Nonparametric estimation of homogeneous functions

Consider the regression y = f(x̃) + ε, where double-struck E sign(ε|x̃) = 0 and the exact functional form of f is unknown, although we do know that f is homogeneous of known degree r. Using a local linear approach, we examine two ways of nonparametrically estimating f: (i) a "direct" approach and (ii) a "projection based" approach. We show that depending upon the nature of the conditional variance var (ε|x̃), one approach may be asymptotically better than the other. Results of a small simulation experiment are presented to support our findings.