  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.
