lecture15b

# lecture15b - ECON 103 Lecture 15B Instrumental Variables II...

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Unformatted text preview: ECON 103, Lecture 15B: Instrumental Variables II Maria Casanova May 28th (version 0) Maria Casanova Lecture 15B Requirements for this lecture: Chapter 12 of Stock and Watson Maria Casanova Lecture 15B 1. IV regression with 1 regressor and 1 instrument We start from the population regression: Y i = β + β 1 X i + ε i , where X i and ε i are correlated. IV regression uses an instrumental variable Z to isolate the variation in X that is not correlated with ε . For an instrumental variable Z to be valid, it must satisfy two conditions: 1 Instrumental relevance: variation in the instrument is related to variation in X i , → Corr ( Z i , X i ) 6 = 0 2 Instrumental exogeneity: the part of variation in X i captured by the instrumental variable is exogenous, i.e. not correlated with the error term. → Corr ( Z i , ε i ) = 0 Maria Casanova Lecture 15B 1. IV regression with 1 regressor and 1 instrument If Z satisfies the two conditions to be a valid instrument, then β 1 can be estimated using an IV estimator called two-stage least squares (TSLS). As its name suggests, TSLS proceeds in two stages: 1 First, we isolate the part of X that is uncorrelated with ε by regressing X on Z using OLS: X i = π + π 1 Z i + v i Because Z i is uncorrelated with ε i , also π + π 1 Z i is uncorrelated with ε i . We don’t know π and π 1 , so we estimate them and then compute the predicted values of X i , i.e. ˆ X i = ˆ π + ˆ π 1 Z i Maria Casanova Lecture 15B 1. IV regression with 1 regressor and 1 instrument 2 Second, we replace X i with ˆ X i in the regression of interest, and regress Y i on ˆ X i using OLS: Y i = β + β 1 ˆ X i + ε i Because ˆ X i is uncorrelated with ε i , the first least squares assumption holds. Thus the estimate ˆ β 1 obtained by OLS in the second regression is consistent....
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lecture15b - ECON 103 Lecture 15B Instrumental Variables II...

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