Optimal Categorical Instrumental Variables

TL;DR

This paper introduces a new estimator for categorical instrumental variables that is asymptotically efficient and robust in settings with limited observations per category, outperforming traditional methods in practical applications.

Contribution

The paper proposes the CIV estimator, which leverages a regularization assumption to achieve semiparametric efficiency and asymptotic normality in categorical IV settings with few observations.

Findings

CIV is root-n asymptotically normal when the support size is known.

CIV achieves the same asymptotic variance as the oracle IV estimator.

CIV performs favorably compared to jackknife IV estimators in empirical application.

Abstract

This paper discusses estimation with a categorical instrumental variable in settings with potentially few observations per category. The proposed categorical instrumental variable estimator (CIV) leverages a regularization assumption that implies existence of a latent categorical variable with fixed finite support achieving the same first stage fit as the observed instrument. In asymptotic regimes that allow the number of observations per category to grow at arbitrary small polynomial rate with the sample size, I show that when the cardinality of the support of the optimal instrument is known, CIV is root-n asymptotically normal, achieves the same asymptotic variance as the oracle IV estimator that presumes knowledge of the optimal instrument, and is semiparametrically efficient under homoskedasticity. Under-specifying the number of support points reduces efficiency but maintains asymptotic normality. In an application that leverages judge fixed effects as instruments, CIV compares favorably to commonly used jackknife-based instrumental variable estimators.