A Smoothed-Distribution Form of Nadaraya-Watson Estimation
Ralph W. Bailey
University of Birmingham
John T. Addison
University of South Carolina, Queen’s University Belfast, and University of Coimbra
Given observation-pairs (xi ,yi ), i = 1,...,n , taken to be independent observations of the random pair (X ,Y), we sometimes want to form a nonparametric estimate of m(x) ≡ E(Y │X = x). Let YE have the empirical distribution of the yi , and let (XS ,YS ) have the kernel-smoothed distribution of the (xi ,yi ). Then the standard estimator, the Nadaraya-Watson form mNW(x) can be interpreted as E(YE│XS = x). The smoothed-distribution estimator ms (x) ≡E(YS│XS = x) is a more general form than mNW (x) and often has better properties. Similar considerations apply to estimating Var(Y│X = x), and to local polynomial estimation. The discussion generalizes to vector (xi ,yi ).
JEL Classification: C140.
Keywords: Nonparametric regression, Nadaraya-Watson, kernel density, conditional expectation estimator, conditional variance estimator, local polynomial estimator.