Estimating the Local Mean Function

The linear regression model

Y

t

=m

X

t

+b+σ

ϵ

t

relies on a number of crucial assumptions, the most important of which is a linear relationship between the covariate

X

t

and the dependent variable

Y

t

. One way of generalizing the standard linear regression model is to extend it to the nonlinear form

Y

t

=m(

X

t

)+σ(

X

t

)

ϵ

t

, where

m(·)

and

σ(·)

are real-valued functions and

ϵ

t

is an independent, identically distributed sequence of standard normal random variables. We study five different sets of data (three "real" and two simulated) to illustrate how kernel-weighted least-squares regression can be used to estimate the local mean function

m(x)=E[

Y

t

|

X

t

=x]

.