1. BASICS

The partially linear model is defined by

 image
where X and T are (possibly) multidimensional regressors, image a vector of unknown parameters, image an unknown smooth function from image to image and image an error term with mean zero conditional on X and T.

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).

image

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 image based on the nonparametric component estimated by a Nadaraya-Waston kernel estimator.

Speckman (1988) estimated the nonparametric component by image, where image is a imagematrix of full rank and image is an additional parameter. The partially linear model (1) is rewritten as a matrix form

 image
The estimator of image based on (2) is
 image
where image is a projection matrix and image is a p x p identity matrix. Green, Jennion and Seheult (1985) proposed another class of estimates as follows.
image
by replacing image in (3) by a smoother operator.

Engle, Granger, Rice and Weiss (1986), Heckman (1986) and Rice (1986) used spline smoothing and defined the estimators of image and g as the solution of

 image
Rice (1986) and Heckman (1986) proposed some asymptotic results for a case of (1).

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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