Generalised least squares: from regression coefficients to correlation coefficients? 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?
 A: The answer is yes, the linear regression coefficients are the correlations of the predictors with the response, but only if you use the correct coordinate system.
To see what I mean, recall that if $x_1, x_2, \ldots, x_n$ and $y$ are centered and standardized, then the correlation between each $x_i$ and $y$ is just the dot product $x_i^t y$.  Also, the least squares solution to linear regression is
$$ \beta = (X^t X)^{-1} X^t y $$
If it so happens that $X^{t} X = I$ (the identity matrix) then 
$$ \beta = X^t y $$
and we recover the correlation vector.  It is often attractive to recast a regression problem in terms of predictors $\tilde{x}_i$ that satisfy $\tilde{X}^t \tilde{X} = I$ by finding appropriate linear combinations of the original predictors that make this relation true (or equivalently, a linear change of coordinates); these new predictors are called the principal components.
So overall, the answer to your question is yes, but only when the predictors are themselves uncorrelated.  Otherwise, the expression
$$X^t X \beta = X^t y$$
shows that the betas must be mixed together with the correllations between the predictors themselves to recover the predictor-response correlations.
As a side note, this also explains why the result is always true for one variable linear regression.  Once the predictor vector $x$ is standardized, then:
$$ x_0^t x = \sum_i x_{i} = 0 $$
where $x_0$ is the intercept vector of all ones.  So the (two column) data matrix $X$ automatically satisfies $X^t X = I$, and the result follows.
