[Paper Review] Inference with Many Weak Instruments

Anna Mikusheva, Liyang Sun|arXiv (Cornell University)|Apr 26, 2020

Economic Policies and Impacts20 references4 citations

TL;DR

This paper develops a jackknifed Anderson-Rubin test statistic robust to weak identification and heteroscedasticity in linear IV models with many instruments, where the number of instruments grows proportionally with sample size. It introduces a novel cross-fitting variance estimator and a pre-test for weak identification, ensuring valid size control and good power even under weak identification or heteroscedasticity.

ABSTRACT

We develop a concept of weak identification in linear IV models in which the number of instruments can grow at the same rate or slower than the sample size. We propose a jackknifed version of the classical weak identification-robust Anderson-Rubin (AR) test statistic. Large-sample inference based on the jackknifed AR is valid under heteroscedasticity and weak identification. The feasible version of this statistic uses a novel variance estimator. The test has uniformly correct size and good power properties. We also develop a pre-test for weak identification that is related to the size property of a Wald test based on the Jackknife Instrumental Variable Estimator (JIVE). This new pre-test is valid under heteroscedasticity and with many instruments.

Motivation & Objective

• Address the challenge of valid inference in linear IV models when the number of instruments grows proportionally with sample size and identification is weak.
• Define weak identification in the context of many instruments as a bounded concentration parameter relative to the square root of the number of instruments.
• Develop a test statistic that maintains correct size and good power under weak identification and heteroscedasticity.
• Propose a pre-test for weak identification that enables a two-step procedure switching between robust jackknifed AR and efficient JIVE-Wald tests.
• Ensure size control of the two-step procedure under heteroscedasticity and many instruments, overcoming limitations of prior pre-tests like Stock and Yogo (2005).

Proposed method

• Propose a jackknifed version of the classical Anderson-Rubin (AR) test statistic to improve robustness to weak identification and heteroscedasticity.
• Introduce a novel cross-fitting-based variance estimator for quadratic forms in the CLT approximation, removing bias from correlated error variance proxies.
• Use cross-fitting to generate unbiased estimates of individual error variances, adjusting the quadratic form to correct for estimation-induced correlation.
• Establish consistency of the variance estimator under the null and local alternatives across a wide range of identification scenarios.
• Develop a pre-test statistic based on the Jackknife Instrumental Variable Estimator (JIVE) that detects weak identification under heteroscedasticity and many instruments.
• Construct a two-step testing procedure: use the jackknifed AR test if identification is weak (pre-test rejects), otherwise switch to the JIVE-Wald test for higher efficiency.

(a) sparse, $\frac{\mu^{2}}{\sqrt{K}}=2.5$

Experimental results

Research questions

• RQ1 How should weak identification be defined in linear IV models with many instruments, where the number of instruments grows proportionally with sample size?
• RQ2 What test statistic ensures valid size and good power under weak identification and heteroscedasticity when the number of instruments is large?
• RQ3 Can a pre-test for weak identification be constructed that is valid under heteroscedasticity and many instruments, enabling a two-step inference procedure?
• RQ4 Does the proposed two-step procedure (jackknifed AR or JIVE-Wald based on pre-test) maintain correct size under heteroscedasticity and many instruments?
• RQ5 How does the performance of the jackknifed AR test compare to standard tests (e.g., 2SLS, LIML, JIVE) in terms of size, power, and confidence interval length under varying identification strength?

Key findings

• The jackknifed AR test controls size at the nominal level (5.1% to 7.2%) across all simulation scenarios, including under weak identification and heteroscedasticity.
• The proposed pre-test correctly identifies strong identification in Angrist and Krueger (1991), rejecting weak identification even with up to 1,530 instruments.
• The cross-fitting variance estimator improves power and reduces the frequency of unbounded confidence intervals compared to the naive estimator, especially under weak identification.
• Under weak identification, the jackknifed AR confidence intervals are longer (e.g., 6.90 vs. 0.24 in length) but remain bounded, while the probability of infinite length increases to 49.6% at small sample sizes.
• The two-step procedure based on the pre-test maintains correct size (5.8% to 7.2%) and achieves better efficiency than the jackknifed AR alone when identification is strong.
• The naive variance estimator leads to higher power loss and more unbounded confidence intervals (e.g., 77.3% infinite CI at N=796, K=77) compared to the cross-fitting estimator (74.4%).

(b) dense, $\frac{\mu^{2}}{\sqrt{K}}=2.5$