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

[Thomas Cornelissen <cornelissen@ewifo.uni-hannover.de>]

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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