The partially linear model is defined by
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Theoretically, the partially linear model is a special case of the additive regression models (Hastie and Tibshrani (1990) and Stone (1985)), which allows easier interpretation of the effect of each variables and may be preferable to a completely nonparametric regression because of the well- known ``curse of dimensionality". On the other hand, partially linear model is more flexiable than the standard linear model since it combines both parametric and nonparametric components when it is believed that the response depends on some variable in linear relationship but is nonlinearly related to other particular independent variable.
To analyze the relationship between temperature and electricity usage, Engle, Granger, Rice and Weiss (1986) introduced the partially linear model and therefore provided a convenient framework for analysis of this problem. They have demonstrated that the partially linear model can be viewed as a way to flexibly correct misspecified parametric models. The following picture presents the nonparametric estimates of the weather-sensitive load for St. Louis as the solid curve and two sets of parametric estimates as the dashed curves. More detailed discussions are referred to Engle, Granger, Rice and Weiss (1986).
Since the work of Engle,
Granger, Rice and Weiss (1986), several methods are proposed to consider
the partially linear models. Robinson (1988)
constructs a feasible least squares estimator of
based on the nonparametric component estimated by a Nadaraya-Waston kernel
estimator.
Speckman (1988)
estimated the nonparametric component by ,
where
is a
matrix
of full rank and
is an additional parameter. The partially linear model (1)
is rewritten as a matrix form
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Engle,
Granger, Rice and Weiss (1986), Heckman
(1986) and Rice (1986) used spline
smoothing and defined the estimators of
and g as the solution of
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Chen (1988) proposed a piecewise polynomial to approximate nonparametric function and then derived the least squares estimator which is the same form as (3).
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