Cui Rui, Li Yuhao, Song Xiaojun
arXiv 5 Jun 2025 · Econometrics
arXiv:2506.04900 · PDF · DOI · OpenAlex · Extracted main text
This paper provides a new theoretical lens for understanding the finite-sample performance of kernel-based specification tests, such as the Kernel Conditional Moment (KCM) test. Rather than introducing a fundamentally new test, we isolate and rigorously analyze the finite-sample distortion arising from the discrepancy between the empirical and population eigenspaces of the kernel operator. Using perturbation theory for compact operators, we demonstrate that the estimation error in directional components is governed by local eigengaps: components associated with small eigenvalues are highly unstable and contribute primarily noise rather than signal under fixed alternatives. Although this error vanishes asymptotically under the null, it can substantially degrade power in finite samples. This insight explains why the effective power of omnibus kernel tests is often concentrated in a low-dimensional subspace. We illustrate how truncating unstable high-frequency components--a natural consequence of our analysis--can improve finite-sample performance, but emphasize that the core contribution is the diagnostic understanding of why and when such instability occurs. The analysis is largely non-asymptotic and applies broadly to reproducing kernel Hilbert space-based inference.
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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 | Juan Carlos Escanciano (2024) A gaussian process approach to model checks | 0.941 | 6 | 3 | 83% |
| 2 | Miguel A Delgado, Manuel A Domńguez, and Pascal Lavergne (2006) Consistent tests of conditional moment restrictions | 0.843 | 4 | 3 | 75% |
| 3 | Lorenzo Rosasco, Mikhail Belkin, and Ernesto De Vito (2010) On learning with integral operators | 0.811 | 4 | 2 | 100% |
| 4 | J Carlos Escanciano (2009) On the lack of power of omnibus specification tests | 0.693 | 6 | 1 | 100% |
| 5 | Tailen Hsing and Randall L Eubank (2015) Theoretical foundations of functional data analysis, with an introduction to linear operators, volume 997 | 0.644 | 3 | 2 | 67% |
| 6 | Mikhail Belkin (2018) Approximation beats concentration? an approximation view on inference with smooth radial kernels | 0.644 | 2 | 2 | 100% |
| 7 | Herman J Bierens and Werner Ploberger (1997) Asymptotic theory of integrated conditional moment tests | 0.644 | 2 | 2 | 100% |
| 8 | Juan Carlos Escanciano and Sze-Chuan Goh (2014) Specification analysis of linear quantile models | 0.511 | 2 | 2 | 50% |
| 9 | Pedro HC Sant’Anna and Xiaojun Song (2019) Specification tests for the propensity score | 0.511 | 2 | 2 | 50% |
| 10 | J Carlos Escanciano (2006) A consistent diagnostic test for regression models using projections | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 24 scored citations.