# A specification test for the strength of instrumental variables

## TL;DR

This paper introduces a new statistical test to assess the strength of many instruments in instrumental variable analysis, especially when the number of instruments is large relative to the sample size.

## Contribution

It develops a novel specification test based on the asymptotic behavior of the difference between 2SLS and OLS estimators, applicable to many weak instruments.

## Key findings

- The test accurately detects weak instruments in simulations.
- The method performs well with multiple endogenous variables.
- Empirical analysis confirms the test's reliability on real data.

## Abstract

This paper develops a new specification test for the instrument weakness when the number of instruments $K_n$ is large with a magnitude comparable to the sample size $n$. The test relies on the fact that the difference between the two-stage least squares (2SLS) estimator and the ordinary least squares (OLS) estimator asymptotically disappears when there are many weak instruments, but otherwise converges to a non-zero limit. We establish the limiting distribution of the difference within the above two specifications, and introduce a delete-$d$ Jackknife procedure to consistently estimate the asymptotic variance/covariance of the difference. Monte Carlo experiments demonstrate the good performance of the test procedure for both cases of single and multiple endogenous variables. Additionally, we re-examine the analysis of returns to education data in Angrist and Keueger (1991) using our proposed test. Both the simulation results and empirical analysis indicate the reliability of the test.