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From |
Yuval Arbel <yuval.arbel@gmail.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: re: |

Date |
Sun, 23 Oct 2011 08:12:30 +0200 |

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 > > > * > * For searches and help try: > * http://www.stata.com/help.cgi?search > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ > -- Dr. Yuval Arbel School of Business Carmel Academic Center 4 Shaar Palmer Street, Haifa, Israel e-mail: yuval.arbel@gmail.com * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**st: re:***From:*Christopher Baum <kit.baum@bc.edu>

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