Nonparametric Estimation of Regression Functions under Restrictions on Partieal Derivatives

Economic theory often provides us with qualitative information on the properties of the functions in a model but rarely indicates their explicit functional form. Among these properties one can find monotonicity, concavity and supermodularity, which involve restricting the sign of the regression's partial derivatives. This paper focuses on such restrictions and provides a sieve estimator based on nonparametric least squares. The estimator enjoys three main advantages: it can handle a variety of restrictions, separately or simultaneously; it is easy to implement; and its geometric interpretation highlights the small sample benefits from using prior information on the shape of the regression function. The last is achieved by evaluating the metric entropy of the space of shape-restricted functions. The small sample efficiency gains are approximated.