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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 06:44:32 +0200

```Having thought about it again, according to the definition of
instrumental variable, Z2 will not be a good instrument to X - because
Z2 and X are uncorrelated. So I guess your definition of the model is
better than mine

On Sun, Oct 23, 2011 at 6:16 AM, Yuval Arbel <yuval.arbel@gmail.com> wrote:
> 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
> 4 Shaar Palmer Street, Haifa, Israel
> e-mail: yuval.arbel@gmail.com
>

--
Dr. Yuval Arbel