<p>Linear regression models have been extensively considered in the literature. However, in some practical applications they may not be appropriate all over the range of the covariate. In this paper, a more flexible model is proposed, introducing a response variable that is a function of the covariate plus a random error term. This function is assumed linear for large values of the covariate, specifically for values greater than a certain threshold. The penalized procedure estimates this threshold, focusing on a semiparametric approach with no parametric model assumed for the regression function for values smaller than the threshold. Consistency properties for the threshold estimator and the estimators of the linear model’s parameters are derived under mild assumptions. A numerical study explores small sample properties and underscores the importance of penalization. Real dataset analyses demonstrate the utility of penalized estimators.</p>