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

Jesse Hoekstra, Frank Windmeijer

TL;DR

The paper advances weak-instrument robust inference for subvector AR tests by showing that conditioning on the second-smallest eigenvalue of the AR concentration matrix yields valid size control and strictly higher power than conditioning on the largest eigenvalue, particularly when $m_W>1$. It establishes that the same conditional critical-value function from GKM applies with this alternative conditioning, via a two-eigenvalue reduction to a $2\times2$ Wishart structure, and demonstrates monotonicity and size properties through analytic results and simulations. The approach extends to feasible AR statistics under random instruments and to approximate Kronecker-product heteroskedasticity (AKP) as in GKM24, preserving power gains and computational simplicity. Collectively, the results identify a practically superior, computationally tractable, and robust conditional testing strategy for subvector inference in IV models with weak instruments, with broad applicability including AKP settings.

Abstract

For subvector inference in the linear instrumental variables model under homoskedasticity but allowing for weak instruments, Guggenberger, Kleibergen, and Mavroeidis (2019) (GKM) propose a conditional subvector Anderson and Rubin (1949) (AR) test that uses data-dependent critical values that adapt to the strength of the parameters not under test. This test has correct size and strictly higher power than the test that uses standard asymptotic chi-square critical values. The subvector AR test is the minimum eigenvalue of a data dependent matrix. The GKM critical value function conditions on the largest eigenvalue of this matrix. We consider instead the data dependent critical value function conditioning on the second-smallest eigenvalue, as this eigenvalue is the appropriate indicator for weak identification. We find that the data dependent critical value function of GKM also applies to this conditioning and show that this test has correct size and power strictly higher than the GKM test when the number of parameters not under test is larger than one. Our proposed procedure further applies to the subvector AR test statistic that is robust to an approximate kronecker product structure of conditional heteroskedasticity as proposed by Guggenberger, Kleibergen, and Mavroeidis (2024), carrying over its power advantage to this setting as well.

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

TL;DR

The paper advances weak-instrument robust inference for subvector AR tests by showing that conditioning on the second-smallest eigenvalue of the AR concentration matrix yields valid size control and strictly higher power than conditioning on the largest eigenvalue, particularly when . It establishes that the same conditional critical-value function from GKM applies with this alternative conditioning, via a two-eigenvalue reduction to a Wishart structure, and demonstrates monotonicity and size properties through analytic results and simulations. The approach extends to feasible AR statistics under random instruments and to approximate Kronecker-product heteroskedasticity (AKP) as in GKM24, preserving power gains and computational simplicity. Collectively, the results identify a practically superior, computationally tractable, and robust conditional testing strategy for subvector inference in IV models with weak instruments, with broad applicability including AKP settings.

Abstract

For subvector inference in the linear instrumental variables model under homoskedasticity but allowing for weak instruments, Guggenberger, Kleibergen, and Mavroeidis (2019) (GKM) propose a conditional subvector Anderson and Rubin (1949) (AR) test that uses data-dependent critical values that adapt to the strength of the parameters not under test. This test has correct size and strictly higher power than the test that uses standard asymptotic chi-square critical values. The subvector AR test is the minimum eigenvalue of a data dependent matrix. The GKM critical value function conditions on the largest eigenvalue of this matrix. We consider instead the data dependent critical value function conditioning on the second-smallest eigenvalue, as this eigenvalue is the appropriate indicator for weak identification. We find that the data dependent critical value function of GKM also applies to this conditioning and show that this test has correct size and power strictly higher than the GKM test when the number of parameters not under test is larger than one. Our proposed procedure further applies to the subvector AR test statistic that is robust to an approximate kronecker product structure of conditional heteroskedasticity as proposed by Guggenberger, Kleibergen, and Mavroeidis (2024), carrying over its power advantage to this setting as well.
Paper Structure (14 sections, 2 theorems, 73 equations, 8 figures, 1 table)

This paper contains 14 sections, 2 theorems, 73 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Finite Sample Analysis
  3. Concentration Parameter
  4. Conditioning on Largest Eigenvalue
  5. $m_{W}=1$
  6. $m_{W}>1$
  7. Conditioning on Second-Smallest Eigenvalue
  8. Feasible $AR$ Test Asymptotic Power Comparisons
  9. Concluding Remarks
  10. Proof of Proposition 1
  11. Conditional Distribution of $\widehat{\kappa}_{p}|\widehat{\kappa}_{p-1}$
  12. Monte Carlo Parameter Values
  13. Conditional Heteroskedasticity with Kronecker Product Structure
  14. Main Deviation from the GKM24 Proof

Key Result

Proposition 1

Let $\Xi^{\prime}\Xi\sim\mathcal{W}_{p}\left(k,I_{p},\mathcal{M}^{\prime}\mathcal{M}\right)$, with $p=m_{W}+1>2$, and let $\widehat{\kappa}_{1}\geq\widehat{\kappa}_{2}\geq\ldots\geq\widehat{\kappa}_{p}$ denote the ordered eigenvalues of $\Xi^{\prime}\Xi$. The joint distribution of $\left(\widehat{\k

Figures (8)

  • Figure 1: Left panel, simulated conditional cdf. Right panel including the calculated points as in Table \ref{['tab:CondCDF']}.
  • Figure 2: Null rejection probabilities at 5% level of $\phi_{\chi^{2}}$, $\phi_{c_{1}}$ and $\phi_{c_{p-1}}$ for different values of $m_{W}$ as a function of $\kappa_{p-1}$, with $k-m_{W}=4$. For $m_{W}>1$, $\kappa_{j}=100,000$ for $j<m_{W}$. Critical values for $\phi_{c_{1}}$ and $\phi_{c_{p-1}}$ obtained from Table 6 in GKM, together with linear interpolation, with for $\phi_{c_{p-1}}$ the conditioning largest eigenvalue $\widehat{\kappa}_{1}$ substituted by the second-smallest one, $\widehat{\kappa}_{p-1}$.
  • Figure 3: NRPs at 5% level of $\phi_{\chi^{2}}$, $\phi_{c_{1}}$ and $\phi_{c_{2}}$ for $m_{W}=2$ and $k-m_{W}=4$. Left panel as a function of $\kappa_{2}$, with $\kappa_{1}=\kappa_{2}$. Right panel as a function of $\kappa_{1}$ for a fixed value of $\kappa_{2}=10$. See further notes to Figure \ref{['fig:NRPlarge']}.
  • Figure 4: NRPs at 5% level of $\phi_{\chi^{2}}$, $\phi_{c_{1}}$ and $\phi_{c_{3}}$ for $m_{W}=3$ and $k-m_{W}=4$. First graph as a function of $\kappa_{3}$, with $\kappa_{1}=\kappa_{2}=\kappa_{3}$. Second graph as a function of $\kappa_{2}$ for a fixed value of $\kappa_{3}=10$, with $\kappa_{1}=100,000$. Third graph as a function of $\kappa_{1}=\kappa_{2}$, for a fixed value of $\kappa_{3}=10$. See further notes to Figure \ref{['fig:NRPlarge']}.
  • Figure 5: Power of $\phi_{\chi^{2}}$, $\phi_{1}$ and $\phi_{p-1}$, for testing $H_{0}:\beta=0$ at level $\alpha=0.05$. $m_{W}=3$, $k=7$, $n=250$, for different values of $\kappa=\left(\kappa_{1},\kappa_{2},\kappa_{3}\right)$.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Proposition 5