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The finite sample properties of estimators are usually understood orapproximated using asymptotic theories. Two main asymptotic constructions havebeen used to characterize the presence of many instruments. The first assumesthat the number of instruments increases with the sample size. I demonstratethat in this case, one of the key assumptions used in the asymptoticconstruction may imply that the number of ``effective" instruments should befinite, resulting in an internal contradiction. The second asymptoticrepresentation considers that the number of instrumental variables (IVs) may befinite, infinite, or even a continuum. The number does not change with thesample size. In this scenario, the regularized estimator obtained depends onthe topology imposed on the set of instruments as well as on a regularizationparameter. These restrictions may induce a bias or restrict the set ofadmissible instruments. However, the assumptions are internally coherent. Thelimitations of many IVs asymptotic assumptions provide support for finitesample distributional studies to better understand the behavior of many IVestimators.

A Note on the Topology of the First Stage of 2SLS with Many Instruments