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st: RE: Do 2sls, ivreg etc. test the rank condition of identification?


From   Thomas Cornelissen <cornelissen@ewifo.uni-hannover.de>
To   statalist@hsphsun2.harvard.edu
Subject   st: RE: Do 2sls, ivreg etc. test the rank condition of identification?
Date   Tue, 02 Oct 2007 17:40:00 +0200

Mark, thanks a lot for the helpful reply. I have still some ongoing queries.
When I asked my question I was thinking about the rank condition for identification of an equation of a simultaneous equation model, such as the conditions stated in Wooldridge 2002 p.218 equation 9.19 or in Greene 2003, page 393, second equation. To my understanding, a violation of these can be asserted by looking at the structure of the model, without needing a statistical test. Am I mistaken here?

Wooldridge 2002, Example 9.3 (p. 219) states a three-equation model which meets the order condition but not the rank condition of identification. I replicated the model structure with variables I have available in a data set and estimated it:

. reg3 (w tenure schooling male) (tenure w) (schooling pubsec male expft), 2sls

Two-stage least-squares regression
----------------------------------------------------------------------
Equation Obs Parms RMSE "R-sq" F-Stat P
----------------------------------------------------------------------
w 9964 3 2362.058 -0.1637 454.83 0.0000
tenure 9964 1 12.35172 -0.6208 1083.72 0.0000
schooling 9964 3 2.712323 0.0423 146.47 0.0000
----------------------------------------------------------------------

------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w |
tenure | 96.01946 4.283876 22.41 0.000 87.62288 104.416
schooling | -210.2199 42.83895 -4.91 0.000 -294.1861 -126.2537
male | 1236.24 48.08673 25.71 0.000 1141.988 1330.492
_cons | 3604.489 535.9666 6.73 0.000 2553.971 4655.007
-------------+----------------------------------------------------------------
tenure |
w | .0046603 .0001416 32.92 0.000 .0043828 .0049378
_cons | -1.5389 .3930236 -3.92 0.000 -2.309244 -.7685571
-------------+----------------------------------------------------------------
schooling |
pubsec | 1.297164 .062117 20.88 0.000 1.175412 1.418916
male | .2680275 .0577517 4.64 0.000 .1548317 .3812234
expft | -.0083658 .0025288 -3.31 0.001 -.0133224 -.0034092
_cons | 12.35305 .0537434 229.85 0.000 12.24771 12.45839
------------------------------------------------------------------------------
Endogenous variables: w tenure schooling
Exogenous variables: male pubsec expft
------------------------------------------------------------------------------

Although the model should not be identified due to failure of the rank condition (see Wooldridge 2002 p. 220 eq. 9.24), Stata estimated the model parameters. Did I overlook something? Does it imply that Stata checks the order condition of identification, but not the rank condition?

To me it seems that the rank condition in Wooldridge 2002 p.218 equation 9.19 is only dependent on the structure of the model (coefficients and restrictions), and not on the actual data. Whereas the condition mentioned by Mark "E(Z'X) has full rank" seems to me dependent on the data. Therefore I lack understanding on whether these different rank conditions mean similar things, or how they are connected.

Thanks for any suggestions.
Thomas

-------- Mark Schaffer wrote:
Thomas,
> I am wondering how Stata would react if I trid to estimate an
> unidentified equation of a simultaneous equations model.
>
> I tried it out, and got "Equation is not identified -- does
> not meet order conditions".
>
> Do -2sls-, -3sls- and -ivreg- also test the rank condition?
> (In case the order condition is met, but not the rank
> condition.) In any case, I imagine I wouldn't get any
> estimation results if the model is not identified. So, at
> least implicitly the rank condition must be checked.

The rank condition is not deterministic, like the order condition. It's
formulated in terms of expectations - E(Z'X) has full rank - and whereas
the number of columns is observable (order condition), the true value of
this expectation is not. Instead, you formulate a null hypothesis and
see if the data reject at some p value. The null is that the sample
counterpart to the expectation - 1/n Z'X - is rank-deficient (has
rank=#columns minus 1), and if you can reject the null, you conclude the
rank condition is satisfied with some probability p.

The rank condition is not automatically checked by -ivreg- (Stata 9.2
and earlier). -estat firststage- after -ivregress- (Stata 10) will
report identification statistics, but these are valid only for the
i.i.d. case and not if you are using some sort of robust vcv. -ivreg2-
reports identification tests based on Anderson's canonical correlation
statistic (F form = Cragg-Donald statistic) in the i.i.d. case, and the
Kleibergen-Paap rk statistic for robust case.

-ivreg2- calls -ranktest- to do this, but you can use -ranktest- to do
the test by hand if, e.g., you are doing a 3sls estimation. -help
ranktest- is perhaps worth reading - the examples show the equivalence
between tests of the rank condition and various regression formulations.

Cheers,
Mark

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