AI 中文总结
该研究针对非参数面板回归模型，通过奇异值分解（SVD）等方法解决双向未观测异质性的非参数识别问题，规避了全局雅可比条件，为有限主导本征函数的单射性提供了理论支撑。
AI 中文摘要
我们研究非参数面板回归$G_{it}=g(\alpha_i,\gamma_t)+\varepsilon_{it}$中双向未观测异质性的识别问题，其中潜在类型的识别可简化为为其构造可识别的单射代理变量。为此，我们考虑双变量回归函数$g(\alpha,\gamma)$在乘积域$\Omega_\alpha\times\Omega_\gamma$上的奇异值分解（SVD），其左奇异函数集$\{u_r\}$用作未观测异质性参数$\alpha$的代理变量，右奇异函数集$\{v_r\}$对$\gamma$而言同理。我们在“观测等价简化”假设下开展研究：两个诱导相同条件响应$g(\alpha,\cdot)$的$\alpha$值是可识别的，因此响应映射$\alpha\mapsto g(\alpha,\cdot)$在构造上是单射的。我们证明两点：第一，这种简化等价于全部左奇异本征函数集的单射性，在无限集$\{u_r\}_{r\ge1}$上无需额外条件；第二，在单一附加的“局部单射性”条件下，对于所有足够大的$R$，有限的主导本征函数集$U_R=(u_1^\top,\dots,u_R^\top)^\top$对$\alpha$是单射的。该证明将全局单值性问题简化为局部一阶条件加拓扑紧性论证，规避了通常所需的全局雅可比条件。
英文摘要
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, \emph{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 \emph{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 \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.
