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From |
Joana Quina <joana.quina@gmail.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: hausman and xthausman after panel fe, re - DROPPED MEAN/DIFF |

Date |
Thu, 25 Aug 2005 09:14:00 +0100 |

Dear Vince, I used your code for the augmented Hausman test, but whenever I include time-invariant variables (or time dummies interacted with time-invariant variables), it does note work - Stata drops the mean for the time-invariant variable (or the diff for the interacted terms). I notice from Eric's varlist that there seem to be time-invariant variables (latin/ssa) - is that correct? Any help would be much appreciated. Thanks, Joana On 23/08/05, Vince Wiggins, StataCorp <vwiggins@stata.com> wrote: > 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. > > 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 > > * > * For searches and help try: > * http://www.stata.com/support/faqs/res/findit.html > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ > * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**Re: st: hausman and xthausman after panel fe, re***From:*vwiggins@stata.com (Vince Wiggins, StataCorp)

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