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From |
Christopher Baum <kit.baum@bc.edu> |

To |
"statalist@hsphsun2.harvard.edu" <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 * * 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/

**Follow-Ups**:**Re: st: re:***From:*Yuval Arbel <yuval.arbel@gmail.com>

**Re: st: re:***From:*Yuval Arbel <yuval.arbel@gmail.com>

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