arXiv 2 Jul 2026 · Econometrics
arXiv:2607.02095 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Douglas Staiger and James H Stock (1997) Instrumental Variables Regression with Weak Instruments | 1.000 | 11 | 5 | 100% |
| 2 | Xavier Gabaix and Ralph S J Koijen (2024) Granular Instrumental Variables | 1.000 | 8 | 5 | 100% |
| 3 | Bertille Antoine and Eric Renault (2021) GMM with Nearly-Weak Identification | 1.000 | 7 | 4 | 100% |
| 4 | T Anderson and H Rubin (1949) Estimators for the parameters of a single equation in a complete set of stochastic equations | 1.000 | 5 | 3 | 100% |
| 5 | Jushan Bai (2003) Inferential Theory for Factor Models of Large Dimensions | 1.000 | 5 | 3 | 100% |
| 6 | Saman Banafti and Tae-Hwy Lee (2022) Inferential Theory for Granular Instrumental Variables in High Dimensions | 0.679 | 16 | 4 | 31% |
| 7 | Seung C Ahn and Alex R Horenstein (2013) Eigenvalue ratio test for the number of factors | 0.644 | 2 | 2 | 100% |
| 8 | Timothy B Armstrong (1961) Large market asymptotics for differentiated product demand estimators with economic models of supply | 0.644 | 2 | 2 | 100% |
| 9 | Jushan Bai and Serena Ng (2002) Determining the Number of Factors in Approximate Factor Models | 0.644 | 2 | 2 | 100% |
| 10 | Xavier Gabaix (2011) The Granular Origins of Aggregate Fluctuations | 0.511 | 2 | 2 | 50% |
Showing the top 10 of 39 scored citations.