  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 introduced
by considering a regression model $Y=r(X)+\varepsilon$ where the regression
function $r(\cdot)$ is assumed to be linear for large values in the domain of
the predictor variable $X$. More precisely, we assume that
$r(x)=\alpha_0+\beta_0 x$ for $x> u_0$, where the value $u_0$ is identified as
the smallest value satisfying such a property. A penalized procedure is
introduced to estimate the threshold $u_0$. The considered proposal focusses on
a semiparametric approach since no parametric model is assumed for the
regression function for values smaller than $u_0$. Consistency properties of
both the threshold estimator and the estimators of $(\alpha_0,\beta_0)$ are
derived, under mild assumptions. Through a numerical study, the small sample
properties of the proposed procedure and the importance of introducing a
penalization are investigated. The analysis of a real data set allows us to
demonstrate the usefulness of the penalized estimators.
