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Re: st: hausman and xthausman after panel fe, re


From   "Mark Schaffer" <M.E.Schaffer@hw.ac.uk>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: hausman and xthausman after panel fe, re
Date   Tue, 23 Aug 2005 16:55:03 +0100

An exceedingly rare correction to a posting by Vince - see bottom, 
where my name comes up.

From:           	vwiggins@stata.com (Vince Wiggins, StataCorp)
To:             	statalist@hsphsun2.harvard.edu
Subject:        	Re: st: hausman and xthausman after panel fe, re
Date sent:      	Tue, 23 Aug 2005 10:17:26 -0500
Send reply to:  	statalist@hsphsun2.harvard.edu

> Carl Nelson <chnelson3@insightbb.com> asks why he gets different results from
> the -hausman- command and the deprecated -xthausman- command.
> 
> > This question concerns problem 10.9 in Jeff Wooldridge's book
> > Econometric Analysis of Cross Section and Panel Data. In this
> > exercise, which I gave to some students in a course this summer,
> > using Cornwell.dat students are asked to estimate xtreg, fe and
> > xtreg, re and perform the hausman test.  Using the old xthausman
> > syntax the result is a significant test statistic (approximately 121
> > for a chisquared(11) rv). Using the newer hausman syntax the result
> > is a negative chisquared statistic and warning about violation of
> > assumptions.  I constructed the statistic from the saved results
> > e(b) and e(V) and I got the same result as the newer hausman syntax.
> > [...]
> 
> It is rare that -hausman- and -xthausman- produce different statistics, but I
> recommend that Carl believe the results from -hausman- and not -xthausman-.
> The main reason -xthausman- was undocumented (and now works only under version
> control) was that that it could be fooled by non positive definite (PD)
> differenced covariance matrices or by variables with degenerate panel
> behavior.
> 
> I posted a rather lengthy discussion of the issues back in March of 2002.
> This post predates some of the statalist archives, so at the risk of being
> long-winded yet again, let me quote from that posting.
> 
> ---------------------------------- Begin excerpts --------------------------
> 
> Eric Neumayer <E.Neumayer@lse.ac.uk> asks why he is getting different results
> from -xthaus- and -hausman- when testing for fixed vs. random effects after
> estimation with -xtreg-. [...]
> 
> I believe there are open questions about Hausman tests in situations like
> Eric's, see the explanation that follows.
> 
> 
> Preliminaries
> -------------
> 
> It is hard to discuss the Hausman test without being specific about how the
> test is performed.  Let B be the parameter estimates from a fully efficient
> estimator (random-effects regression in this case) and b be the estimates from
> a less efficient estimator (fixed-effects regression), but one that is
> consistent in the face of one or more violated assumptions, in this case that
> the effects are correlated with one or more of the regressors.  If the
> assumption is violated then we expect that the estimates from the two
> estimators will not be the same, b~=B.
> 
> The Hausman test is essentially a Wald test that (b-B)==0 for all coefficients
> where the covariance matrix for b-B is taken as the difference of the
> covariance matrices (VCEs) for b and B.  What is amazing about the test is
> that we can just subtract these two covariance matrices to get an estimate of
> the covariance matrix of (b-B) without even considering that the VCEs of the
> two estimators might be correlated -- they are after all estimated on the same
> data.  We can just subtract, but only because the the VCE of the fully
> efficient estimator is uncorrelated with the VCEs of all other estimators, see
> Hausman and Taylor (1981), "panel data and unobservable individual effects",
> econometrica, 49, 1337-1398).  The VCE of the efficient estimator will also be
> smaller than the less efficient estimator.  Taken together, these results
> imply that the subtraction of the two VCE (V_b-V_B) will be positive definite
> (PD) and that we need not consider the covariance between the two VCEs.
> 
> These results, however, hold only asymptotically.  For any given finite sample
> we have no reason to believe that (V_b-V_B) will be PD.  So, it is amazing
> that we can just subtract these two matrices, but the price we pay is that we
> can only do so safely if we have an infinite amount of data.  The Hausman
> test, unlike most tests, relies on asymptotic arguments not only for its
> distribution, but for its ability to be computed!  Let's discuss what we do
> what we do when (V_b-V_B) in not PD in the context of Eric's results.
> 
> Aside:  If anyone is interested in a Hausman-like test that drops the
> assumption that either estimator is fully efficient, actually estimates the
> covariance between the VCEs, and can always be computed, see Weesie (2000)
> "Seemingly unrelated est. and cluster-adjusted sandwich estimator", STB
> Reprints Vol 9, pp 231-248.  The test unfortunately requires the scores from
> the estimator, and -xtreg, fe- does not directly produce these.
> 
> <Note, a version of -suest- command is now official, but is still unavailable
