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

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 >> >> * >> * 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 > -- 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**:**Re:Re: RE: st: Multiple endogenous regressors***From:*Christopher Baum <kit.baum@bc.edu>

**Re: Re: RE: st: Multiple endogenous regressors***From:*Yuval Arbel <yuval.arbel@gmail.com>

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