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

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Feb 14, 2016

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It is quantitative

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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 " #Cov =# "Covariance Matrix

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