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
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
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)
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.
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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