Semiparametric regression

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
where formula_2 is the dependent variable, formula_3 is a formula_4 vector of explanatory variables, formula_5 is a formula_4 vector of unknown parameters and formula_7. The parametric part of the partially linear model is given by the parameter vector formula_5 while the nonparametric part is the unknown function formula_9. The data is assumed to be i.i.d. with formula_10 and the model allows for a conditionally heteroskedastic error process formula_11 of unknown form. 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 formula_12 consistent estimator of formula_5 and then deriving an estimator of formula_9 from the nonparametric regression of formula_15 on formula_16 using an appropriate nonparametric regression method.
Index models.
A single index model takes the form
where formula_18, formula_19 and formula_20 are defined as earlier and the error term formula_21 satisfies formula_22. The single index model takes its name from the parametric part of the model formula_23 which is a "scalar" single index. The nonparametric part is the unknown function formula_24.
Ichimura's method.
The single index model method developed by Ichimura (1993) is as follows. Consider the situation in which formula_25 is continuous. Given a known form for the function formula_24, formula_20 could be estimated using the nonlinear least squares method to minimize the function
Since the functional form of formula_24 is not known, we need to estimate it. For a given value for formula_5 an estimate of the function
using kernel method. Ichimura (1993) proposes estimating formula_32 with
the leave-one-out nonparametric kernel estimator of formula_34.
Klein and Spady's estimator.
If the dependent variable formula_25 is binary and formula_3 and formula_37 are assumed to be independent, Klein and Spady (1993) propose a technique for estimating formula_5 using maximum likelihood methods. The log-likelihood function is given by
where formula_40 is the leave-one-out estimator.
Smooth coefficient/varying coefficient models.
Hastie and Tibshirani (1993) propose a smooth coefficient model given by
where formula_3 is a formula_43 vector and formula_44 is a vector of unspecified smooth functions of formula_16.
formula_46 may be expressed as