Representation Theorem for Convex Nonparametric Least Squares

18 Pages Posted: 14 Jul 2008  

Timo Kuosmanen

Aalto University School of Business

Date Written: 2008-01

Abstract

We examine a nonparametric least-squares regression model that endogenously selects the functional form of the regression function from the family of continuous, monotonic increasing and globally concave functions that can be nondifferentiable. We show that this family of functions can be characterized without a loss of generality 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 us 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 modelling. 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.

Suggested Citation

Kuosmanen, Timo, Representation Theorem for Convex Nonparametric Least Squares (2008-01). Econometrics Journal, Vol. 11, Issue 2, pp. 308-325, July 2008. Available at SSRN: https://ssrn.com/abstract=1158681 or http://dx.doi.org/10.1111/j.1368-423X.2008.00239.x

Timo Kuosmanen (Contact Author)

Aalto University School of Business ( email )

P.O. Box 1210
Runeberginkatu 22-24
Helsinki, Finland 00101
Finland

HOME PAGE: http://www.aalto.fi

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