Abstract: Component functions of an additive model can be estimated at univariate rate of convergence, by such methods as backfitting, marginal integration, etc. An alternative direct method is developed when the components are proportional. This new direct local polynomial estimator requires as little computing as a univariate estimator, less than the integration method by a factor of the sample size. Combination with one-step backfitting yields an improved estimator with univariate rate of convergence and ``oracle'' efficiency, and retains comparable computational efficiency. Monte-Carlo results indicate good performance of both estimators, which work much better than the integration method. The direct method is applied to a GARCH type model, illustrated by an analysis of the daily returns of Deutsche Mark/British Pound (DEM/GBP).
Key words and phrases: Coefficient parameter, DEM/GBP daily returns, efficient estimator, equivalent kernel, local polynomial, nonparametric GARCH, rule-of-thumb bandwidth.