Nonparametric Identification of Two-Way Unobserved Heterogeneity

We study identification of two-way unobserved heterogeneity in the nonparametric panel regression $G_{it}=g(\alpha_i,\gamma_t)+\varepsilon_{it}$, where identification of the latent types reduces to constructing identified, \emph{injective} proxies for them. To this end we consider the singular value decomposition (SVD) of the bivariate regression function $g(\alpha,\gamma)$ on a product domain $\Omega_\alpha\times\Omega_\gamma$, whose left singular functions $\{u_r\}$ serve as proxies for the unobserved heterogeneity parameter $\alpha$. The arguments are symmetric for $\{v_r\}$ vis-\`a-vis $\gamma$. We work under an \emph{observational-equivalence simplification}: two values of $\alpha$ that induce the same conditional response $g(\alpha,\cdot)$ are identified, so that the response map $\alpha\mapsto g(\alpha,\cdot)$ is injective by construction. We show two things. First, this reduction is \emph{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 \emph{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.