Kosuke et al.,
Having 15 dead minutes between meetings, I want to weigh in on this very interesting subject to maybe start a discussion and generate some response to thoughts & questions of my own on the topic:
What Kosuke, and Freedman as Kosuke cites him, say is of course correct.
Rendering estimated variance-covariances of estimated parameters consistent to certain forms of non-sphericity in error components does nothing to address anything that might be wrong with the parameter estimates. In the general cases where we usually tend to adopt MLE strategies, this can be as unhelpful as Freedman suggests, but I think the key issue in this regard is not ML or something other, per se, and not that we necessarily have the "wrong" parameter any time we might want to use such "robust" s.e. estimators. The key issue that triggers the more serious concerns of Freedman & of Imai, it seems to me, is the additive separability or non-separability of the error component in the model. When the error component is additively separable, such as in the linear regression model:
Y=XB+e
or the nonlinear one
Y=f(X,B)+e,
it remains quite possible that parameter estimates, Bhat, by least-squares or ML say, could be consistent or unbiased and consistent (but inefficient--always or usually? I'm not sure, but I think always) while standard errors could be just plain wrong in a way to which the appropriate "sandwich" estimator would be consistent ("robust").
In this case, it may well be a quite defensible strategy to content oneself with the consistency or unbiasedness & consistency of the parameter estimates, accept their inefficiency, and report consistent standard errors, rather than alternatively, or in addition to, attempting to parameterize the error v-cov mat such that one could gain some _asymptotic_ efficiency and unbiasedness in parameter and parameter-standard-deviation estimation--IF that parameterization is accurate and sufficient degrees of freedom remain thereafter for asymptotic properties to be consoling.
So, in the hierarchical _linear_ model, for example, a correctly specified E(y) component leaves parameter-standard-deviation estimation (and parameter-estimate efficiency) the only remaining issue(s).
However, if the stochastic component is not additively separable from the systematic one, then the possibility of unbiased &/or consistent systematic-component parameter-estimation while simultaneously having incorrect variance-covariance formula can (or necessarily does? I'm not sure, but I think necessarily does) vanish.
Beck & Katz's highly influential suggestion of PCSE's for TSCS data analysis seems an instructive example to me. If the spatial correlation to which PCSE's are consistent occurs only in the _additively_separable_ error component of the model--y=XB+e, where e=rWe+nu or some such--then PCSE application fits the stronger case for sandwich-estimator usage. If, on the other hand, the correlation arises because the model has spatial dynamics in the dependent variable--y=rWy+XB+e--then PCSE fits the problematic case because r and B are inconsistently estimated by LS and the stochastic component, e, being part of y right along with X in the rWy part, is not additively separable (despite the additive-lookingness of the equation).
Thoughts, Comments, Reactions? Answers to my implicit queries about "mays" vs. "musts"?
Thanks all, my 15 minutes is up,
Rob
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Robert (Rob) J. Franzese, Jr. US Mail: (ISR Room 4246)
Assoc. Prof. Political Science P.O. Box 1248
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Interim Director, Center for Political Studies,
Institute for Social Research,
University of Michigan, Ann Arbor: Sep 2006-May 2007
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> -----Original Message-----
> From: Political Methodology Society [mailto:[log in to unmask]] On
> Behalf Of Kosuke Imai
> Sent: Thursday, October 26, 2006 1:06 PM
> To: [log in to unmask]
> Subject: [POLMETH] "robust" standard errors
>
> I have always felt that many of my colleagues and students use the
> Huber
> robust standard errors without knowing what they actually mean because
> that's what everyone in the discipline does. In my intermediate method
> course, I try to teach my students that the confidence interval based on
> the robust standard error covers the ``wrong'' parameter with the
> ``correct'' coverage probability, and tell them that maybe they should be
> worried first about how wrong your estimator is.
>
> In any event, I came across an article by David Freedman in the most
> recent issue of the American Statistician gives a very nice discussion on
> this point, and thought it may be of interest to some people on the
> mailing list.
>
> On The So-Called "Huber Sandwich Estimator" and "Robust Standard Errors"
> Author: Freedman, David A.1
> Source: The American Statistician, Volume 60, Number 4, November 2006, pp.
> 299-302(4)
>
> Abstract: The "Huber Sandwich Estimator" can be used to estimate the
> variance of the MLE when the underlying model is incorrect. If the model
> is nearly correct, so are the usual standard errors, and robustification
> is unlikely to help much. On the other hand, if the model is seriously in
> error, the sandwich may help on the variance side, but the parameters
> being estimated by the MLE are likely to be meaninglessâexcept perhaps as
> descriptive statistics.
>
> Best,
> Kosuke
>
> -----------------------------------------------------
> Kosuke Imai Office: Corwin Hall 041
> Assistant Professor Phone: 609-258-6601
> Department of Politics eFax: 973-556-1929
> Princeton University Email: [log in to unmask]
> Princeton, NJ 08544-1012 http://imai.princeton.edu
> -----------------------------------------------------
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