Semiparametric Estimation under Shape Constraints

Economic theory provides the econometrician with substantial structure and restrictions necessary to give economic interpretation to empirical findings. In many settings, such as those in consumer demand and production studies, these restrictions often take the form of monotonicity and curvature constraints. Although such restrictions may be imposed in certain parametric empirical settings in a relatively straight-forward fashion by utilizing parametric restrictions or particular parametric functional forms (Cobb-Douglas, CES, etc.), imposing such restrictions in semiparametric models is often problematic. Our paper provides one solution to this problem by incorporating penalized splines, where monotonicity and curvature constraints are maintained via integral transformations of spline basis expansions. We derive the estimator, algorithms for its solution, and its large sample properties. Inferential procedures are discussed as well as methods for selecting the smoothing parameter. We also consider multiple regressions under the framework of additive models. We conduct a series of Monte Carlo simulations to illustrate the finite sample properties of the estimator. We apply the proposed methods to estimate two canonical relationships, one in consumer behavior and one in producer behavior. These two empirical settings examine the relationship between individuals' degree of optimism and risk tolerance and a production function with multiple inputs.