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# Re: Re: RE: st: Multiple endogenous regressors

 From Yuval Arbel To statalist@hsphsun2.harvard.edu Subject Re: Re: RE: st: Multiple endogenous regressors Date Sun, 23 Oct 2011 06:16:58 +0200

```Kit,

Thanks for the proof, which made me see where do we fail to understand
each other:

You assumed that Xhat=a+bZ, i.e., Xhat is a linear function of Z.

I was referring to the model Xhat=a+bZ1 and Z2, where Z1 and Z2 are
different variables. Clearly, in my model the IV and 2SLS estimators
yield different numbers, because you are talking about two different
instruments.

To summarize, it is not a matter of wrong intuition, but of different
definitions of the model

On Sun, Oct 23, 2011 at 3:30 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