is conditionally independent of X Y gi g K X k 1 X ik gk gi g A B 1 where E gi

Is conditionally independent of x y gi g k x k 1 x ik

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is conditionally independent of X : Y gi = ± g 0 + K X k =1 X ik ± gk + ° gi ; g = A; B; (1) where E ( ° gi j X i ) = 0 , and X is the vector of covariates ( X i = [ X i 1 ; ::; X iK ] ). As is well known, the overall di/erence in average outcomes between group B and A , b ° ° O = Y B ° Y A ; 3
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can be written as: 3 b ° ° O = ( b ± B 0 ° b ± A 0 ) + K X k =1 X Bk ° b ± Bk ° b ± Ak ± | {z } b ° ° S (Unexplained) + K X k =1 ² X Bk ° X Ak ³ b ± Ak | {z } b ° ° X (Explained) where b ± g 0 and b ± gk ( k = 1 ; ::; K ) are the estimated intercept and slope coe¢ cients, re- spectively, of the regression models for groups g = A; B . The °rst term in the equation is what is usually called the ±unexplained² e/ect in Oaxaca decompositions. Since we mostly focus on wage decompositions in this chapter, we typically refer to this °rst ele- ment as the ±wage structure²e/ect ( ° ° S ). The second component, ° ° X , is a composition e/ect, which is also called the ±explained² e/ect (by di/erences in covariates) in OB decompositions. In the above decomposition, it is straightforward to compute both the overall composi- tion and wage structure e/ects, and the contribution of each covariate to these two e/ects. Following the existing literature on decompositions, we refer to the overall decomposition (separating ° ° O in its two components ° ° S and ° ° X ) as an aggregate decomposition . The detailed decomposition involves subdividing both ° ° S , the wage structure e/ect, and ° ° X , the composition e/ect, into the respective contributions of each covariate, ° ° S;k and ° ° X;k , for k = 1 ; ::; K . The chapter is organized around the following ±take away²messages: A. The wage structure e/ect can be interpreted as a treatment e/ect This point is easily seen in the case where group B consists of union workers, and group A consists of non-union workers. The raw wage gap ° ° can be decomposed as the sum of the ±e/ect² of unions on union workers, ° ° S , and the composition e/ect linked to di/erences in covariates between union and non-union workers, ° ° X . We can 3 The decomposition can also be written by exchanging the reference group used for the wage structure and composition e/ects as follows: b ° ° O = ´ ( b ° B 0 ° b ° A 0 ) + K P k =1 X Ak ° b ° Bk ° b ° Ak ± µ + ´ K P k =1 ² X Bk ° X Ak ³ b ° Bk µ . Alternatively, the so-called three-fold decomposition uses the same reference group for both ef- fects, but introduces a third interaction term: b ° ° O = ´ ( b ° B 0 ° b ° A 0 ) + K P k =1 X Ak ° b ° Bk ° b ° Ak ± µ + ´ K P k =1 ² X Bk ° X Ak ³ b ° Ak µ + ´ K P k =1 ° X Bk ° X Ak ) ° b ° Bk ° b ° Ak ±± µ . While these various versions of the basic decomposition are used in the literature, using one or the other does not involve any speci°c esti- mation issues. For the sake of simplicity, we thus focus on the one decomposition introduced in the text for most of the chapter.
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