Nonparametric Estimation of a Generalized Additive Model with an Unknown Link Function

This paper is concerned with estimating the mean of a random variable Y conditional on a vector of covariates X under weak assumptions about the form of the conditional mean function. Fully nonparametric estimation is usually unattractive when X is multidimensional because estimation precision decreases rapidly as the dimension of X increases. This problem can be overcome by using dimension reduction methods such as single-index, additive, multiplicative, and partially linear models. These models are non-nested, however, so an analyst must choose among them. If an incorrect choice is made, the resulting model is misspecified and inferences based on it may be misleading. This paper describes an estimator for a new model that nests single-index, additive, and multiplicative models. The new model achieves dimension reduction without the need for choosing between single-index, additive, and multiplicative specifications. The centered, normalized estimators of the new model’s unknown functions are asymptotically normally distributed. An extension of the new model nests partially linear models.