Re: st: re:

[Yuval Arbel <yuval.arbel@gmail.com>]

Kit,

I would like to thank you very much for the illuminating proof. It
really made me understand where did I go wrong and to gain a better
insight of the subject. I guess we should have start the discussion
from there.

Yuval

On Sun, Oct 23, 2011 at 3:28 AM, Christopher Baum <kit.baum@bc.edu> wrote:
> <>
> Yusal said
>
>> 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.
>
>
>
> Yusal's intuition fails here. Not only are the numbers the same computationally, as I have demonstrated, but a bit of undergraduate statistical theory and the definition of OLS regression proves that they refer to the same quantity:
>
> Yusal wants a proof that in the exactly identified equation
>
> y = alpha + beta X + U
>
> with single instrument Z, uncorrelated with U, defining the first stage regression
>
> Xhat = a + b Z   where the OLS coefficient b = cov(X,Z) / var(Z)
>
> The expression for the IV slope coefficient,  betahat = cov(y, Z) / cov(X, Z)
> which corresponds to the matrix expression
>
> (Z'X)^-1 Z'y
>
> will yield the same point estimate as doing 2SLS 'by hand', that is, computing Xhat
> and running the second-stage OLS regression of y on Xhat. That regression has, let's say,
> slope coefficient
>
> gamma = cov(Y, Xhat) / var(Xhat).
>
>
> The proof:
>
> gamma =       cov(Y, Xhat) / var(Xhat) = cov(Y, a + b Z) / var(a + b Z)
>
>                         = cov(Y, b Z) / var(b Z)
>
>                         = b cov(Y, Z) / b^2 var(Z)
>
>                         = cov(Y, Z) / b var(Z)
>
>                         = cov(Y, Z) / [cov(X,Z) / var(Z)] var(Z)
>
>                         = cov(Y, Z) / cox(X, Z)  =  beta
>
>                         Q.E.D.
>
>
> 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
School of Business
Carmel Academic Center
4 Shaar Palmer Street, Haifa, Israel
e-mail: yuval.arbel@gmail.com

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