1
WLS
WLS estimates regression model with different weights for different cases.
Notation
The following notation is used throughout this chapter unless otherwise stated:
n The number of cases
p The number of parameters for the model
y n1 vector with element yi, which represents the observed
dependent variable for case i
X np matrix with element xij , which represents the observed
value of the ith case of the jth independent variable
β p1 vector with element
β
j, which represents the regression
coefficient of the jth independent variable
w n1 vector with element wi, which represents the weight for
case i
Model
The linear regression model has the form of
yin
iii
=+ =X’,,,β
ε
1K 1
16
where Xi is the vector of covariates for the ith case, E
ε
i
16
=0, and
var
εσ
ii
w
16
=−12
. Assuming that
εε
1,,Kn follow a normal distribution, the log-
likelihood function of model (1) is
