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)numbered Display Equation

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

(4.2)numbered Display Equation

(4.3)numbered Display Equation

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

(4.4)

(4.5)

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

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