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

I develop the asymptotic theory of instrument strength 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.

I formalise what dominance means and characterise three regimes of instrument strength.

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.

When large units stand out but do not dominate, the instrument weakens.

But I show that the parameter of interest remains recoverable.

The GIV estimator remains consistent and asymptotically normal, now at a rate slower than $\sqrt{T}$.

When units are comparable in size and none stands out, the instrument is weak in the standard sense.

The GIV estimator is inconsistent and has a non-standard distribution.

Wald inference is reliable only outside the weak regime.

When the instrument is weak, I recommend Anderson-Rubin confidence sets.

In practice, the instrument must be constructed in a first stage.

I show that the feasible estimator attains the same rate, but its asymptotic variance picks up an additional term from the first-stage estimation.

Valid inference must use standard errors that account for this term.

I apply the GIV estimator with the correct standard errors to recover the short-run demand elasticities of three commodities: refined copper, crude oil, and natural gas.