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.

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 ${\displaystyle Y_{i}}$ is the dependent variable, ${\displaystyle X_{i}}$ and ${\displaystyle Z_{i}}$ are ${\displaystyle p\times 1}$ vectors of explanatory variables, ${\displaystyle \beta }$ is a ${\displaystyle p\times 1}$ vector of unknown parameters and ${\displaystyle Z_{i}\in \operatorname {R} ^{q}}$. The parametric part of the partially linear model is given by the parameter vector ${\displaystyle \beta }$ while the nonparametric part is the unknown function ${\displaystyle g\left(Z_{i}\right)}$. The data is assumed to be i.i.d. with ${\displaystyle E\left(u_{i}|X_{i},Z_{i}\right)=0}$ and the model allows for a conditionally heteroskedastic error process ${\displaystyle E\left(u_{i}^{2}|x,z\right)=\sigma ^{2}\left(x,z\right)}$ of unknown form. This type of model was proposed by Robinson (1988) and extended to handle categorical covariates by Racine and Liu (2007).

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