On Tue, 2 Feb 2021, Sven Schreiber wrote:

> Well, in a sense that's the classical example why the simple algebraic
> formulas shouldn't be directly applied in numerical computations. Given
> floating-point representation the collinearity is not "so" exact that
> the inverse-computing algorithm must fail. So it returns an inverse with
> absurdly large numbers (check invpd(x'x)), and the whole thing is of
> course numerically unstable, because we know that it isn't even
> identified in theory.
>
> So that's exactly why mols(), svd() and so forth exist; not only in
> gretl, but in any numerical algebra software.

On top of this, you didn't try the coolest variant:

# Model 1
ols GPA xlist --simple
b1 = $coeff
yh1 = {$yhat}
# Model 4
b4 = x\y
yh4 = x * b4
<hansl>

you may check that the fitted values are the same ;)