Ch. 12 Instrumental Variables Regression

Ch 12 Instrumental - INSTRUMENTAL VARIABLES REGRESSION(Ch 12 The recommended exercise questions from the textbook Chapter 12 All

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1 | Instrumental Variables INSTRUMENTAL VARIABLES REGRESSION (Ch. 12) The recommended exercise questions from the textbook: Chapter 12: All except (12.3), (12.4), (12.6). [1] Motivation • Consider the following simple regression model: Y = β 0 + β 1 X 1 + u . • Least Square Assumption 1 requires E( u | X 1 ) = 0. This condition is violated if X 1 and u are correlated. • What happens if X and u are correlated? 1 11 1 cov( , ) ˆ var( ) p Xu X   .
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2 | Instrumental Variables • When can X 1 and u be correlated? • Omitted regressors: • True model: Y = β 0 + β 1 X 1 + β 2 X 2 + u * . • The model you estimate: Y = β 0 + β 1 X 1 + ( β 2 X 2 + u * ). 1 ˆ from your model (with the omitted variable X 2 ) * 12 2 2 11 1 cov( , ) cov( , ) var( ) var( ) cov( , ) . var( ) p X Xu X X XX X X X   
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3 | Instrumental Variables • Measurement errors in the data on X ( X i ’s). • True model: Y = β 0 + β 1 X + u . O b s e r v e Z = X + v where v is measurement error. • Model you estimate when you use Z instead of X : Y = β 0 + β 1 ( Z - v ) + u = β 0 + β 1 Z + ( u - β 1 v ). This model violates the LS Assumption 1. For this case, 2 11 22 ˆ X p X v  .
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4 | Instrumental Variables • Simultaneous equation bias. • Example: ( 1 ) Y = β 0 + β 1 X + u (Demand: Y = Q and X = P ) ( 2 ) X = γ 0 + γ 1 Y + γ 2 Z + ξ ( ξ is the error in this model) ( I n v e r s e S u p p l y ) ( 1 ) (2): X = γ 0 + γ 1 ( β 0 + β 1 X + u ) + γ 2 Z + ξ X = ( γ 0 + γ 1 β 0 )/(1- γ 1 β 1 )+ γ 2 Z /(1- γ 1 β 1 )+( ξ + γ 1 u )/(1- γ 1 β 1 ). X and u are correlated. Equation (1) violates the LS Assumption 1.
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5 | Instrumental Variables [2] The IV estimator with a Single X and a Single Instrument (1) Basic Model: Y = β 0 + β 1 X + u , where cov( X , u ) may not be equal to zero. • Suppose that a variable Z is uncorrelated with u ; that is, cov( Z , u ) = 0. • When a variable, say Z , is uncorrelated with the error term u , Z is called exogenous . • Conversely, if a variable Z is correlated with u , it is called endogenous . • Two conditions for a valid instrumental variable Z : • Instrument Exogeneity: cov( Z , u ) = 0. • Instrument Relevance: cov( Z , X ) 0.
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6 | Instrumental Variables (2) Two-Stage Least Squares (TSLS) Estimation. 1 st Step: Regress the model X i = γ 0 + γ 1 Z i + ν i , and get 01 ˆ ˆˆ ii X Z  . 2 nd Step: Using ˆ i X instead of X i as a regressor, estimate β 0 and β 1 by OLS on ˆ i YX e r r o r  .
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7 | Instrumental Variables (3) Intuition for TSLS • Suppose we wish to estimate the demand for butter: ,0 1 , i Butter i Butter i QP u  . • Suppose the supply function of butter is given: 1 ,2 iBu t ter t i i P QZ  , w h e r e Z i = average daily-region rainfall.
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8 | Instrumental Variables Can Z be used as an instrument to estimate the demand curve? Z is uncorrelated with u ? (Why or why not?) Z is correlated with P ? (Why or why not?)
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9 | Instrumental Variables (4) Sampling distribution of the TSLS estimators when n is large: Let 0 and 1 be the TSLS estimators.
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Ch 12 Instrumental - INSTRUMENTAL VARIABLES REGRESSION(Ch 12 The recommended exercise questions from the textbook Chapter 12 All

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