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15 93 identication food for thought ii step 3 take

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Unformatted text preview: ; indeed you can verify it: E [U + g (X ) Melissa Tartari (Yale) G (X ) jX ] = E [U + Y U G (X ) jX ] = E [Y G (X ) jX ] = E [Y jX ] G (X ) = 0 Econometrics 14 / 93 Identi…cation: Food for Thought II - step 3 Take (3) and add and subtract g (X ) on the RHS: Y = g (X ) + ( U + G (X ) Melissa Tartari (Yale) Econometrics g (X )) 15 / 93 Identi…cation: Food for Thought II - step 3 Take (3) and add and subtract g (X ) on the RHS: Y = g (X ) + ( U + G (X ) g (X )) From (2) we see that the term in brackets must equal U . What is then E [U jX ]? We see that E [U jX ] needs not be zero: E [ U + G (X ) Melissa Tartari (Yale) g (X ) jX ] = E [ U jX ] + G (X ) = 0 + G (X ) Econometrics g (X ) g (X ) 15 / 93 Identi…cation: Food for Thought II - step 3 Take (3) and add and subtract g (X ) on the RHS: Y = g (X ) + ( U + G (X ) g (X )) From (2) we see that the term in brackets must equal U . What is then E [U jX ]? We see that E [U jX ] needs not be zero: E [ U + G (X ) g (X ) jX ] = E [ U jX ] + G (X ) = 0 + G (X ) g (X ) g (X ) E [U jX ] is zero only when G (X ) = g (X ); thus, the ZCMA is equivalent to assuming that G (X ) = g (X ) in which case we also have U = U . Melissa Tartari (Yale) Econometrics 15 / 93 Identi…cation: an Example - part I Suppose that Y = wages , X = education, and U = ability , and an economic model tells us that: w = g (ed ) + abil = ( βo + β1 ed ) + abil . Melissa Tartari (Yale) Econometrics 16 / 93 Identi…cation: an Example - part I Suppose that Y = wages , X = education, and U = ability , and an economic model tells us that: w = g (ed ) + abil = ( βo + β1 ed ) + abil . Consider the mean of wages conditional on education, call it G (ed ): G (ed ) E [w jed ] = βo + β1 ed + E [abil jed ] = g (ed ) + E [abil jed ] Melissa Tartari (Yale) Econometrics 16 / 93 Identi…cation: an Example - part I Suppose that Y = wages , X = education, and U = ability , and an economic model tells us that: w = g (ed ) + abil = ( βo + β1 ed ) + abil . Consider the mean of wages conditional on education, call it G (ed ): G (ed ) E [w jed ] = βo + β1 ed + E [abil jed ] = g (ed ) + E [abil jed ] We can equ...
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