5

with null hypothesis $A$6. Under homoskedastic normal reduced-form errors, rows of $A$7 are i.i.d. $A$8 with unknown positive definite $A$9. The source studies regimes in which the number of instruments $b$0 increases with $b$1, including a dense regime with $b$2 and a sparse regime with $b$3 (Ayyar et al., 2022).

With known $b$4, the classical CLR statistic can be written in terms of invariant statistics $b$5 and $b$6, and conditioning on $b$7 yields exact similarity because $b$8 is sufficient for the nuisance $b$9 and $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$0 is independent of $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$1 under the null. When $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$2 is unknown, however, plugging in $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$3 and reusing the known-variance critical value is no longer valid. The paper states that the conventional CLR test with estimated error variance loses exact similarity and is asymptotically invalid in many weak instrument regimes because the feasible LR statistic depends on additional sample randomness not captured by $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$4 (Ayyar et al., 2022).

The proposed remedy is a modified critical value function based on a four-statistic representation. Besides the projection-space statistics $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$5 and $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$6, the unknown-variance problem introduces residual-space analogues $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$7 and $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$8, because variance estimation uses the residualized data $$Z(\theta) := \sqrt{n}\, \overline{m}_n(\theta), \qquad \Sigma(\theta) := \operatorname{Var}\!\left(\sqrt{n}\,\overline{m}_n(\theta)\right).$$9. Under $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$0 and normality, the block matrix built from $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$1 and $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$2 is Wishart and is independent of $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$3 and $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$4. The feasible LR statistic can therefore be represented as a function of six inner products, and the critical value $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$5 is defined as the conditional $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$6-quantile of that representation given $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$7 (Ayyar et al., 2022).

This produces an infeasible test with exact similarity when conditioning on the true $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$8, and a feasible modified CLR (MCLR) test obtained by replacing $$Q(\theta) := \min_{\mu \in \mathbb{R}^{d_m}:\, A\mu \le b} \big(Z(\theta) - \mu\big)'\, \Sigma(\theta)^{-1} \, \big(Z(\theta) - \mu\big).$$9 with its plug-in estimator $Z(\theta)$0. The paper proves that, under normality and many weak instruments, the feasible MCLR test is asymptotically valid; in the sparse regime, the same convergence holds under a mild projection-moment condition even without normality. In the terminology section of the paper, this is the sense in which the procedure is a conditional quasi-LR test: it retains the LR form but adjusts the conditional critical value to account for the estimation of the nuisance variance (Ayyar et al., 2022).

The comparison results are explicit. The conventional feasible CLR that uses known-variance critical values is not similar and over-rejects when $Z(\theta)$1 is non-negligible. By contrast, simulations reported in the paper show that MCLR has empirical rejection rates closest to $Z(\theta)$2 across the scenarios considered and power uniformly higher than J-AR and mKLM in the designs considered, especially when identification per instrument $Z(\theta)$3 is small (Ayyar et al., 2022).

6. Exact conditional distribution and spectrum-aware conditioning

A further development in the IV literature derives the exact asymptotic distribution of the CLR statistic with multiple endogenous variables under weak-instrument asymptotics. The model is

$Z(\theta)$4

with $Z(\theta)$5 fixed and of full column rank. The central object is the sample concentration matrix

$Z(\theta)$6

whose eigenvalues $Z(\theta)$7 estimate the eigenvalues of the population concentration matrix. The CLR test conditions on these realized eigenvalues (Londschien, 4 Sep 2025).

The theorem in the paper states that, conditional on the full vector $Z(\theta)$8, the LR statistic obeys the exact asymptotic law

$Z(\theta)$9

where $A\, E_F[\overline{m}_n(\theta)] \le b,$00, $A\, E_F[\overline{m}_n(\theta)] \le b,$01 are mutually independent, and $A\, E_F[\overline{m}_n(\theta)] \le b,$02 is the smallest root of an arrowhead polynomial. In the scalar case $A\, E_F[\overline{m}_n(\theta)] \le b,$03, this reduces to Moreira’s conditional distribution. For $A\, E_F[\overline{m}_n(\theta)] \le b,$04, the result shows that conditioning on the full spectrum, rather than only on the smallest eigenvalue, yields the exact conditional asymptotic law (Londschien, 4 Sep 2025).

This distinction is substantively important when endogenous regressors are identified with different strengths. Earlier upper-bound results based only on $A\, E_F[\overline{m}_n(\theta)] \le b,$05 are sharp only when all eigenvalues are equal. When $A\, E_F[\overline{m}_n(\theta)] \le b,$06, conditioning only on the weakest direction produces conservative critical values, whereas conditioning on the whole spectrum recognizes that most directions are well identified. The paper reports that, in Gaussian simulations with heterogeneous eigenvalues, tests using smallest-eigenvalue upper-bound critical values have empirical sizes well below the nominal level, while the exact conditional test maintains empirical size equal to $A\, E_F[\overline{m}_n(\theta)] \le b,$07 up to simulation error and achieves gains up to about $A\, E_F[\overline{m}_n(\theta)] \le b,$08 for $A\, E_F[\overline{m}_n(\theta)] \le b,$09 and up to about $A\, E_F[\overline{m}_n(\theta)] \le b,$10 for $A\, E_F[\overline{m}_n(\theta)] \le b,$11 at $A\, E_F[\overline{m}_n(\theta)] \le b,$12 (Londschien, 4 Sep 2025).

Implementation is based on Monte Carlo draws of the chi-squared variables and root finding for $A\, E_F[\overline{m}_n(\theta)] \le b,$13. For $A\, E_F[\overline{m}_n(\theta)] \le b,$14, the transformed polynomial $A\, E_F[\overline{m}_n(\theta)] \le b,$15 is strictly increasing, so either bisection or Newton’s method is available; the paper notes that the ivmodels Python package implements these methods. The resulting p-value is conditional on the observed eigenvalue vector and is exact in large samples under the weak-IV asymptotics stated in the paper. In this sense, the modern CLR/CQLR literature extends the original conditional argument from a scalar concentration parameter to a spectrum-aware conditioning scheme that preserves size while improving power under differential identification (Londschien, 4 Sep 2025).