The Ridge Path Estimator for Linear Instrumental Variables

TL;DR

This paper introduces a ridge regression-based IV estimator with an empirically selected tuning parameter, demonstrating its asymptotic properties and potential advantages over traditional methods through simulations.

Contribution

It develops an asymptotic framework for a ridge IV estimator with data-driven tuning, linking regularization to estimator performance and distribution.

Findings

The optimal tuning parameter minimizes the IV objective on test data.

The asymptotic distribution of the tuning parameter is a nonstandard mixture.

The ridge IV estimator can outperform two-stage least squares in simulations.

Abstract

This paper presents the asymptotic behavior of a linear instrumental variables (IV) estimator that uses a ridge regression penalty. The regularization tuning parameter is selected empirically by splitting the observed data into training and test samples. Conditional on the tuning parameter, the training sample creates a path from the IV estimator to a prior. The optimal tuning parameter is the value along this path that minimizes the IV objective function for the test sample. The empirically selected regularization tuning parameter becomes an estimated parameter that jointly converges with the parameters of interest. The asymptotic distribution of the tuning parameter is a nonstandard mixture distribution. Monte Carlo simulations show the asymptotic distribution captures the characteristics of the sampling distributions and when this ridge estimator performs better than two-stage least squares.