>  after -xtreg->
> 
> 
> Of Inverses and Hausman Statistics
> ----------------------------------
> 
> The reason that -xthaus- and -hausman- produce different statistics on Eric's
> models is that they take different inverses of this non-PD matrix.  -xthaus-
> uses Stata's -syminv()- which zeros out columns and rows to form a sub-matrix
> that is PD and inverts that matrix, whereas -hausman- uses a Moore-Penrose
> generalized inverse.  Most of the literature on Hausman tests suggests that a
> generalized inverse such as Moore-Penrose be used when the matrix is not PD,
> however, I have not seen a foundation of this suggestion (and would
> appreciation a reference if anyone knows of one).
> 
> Two of us at Stata have independently run some informal simulations, where
> non-PD matrices are common, to determine if either of these inverses has
> nominal coverage for a true null.  While these simulations are not complete
> enough to share or publish, we both found that neither inverse performs well.
> This doesn't seem too surprising to me, if the information in our sample is
> insufficient to produce a PD "VCE" then the basis of the test would seem to be
> in question.
> 
> -xthaus- does not make it clear when the matrix is not PD.  I recall having
> read, though I cannot now find the reference, that in the case of random vs.
> fixed effects that the matrix was either always PD.  This may have been the
> thinking in excluding this check from -xthausman-.  Regardless, it is clearly
> not impossible and is not even unlikely.  Simulations show that non-PD
> matrices are quite common.
> 
> 
> An Alternative
> --------------
> 
> Even in their early work, Hausman and Taylor (1981) discuss an asymptotically
> equivalent test for random vs. fixed effects using an augmented regression.
> There are actually several forms of the augmented regression, all of which are
> asymptotically equivalent to the Hausman test.  All of these augmented
> regression tests are based on estimating an augmented regression that nests
> both the random- and fixed-effects models.  They are parameterized in such a
> way that we can perform a simple Wald test of a set of the jointly estimated
> coefficients.  They have fewer of the mechanical and interpretation problems
> associated with the Hausman test.  Their results will differ numerically from
> the Hausman test in finite samples because they are only asymptotically
> equivalent.
> 
> I have include below a block of code that will perform an augmented regression
> test for Eric's model (it also performs the Hausman test using -xthaus- and
> -hausman-).  It can easily be adapted to any model by changing the depvar and
> varlist macros.
> 
> If I have given the impression that I don't much care for the Hausman test,
> good.  I don't.  In ad hoc simulations I have found that in addition to its
> proclivity to be uncomputable, the test has low power for the current problem,
> for tests of endogeneity in instrumental variables regression, and for tests
> of independence of irrelevant alternatives (IIA) in choice models.
> 
> Regardless, the test is a staple in econometrics and it will stay in Stata.
> 
> 
> <Note:  Carl should be able to easily adapt this code by specifying the id
>  variable, dependent variable, and varlist.>
> 
> ---------------------------------- BEGIN --- foreric.do --- CUT HERE -------
> local id myid
> local depvar lnuncs
> local varlist lngdp ecrise ecfall urban lnhouse femalepa male1544        /*
> 	*/ lndiscr lnfree lnpts latin ssa deathp rulelaw protest cathol  /*
> 	*/ muslim transiti lnethv oecd war year89 year92 year95
> 
> xtreg `depvar' `varlist', re
> hausman, save
> version 7: xthausman
> 
> xtreg `depvar' `varlist', fe
> hausman, less
> 
> tokenize `varlist'
> local i 1
> while "``i''" != "" {
> 	qui by `id':  gen double mean`i' = sum(``i'') / _n
> 	qui by `id':  replace mean`i' = mean`i'[_N]
> 	qui by `id':  gen double diff`i' = ``i'' - mean`i'
> 	local newlist `newlist' mean`i' diff`i'
> 
> 	local i = `i' + 1
> }
> 
> xtreg `depvar' `newlist' , re
> tempname b
> matrix `b' = e(b)
> 
> qui test mean1 = diff1 , notest		/* clear test */
> local i 2
> while "``i''" != "" {
> 	if `b'[1,colnumb(`b', "mean`i'")] != 0 & 	/*
> 	*/ `b'[1,colnumb(`b', "diff`i'")]  != 0 {
> 		qui test mean`i' = diff`i' , accum notest
> 	}
> 	local i = `i' + 1
> }
> test
> 
> ----------------------------------   END --- foreric.do --- CUT HERE -------
> 
> ---------------------------------- End   excerpts --------------------------
> 
> As noted in the excerpt, When -xthausman- was written we were swayed by
> published "proofs" that the difference matrix was required mathematically to
> be positive definite when comparing FE and RE linear regression.  As Eric's
> and Carl's examples show, this is not true.  I would like to thank Mark
> Schaffer <M.E.Schaffer@hw.ac.uk> for reminding me of one of the "proofs",
> 
> 
>         "This appendix proves that the Avar(q_hat) in (5.2.21) is
>         positive definite and the Hausman statistic (5.2.22) is
>         guaranteed to be nonnegative in any finite samples."
> 
>         Hayashi, Econometrics (2000), Appendix 5.A, pp. 346-349 and 334-335.

