7. Generalized Partial Linear Models

As indicated in the overview in Chapter 5, a partial linear model (PLM) consists of two additive components, a linear and a nonparametric part:

$$E(Y\vert{\boldsymbol{U}},{\boldsymbol{T}})={\boldsymbol{U}}^\top {\boldsymbol{\beta}}+ m({\boldsymbol{T}})$$

where ${\boldsymbol{\beta}}=(\beta_1,\ldots,\beta_p)^\top$ is a finite dimensional parameter and $m(\bullet)$ a smooth function. Here, we assume again a decomposition of the explanatory variables into two vectors, ${\boldsymbol{U}}$ and ${\boldsymbol{T}}$. The vector ${\boldsymbol{U}}$ denotes a $p$-variate random vector which typically covers categorical explanatory variables or variables that are known to influence the index in a linear way. The vector ${\boldsymbol{T}}$ is a $q$-variate random vector of continuous explanatory variables which is to be modeled in a nonparametric way. Economic theory or intuition should guide you as to which regressors should be included in ${\boldsymbol{U}}$ or ${\boldsymbol{T}}$, respectively.

Obviously, there is a straightforward generalization of this model to the case with a known link function $G(\bullet)$. We denote this semiparametric extension of the GLM

$$E(Y\vert{\boldsymbol{U}},{\boldsymbol{T}})=G\{{\boldsymbol{U}}^\top{\boldsymbol{\beta}}+m({\boldsymbol{T}})\}$$ (7.1)

as generalized partial linear model (GPLM).