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.
Many different semiparametric regression methods have been proposed and developed.
The most popular methods are the partially linear, index and varying coefficient models.
A partially linear model is given by where
is the dependent variable,
vector of explanatory variables,
vector of unknown parameters and
The parametric part of the partially linear model is given by the parameter vector
while the nonparametric part is the unknown function
and the model allows for a conditionally heteroskedastic error process
This type of model was proposed by Robinson (1988) and extended to handle categorical covariates by Racine and Li (2007).
This method is implemented by obtaining a
using an appropriate nonparametric regression method.
[1] A single index model takes the form where
are defined as earlier and the error term
The single index model takes its name from the parametric part of the model
which is a scalar single index.
The nonparametric part is the unknown function
The single index model method developed by Ichimura (1993) is as follows.
Given a known form for the function
could be estimated using the nonlinear least squares method to minimize the function Since the functional form of
an estimate of the function using kernel method.
Ichimura (1993) proposes estimating
with the leave-one-out nonparametric kernel estimator of
are assumed to be independent, Klein and Spady (1993) propose a technique for estimating
using maximum likelihood methods.
Hastie and Tibshirani (1993) propose a smooth coefficient model given by where
is a vector of unspecified smooth functions of