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

${\displaystyle Y_{i}=X'_{i}\beta +g\left(Z_{i}\right)+u_{i},,円\quad i=1,\ldots ,n,,円}${\displaystyle Y_{i}=X'_{i}\beta +g\left(Z_{i}\right)+u_{i},,円\quad i=1,\ldots ,n,,円}

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

This method is implemented by obtaining a ${\displaystyle {\sqrt {n}}}${\displaystyle {\sqrt {n}}} consistent estimator of ${\displaystyle \beta }${\displaystyle \beta } and then deriving an estimator of ${\displaystyle g\left(Z_{i}\right)}${\displaystyle g\left(Z_{i}\right)} from the nonparametric regression of ${\displaystyle Y_{i}-X'_{i}{\hat {\beta }}}${\displaystyle Y_{i}-X'_{i}{\hat {\beta }}} on ${\displaystyle z}${\displaystyle z} using an appropriate nonparametric regression method.^[1]

A single index model takes the form

${\displaystyle Y=g\left(X'\beta _{0}\right)+u,,円}${\displaystyle Y=g\left(X'\beta _{0}\right)+u,,円}

where ${\displaystyle Y}${\displaystyle Y}, ${\displaystyle X}${\displaystyle X} and ${\displaystyle \beta _{0}}${\displaystyle \beta _{0}} are defined as earlier and the error term ${\displaystyle u}${\displaystyle u} satisfies ${\displaystyle E\left(u|X\right)=0}${\displaystyle E\left(u|X\right)=0}. The single index model takes its name from the parametric part of the model ${\displaystyle x'\beta }${\displaystyle x'\beta } which is a scalar single index. The nonparametric part is the unknown function ${\displaystyle g\left(\cdot \right)}${\displaystyle g\left(\cdot \right)}.

The single index model method developed by Ichimura (1993) is as follows. Consider the situation in which ${\displaystyle y}${\displaystyle y} is continuous. Given a known form for the function ${\displaystyle g\left(\cdot \right)}${\displaystyle g\left(\cdot \right)}, ${\displaystyle \beta _{0}}${\displaystyle \beta _{0}} could be estimated using the nonlinear least squares method to minimize the function

${\displaystyle \sum _{i=1}\left(Y_{i}-g\left(X'_{i}\beta \right)\right)^{2}.}${\displaystyle \sum _{i=1}\left(Y_{i}-g\left(X'_{i}\beta \right)\right)^{2}.}

Since the functional form of ${\displaystyle g\left(\cdot \right)}${\displaystyle g\left(\cdot \right)} is not known, we need to estimate it. For a given value for ${\displaystyle \beta }${\displaystyle \beta } an estimate of the function

${\displaystyle G\left(X'_{i}\beta \right)=E\left(Y_{i}|X'_{i}\beta \right)=E\left[g\left(X'_{i}\beta _{o}\right)|X'_{i}\beta \right]}${\displaystyle G\left(X'_{i}\beta \right)=E\left(Y_{i}|X'_{i}\beta \right)=E\left[g\left(X'_{i}\beta _{o}\right)|X'_{i}\beta \right]}

using kernel method. Ichimura (1993) proposes estimating ${\displaystyle g\left(X'_{i}\beta \right)}${\displaystyle g\left(X'_{i}\beta \right)} with

${\displaystyle {\hat {G}}_{-i}\left(X'_{i}\beta \right),,円}${\displaystyle {\hat {G}}_{-i}\left(X'_{i}\beta \right),,円}

the leave-one-out nonparametric kernel estimator of ${\displaystyle G\left(X'_{i}\beta \right)}${\displaystyle G\left(X'_{i}\beta \right)}.

