I think we can blame Tim Hesterberg for the confusion:
He writes
"
I'll add:
* inverse-variance weights, where var(y for observation) = 1/weight (as
opposed to just being inversely proportional to the weight) *
"
And, although I'm not a native English speaker, I think there's a spurious
comma in there. The intention was clearly to have this as a 4th type of weight
which is a special case of inverse-variance weights, not as an elaboration on
the definition of inv.var. weights.
I.e., it is the difference between
Motorists who are reckless drivers...
and
Motorists, who are reckless drivers...
-pd
In fact, that wasn't what caused the confusion as I have understood what
he meant despite the problem with the comma.
But I got the idea now, R uses weighted least squares and:
"Var[\epsilon | X] = \Omega" (equal, not proportion)
(https://en.wikipedia.org/wiki/Generalized_least_squares) (since WLS is
just a special case of GLS)
"The weights should, ideally, be equal to the reciprocal of the variance
of the measurement."
(https://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)#Weighted_linear_least_squares)
I guess I need to find another strategy to use proportional weights
(weights know up to a constant, as John says).
So, thank you much to you all, and sorry the inconvenience I caused.
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