Vince et al.,
A couple of follow-up questions to Paula's case and Vince's reply:
From: [email protected] (Vince Wiggins, StataCorp)
To: [email protected]
Subject: Re: st: panel data hausman negative
Date sent: Wed, 01 Oct 2003 17:20:49 -0500
Send reply to: [email protected]
> Paula Garcia <[email protected]> reports getting different
> results from -hausman- and -xthausman-. I recommend that Paula
> believe the results from -hausman- and not -xthausman-. The main
> reason -xthausman- was undocumented was that it was too easily fooled
> by non positive definite (PD) differenced covariance matrices or by
> variables with degenerate panel behavior.
1. Vince recommends -hausman- over -xthausman-. In Paula's case,
however, -hausman- gives a negative statistic whereas -xthausman-
gives a positive one. Is this because -xthausman- is being "fooled"?
2. The degrees of freedom in the fixed vs. random effects version is
the number of regressors. In Paula's case, -xthausman- gave the
(apparently) correct number, 5, whereas -hausman- reported the
degrees of freedom to be 4. Is this because of "degenerate panel
behaviour" that is again fooling -xthausman-?
I suppose another way of asking the question is whether there are
circumstances where -hausman- is fooled but -xthausman- isn't.
--Mark
> Paula notes that -suest- cannot be easily run because -xtreg- does not
> produce scores. If Paula wants another test, I suggest an augmented
> regression that is asymptotically equivalent to the Hausman test (see
> example below).
>
> Let me take some wholesale excerpts from an earlier statalist post (I
> don't recall the exact day). This post addressed a substantially
> similar question from Eric Neumayer.
>
> ---------------------------------- Begin excerts
> --------------------------
>
> Eric Neumayer <[email protected]> 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 asymtotically. 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>
>
>
> 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.
>
> I have include below my signature a block of code that will perform an
> augmented regression test for Eric's model (it also performs the
> Hausman test using -xthause- 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 engodeniety in instrumental
> variables regression, and for tests of independence of irrelvant
> alternatives (IIA) in choice models.
>
> Regardless, the test is a staple in econometrics and it will stay in
> Stata.
>
>
>
> -- Vince
> [email protected]
>
>
> Note: Paula should be able to easily adapt this code.
>
> ---------------------------------- 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
> xthaus
>
> 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
>
> qui test mean1 = mean1 , 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
> -------
>
> *
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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
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