Generalised least squares: from regression coefficients to correlation coefficients?

Hi All, I asked this on Stack Exchange, but it seems no-one there knows the answer. I wonder if anyone on talkstats can shed some light on it.

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.

Thanks for considering this