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An Improved Inference for IV Regressions

Liyu Dou, Pengjin Min, Wenjie Wang, Yichong Zhang

TL;DR

This paper tackles the inefficiency in IV inference when researchers report results using both low-dimensional instruments and many base IVs in clustered samples. It develops a combined inference approach that linearly blends a cluster-robust Wald statistic from low-dimensional IVs with leave-one-cluster-out LM and AR statistics from the many IVs, yielding a UMPU-optimal test under strong identification of the low-dimensional IVs. The method automatically adapts to the identification strength of the many IVs and provides costless efficiency gains, with a practical rule-of-thumb to anticipate CI-length reductions. An empirical illustration based on shift-share/Bartik instruments and simulation studies confirm substantial finite-sample gains and robust size control, suggesting broad applicability in applied IV work.

Abstract

Empirical instrumental variables (IV) studies often report separate results based on low-dimensional instruments and many base instruments. This paper proposes a combination test that integrates these commonly reported statistics. The test linearly combines a cluster-robust Wald statistic based on low-dimensional IVs with leave-one-cluster-out Lagrangian Multiplier (LM) and Anderson-Rubin (AR) statistics constructed from many IVs. Under strong identification of the low-dimensional IVs, we establish joint asymptotic normality and asymptotic optimality of the proposed test. The procedure yields costless efficiency improvements, automatically adapts to weak identification of many instruments, and is accompanied by a practical rule of thumb for assessing efficiency gains.

An Improved Inference for IV Regressions

TL;DR

This paper tackles the inefficiency in IV inference when researchers report results using both low-dimensional instruments and many base IVs in clustered samples. It develops a combined inference approach that linearly blends a cluster-robust Wald statistic from low-dimensional IVs with leave-one-cluster-out LM and AR statistics from the many IVs, yielding a UMPU-optimal test under strong identification of the low-dimensional IVs. The method automatically adapts to the identification strength of the many IVs and provides costless efficiency gains, with a practical rule-of-thumb to anticipate CI-length reductions. An empirical illustration based on shift-share/Bartik instruments and simulation studies confirm substantial finite-sample gains and robust size control, suggesting broad applicability in applied IV work.

Abstract

Empirical instrumental variables (IV) studies often report separate results based on low-dimensional instruments and many base instruments. This paper proposes a combination test that integrates these commonly reported statistics. The test linearly combines a cluster-robust Wald statistic based on low-dimensional IVs with leave-one-cluster-out Lagrangian Multiplier (LM) and Anderson-Rubin (AR) statistics constructed from many IVs. Under strong identification of the low-dimensional IVs, we establish joint asymptotic normality and asymptotic optimality of the proposed test. The procedure yields costless efficiency improvements, automatically adapts to weak identification of many instruments, and is accompanied by a practical rule of thumb for assessing efficiency gains.

Paper Structure

This paper contains 36 sections, 18 theorems, 459 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Rule of Thumb and Empirical Illustration
  3. Model and Preliminaries
  4. Setup
  5. Test Statistics
  6. Combination Test
  7. Large-Sample Theory
  8. General Assumptions
  9. Asymptotic Efficiency Properties of Combination Test
  10. Quantifying Efficiency Improvement in Large Samples
  11. Simulation Study
  12. Conclusion
  13. Combination Test with Weak Low-Dimensional IVs
  14. Technical Lemmas
  15. Proofs of Technical Lemmas
  16. ...and 21 more sections

Key Result

Proposition 3.1

Suppose one observes $(\mathcal{N}_1,\mathcal{N}_2,\mathcal{N}_3)$, which follows the limiting distribution in (eq:weak_convergence) with $\rho_{1}^2 + \rho_{2}^2 < 1$ and wants to test $\mathcal{H}_0: \delta = 0$ against $\mathcal{H}_1: \delta \neq 0$ for known values of $(a_1,a_2,\rho_1,\rho_2)$, where $\mathbb C_\alpha$ is the $(1-\alpha)$ percentile of a chi-squared random variable with one d

Figures (5)

  • Figure 1: Theoretical lower bounds for percentage reduction in confidence interval length.
  • Figure 2: Realized percentage reduction in confidence interval length: immigrant enclave.
  • Figure 3: Power curves of the combination, Wald, and jackknife LM tests.
  • Figure 4: Power Curve of the combination, Wald, and jacknife LM tests.
  • Figure 5: Realized percentage reduction in confidence interval length: returns to education

Theorems & Definitions (28)

  • Proposition 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.1
  • Remark 4.5
  • Remark 4.6
  • Theorem 4.2
  • Remark 4.7
  • ...and 18 more