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18 Simultaneous Equation Models Two Stage Least Squares Estimation

18 Simultaneous Equation Models Two Stage Least Squares Estimation

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Unformatted text preview: Economics 140A Two-Stage 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 two-stage 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 &rst-stage 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 reduced-form 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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18 Simultaneous Equation Models Two Stage Least Squares Estimation

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