  This paper provides ANOVA inference for nonparametric local polynomial
regression (LPR) in analogy with ANOVA tools for the classical linear
regression model. A surprisingly simple and exact local ANOVA decomposition is
established, and a local R-squared quantity is defined to measure the
proportion of local variation explained by fitting LPR. A global ANOVA
decomposition is obtained by integrating local counterparts, and a global
R-squared and a symmetric projection matrix are defined. We show that the
proposed projection matrix is asymptotically idempotent and asymptotically
orthogonal to its complement, naturally leading to an $F$-test for testing for
no effect. A by-product result is that the asymptotic bias of the ``projected''
response based on local linear regression is of quartic order of the bandwidth.
Numerical results illustrate the behaviors of the proposed R-squared and
$F$-test. The ANOVA methodology is also extended to varying coefficient models.