The proof is correct.  The issue is that it depends on the use of a 
single estimate of the error variance for both the random effects VCE 
and the fixed effects VCE.  Asymptotically, it doesn't matter which 
one is used, and either will guarantee a positive test statistic.  
What this means in practice is that, say, the random effects VCE is 
multiplied by s2_fe/s2_re, so that it incorporates the fixed effects 
s2 instead of the original random effect s2, before the Hausman test 
is applied.

In fact, if the panel is balanced, using the fixed effects estimate 
of the error variance in a Hausman test is numerically equivalent to 
the artificial regression version of the test when group means are 
added as regressors.

The situation is very similar to an IV endogeneity test (discussed in 
the Stata manuals, if I recall).

The old xthausman didn't use a single estimate of the error variance -
the untransformed VCEs were used, and this could sometimes generate 
negative test stats.  The same applied to the hausman command as of 
version 8.2.  The problem was that the -sigmamore- or -sigmaless- 
hausman options, which force the usage of a single error variance in 
a hausman test, wouldn't work when the estimation results were done 
by xtreg.  I *still* don't have Stata 9 - sigh - and can't check if 
this has been rectified.

In any case, the artificial regression approach will always work.

Cheers,
Mark

> To avoid breaking user's do-files, we were reluctant to remove -xthausman-
> when -hausman- was first introduced.  Sufficient time has passed, and as of
> version 9 of Stata, -xthausman- works only when your version is set to 8 or
> lower.
> 
>  
> -- Vince
>    vwiggins@stata.com
> 
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Prof. Mark E. Schaffer
Director
Centre for Economic Reform and Transformation
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS  UK
44-131-451-3494 direct
44-131-451-3296 fax
44-131-451-3485 CERT administrator
http://www.sml.hw.ac.uk/cert

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