  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.
