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Best Feasible Conditional Critical Values for a More Powerful Subvector Anderson-Rubin Test

This paper proposes a more powerful subvector Anderson-Rubin test for weak instruments by replacing the Guggenberger, Kleibergen, and Mavroeidis (2019) critical value conditioning on the largest eigenvalue with one based on the second-smallest eigenvalue, thereby achieving strictly higher power when the number of parameters not under test exceeds one while maintaining correct size and extending to settings with approximate Kronecker product heteroskedasticity.

Original authors: Jesse Hoekstra, Frank Windmeijer

Published 2026-01-27

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Jesse Hoekstra, Frank Windmeijer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery. You have a suspect (a specific number or parameter you want to test) and a team of witnesses (data points called "instruments") who can help you figure out if the suspect is guilty or innocent.

In the world of statistics, this is called a Linear Instrumental Variables model. The goal is to test if your suspect is actually innocent (the null hypothesis).

The Problem: Weak Witnesses

Sometimes, your witnesses are shaky. They might be confused, or they might not have seen the crime very clearly. In statistics, we call this "weak identification."

If your witnesses are weak, the standard way of judging the suspect (using a fixed rulebook) often fails. It might either:

False Alarm: Accuse an innocent person (too many false positives).
Miss the Culprit: Let a guilty person go free (low power).

The Old Solution: The "Biggest Witness" Rule

A previous group of researchers (GKM, 2019) came up with a clever fix. They said, "Let's look at our strongest witness (the one with the biggest eigenvalue) to decide how strict our rulebook should be."

The Logic: If the strongest witness is very shaky, we loosen the rules slightly to avoid false alarms. If the strongest witness is solid, we stick to the standard rules.
The Catch: This method works, but it's not perfect. Just because your best witness is strong doesn't mean the rest of the team is strong. You could have one superstar witness and a whole team of confused ones. The old method might miss the fact that the team as a whole is struggling.

The New Solution: The "Second-Weakest Link" Rule

This paper proposes a better way to judge the team. Instead of looking at the strongest witness, the authors suggest looking at the second-smallest witness (the second-worst link in the chain).

Think of it like a chain holding a heavy weight. The chain is only as strong as its weakest link.

The old method checked the strongest link to guess the chain's strength.
The new method checks the second-weakest link. Why? Because in a team of witnesses, the second-worst one is usually the best indicator of whether the whole group is struggling with the case.

The Magic Trick

The authors discovered something surprising: You can use the exact same rulebook (critical values) that the old method used, but you just swap the input.

Instead of feeding the rulebook the strength of the "Biggest Witness," you feed it the strength of the "Second-Weakest Witness."

Why it works: Mathematically, they proved that if you look at the second-weakest link, the math behaves exactly the same way as the old method, just with a slight adjustment.
The Result: Because the "Second-Weakest" number is usually smaller (weaker) than the "Biggest" number, the rulebook tells you to be slightly more lenient with your threshold. This means you are less likely to miss a guilty suspect (higher power) while still keeping the false alarm rate exactly where it should be (correct size).

The Takeaway

The authors have built a better detector for weak witnesses.

It's Fair: It doesn't accuse innocent people more often than the old method.
It's Sharper: It catches guilty people more often, especially when the team of witnesses is mixed (some strong, some weak).
It's Simple: You don't need a supercomputer to do it. You just take the existing, trusted rulebook and plug in a different number (the second-smallest one instead of the largest).

They also showed this trick works even when the weather is stormy (when data is messy or "heteroskedastic"), proving that this new way of looking at the "second-weakest link" is the most practical and powerful tool available for this specific type of statistical detective work.

Technical Summary: Best Feasible Conditional Critical Values for a More Powerful Subvector Anderson-Rubin Test

Problem Statement
In the linear instrumental variables (IV) model under homoskedasticity, inference on a subset of parameters (subvector inference) is complicated by the potential weakness of instruments. Guggenberger, Kleibergen, and Mavroeidis (2019) (GKM) proposed a conditional subvector Anderson-Rubin (AR) test that utilizes data-dependent critical values to adapt to the identification strength of nuisance parameters (those not under test). The GKM test conditions on the largest eigenvalue of a non-central Wishart distributed matrix derived from the AR statistic. While this approach ensures correct size and strictly higher power than standard asymptotic $\chi^2$ tests, the authors argue that the largest eigenvalue is not the optimal indicator for weak identification.

