Question

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

Answer 1

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 " `Cov =` "Covariance Matrix

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