Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.

Re: RE: st: Multiple endogenous regressors

 From Yuval Arbel To statalist@hsphsun2.harvard.edu Subject Re: RE: st: Multiple endogenous regressors Date Sat, 22 Oct 2011 20:35:37 +0200

```I see you did not get my point: the question is how did you define
Xhati??? after all you are controlling the program.

Lets try another example. Suppose you have the system:

Yi=a+bX1i+u1i

X1i=c+dYi+eX2i+u2i

where X2i is exogenous. The solution to this system is:

X1i=c'+e'X2i+u'2i

from which you prduce X1hati

Now suppose you have an instrument Zi, with numerical figures which
are totally different from X1hati. You want to tell me that
cov(Zi,Yi)/cov(Zi,X1i) and cov(X1hati,Yi)/Var(X1hati) yield identical
numbers??? I don't buy that

On Sat, Oct 22, 2011 at 7:49 PM, Christopher Baum <kit.baum@bc.edu> wrote:
> <>
> 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
>
>
> *
> *   For searches and help try:
> *   http://www.stata.com/help.cgi?search
> *   http://www.stata.com/support/statalist/faq
> *   http://www.ats.ucla.edu/stat/stata/
>

--
Dr. Yuval Arbel