Nonparametric Identification of Two-Way Unobserved Heterogeneity

Hugo Freeman, Dennis Kristensen

arXiv 27 Aug 2026 · Econometrics

arXiv:2608.27155 · PDF · Extracted main text

Abstract

We study identification of two-way unobserved heterogeneity in the nonparametric panel regression $G_{it}=g(α_i,γ_t)+\varepsilon_{it}$, where identification of the latent types reduces to constructing identified, injective proxies for them. To this end we consider the singular value decomposition (SVD) of the bivariate regression function $g(α,γ)$ on a product domain $Ω_α\timesΩ_γ$, whose left singular functions ${u_r}$ serve as proxies for the unobserved heterogeneity parameter $α$. The arguments are symmetric for ${v_r}$ vis-à-vis $γ$. We work under an observational-equivalence simplification: two values of $α$ that induce the same conditional response $g(α,\cdot)$ are identified, so that the response map $α\mapsto g(α,\cdot)$ is injective by construction. We show two things. First, this reduction is equivalent to injectivity of the full collection of left singular eigenfunctions, so no further condition is needed over the infinite collection ${u_r}_{r\ge1}$. Second, under a single additional local injectivity condition, a finite collection of leading eigenfunctions $U_R=(u_1^{\top},\dots,u_R^{\top})^{\top}$ is injective for all sufficiently large $R$. The proof reduces a global univalence question to a local first-order condition plus a topological compactness argument, bypassing the global Jacobian conditions usually required.

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