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Econometrics-I-13

Econometrics-I-13 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 13. Instrumental Variables
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The Problem 1 2 Cov( , ) , K variables Cov( , ) , K variables is OLS regression of y on ( , ) cannot estimate ( , ) consistently. Some other estimator is needed. Additional structure: = + wh = + + = endogenous y X Y X 0 Y 0 Y X Y Y Z V β δ ε ε ε β δ Π ere Cov( , )= . An estimator based on ( , ) may be able to estimate ( , ) consistently. instrumental variable ( Z 0 X IV) Z ε β δ
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Instrumental Variables Framework:  y    =    X β   +   ε , K variables in  X . There exists a set of K variables,  such that             plim( Z’X/n   0   but  plim( Z’ ε /n) =  0 The variables in  Z  are called instrumental variables. An alternative (to least squares) estimator of  β  is              b IV   =  ( Z’X ) -1 Z’y We consider the following: Why use this estimator? What are its properties compared to least squares? We will also examine an important application
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The First IV Study (Snow, J., On the Mode of Communication of Cholera, 1855) London Cholera epidemic, ca 1853-4 Cholera = f(Water Purity,u)+ ε . Effect of water purity on cholera? Purity=f(cholera prone environment (poor, garbage  in streets, rodents, etc.). Regression does not work.     Two London water companies      Lambeth                      Southwark ======|||||======            Main sewage discharge Paul Grootendorst: A Review of Instrumental Variables Estimation of Treatment Effects… http://individual.utoronto.ca/grootendorst/pdf/IV_Paper_Sept6_2007.pdf
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IV Estimation Cholera=f(Purity,u)+ ε Z = water company Cov(Cholera,Z)= δ Cov(Purity,Z) Z is randomly mixed in the population (two full  sets of pipes) and uncorrelated with behavioral  unobservables, u) Cholera= α + δ Purity+u+ ε Purity = Mean+random variation+ λ u Cov(Cholera,Z)=  δ Cov(Purity,Z)
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IV Estimators Consistent b IV  = ( Z’X ) -1 Z’y            = ( Z’X /n) -1  ( Z’X /n) β + ( Z’X /n) -1 Z’ ε /n          =  β + ( Z’X /n) -1 Z’ ε /n    β Asymptotically normal (same approach to proof as  for OLS) Inefficient – to be shown.
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LS as an IV Estimator The least squares estimator is         ( X X ) -1 X y   =  ( X X ) -1 Σ i x i y i                            =   β    + ( X X ) -1 Σ i x i ε i   If plim( X’X /n) =  Q  nonzero    plim( X’ ε /n)  =  0   Under the usual assumptions LS is an IV estimator    X  is its own instrument.
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IV Estimation Why use an IV estimator ?  Suppose that  X  and  ε   are  not  uncorrelated.  Then least squares is  neither unbiased nor consistent.
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