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Grupo de Estudos Monetários e Financeiros

Estudos do GEMF, N.º 01 de 2011


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(YEXS = x). The smoothed-distribution estimator ms (x) ≡E(YSXS = 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.

Nonparametric regression, Nadaraya-Watson, kernel density, conditional expectation estimator, conditional variance estimator, local polynomial estimator.

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