Specifically, a small largest eigenvalue correctly signals weak identification, but a large largest eigenvalue does not necessarily imply strong identification for all nuisance parameters. In scenarios where the largest eigenvalue is large but the second-smallest eigenvalue is small, the nuisance parameters may still suffer from weak identification. Conditioning on the largest eigenvalue in such cases fails to adjust the critical values sufficiently, potentially leading to conservative tests with lower power than necessary.

Methodology
The paper proposes conditioning the subvector AR test on the second-smallest eigenvalue ( $\hat{\kappa}_{p-1}$ ) of the $p \times p$ non-central Wishart matrix, where $p = m_W + 1$ ( $m_W$ is the number of nuisance parameters). The AR statistic itself corresponds to the minimum eigenvalue ( $\hat{\kappa}_p$ ).

The core theoretical contribution relies on the properties of the joint distribution of the eigenvalues of a non-central Wishart matrix $\Xi'\Xi \sim W_p(k, I_p, M'M)$ :

Asymptotic Equivalence: The authors demonstrate that when the $p-2$ largest eigenvalues of the non-centrality matrix $M'M$ are large (approaching infinity), the joint distribution of the two smallest eigenvalues ( $\hat{\kappa}_{p-1}, \hat{\kappa}_p$ ) is identical to the joint distribution of the eigenvalues of a $2 \times 2$ non-central Wishart matrix.
Degrees of Freedom Adjustment: In this limiting case, the $2 \times 2$ matrix has degrees of freedom equal to $k - m_W + 1$ (where $k$ is the number of instruments), and the non-centrality parameters correspond to the two smallest eigenvalues of the original $p \times p$ matrix.
Critical Value Transfer: This result implies that the critical value functions tabulated by GKM (which condition on the largest eigenvalue for a $2 \times 2$ system) can be directly applied to the proposed test by simply substituting the conditioning variable: replacing the largest eigenvalue $\hat{\kappa}_1$ with the second-smallest eigenvalue $\hat{\kappa}_{p-1}$ .

To address cases where the $p-2$ largest eigenvalues are finite (not approaching infinity), the authors derive the conditional probability density function of the two smallest eigenvalues based on results from James (1964) and Muirhead (2009). They prove that the rejection probability of the test is monotonically decreasing in the values of the larger eigenvalues. Consequently, the size of the test remains controlled (bounded by the nominal level $\alpha$ ) even when the largest eigenvalues are not infinite, as the rejection probability is strictly lower than in the limiting case where they are infinite.

Key Contributions and Results

Correct Size: The proposed test, which conditions on the second-smallest eigenvalue, maintains correct size control under the null hypothesis for all configurations of the non-centrality parameters.
Strictly Higher Power: For $m_W > 1$ , the critical value derived from conditioning on the second-smallest eigenvalue is strictly smaller than the critical value derived from conditioning on the largest eigenvalue (for the same observed statistic). Because the critical value is lower, the proposed test ( $\phi_{c_{p-1}}$ ) has strictly higher power than the original GKM test ( $\phi_{c_1}$ ).
Computational Feasibility: The method retains the computational simplicity of the GKM approach. It avoids the need to condition on a larger subset of eigenvalues, which would increase power but at the cost of prohibitive computational complexity.
Extension to Heteroskedasticity: The authors show that their findings extend to the setting proposed by Guggenberger, Kleibergen, and Mavroeidis (2024), where the model allows for conditional heteroskedasticity with an approximate Kronecker product structure. The power advantage of conditioning on the second-smallest eigenvalue holds in this robust setting as well.

Significance
The paper claims to provide the "best feasible" subset AR test for the linear IV setting under homoskedasticity. By identifying the second-smallest eigenvalue as the appropriate indicator for weak identification of nuisance parameters, the proposed procedure properly incorporates weak identification information that the GKM test misses. This results in a test that is both computationally simple and statistically more powerful, particularly when the number of nuisance parameters exceeds one. The authors emphasize that while conditioning on more eigenvalues could theoretically yield further power gains, the computational cost would be excessive; thus, conditioning on the second-smallest eigenvalue represents the optimal trade-off between feasibility and power.