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Unformatted text preview: Economics 140A TwoStage Least Squares Estimation in Simultaneous Equation Models Given that the parameters of an equation in a simultaneous equation model are identi&ed, one then turns to estimation of the parameters. Consider the identi&ed simultaneous equation model Y 1 t = & 1 + & 2 Y 2 t + & 3 X 1 t + U 1 t ; Y 2 t = ¡ 1 + ¡ 2 Y 1 t + ¡ 3 X 2 t + U 2 t ; for which Y 1 t and Y 2 t are the jointly endogenous variables. The structural equa tions contain regressors that are correlated with the error. To understand the correlation: If U 1 t increases, then Y 1 t increases ) Y 2 t increases (assuming ¡ 2 > ) so U 1 t and Y 2 t are (positively) correlated. In essence, when U 1 t is positive, Y 2 t tends to be above its mean and both increase Y 1 t (assuming & 2 > ). Because U 1 t is unobserved, we attribute all of the increase in Y 1 t to Y 2 t , thereby overestimating & 1 . Because the source of the bias is the simultaneous determination of Y 1 t and Y 2 t , the bias is referred to as simultaneity bias. (The OLS estimators are not only biased, they are inconsistent.) The problem of endogenous regressors is also revealed by attempting to in terpret the coe¢ cients. The coe¢ cient & 2 is designed to capture the e/ect of a small change in Y 2 t holding X 1 t constant. Yet a change in Y 2 t (caused, say, by a change in U 2 t ) leads to a change in Y 1 t , which then feeds back on Y 2 t through the second equation, which again a/ects Y 1 t and so on. We see that & 2 captures the e/ects of all the feedbacks and so represents some mix of the e/ect of Y 2 t on Y 1 t and the e/ect of Y 1 t on Y 2 t . Further, consider & 3 , which is designed to capture the e/ect of a small change in X 1 t on Y 1 t holding Y 2 t constant. Yet Y 2 t cannot be held constant as Y 1 t changes, implying that all the coe¢ cients estimators are biased by simultaneity. As noted earlier, a natural way to mitigate the bias would be to replace the endogenous regressors with instruments. Because a good instrument is hard to &nd, the idea is to construct the instruments from the predetermined regressors and then form IV estimators. The method is termed twostage least squares (2SLS) estimation, in which the &rst stage constructs the instruments and the second stage constructs IV estimators of the parameters of interest. To begin, we must create instruments. A natural set of variables from which to construct the instruments is the set of predetermined regressors in the model ( X 1 t ;X 2 t ) . A natural way to form the instruments is to select the linear com bination of the predetermined regressors that is most highly correlated with the endogenous regressor. To do so, we estimate the &rststage equations Y 1 t = & 1 + & 2 X 1 t + & 3 X 2 t + V 1 t ; Y 2 t = ¡ 1 + ¡ 2 X 1 t + ¡ 3 X 2 t + V 2 t : The two equations from the &rst stage are termed reducedform equations, as they express the endogenous variables wholly in terms of predetermined regressors.express the endogenous variables wholly in terms of predetermined regressors....
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This note was uploaded on 09/04/2011 for the course ECON 140a taught by Professor Staff during the Fall '08 term at UCSB.
 Fall '08
 Staff
 Economics

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