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re:RE: st: Multiple endogenous regressors


From   Christopher Baum <kit.baum@bc.edu>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   re:RE: st: Multiple endogenous regressors
Date   Sat, 22 Oct 2011 13:49:30 -0400

<>
Cam said

> Like Kit, I got a bit of a a surprise (and chuckle) about the example. In the Keynesian model, 2SLS, ILS, and the simple IV estimator yield identical results when instrumenting Y_t with I_t. See Chapter 11 in:
> 
> Batalgi, B.H. (2008). Econometrics (3rd. ed.). Berlin - Heidelberg: Springer-Verlag. 
> 


I don't thinl Badi has to worry too much about Yuval's challenge to his book. Yuval said

> Suppose Yi and Xi are endogenous, Zi is an instrumental variable and
> Xhati is the projected values of Xi obtained from the solution
> equation (in which all the right-hand-side variables are exogenous).
> 
> The plim of the IV esimator for b is: cov(Zi,Yi)/cov(Zi,Xi). Note that
> to generate the IV estimator you are using all the 3 variables (Xi, Yi
> and Zi). I suppose this is what STATA estimated in Kit's example
> 
> On the other hand, the plim of the 2SLS estimator for b is:
> cov(Xhati,Yi)/Var(Xhati). The 2SLS estimator uses just Xhati and Yi,
> because you are literally replacing Xi by Xhati.
> …
> Note, that for small samples, the two estimators are by no mean
> identical. I suppose, that for large sample they are both consistent

Strangely enough, the two quantities he speaks of computing are exactly the same to 8 decimals. This is hardly relying on asymptotics, as N=21. (I suppose by the "solution equation" Yuval means what the rest of the world calls a first stage regression).  From the Klein regression in my last posting:

. corr consump invest totinc inchat,cov
(obs=22)

             |  consump   invest   totinc   inchat
-------------+------------------------------------
     consump |  53.9893
      invest |   10.634  12.1089
      totinc |  76.5988  23.1506    117.8
      inchat |  20.3308  23.1506  44.2607  44.2607


. mata
------------------------------------------------- mata (type end to exit) --------
: cov=st_matrix("r(C)")

: cov[2,1] / cov[3,2]     <== cov (Z,Y) / cov(Z,X)
  .4593424594

: cov[4,1] / cov[4,4]     <== cov(Xhat,Y) / var(Xhat)
  .4593424594

I'm not sure what criterion Yuval would use to define "no mean identical", but they sure look the same to me… as econometric theory demands, as they are the same quantities. Think about the fact that covariance is a linear operator, and Xhat is a deterministic linear function of X...


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


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