Representation theorem for convex nonparametric least squares

Abstract

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.

Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints

“…When all inequalities of (6) are satisfied, we can employ the Afriat's Theorem to show that there exist a continuous, monotonic increasing and concave functionĝ that satisfies y i ¼ĝðx i Þ þ t i for all i = 1,…,n. As Kuosmanen (2008) emphasizes, the Afriat inequalities are the key to modeling the concavity axiom in the general multiple regression setting where there is no unambiguous way of sorting input vectors x. For estimating the shape of the production function, the coefficients (a i ,b i ) have a compelling economic interpretation: vector b i can be interpreted as the subgradient vector rgðx i Þ, and thus it represents the vector of marginal products of inputs at point x i .…”

“…Since conventional DEA literature emphasizes the fundamental philosophical difference between DEA and the regression techniques (e.g., Cooper et al 2004), the intimate links between DEA and regression analysis may not have attracted sufficient attention. In this respect, the recent studies Kuosmanen (2008) and Kuosmanen and Johnson (2010) have shown that DEA can be understood as a constrained special case of nonparametric least squares subject to shape constraints. More specifically, Kuosmanen and Johnson (2010) prove formally that the classic outputoriented DEA estimator can be computed in the singleoutput case by solving the convex nonparametric least squares (CNLS) problem (Hildreth 1954;Hanson and Pledger 1976;Groeneboom et al 2001a,b;Kuosmanen 2008) subject to monotonicity and concavity constraints that characterize the frontier, and a sign constraint on the regression residuals.…”

“…In this respect, the recent studies Kuosmanen (2008) and Kuosmanen and Johnson (2010) have shown that DEA can be understood as a constrained special case of nonparametric least squares subject to shape constraints. More specifically, Kuosmanen and Johnson (2010) prove formally that the classic outputoriented DEA estimator can be computed in the singleoutput case by solving the convex nonparametric least squares (CNLS) problem (Hildreth 1954;Hanson and Pledger 1976;Groeneboom et al 2001a,b;Kuosmanen 2008) subject to monotonicity and concavity constraints that characterize the frontier, and a sign constraint on the regression residuals. Thus, DEA can be naturally viewed as a nonparametric counterpart to the parametric programming approach of Aigner and Chu (1968).…”

“…The CNLS problems (4) and (6) are equivalent in the following sense (see Kuosmanen 2008, ''Appendix'', for a formal proof).…”

“…However, the input vector x 2 < m þ cannot be sorted prior to the estimation such that g(x i ) C g(x i-1 ), i = 2,…,n. To resolve this challenge, Kuosmanen (2008) has shown that the infinite dimensional CNLS problem (4) has an equivalent finite dimensional representation, which can be stated as the following quadratic programming (QP) problem…”