Some Words about Nonparametrics

If m is believed to be smooth, then the observations at Xi near x should contain information about the value of m at x. Thus it should be possible to use something like a local average of the data near x to construct an estimator of m(x).   --R. Eubank (1988. p.7)


Parametric models are fully determined up to a parameter (vector). The fitted models can easily be interpreted and estimated accurately if the underlying assumptions are correct. If, however, they are violated then parametric estimates may be inconsistent and give a misleading picture of the regression relationship.
Nonparametric models avoid restrictive assumptions of the functional form of the regression function m. However, they may be difficult to interpret and yield inaccurate estimates if the number of regressors is large. This has been appropriately  termed The Curse of Dimensionality. Semiparametric models combine components of parametric and nonparametric models, keeping the easy interpretability of the former and retaining some  of the flexibility of the latter.


Note: Nonparametric regression estimators are very flexible but their statistical precision decreases greatly if we include several explanatory variables in the model. The latter Caveat has been appropriately termed the curse of dimensionality. Consequently, researchers have tried to develop models and estimators which offer more flexibility than standard parametric regression but overcome the curse of dimensionality by employing some form of dimension reduction. Such methods usually combine features of parametric and nonparametric techniques. As a consequence, they are usually referred to as semiparametric methods. Further advantages of semiparametric methods are the possible inclusion of categorical variables (which can often only be included in a parametric way), an easy (economic) interpretation of the results, and the possibility of a part specification of a model.          --Wolfgang Hardle (2004)