Weak-instrument-robust subvector inference in instrumental variables regression: A subvector Lagrange multiplier test and properties of subvector Anderson-Rubin confidence sets

We propose a weak-instrument-robust subvector Lagrange multiplier test for instrumental variables regression. We show that it is asymptotically size-correct under a technical condition or as the number of instruments grows to infinity. This is the first weak-instrument-robust subvector test for instrumental variables regression to recover the degrees of freedom of the commonly used non-weak-instrument-robust Wald test. Additionally, we provide a closed-form solution for subvector confidence sets obtained by inverting the subvector Anderson-Rubin test. We show that they are centered around a k-class estimator. We show that the subvector confidence sets for single coefficients of the causal parameter are jointly bounded if and only if Anderson’s likelihood-ratio test rejects the null hypothesis that the first-stage regression parameter is of reduced rank, that is, that the causal parameter is not identified. Finally, we show that if a confidence set obtained by inverting the Anderson-Rubin test is bounded and nonempty, it is equal to a Wald-based confidence set with a data-dependent confidence level. We explicitly compute this Wald-based confidence set and its confidence level.

💡 Research Summary

This paper tackles the problem of conducting reliable inference on individual coefficients in instrumental variables (IV) regressions when instruments may be weak, and when there are multiple endogenous regressors and possibly many instruments. The authors introduce two main contributions: a weak‑instrument‑robust sub‑vector Lagrange‑multiplier (LM) test and a closed‑form characterization of sub‑vector confidence sets obtained by inverting the Anderson‑Rubin (AR) test.

Model and assumptions
The setting is a linear IV model with endogenous variables split into those of interest (X, dimension mₓ) and those that are not (W, dimension m_w). Instruments Z have dimension k ≥ mₓ + m_w. The first‑stage coefficients Πₓ and Π_w are allowed to shrink at the √n rate (weak‑instrument asymptotics). Assumption 1 requires a joint central limit theorem for the sample moments Zᵀε, ZᵀVₓ, ZᵀV_w, with limiting covariance Ω and a positive‑definite matrix Q for ZᵀZ/n. The assumption is standard in the weak‑instrument literature but excludes general heteroskedasticity.

Sub‑vector LM test
Define the projection matrices P_A = A(AᵀA)†Aᵀ and its orthogonal complement M_A = I – P_A. For any β and γ, construct ˜S(β,γ) = (X,W)ᵀM_Z(y – Xβ – Wγ)(y – Xβ – Wγ)ᵀM_Z(X,W). The test statistic is

 LM(β) = (n – k) · min_{γ∈ℝ^{m_w}} (y – Xβ – Wγ)ᵀ P_Z ˜S(β,γ) (y – Xβ – Wγ).

When m_w = 0 this collapses to Kleibergen’s (2002) LM test. The key novelty is the minimisation over γ, which yields a statistic that, under a technical condition, is bounded above by a χ²_{mₓ} random variable. Technical Condition 1 postulates the existence of a γ* satisfying Π_Wᵀ Zᵀ P_Z ˜S(β₀,γ*) ε = 0. Under this condition (Theorem 1) the LM statistic converges in distribution to χ²_{mₓ}, so a critical value based on this χ² restores the exact asymptotic size α, regardless of instrument strength. The authors acknowledge that proving the condition holds with probability one is difficult, but they provide extensive Monte‑Carlo evidence that it does in practice. Moreover, when the number of instruments grows with the sample size (k → ∞), Theorem 2 shows that the same χ²_{mₓ} limit holds without any extra condition, because the projection matrices converge and the dependence structure simplifies.

Thus the proposed sub‑vector LM test recovers the degrees of freedom of the familiar Wald test (mₓ) even when many instruments are present, a property lacking in existing sub‑vector tests (e.g., Kleibergen 2021) which inherit the larger degrees of freedom k – m_w.

Sub‑vector Anderson‑Rubin confidence sets
The AR test statistic for a given β is (y – Xβ)ᵀ P_Z (y – Xβ). Inverting this test yields a confidence region for β. Prior work noted that the resulting region is a quadratic (ellipsoid) that can be difficult to interpret, especially after projection onto a single coefficient. Using the refined critical values from Guggenberger et al. (2012), the authors derive a closed‑form expression for the inverted AR set. They show that the set is centred on a k‑class estimator

 β̂(κ) = (XᵀM_ZX)⁻¹ XᵀM_Zy + κ · (XᵀM_ZX)⁻¹ Π̂ ZᵀM_Zy,

where κ is a scalar that depends on the first‑stage reduced‑form estimates. Two important theoretical results follow:

1. Boundedness equivalence – The confidence interval for a single coefficient is jointly bounded (i.e., does not explode to ±∞) if and only if Anderson’s (1951) likelihood‑ratio test rejects the null that the first‑stage matrix Π has reduced rank. In other words, boundedness is exactly the condition that the model is identified.

2. Equivalence to a Wald set – If the inverted AR set is bounded and non‑empty, it coincides with a Wald confidence set constructed around the same k‑class estimator, but with a data‑dependent confidence level α̂. The authors provide explicit formulas for α̂ and for the κ parameter that defines the centre of the set.

Consequently, practitioners can compute a familiar Wald‑type interval while retaining the weak‑instrument robustness of the AR inversion, and they can diagnose identification problems simply by checking whether the AR set is unbounded.

Implementation and simulations
All methods are implemented in the open‑source Python package ivmodels. The package includes functions for the sub‑vector LM test, the sub‑vector AR test, the conditional likelihood‑ratio test, k‑class estimators, and the inversion routine that returns the closed‑form confidence set. The authors supply reproducible code and simulation scripts on GitHub.

Monte‑Carlo experiments varying n, k, mₓ, and m_w confirm that:

• The sub‑vector LM test maintains nominal size under both strong and weak instrument asymptotics, even when k is large relative to n.
• The AR‑based confidence sets shrink as the number of instruments grows, reflecting improved identification, and their coverage matches that of the Wald set with the data‑dependent level.
• Compared with existing sub‑vector tests, the proposed LM test enjoys higher power because it uses the correct (mₓ) degrees of freedom.

Conclusion
The paper fills a gap in the weak‑instrument literature by delivering a sub‑vector test that is both robust to weak instruments and retains the familiar Wald degrees of freedom, and by providing a transparent, closed‑form description of sub‑vector Anderson‑Rubin confidence sets that are directly linked to identification. The theoretical results, together with the publicly released software, make the methodology readily applicable to empirical work involving multiple endogenous regressors and many instruments.