Lecture_20_Prof._Arkonac's_Slides_(Ch_12.2_on)_Fall_10

Lecture_20_Prof._Arkonac's_Slides_(Ch_12.2_on)_Fall_10 -...

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Instrumental Variable Regression II (Fall 2010) Lecture 20 Seyhan Erden Arkonac, PhD Problem Set 8 is posted, it is due on Tuesday, November 23rd 1
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2 The General IV Regression Model (SW Section 12.2) So far we have considered IV regression with a single endogenous regressor ( X ) and a single instrument ( Z ). We need to extend this to: multiple endogenous regressors ( X 1 ,…, X k ) multiple included exogenous variables ( W 1 ,…, W r ) These need to be included for the usual OV reason multiple instrumental variables ( Z 1 ,…, Z m ) More (relevant) instruments can produce a smaller variance of TSLS: the R 2 of the first stage increases, so you have more variation in ˆ X . Terminology: identification & overidentification
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3 Identification In general, a parameter is said to be identified if different values of the parameter would produce different distributions of the data. In IV regression, whether the coefficients are identified depends on the relation between the number of instruments ( m ) and the number of endogenous regressors ( k ) Intuitively, if there are fewer instruments than endogenous regressors, we can’t estimate 1 ,…, k For example, suppose k = 1 but m = 0 (no instruments)!
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4 Identification, ctd. The coefficients 1 ,…, k are said to be: exactly identified if m = k . There are just enough instruments to estimate 1 ,…, k . overidentified if m > k . There are more than enough instruments to estimate 1 ,…, k . If so, you can test whether the instruments are valid ( a test of the “overidentifying restrictions ”) – we’ll return to this later underidentified if m < k . There are too few instruments to estimate 1 ,…, k . If so, you need to get more instruments!
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5 The general IV regression model: Summary of jargon Y i = 0 + 1 X 1 i + … + k X ki + k +1 W 1 i + … + k+r W ri + u i Y i is the dependent variable X 1 i ,…, X ki are the endogenous regressors (potentially correlated with u i ) W 1 i ,…, W ri are the included exogenous variables or included exogenous regressors (uncorrelated with u i ) 0 , 1 ,…, k+r are the unknown regression coefficients Z 1 i ,…, Z mi are the m instrumental variables (the excluded exogenous variables ) The coefficients are overidentified if m > k ; exactly identified if m = k; and underidentified if m < k .
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6 TSLS with a single endogenous regressor Y i = 0 + 1 X 1 i + 2 W 1 i + … + 1 +r W ri + u i m instruments: Z 1 i ,…, Z m First stage Regress X 1 on all the exogenous regressors: regress X 1 on W 1 ,…, W r , Z 1 ,…, Z m by OLS Compute predicted values 1 ˆ i X , i = 1,…, n Second stage Regress Y on 1 ˆ X , W 1 ,…, W r by OLS The coefficients from this second stage regression are the TSLS estimators, but SE s are wrong To get correct SE s, do this in a single step
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7 Example : Demand for cigarettes ln( cigarettes i Q ) = 0 + 1 ln( cigarettes i P ) + 2 ln( Income i ) + u i Z 1 i = general sales tax i Z 2 i = cigarette-specific tax i Endogenous variable: ln( cigarettes i P ) (“one
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This note was uploaded on 11/10/2011 for the course ECON 3142 taught by Professor Arkonac during the Spring '11 term at Columbia.

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Lecture_20_Prof._Arkonac's_Slides_(Ch_12.2_on)_Fall_10 -...

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