# Is the difference between an OLS regression and a GLS regression qualitative or quantitative?

Feb 14, 2016

It is quantitative

#### Explanation:

Ordinary least squares is a technique for estimating unknown parameters in a linear regression model. OLS yield the maximum likelihood in a vector β, assuming the parameters have equal variance and are uncorrelated, in a noise ε - homoscedastic.

vec(y)=Xvec(β)+vec(ε)

Generalized least squares allows this approach to be generalized to give the maximum likelihood estimate β when the noise is of unequal variance (heteroscedasticity). Typically this leads to mathematical treatment that presents the two as follows:
OLS: vecY=Xbeta+ε " where " ε~N(0,σ²I)
GLS: vecY=Xbeta+η " where " η~N(0,σ²Cov)
Note the formulation for the two approaches results in real structural and quantitative difference. Notice the two are governed by two different Gauss distribution the
N(0, sigma^2M); M = I " or " M=Cov part
Where: $I =$ "Identity Matrix and " $C o v =$ "Covariance Matrix

Cheers!