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Lecture 21 Multiple linear IV model and 2-stage least squares

# Lecture 21 Multiple linear IV model and 2-stage least squares

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Economics 326 Methods of Empirical Research in Economics Lecture 21: Multiple linear IV model and two-stage least squares (2SLS) Vadim Marmer University of British Columbia May 5, 2010

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Multiple linear IV model I In empirical research, we often have to estimate models that include multiple endogenous and exogenous regressors. I Example: ln Wage i = γ 0 + γ 1 Age i + γ 2 Sex i + β 1 Educ i + β 2 Children i + U i . I Exogenous regressors: age, sex, and a constant. I Endogenous regressors: education and children (family size). 1/11
Multiple linear IV model I Consider the following model: y i = γ 0 + γ 1 X 1 , i + . . . + γ k X k , i + β 1 Y 1 , i + . . . + β m Y m , i + U i , where I y i is the dependent variable. I γ 0 is the coe¢ cient on the constant regressor: EU i = 0 . I X 1 , i , . . . , X k , i are the k exogenous regressors: Cov ( X 1 , i , U i ) = . . . = Cov ( X k , i , U i ) = 0 . I Y 1 , i , . . . , Y m , i are the m endogenous regressors: Cov ( Y 1 , i , U i ) 6 = 0 , . . . , Cov ( Y k , i , U i ) 6 = 0 . 2/11

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Identi°cation problem I There are k + 1 + m unknown coe¢ cients y i = γ 0 + γ 1 X 1 , i + . . . + γ k X k , i + β 1 Y 1 , i + . . . + β m Y m , i + U i . I The exogeneity conditions EU i = 0 and Cov ( X 1 , i , U i ) = . . . = Cov ( X k , i , U i ) = 0 give us only k + 1 equations: E [ y i ° γ 0 ° γ 1 X 1 , i ° . . . ° γ k X k , i ° β 1 Y 1 , i ° . . . ° β m Y m , i ] = 0 , E [ X 1 , i ( y i ° γ 0 ° γ 1 X 1 , i ° . . . ° γ k X k , i ° β 1 Y 1 , i ° . . . ° β m Y m , i )] = 0 , . . . E [ X k , i ( y i ° γ 0 ° γ 1 X 1 , i ° . . . ° γ k X k , i ° β 1 Y 1 , i ° . . . ° β m Y m , i )] = 0 . I There are more unknowns than equations. Thus, the knowledge of the true covariances between X ±s, Y ±s and y is not su¢ cient to recover the unknown coe¢ cients γ 0 , γ 1 , . . . , γ k , β 1 , . . . , β m .
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