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Re: st: The use of pweights with regress


From   Stas Kolenikov <skolenik@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: The use of pweights with regress
Date   Mon, 8 Nov 2004 13:45:40 -0500

> This is another way to remember the difference bewteen aweights and
> pweights.  aweights can be used to remove, so to speak,
> heteroskedasticity.  pweights are supposed to help you reduce
> sampling biases in the coefficient estimates, but their use may
> actually introduce heteroskedasticity, and hence automatic use of the
> robust covariance estimator is triggered.  (I hope I got this right!)

Sandwich covariance estimator is in fact a fundamental estimator: even
you have b = (X'X)^{-1} X'Y, and you are not willing to assume much
about the distribution of Y's conditional on X, then you are led to
Var[b] = Var[ (X'X)^{-1} X'Y ] = (X'X)^{-1} X' Var[Y] X' (X'X)^{-1}
which is the sandwich (or robust, which I think is a not very happy
choice of the name) variance estimator.

Another, and may be more appropriate, way to derive it is through the
estimating equations approach (which economists would call "moment
conditions"): the normal equations for b are X'e = 0, and the variance
of the estimators obtained from estimating equations / moment
conditions is (E[derivative of the estimating equations])^{-1} times
E[outer product of estimating equations] times (E[derivative of the
estimating equations])^{-1}, which again leads to the robust / White /
you name it formula. Now, in computing those expectations, it is
important to use the sampling weights (pweights). So I'd say it has
nothing to do with heteroskedasticity, although it is one of the most
common model misspecifications with which people tend to deal with
this sandwich estimator, at least within econometrics (survey
statisticicans would raise a brow if you'd say that the sandwich
estimator is only good to deal with heteroskedasticity). Rather, this
is a generic variance estimation problem, and a general approach to
solve it.

-- 
Stas Kolenikov
http://stas.kolenikov.name
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