singleequationgmmslides

singleequationgmmslides - Single Equation Linear GMM...

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Single Equation Linear GMM Consider the linear regression model y t = z 0 t δ 0 + ε t ,t =1 ,...,n z t = L × 1 vector of explanatory variables δ 0 = L × 1 vector of unknown coe cients ε t = random error term Engodeneity The model allows for the possibility that some or all of the elements of z t may be correlated with the error term ε t (i.e., E [ z tk ε t ] 6 =0 for some k ) . If E [ z tk ε i ] 6 =0 , then z tk is called an endogenous variable . If z t contains endogenous variables, then the least squares estimator of δ 0 in is biased and inconsistent.
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Instruments It is assumed that there exists a K × 1 vector of instrumental variables x t that may contain some or all of the elements of z t . Let w t represent the vector of unique and nonconstant elements of { y t , z t , x t } . It is assumed that { w t } is a stationary and ergodic stochastic process.
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Moment Conditions for General Model De f ne g t ( w t , δ 0 )= x t ε t = x t ( y t z 0 t δ 0 ) It is assumed that the instrumental variables x t satisfy the set of K orthogo- nality conditions E [ g t ( w t , δ 0 )] = E [ x t ε t ]= E [ x t ( y t z 0 t δ 0 )] = 0 Expanding gives the relation Σ xy ( K × 1) = Σ xz ( K × L ) δ 0 ( L × 1) where Σ xy = E [ x t y t ] and Σ xz = E [ x t z 0 t ] .
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Example: Demand-Supply model with supply shifter demand: q d i = α 0 + α 1 p i + u i supply: q s i = β 0 + β 1 p i + β 2 temp i + v i equilibrium: q d i = q s i = q i E [ temp i u i ]= E [ temp i v i ]=0 Goal: Estimate demand equation; α 1 = demand elasticity if data are in logs Remark Simultaneity causes p i to be endogenous in demand equation: E [ p i u i ] 6 =0 Why? Use equilibrium condition, solve for p i and compute E [ p i u i ]
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Demand/Supply model in Hayashi notation y i = z 0 i δ + ε i y i = q i , z i =(1 ,p i ) 0 , δ =( α 0 1 ) 0 x i =( 1 , temp i ) 0 , w i =( q i ,p i , temp i ) 0 L =2 ,K =2 Note: 1 is common to both z i and x i .
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Example: Wage Equation ln W i = δ 1 + δ 2 S i + δ 3 EXPR i + δ 4 IQ i + ε i W i = wages S i = years of schooling EXPR i = years of experience IQ i = score on IQ test δ 2 = rate of return to schooling Assume E [ S i ε i ]= E [ EXPR i ε i ]=0 Endogeneity: IQ i is a proxy for ability but is measured with error IQ i = π 0 + π 1 ABILITY i + error i E [ IQ i ε i ] 6 =0
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Instruments: AGE i = age in years MED i = years of mother’s education E [ AGE i ε i ]= E [ MED i ε i ]=0
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y i = z 0 i δ + ε i y i =l n W i , z i =( 1 ,S i ,EXPR i ,IQ i ) 0 x i =( 1 ,S i ,EXPR i ,AGE i ,MED i ) 0 w i =( l n W i ,S i ,EXPR i ,IQ i ,AGE i ,MED i ) 0 L =4 ,K =5 Remarks 1. 1 ,S i ,EXPR i are often called included exogenous variables. That is, they are included in the behavioral wage equation. 2.
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This note was uploaded on 08/23/2010 for the course ECON 583 taught by Professor Zivot during the Fall '09 term at W. Alabama.

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singleequationgmmslides - Single Equation Linear GMM...

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