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From |
"ALICE DOBSON" <alice_dobson@hotmail.com> |

To |
statalist@hsphsun2.harvard.edu |

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

Date |
Wed, 24 Aug 2005 08:07:41 -0400 |

Hi Vince,

I have a similar problem. I obtain this output after using Hausman.

Since, suest is unavailable after xtreg, can I simply go ahead and present the results from xtreg, re?

. hausman fixed random

---- Coefficients ----

| (b) (B) (b-B) sqrt(diag(V_b-V_B))

| fixed random Difference S.E.

-------------+----------------------------------------------------------------

age | -6519.588 -6822.915 303.3273 .

v | -411.6388 -31331.31 30919.67 .

totlsa | 3393.901 3503.643 -109.7421 .

numcicd | -61109.29 -63309.05 2199.759 .

------------------------------------------------------------------------------

b = consistent under Ho and Ha; obtained from xtreg

B = inconsistent under Ha, efficient under Ho; obtained from xtreg

Test: Ho: difference in coefficients not systematic

chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B)

= -6.22 chi2<0 ==> model fitted on these

data fails to meet the asymptotic

assumptions of the Hausman test;

see suest for a generalized test

Best, Alice

From: vwiggins@stata.com (Vince Wiggins, StataCorp)_________________________________________________________________

Reply-To: statalist@hsphsun2.harvard.edu

To: statalist@hsphsun2.harvard.edu

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

Date: Tue, 23 Aug 2005 10:17:26 -0500

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

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

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