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st: re:

 From Christopher Baum To "statalist@hsphsun2.harvard.edu" Subject st: re: Date Sat, 22 Oct 2011 21:28:59 -0400

```<>
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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