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.

翻译：本文探讨了在每类观测值可能较少的情况下，使用分类工具变量进行估计的方法。提出的分类工具变量估计量（CIV）利用正则化假设，该假设意味着存在一个具有固定有限支撑的潜在分类变量，其第一阶段的拟合效果与观测工具相同。在渐近框架下，允许每类观测数量以任意小的多项式速率随样本量增长，本文证明：当最优工具的支持基数已知时，CIV估计量是根号n-渐近正态的，其渐近方差与预知最优工具的工具变量估计量相同，并且在同方差条件下达到半参数有效。低估支持点的数量会降低效率，但保持渐近正态性。