# Re: R: st: Testing for endogeneity with xtabond

 From Mark Schaffer <[email protected]> To [email protected], M Quagliariello <[email protected]> Subject Re: R: st: Testing for endogeneity with xtabond Date Sat, 03 Jul 2004 21:36:19 +0100 (BST)

```Mario,

Quoting M Quagliariello <[email protected]>:

> Dear statalister, Mark, I have read Baum, Schaffer, Stillman (2003)
> paper,
> but I still don't understand why I should include the same set of
> instruments in both the "consistent" and the "efficient" model when
> I test
> for the endogeneity of a subset of regressors. Does someone has an
> easy
> explanation?

Under the null, the set of exogenous instruments (including exogenous
regressors) is big.  In GMM terms, each of these gives you an
orthogonality condition that is imposed when you estimate the model.
Estimate the model and you get a Sargan-Hansen statistic; call it S0.

Under the alternative, a subset of the original set of instruments is
endogenous, and the remainder are still exogenous.  In GMM terms you have
a smaller set of orthogonality conditions that you use when you estimate
the model. Estimate the same model, but treat this subset of instruments
as endogenous.  That means (a) any regressors (included instruments) in
this subset are considered endogenous in the new estimation, and (b) any
excluded instruments are dropped completely.  This estimation gives you
another Sargan-Hansen statistic; call it S1.

Under the null, S0 is distributed as chi-square with dof=number of
(included and excluded) instruments - number of regressors.

Under the null, S1 is also distributed as chi-square with dof=number of
IV - number of regressors.  The dof are, of course, smaller than the dof
for S0 above, and indeed smaller by the number of regressors you are
testing for endogeneity.

Under the alternative, S1 is still distributed as chi-square as above, but
S0 isn't - it will be "too big".

The endogeneity test is simply a test of S0-S1.  The test statistic (S0-
S1) should be distributed as chi-square in the number of regressors being
tested for endogeneity.  If they're endogenous, then (S0-S1) will be big
because S0 is big but S1 isn't.

The above should make clear why you need to use the same maintained set of
exogenous instruments for both estimations.  If you are testing whether or
not the regressors of interest are endogenous, you have to be
saying "ceteris paribus", and that means assuming all the other
orthogonality conditions hold, i.e., all the other included and excluded
instruments remain exogneous.

> In any case, when I use -xtabond- how can I retrieve
> the set
> of instruments to plug in in the "efficient" model?

This could be a little tricky.

Say that x_it is strictly exogenous.  This is the estimation that will
give you S0.  You want to test whether or not it's endogenous.  The
question is, what is the specification - what are the orthogonality
conditions, what is being treated as exogenous - when you estimate and
treat x_it as endogenous?

The key point is that whatever instruments constructed using x and its
lags in this latter estimation also need to be used as instruments in the
estimation that gives you S0.  If the 3 lagged first difference of x is an
instrument here, it also needs to be an instrument in the first
estimation - even though x is there being treated as exogenous.

Probably David Roodman's xtabond2 gives you enough control over the
instrument sets to do this, but I have to confess I haven't tried this
myself.

Hope this helps.

--Mark

> Thanks a lot.
>
> Mario
>
>
> ----- Original Message -----
> From: Mark Schaffer
> To: [email protected] ; M Quagliariello
> Sent: Thursday, June 17, 2004 9:59 PM
> Subject: Re: st: Testing for endogeneity with xtabond
>
>
> Mario,
>
> Quoting M Quagliariello <[email protected]>:
>
> > Hallo!
> >
> > I hope someone can help me.
> > Suppose I want to estimate a dynamic panel with xtabond having
> > three
> > sets of variables:
> > A) surely endogenous
> > B) surely exogenous
> > C) maybe endogenous
> >
> > I was thinking to test for the endogenity of variables C) in
> this
> > way:
> > 1) estimate a model in which variables C are considered as
> > exogenous.
> > The estimated coefficients should be consistent and efficient if
> > the
> > variables are actually exogenous, but inconsistent if the
> variables
> > C)
> > are endogenous (model1).
> > 2) re-estimate the model considering the variables C) as
> endogenous
> > and
> > instrumenting them with their lagged levels (as for the lagged
> > dependent
> > variable and the other enedogenous regressors). The estimated
> > coefficients should be always consistent (model2).
> > 3) test for endogeneity using -hausman model2 model1-
> >
> > Is it reasonable?
>
> You don't say so explicitly, but it looks like your category C
> variables
> are regressors.  In that case, what you want is a test of a subset
> of
> orthogonality conditions.  Because the suspect instruments are
> regressors,
> the usual GMM "difference-in-Sargan" or "C test" for a subset of
> orthogonality conditions is numerically equivalent to a Hausman
> test.  See
> Hayashi's (2000) textbook, pp. 233-34, or the 2003 Stata Journal
> paper I
> did with Kit Baum and Steve Stillman.
>
> You can proceed as you describe, or, since you're using xtabond2,
> even
> more easily, just calculate the difference of the Sargan-Hansen
> statistics
> of the two estimations.  This will be distributed as chi-sq with
> dof=number of suspect instruments.
>
> IMPORTANT - you must make sure that every instrument that appears in
> model
> 2 is also an instrument in model 1.  You want to use lagged C
> variables as
> instruments when estimating model 2, so that means that you have to
> use
> exactly these lagged variables as instruments when estimating model
> 1 as
> well (even though the C variables are being treated as exogenous).
>
> --Mark
>
> >
> > Thanks a lot,
> >
> > Mario
> >
> > *
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> >
>
>
>
> Prof. Mark Schaffer
> Director, CERT
> Department of Economics
> School of Management & Languages
> Heriot-Watt University, Edinburgh EH14 4AS
> tel +44-131-451-3494 / fax +44-131-451-3008
> email: [email protected]
> web: http://www.sml.hw.ac.uk/ecomes
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Prof. Mark Schaffer
Director, CERT
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS
tel +44-131-451-3494 / fax +44-131-451-3008
email: [email protected]
web: http://www.sml.hw.ac.uk/ecomes
________________________________________________________________

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Watt University does not accept liability or responsibility
for changes made to this e-mail after it was sent, or for
viruses transmitted through this e-mail.  Opinions, comments,
conclusions and other information in this e-mail that do not
relate to the official business of Heriot Watt University are
not endorsed by it.
________________________________________________________________
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```