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Re: st: qreg SEs

From (Roberto G. Gutierrez, StataCorp.)
Subject   Re: st: qreg SEs
Date   Thu, 27 Feb 2003 14:05:19 -0600

In the midst of the fray of posts on -qreg- standard errors, Scott Merryman
<> offers:

> "Their results [Koenker and Bassett (1978,1982), Huber (1967) Rogers(1993)
> and Stata (1997)] suggest an estimator for the asymptotic covariance matrix
> of the quantile regression estimator,

>             Est. Asy. Var[b] = (X'X)^-1(X'DX)(X'X)^-1

> Where D is a diagonal matrix containing weights

>  d = [q/f(0)]^2 if y - xB > 0 and [(1-q)/f(0)]^2 otherwise

> and f(0) is the true density of the disturbances evaluated at 0.  There is,
> at this point, a rather large hole in the theory. How one is to know f(0) is
> unclear.  Moreover, if one knew the true density, then the maximum
> likelihood estimator would be a preferable, and available, estimator.
> ......The bootstrap method of inferring statistical properties is well
> suited for this application.  Since the efficacy of the bootstrap has been
> established for this purpose, the search for a formula for standard errors
> of the LAD estimator is not really necessary."

> Koenker has a couple papers on quantile regression you may want to take a
> look at: (also, I
> believe, volume 26 of Empirical Economics was devoted to applications of
> quantile regression).

> In Koenker and Hallock's paper "Quantile Regression: An Introduction" (the
> longer version) they write (page 16):

> "Stata's command qreg also produces estimates of asymptotic standard errors
> based on iid error assumptions. Although they are designated as
> "Koenker-Bassett standard errors" the method bears little resemblance to the
> histospline approach of the cited reference. As described by Rogers (1993)
> the qreg's standard errors appear to be a variant of the iid Siddiqui method
> with a rather unfortunate choice of bandwidth.[see footnote 6] A consequence
> of the undersmoothing implied by the Stata rule is that the resulting
> standard errors are frequently considerably smaller than would be obtained
> with a more conventional bandwidth selection rule. This conclusion is
> supported by the Monte Carlo comparison reported in Rogers (1992)."

> This is discussed briefly in the reference manual [R] qreg (page 276).
> Koenker uses the Hall and Sheather (1988) bandwidth rule in his S and R
> implementation.

Scott pretty much gives the whole story here, and as a result has made my post
much shorter than what was originally planned.  I'll simply add that it has
been our plan for some time to improve on our ad hoc estimate of f(0), as
implemented in -qreg-.  This remains on our list of things to do.

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