st: re:

[Christopher Baum <kit.baum@bc.edu>]

<>
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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