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# Re: st: RE: Hausman test (via a Wald test)

 From "Brian P. Poi" To statalist@hsphsun2.harvard.edu Subject Re: st: RE: Hausman test (via a Wald test) Date Sun, 01 May 2011 13:01:10 -0400

```
On 05/01/2011 12:01 PM, John Antonakis wrote:
```
```Right, Eric.

But some estimators are more efficient; take the case where you compare
IV estimates (consistent) versus OLS estimates (efficient). The Hausman
test will suggest that an efficient estimator is not consistent in the
case where, for example, a regressor is endogenous, because the
consistent vs efficient estimates will be significantly different, as
tested by the usual Hausman test:

(d_IV - d_OLS)/(SE_d^2_IV - SE_d^2_OLS)^-1.

Thus, I am suggesting that one uses the Wald postestimation test
following the OLS model to determine whether the estimate/s of the OLS
model is/are significantly different from the estimate/s of the IV
model. My question has to do with the fact that the Wald test, in this
case, tests the OLS coefficient/s against a specific value/s (that of
the IV estimator) and ignores the variance in the IV estimate/s--so I
was wondering whether this was econometrically sound to do.

```
```
```
In the case of IV versus OLS, there is a way to use a Wald test after OLS to see whether the IV and OLS estimates are significantly different, though it doesn't escape the need to have instruments available in the first place. In this case, fit the regression
```
y1 = y2*beta + X1*gamma + V*theta + error

```
by OLS where y2 is your endogenous regressor(s), X1 are exogenous variables, and V are the residuals from the regression(s)
```
y2 = X1*pi1 + x2*pi2 + error

```
with extra instruments X2. Test whether theta = 0 using a Wald test; a rejection of H0 suggests that y2 is endogenous and therefore that OLS is not consistent.
```
```
But to reiterate, this presumes you are able to come up with instruments in the first place so that you can add that V term to your OLS regression.
```
>> I was wondering, though, if anyone is aware of literature showing that
>> a Wald test could do too (which would be particularly useful in the
>> case of models where the Hausman test can't be used in Stata); more
>> specifically, the Wald test I am suggesting is to test whether the
>> parameters (of interest) from the efficient estimator are
>> significantly different from those of the consistent estimator. It

```
There's also a way to do a Wald test equivalent to a Hausman test of fixed versus random effects. Instead of doing
```
. xtreg y x1 x2, re

do

. egen x1bar = mean(x1), by(panelvar)
. egen x2bar = mean(x2), by(panelvar)
. xtreg y x1 x2 x1bar x2bar, re
. test x1bar x2bar

Wooldridge derives this test in his cross-sectional and panel textbook.

```
There are probably other examples of Hausman-like tests that can be done as Wald tests, too.
```
-- Brian Poi
-- brian@poiholdings.com

```
```Best,
J.

__________________________________________

Prof. John Antonakis
Department of Organizational Behavior
University of Lausanne
Internef #618
CH-1015 Lausanne-Dorigny
Switzerland
Tel ++41 (0)21 692-3438
Fax ++41 (0)21 692-3305
http://www.hec.unil.ch/people/jantonakis

Associate Editor
__________________________________________

On 01.05.2011 17:03, DE SOUZA Eric wrote:
```
```John,

I don't get your point. Consistency refers to the parameter estimate,
efficiency refers to the variance of the estimate.

The Hausman principle compares two estimators, both of which are
consistent under the null but only one under the alternative.

This is quite general.

Eric de Souza
College of Europe
Brugge (Bruges), Belgium
http://www.coleurope.eu

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of John Antonakis
Sent: 01 May 2011 16:51
To: statalist@hsphsun2.harvard.edu
Subject: st: Hausman test (via a Wald test)

Hi:

Given that sometimes Stata cannot compute the Hausman test (because it
is undefined or because the estimation procedure does not allow it), I
have been thinking about ways to go around this limitation (and find
SUEST to be particularly useful in this regard).

I was wondering, though, if anyone is aware of literature showing that
a Wald test could do too (which would be particularly useful in the
case of models where the Hausman test can't be used in Stata); more
specifically, the Wald test I am suggesting is to test whether the
parameters (of interest) from the efficient estimator are
significantly different from those of the consistent estimator. It
would be very simply to do and could accommodate a large class of
estimators.
However, this test is only a constraint on the coefficient estimate/s
from the efficient estimator and ignores the variance of estimates
from the consistent estimator (thus I don't know to what extent this
test would be useful).

Any thoughts?

Best,
John.

--
__________________________________________

Prof. John Antonakis
Department of Organizational Behavior
University of Lausanne
Internef #618
CH-1015 Lausanne-Dorigny
Switzerland
Tel ++41 (0)21 692-3438
Fax ++41 (0)21 692-3305
http://www.hec.unil.ch/people/jantonakis

Associate Editor
```*