# Ridge regularization for Mean Squared Error Reduction in Regression with Weak Instruments

## TL;DR

This paper introduces a ridge regularization approach for instrumental variable regression, significantly improving mean squared error reduction and stability in the presence of weak instruments compared to traditional 2SLS methods.

## Contribution

It proposes a novel ridge IV estimator that is asymptotically normal with weak instruments, addressing instability issues of classic 2SLS.

## Key findings

- Ridge IV estimator is asymptotically normal with weak instruments.
- Ridge IV reduces mean squared error compared to 2SLS.
- Simulation results validate theoretical improvements.

## Abstract

In this paper, I show that classic two-stage least squares (2SLS) estimates are highly unstable with weak instruments. I propose a ridge estimator (ridge IV) and show that it is asymptotically normal even with weak instruments, whereas 2SLS is severely distorted and un-bounded. I motivate the ridge IV estimator as a convex optimization problem with a GMM objective function and an L2 penalty. I show that ridge IV leads to sizable mean squared error reductions theoretically and validate these results in a simulation study inspired by data designs of papers published in the American Economic Review.