Feasible Multivariate Nonparametric Regression Estimation Using Weak Separability

One of the main practical problems of nonparametric regression estimation is the curse of dimensionality. The curse of dimensionality arises because nonparametric regression estimates are dependent variable averages local to the point at which the regression function is to be estimated. The number of observations ‘local’ to the point of estimation decreases exponentially with the number of dimensions. The consequence is that the variance of unconstrained nonparametric regression estimators of multivariate regression functions is often so great that the unconstrained nonparametric regression estimates are of no practical use. In this paper I propose a new estimation method of weakly separable multivariate nonparametric regression functions. Weak separability is a weaker condition than required by other dimension–reduction techniques, although similar asymptotic variance reductions obtain. Indeed, weak separability is weaker than generalized additivity (see Härdle and Linton, 1996 and Horowitz, 1998). The proposed estimator is relatively easy to compute. Theoretical results in this paper include (i) a uniform law of large numbers for marginal integration estimators, (ii) a uniform law of large numbers for marginal summation estimators, (iii) a uniform law of large numbers for my new nonparametric regression estimator for weakly separable regression functions, (iv) both a uniform strong and weak law of large numbers for U–statistics, and (v) three central limit theorems for my nonparametric regression estimator for weakly separable regression functions. ∗This paper is based on research supported by a UBC Humanities and Social Sciences grant. I thank Don Andrews, Chuck Blackorby, Richard Blundell, Craig Brett, Erwin Diewert, David Green, Nancy Heckman, Joel Horowitz, Oliver Linton, Peter Robinson, Margaret Slade, Thanasis Stengos and seminar participants at the University of British Columbia (statistics and economics), the London School of Economics and Political Science, University College London, the University of Bristol, Yale University and the University of Groningen for useful suggestions.