ABSTRACT
We noted in previous chapters that, if the random error terms εi in the population regression line Yi = βo + β1Xi + εi, i = 1, … ,n, are independent with εi: N(0, σε), then the least-squares estimators βˆo and βˆ1 are maximum-likelihood estimators of βo and β1, respectively. That is, if εi follows the probability density function
f e i ni i( ) ,..., ,ε π σε
1 2
then the likelihood function for the sample random variables Yi, i = 1,…,n, is
L( , , ; , ..., , )β β σ π σ
21 2
= ( ) − σε
= ∑ ,
or the log-likelihood function is
log log log .L = − ( ) − ( ) − =
∑n n iin2 2 1
1 π σ
σ εε
(10.1)
1=∑ . So if the normality assumption concerning εi is tenable, then applying the OLS criterion, choose βˆo and βˆ1 so as to minimize eii
1=∑ is appropriate, where e Y Yi i i= − ˆ is the ith deviation or residual from the sample regression line.
