For least squares with one predictor:
$y = \beta x + \epsilon$
If $x$ and $y$ are standardised prior to fitting (i.e. $\sim N(0,1)$), then:
- $\beta$ is the same as the Pearson correlation coefficient, $r$.
- $\beta$ is the same in the reflected regression: $x = \beta y + \epsilon$
For generalised least squares (GLS), does the same apply? I.e. if I standardise my data, can I obtained correlation coefficients directly from the regression coefficients?
From experimenting with data, the reflected GLS leads to different $\beta$ coefficients and also I'm not sure that I'm believing that the regression coefficients fit with my expected values for correlation. I know people quote GLS correlation coefficients, so I am wondering how they arrive at them and hence what they really mean?