Breaking the Curse of Dimensionality

This paper proposes a new nonparametric estimator for general regression functions with multiple regressors. The method used here is motivated by a remarkable result derived by Kolmogorov (1957) and later tightened by Lorentz (1966). In short, they show that any continuous function of multiple variables can be written as univariate functions. As it stands, this representation is difficult to estimate because of its lack of smoothness. Hence we propose to use a generalization of their representation that allows for the univariate functions to be differentiable. The model will be estimated using B-splines, which have excellent numerical properties. A crucial restriction in this representation is that some of the functions must be increasing. One of the main contributions of this paper is that we develop a method for imposing monotonicity on the cubic B-splines, a priori, such that the estimator is dense in the set of all monotonic cubic B-splines. A simulation experiment shows that the estimator works well when optimization is performed by using the back-fitting algorithm. The monotonic restriction has many other applications besides the one presented here, such as estimating a demand function. With only r + 2 more constraints, it is also possible to impose concavity.