Semi-parametric local variable selection under misspecification

Local variable selection aims to test for the effect of covariates on an outcome within specific regions. We outline a challenge that arises in the presence of non-linear effects and model misspecification. Specifically, for common semi-parametric methods even slight model misspecification can result in a high false positive rate, in a manner that is highly sensitive to the chosen basis functions. We propose a methodology based on orthogonal cut splines that avoids false positive inflation for any choice of knots, and achieves consistent local variable selection. Our approach offers simplicity, handles both continuous and categorical covariates, and provides theory for high-dimensional covariates and model misspecification. We discuss settings with either independent or dependent data. Our proposal allows including adjustment covariates that do not undergo selection, enhancing the model's flexibility. Our examples describe salary gaps associated with various discrimination factors at different ages, and the effects of covariates on functional data measuring brain activation at different times.