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Theory"],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:p>We set up a formal framework to characterize encompassing of nonparametric models through the \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466624000100_inline1.png\"\/><jats:tex-math>\n$L^2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> distance. We contrast it to previous literature on the comparison of nonparametric regression models. We then develop testing procedures for the encompassing hypothesis that are fully nonparametric. Our test statistics depend on kernel regression, raising the issue of bandwidth\u2019s choice. We investigate two alternative approaches to obtain a \u201csmall bias property\u201d for our test statistics. We show the validity of a wild bootstrap method. 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