If the dependent variable ${\displaystyle y}${\displaystyle y} is binary and ${\displaystyle X_{i}}${\displaystyle X_{i}} and ${\displaystyle u_{i}}${\displaystyle u_{i}} are assumed to be independent, Klein and Spady (1993) propose a technique for estimating ${\displaystyle \beta }${\displaystyle \beta } using maximum likelihood methods. The log-likelihood function is given by

${\displaystyle L\left(\beta \right)=\sum _{i}\left(1-Y_{i}\right)\ln \left(1-{\hat {g}}_{-i}\left(X'_{i}\beta \right)\right)+\sum _{i}Y_{i}\ln \left({\hat {g}}_{-i}\left(X'_{i}\beta \right)\right),}${\displaystyle L\left(\beta \right)=\sum _{i}\left(1-Y_{i}\right)\ln \left(1-{\hat {g}}_{-i}\left(X'_{i}\beta \right)\right)+\sum _{i}Y_{i}\ln \left({\hat {g}}_{-i}\left(X'_{i}\beta \right)\right),}

where ${\displaystyle {\hat {g}}_{-i}\left(X'_{i}\beta \right)}${\displaystyle {\hat {g}}_{-i}\left(X'_{i}\beta \right)} is the leave-one-out estimator.

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

${\displaystyle Y_{i}=\alpha \left(Z_{i}\right)+X'_{i}\beta \left(Z_{i}\right)+u_{i}=\left(1+X'_{i}\right)\left({\begin{array}{c}\alpha \left(Z_{i}\right)\\\beta \left(Z_{i}\right)\end{array}}\right)+u_{i}=W'_{i}\gamma \left(Z_{i}\right)+u_{i},}${\displaystyle Y_{i}=\alpha \left(Z_{i}\right)+X'_{i}\beta \left(Z_{i}\right)+u_{i}=\left(1+X'_{i}\right)\left({\begin{array}{c}\alpha \left(Z_{i}\right)\\\beta \left(Z_{i}\right)\end{array}}\right)+u_{i}=W'_{i}\gamma \left(Z_{i}\right)+u_{i},}

where ${\displaystyle X_{i}}${\displaystyle X_{i}} is a ${\displaystyle k\times 1}${\displaystyle k\times 1} vector and ${\displaystyle \beta \left(z\right)}${\displaystyle \beta \left(z\right)} is a vector of unspecified smooth functions of ${\displaystyle z}${\displaystyle z}.

${\displaystyle \gamma \left(\cdot \right)}${\displaystyle \gamma \left(\cdot \right)} may be expressed as

${\displaystyle \gamma \left(Z_{i}\right)=\left(E\left[W_{i}W'_{i}|Z_{i}\right]\right)^{-1}E\left[W_{i}Y_{i}|Z_{i}\right].}${\displaystyle \gamma \left(Z_{i}\right)=\left(E\left[W_{i}W'_{i}|Z_{i}\right]\right)^{-1}E\left[W_{i}Y_{i}|Z_{i}\right].}

• Nonparametric regression
• Effective degree of freedom

• Robinson, P.M. (1988). "Root-n Consistent Semiparametric Regression". Econometrica. 56 (4). The Econometric Society: 931–954. doi:10.2307/1912705. JSTOR 1912705.
• Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. ISBN 978-0-691-12161-1.
• Racine, J.S.; Qui, L. (2007). "A Partially Linear Kernel Estimator for Categorical Data". Unpublished Manuscript, Mcmaster University.
• Ichimura, H. (1993). "Semiparametric Least Squares (SLS) and Weighted SLS Estimation of Single Index Models". Journal of Econometrics. 58 (1–2): 71–120. doi:10.1016/0304-4076(93)90114-K.
• Klein, R. W.; R. H. Spady (1993). "An Efficient Semiparametric Estimator for Binary Response Models". Econometrica. 61 (2). The Econometric Society: 387–421. CiteSeerX 10.1.1.318.4925 . doi:10.2307/2951556. JSTOR 2951556.
• Hastie, T.; R. Tibshirani (1993). "Varying-Coefficient Models". Journal of the Royal Statistical Society, Series B. 55: 757–796.

• Nonparametric statistics

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