Lp-NORM CONSISTENCIES OF NONPARAMETRIC ESTIMATES OF REGRESSION, HETEROSKEDASTICITY AND VARIANCE OF REGRESSION ESTIMATE WHEN DISTRIBUTION OF REGRESSOR IS KNOWN

When dealing with heteroskedastic models Y = μ(X) + c in econometrics and other disciplines, situations often arise (especially with structural models) where the probability distribution of the ((Rd-valued) regressor vector X is known, but postulations about the functional form of the regression μ(x), the heteroskedasticity a'2(x) = var(c I X=x) and the distribution of the disturbance term c are made. These three postulations generally lead to misspecification of • the models. This paper, based on a data set on (Y,X), considers nonparametric kernel estimators μ(x), c3.2(x) and c/(μ(x)), respectively, of the regression μ(x), the heteroskedasticity 2(x) and the asymptotic variances V(i.i(x)) of the regressi cr on estimate ii(x) for situations where only the probability distribution of X, say A is known. For an arbitrary subset A in the interior of the support of A and for 1 p < co, we establish convergences to zero, as the data set gets large, of the L -norms 11μ-μIIP = E I μ(x)-μ(x) I PdX(x) (with A a Rd for p = A •.. 1), 116.-2-cr2IIP (with A a Rd for p = 1) and II'()-V(μ)11P under certain moment conditions on Y but with no assumptions on the joint distribution of (Y,X) or the continuity of μ(x), o'2(x) or the density of X.