Representation Theorem for Convex Nonparametric Least Squares

We examine a nonparametric least squares regression model where the regression function is endogenously selected from the family of continuous, monotonic increasing and globally concave functions that can be nondifferentiable. We show that this family of functions is perfectly represented by a subset of continuous, piece-wise linear functions whose intercept and slope coefficients are constrained to satisfy the required monotonicity and concavity conditions. This representation theorem is useful at least in three respects. First, it enables is to derive an explicit representation for the regression function, which can be used for assessing marginal properties and for the purposes of forecasting and ex post economic modeling. Second, it enables us to transform the infinite dimensional regression problem into a tractable quadratic programming (QP) form, which can be solved by standard QP algorithms and solver software. Importantly, the QP formulation applies to the general multiple regression setting. Third, an operational computational procedure enables us to apply bootstrap techniques to draw statistical inference.