Re: st: Computation of standard errors in an IV setting

[Christopher Baum <kit.baum@bc.edu>]

<>
On Jul 22, 2013, at 2:33 AM, Surithra wrote:

> The original equation of interest is:
> 
> y1 = x1 + x2 + x3 + e, where x2 is defined as negative of the absolute
> value of (0.5 - x1).
> 
> The independent variable, x1 is endogenous and hence, x2 is also endogenous.
> 
> I have an instrument for x1 - say z1.
> 
> Now I would like to estimate the original equation (y1 = x1 + x2 + x3
> + e) using the IV, z1. However, what makes it complicated relative to
> a standard IV application is that x2 is a non-linear function of x1.
> As a result, while z1 has a monotonic effect on x1, z1 has a
> non-monotonic effect on x2.
> 
> As a result I am not sure how I can use Stata to obtain the correct
> standard errors in this case.


This logic is misguided. The fact that one endogenous variable is a nonlinear function of another will not have
any deleterious effect on the computation of IV estimates. This situation would arise, for example, if x2 = x1^2.
As long as you have enough instruments that satisfy the necessary conditions, IV (or IV-GMM, better yet) will 
work fine. E.g., using Mark Schaffer's -xtivreg2- from SSC:

. webuse grunfeld, clear

. g mv2 = mvalue^2

. g ks2 = kstock^2

. xtivreg2 invest (mvalue mv2 = kstock ks2 time), gmm2s robust fe

FIXED EFFECTS ESTIMATION
------------------------
Number of groups =        10                    Obs per group: min =        20
                                                               avg =      20.0
                                                               max =        20

2-Step GMM estimation
---------------------

Estimates efficient for arbitrary heteroskedasticity
Statistics robust to heteroskedasticity

                                                      Number of obs =      200
                                                      F(  2,   188) =    14.29
                                                      Prob > F      =   0.0000
Total (centered) SS     =  2244352.228                Centered R2   =  -0.2833
Total (uncentered) SS   =  2244352.228                Uncentered R2 =  -0.2833
Residual SS             =  2880270.244                Root MSE      =    123.1

------------------------------------------------------------------------------
             |               Robust
      invest |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .2998763   .1260421     2.38   0.017     .0528383    .5469142
         mv2 |   .0000222   .0000193     1.15   0.250    -.0000156    .0000601
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic):             20.011
                                                   Chi-sq(2) P-val =    0.0000
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               10.196
                         (Kleibergen-Paap rk Wald F statistic):          8.751
Stock-Yogo weak ID test critical values: 10% maximal IV size             13.43
                                         15% maximal IV size              8.18
                                         20% maximal IV size              6.40
                                         25% maximal IV size              5.45
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         3.006
                                                   Chi-sq(1) P-val =    0.0830
------------------------------------------------------------------------------
Instrumented:         mvalue mv2
Excluded instruments: kstock ks2 time
------------------------------------------------------------------------------

Kit

Kit Baum   |   Boston College Economics & DIW Berlin   |   http://ideas.repec.org/e/pba1.html
                             An Introduction to Stata Programming  |   http://www.stata-press.com/books/isp.html
  An Introduction to Modern Econometrics Using Stata  |   http://www.stata-press.com/books/imeus.html
                                                                                                   | http://www.crup.com.cn/Item/111779.aspx	


*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/faqs/resources/statalist-faq/
*   http://www.ats.ucla.edu/stat/stata/