LectureCh12

LectureCh12 - Instrumental Variables Regression (Stock...

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12-1 Instrumental Variables Regression (Stock & Watson Chapter 12) We're only covering a few key sections
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12-2 Instrumental Variables Regression (SW Chapter 12) Recall the threats to internal validity from chapter 9. We are worried about cases in which the first OLS assumption is violated: E ( u | X ) 0 When E(u| X ) 0, the OLS estimate of 1 is biased and inconsistent. This chapter is going to introduce an alternate technique for getting around this problem: Instrumental Variables (IV) regression.
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12-3 IV Regression with One Regressor and One Instrument (Section 1) Y i = 0 + 1 X i + u i If E(u|X)=0, then great. If not, then u and X are correlated (bad). IV regression separates X into two parts: a part that might be correlated with u (bad), and a part that is not (good). By isolating the part that is not correlated with u , it is possible to get a consistent (although still possibly biased) estimate 1 .
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12-4 This is done using an instrumental variable , Z i , which is uncorrelated with u i . The instrumental variable detects movements in X i that are uncorrelated with u i , and uses these to estimate 1 . Again, X is correlated with u, which makes our lives difficult. If we can figure out a way to break X into a part that is correlated with u and a part that is not, we can use the part that is not to get a consistent estimate of 1 .
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12-5 Terminology: endogeneity and exogeneity An endogenous variable is one that is correlated with u An exogenous variable is one that is uncorrelated with u So our problem might be that the error term u is correlated with the explanatory variable X, we thus say that X is endogenous. If X is not correlated with the error term, then it is exogenous. If X is endogenous, our main remedy is to find the instrumental variable (aka "instrument") Z.
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12-6 Two conditions for a valid instrument Y i = 0 + 1 X i + u i For an instrumental variable (aka “ instrument ”) Z to be valid, it must satisfy two conditions: 1. Instrument relevance : corr( Z i , X i ) 0 2. Instrument exogeneity : corr( Z i , u i ) = 0 The first condition just says that Z needs to be correlated with X. The second condition says that this same Z needs to also be uncorrelated with u (which implies E(u|Z)=0). Suppose we have found such an instrument Z.
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The IV Estimator for a single X and Z So how is Z going to help in estimating 1 ? Explanation #1: Two Stage Least Squares (TSLS) As it sounds, TSLS has two stages: (1) Isolate the part of X that is uncorrelated with u : regress X on Z using OLS X i = 0 + 1 Z i + v i ( 1 ) Because Z i is uncorrelated with u i , 0 + 1 Z i is uncorrelated with u i . We don’t know 0 or 1 but we have estimated them, so… ˆ ˆ Compute the predicted values of X i , , where = i X i X 0 ˆ + 1 ˆ Z i , for observations i = 1,…, n . 12-7
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Two Stage Least Squares, ctd .
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This note was uploaded on 09/19/2009 for the course ECON 656820 taught by Professor Megerdichian during the Summer '09 term at UCSD.

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LectureCh12 - Instrumental Variables Regression (Stock...

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