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# Re: RE: st: Multiple endogenous regressors

 From Yuval Arbel To statalist@hsphsun2.harvard.edu Subject Re: RE: st: Multiple endogenous regressors Date Sun, 23 Oct 2011 01:35:41 +0200

```Bill,

Thanks for the advice, and I certainly will make an effort to read
these references.

Nevertheless, note that my question relates to the theoretical aspects
of IV and 2SLS estimators. I'm a very curious person (I guess this is
the reason why did I become a researcher) and from time to time I
teach Econometrics classes and work with IV and 2SLS estimators. It is
thus important for me to know (and not for the sake of argument) if
I'm wrong here and if so where is my mistake.

In other words I need a more specific application to a reference,
which provides a mathematical proof that cov(Zi,Yi)/cov(Zi,X1i) and
cov(X1hati,Yi)/Var(X1hati) yield identical numbers (in the case that
I'm wrong here). My intuition says that the number will not be the
same. BTW: It can also be a reference in matrix notation. The question
is whether Baltagi (2008) provides such a proof, in which case I will
make an effort to get it.

One final point: my claim regarding the ILS is not that it is more
useful in research, but that it is useful in teaching Econometrics.

In my opinion, the ILS methodology gives students a sense of why it is
important to identify equations. A discussion and use of the ILS
methodology is given in many textbooks such as: Greene, Kmenta etc.

On Sat, Oct 22, 2011 at 9:18 PM, William Buchanan
<william@williambuchanan.net> wrote:
> Yuval,
>
> Although I personally find this thread to be amazingly hilarious, I thought
> I might try to offer you some help.  Keep in mind, however, that I am by no
> means an economist or econometrician.  But I wanted to point you to some
> more resources that I don't think you've read and which is probably why
> you're continuing to argue with people who are infinitely more knowledgable
> coauthors of an amazingly robust instrumental variables regression program).
>
>
> Angrist, J. D. and Pischke, J. (2009). Mostly Harmless Econometrics: An
> Empiricist's Companion. Ch. 4 "Instrumental Variables in Action: Sometimes
> You Get What You Need". P 113-220. Princeton, NJ: Princeton University
> Press.
>
> Baum, C. F. (2006). An Introduction to Modern Econometrics Using Stata. Ch.
> 8 "Instrumental-Variables Estimators". P. 185-218. College Station, TX:
> Stata Press.
>
> Murnane, R. J. and Willett, J. B. (2011). Methods Matter: Improving Causal
> Inference in Educational and Social Science Research. Ch. 10. "Introducing
> Instrumental-Variables Estimation". P. 203-264. Cambridge, MA: Harvard
> University Press.
>
> Stock, J. H. and Watson, M. W. (2007). Introduction to Econometrics. 2nd Ed.
> Ch 12 "Instrumental Variables Regression". P. 421-467. Boston, MA: Pearson
> Education.
>
> Wooldridge, J. M. (2003). Introductory Econometrics: A Modern Approach. 2nd
> Ed. Ch 15 "Instrumental Variables Estimation and Two Stage Least Squares".
> P. 484-525. Mason, OH: Thomson South-Western Publishing.
>
> These are just the books that I have next to my desk and could get to
> easily.  However, if you look at the help file for -ivreg2- the authors of
> that package (available from SSC) provide a fairly extensive list of
> references.  Additionally, Austin Nichols also has a great paper that you
> can find in the stata journal on causal inference using observational data.
>
>
> While you certainly are free to proclaim the merit that you find in the ILS
> estimator, you have yet to cite any research that supports your case.  On
> the otherhand, everyone else who has been trying to help Elizabeth out with
> her problem has been trying to guide her to resources and answers that will
> that Elizabeth had - and that were directly relevant to the original intent
> of this thread - you could try looking at Angrist and Pishcke's website
> (http://www.mostlyharmlesseconometrics.com/2010/02/multiple-endogenous-varia
> bles-what-now/) where they have a brief discussion specifically regarding
> IVE with multiple endogenous regressors.
>
> - Billy
>
>
>
>
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Yuval Arbel
> Sent: Saturday, October 22, 2011 11:36 AM
> To: statalist@hsphsun2.harvard.edu
> Subject: Re: RE: st: Multiple endogenous regressors
>
> 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
>>
>>
>> *
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>>
>
>
>
> --
> Dr. Yuval Arbel
> 4 Shaar Palmer Street, Haifa, Israel
> e-mail: yuval.arbel@gmail.com
>
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--
Dr. Yuval Arbel