In statistics, semiparametric regression includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. Semiparametric regression models are a particular type of semiparametric modelling and, since semiparametric models contain a parametric component, they rely on parametric assumptions and may be misspecified and inconsistent, just like a fully parametric model.

Methods

Many different semiparametric regression methods have been proposed and developed. The most popular methods are the partially linear, index and varying coefficient models.

Partially linear models

A partially linear model is given by

and the model allows for a conditionally heteroskedastic of unknown form. This type of model was proposed by Robinson (1988) and extended to handle categorical covariates by Racine and Li (2007).

from the nonparametric regression using an appropriate nonparametric regression method.

Index models

A single index model takes the form

which is a scalar.

Ichimura's method

could be estimated using the nonlinear least squares method to minimize the function

an estimate of the function

using kernel with

the leave-one-out nonparametric kernel.

Klein and Spady's estimator

are assumed to be independent using maximum likelihood methods. The log-likelihood function is given by

is the leave-one-out estimator.

Smooth coefficient/varying coefficient models

Hastie and Tibshirani (1993) propose a smooth coefficient model given by

.

may be expressed as

See also

• Nonparametric regression

• Effective degree of freedom

Notes

References