Nonparametric tests for semiparametric regression models

• Published in: Test (Madrid, Spain) Vol. 32; no. 3; pp. 1106 - 1130
• Main Authors: Ferraccioli, Federico, Sangalli, Laura M., Finos, Livio
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-09-2023; Springer Nature B.V
• Subjects: Asymptotic properties; Economics; Finance; Insurance; Management; Mathematics and Statistics; Model testing; Nonparametric statistics; Original Paper; Regression models; Spline functions; Statistical Theory and Methods; Statistics; Statistics for Business; Sign-flip; Smoothing; Functional data analysis; primary 62G10; secondary 62G05; Roughness penalty
• ISSN: 1133-0686; 1863-8260
• DOI: 10.1007/s11749-023-00868-9

Semiparametric regression models have received considerable attention over the last decades, because of their flexibility and their good finite sample performances. Here we propose an innovative nonparametric test for the linear part of the models, based on random sign-flipping of an appropriate transformation of the residuals, that exploits a spectral decomposition of the residualizing matrix associated with the nonparametric part of the model. The test can be applied to a vast class of extensively used semiparametric regression models with roughness penalties, with nonparametric components defined over one-dimensional, as well as over multi-dimensional domains, including, for instance, models based on univariate or multivariate splines. We prove the good asymptotic properties of the proposed test. Moreover, by means of extensive simulation studies, we show the superiority of the proposed test with respect to current parametric alternatives, demonstrating its excellent control of the Type I error, accompanied by a good power, even in challenging data scenarios, where instead current parametric alternatives fail.