Kernel Smoothing

4 Semiparametric Regression

4.1 Partial linear models with constant slope

A partial linear model is a regression model containing a smooth nonparametric component and a linear parametric regression component. It is thus a semiparametric model, where the nonparametric component is unspecified except for some regularity conditions such as continuity, differentiability, etc. Below is an example of a partial linear model:

(4.1)

where is a column vector of explanatory variables and is a column vector of regression parameters, defined as

(4.2)

(4.3)

The nonparametric component m is a smooth function to be estimated in . The u_i are regression errors with zero mean and constant variance. We consider the case when u_i is a stationary long-memory process with a covariance function γ_u and a spectral density ϕ_u:

(4.4)

(4.5)

where for two functions a(v) and b(v), a(v) ∼ b(v) implies a(v)/