Granular Instrumental Variables in Large Panels: Identification and Inference Across Strong, Nearly Weak, and Weak GIV

Abstract

I develop the asymptotic theory for Granular Instrumental Variables (GIV) in large panels with both $N$ and $T$ growing. The strength of the GIV depends on the presence of dominant units with large shares. I find three regimes. First, when a few units dominate the aggregate, the instrument is strong. The GIV estimator is consistent and asymptotically normal at the standard $\sqrt{T}$ rate. Second, when large units stand out but do not dominate, the instrument weakens. But if the sample size ($T$) is larger than the cross-section ($N$), then the GIV estimator remains consistent and asymptotically normal, now at a rate slower than $\sqrt{T}$. Finally, when units are comparable in size and none stands out, the instrument is weak in the standard sense. Here the relative growth of $T$ and $N$ decides the outcome. If $N$ grows in proportion to $T$, the GIV estimator is inconsistent and has a non-standard distribution. Across all three regimes, the first-stage construction of the GIV changes the second-stage asymptotic variance. It replaces the structural error with its factor-residualised version. The correction has no determinate sign. Wald inference with the corrected variance is valid in the first two regimes. In the third I recommend a first-stage-corrected Anderson--Rubin test. Applying this theory to data, I find that copper and natural gas markets fall into the strong-instrument regime while crude oil is in the nearly weak regime, and I recover the short-run demand and supply elasticities of these commodities. Across the six elasticities the correction moves standard errors by up to a fifth, in both directions.

Job market paper. 165 pages, 2 figures. JEL: C33, C36, C55, C38, C12, Q41