econ483lec05endog - 1 Endogeneity and Causality Wooldridge...

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1 Endogeneity and Causality Wooldridge Chapters 15-16 & Berndt Chapter 8 1.1 Endogeneity due to Omitted Variables suppose that wage is determined by log wage i = & 0 + & 1 educ i + & 2 abil i + e i , for i = 1 ; :::; n assume that the error term e i is not correlated with the right hand side variables ( educ i and abil i ) that means e i can be considered as pure luck typically the right hand side variables are correlated each other in this model, it is likely that educ i and abil i are positively correlated (more able person gets more education) in practice, however, suppose that ability is not observed and we decide to estimate log wage i = & 0 + & 1 educ i + u i , for i = 1 ; :::; n in this case, the error term u i contains abil i ( u i = & 2 abil i + e i ) therefore the error term u i is likely to be correlated with educ i when any of the right hand side variables are correlated with the error term, the least squares estimator is not consistent any more (consistency: as the sample size tends to in&nity, the estimator gets closer to the true parameter) to see why the estimator is inconsistent or to see whether b & 1 gets closer to & 1 as n gets large let±s rewrite the above models by y i = & 0 + & 1 x 1 i + & 2 x 2 i + e i , for i = 1 ; :::; n and y i = & 0 + & 1 x 1 i + u i , for i = 1 ; :::; n we know that when we estimate the latter model by the least squares method, we obtain b & 1 = P n i =1 ( x 1 i & x 1 ) y i P n i =1 ( x 1 i & x 1 ) 2 substitute the true model into b & 1 formula b & 1 = P n i =1 ( x 1 i & x 1 ) ( & 0 + & 1 x 1 i + & 2 x 2 i + e i ) P n i =1 ( x 1 i & x 1 ) 2 = & 0 P n i =1 ( x 1 i & x 1 ) P n i =1 ( x 1 i & x 1 ) 2 + & 1 P n i =1 ( x 1 i & x 1 ) x 1 i P n i =1 ( x 1 i & x 1 ) 2 + & 2 P n i =1 ( x 1 i & x 1 ) x 2 i P n i =1 ( x 1 i & x 1 ) 2 + P n i =1 ( x 1 i & x 1 ) e i P n i =1 ( x 1 i & x 1 ) 2 = 0 + & 1 + & 2 c cov ( x 1 i ; x 2 i ) d var ( x 1 i ) + c cov ( x 1 i ; e i ) d var ( x 1 i ) by the law of large numbers (plus some theory) b & 1 ! p & 1 + & 2 cov ( x 1 i ; x 2 i ) var ( x 1 i ) + 0 6 = & 1 what we &nd is that b & 1 is consistent only if & 2 = 0 or cov ( x 1 i ; x 2 i ) = 0 1
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in this example, we expect & 2 > 0 (more able person earns higher wage) and cov ( x 1 i ; x 2 i ) > 0 in consequence, there is an upward bias in b & 1 therefore, when we do not include abil i in the estimation model, we are likely to get an estimate that overstates the true e/ect of education on wage let&s go back to the b & 1 = & 1 + & 2 c cov ( x 1 i ; x 2 i ) d var ( x 1 i ) + c cov ( x 1 i ; e i ) d var ( x 1 i ) we know that c cov ( x 1 i ; x 2 i ) d var ( x 1 i ) is the least squares estimator of a simple regression of x 2 i on x 1 i that is if we construct a regression model x 2 i = ± 0 + ± 1 x 1 i + w i , then, b ± 1 = c cov ( x 1 i ; x 2 i ) d var ( x 1 i ) the model x 2 i = ± 0 + ± 1 x 1 i + w i is called the auxiliary equation and we learn that b & 1 ! p & 1 + & 2 ± 1 another way of calculating the bias, consider y i = & 0 + & 1 x 1 i + & 2 x 2 i + e i = & 0 + & 1 x 1 i + & 2 ( ± 0 + ± 1 x 1 i + w i ) + e i = ( & 0 + & 2 ± 0 ) + ( & 1 + & 2 ± 1 ) x 1 i + ( e i + & 2 w i ) which implies that b & 1 !
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This note was uploaded on 04/04/2011 for the course ECON 401 taught by Professor Staff during the Spring '08 term at University of Washington.

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econ483lec05endog - 1 Endogeneity and Causality Wooldridge...

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