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

["Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>]

Thomas,

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu 
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of 
> Thomas Cornelissen
> Sent: 02 October 2007 16:40
> To: statalist@hsphsun2.harvard.edu
> Subject: st: RE: Do 2sls, ivreg etc. test the rank condition 
> of identification?
> 
> 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?

I oversimplified.  You have system estimation in mind, which is a little
more complex than the 1-equation case I was thinking of.  But the same
principles apply.

There are two ways the rank condition can fail in the system framework.
One way is the one you have in mind.  There is a matrix that depends on
the parameters of the model that has to be full rank in order for the
system to be identified.  It's possible that your model and a priori
restrictions are such that the matrix is not full rank.  This is the
first part of the example that Wooldridge works through on p. 219.  His
equation 9.24 is a 2x3 matrix with two columns of zeros, so it can have
at most rank=1, and so the first equation is unidentified.

The other way the rank condition can fail is the way I described in my
previous post - the values of parameters are such that the matrix is not
full rank.  In other words, the rank of the matrix depends on the values
of the parameters, but these are not known with certainty - you estimate
them.  This is the second part of Wooldridge's example on p. 219: "It is
left to you to show that the rank condition holds if and only if d_13
\ne 0 and at least one of d_32 and d_34 is different from zero."  But
you can't know if these conditions hold for sure - you have to estimate
the rank and test the associated hypothesis that Wooldridge states.

That Stata doesn't complain below means that indeed it isn't checking
the rank condition for you.  It's not too surprising, I suppose.

Hope this helps.

--Mark

> 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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