Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 3. The weighted BO decomposition

3.1  The decomposition

Using the setup in Section 2, the findings of Blinder (1973) and Oaxaca (1973) are summarized in the context of sampling theory, namely by using sampling weights. Assume that in each sample, a linear relationship is suitable between the p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36F0@ characteristics that are available and the logarithm of the wage. A regression is done separately in each subpopulation U h , h = { M , F } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGObaabeaakiaaiYcacaWGObGaaGypamaacmaabaGaamyt aiaaiYcacaWGgbaacaGL7bGaayzFaaGaaiOlaaaa@3F98@ At the subpopulation level, the values of the regression coefficients are given by

β h = ( k U h x k x k ) 1 k U h x k y k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaaBa aaleaacaWGObaabeaakiaai2dadaqadaqaamaaqafabaGaaCiEamaa BaaaleaacaWGRbaabeaakiaahIhadaqhaaWcbaGaam4AaaqaamXvP5 wqSX2qVrwzqf2zLnharyqqYLwySbsvUL2yVrwzG00uaGqbaiaa=jrm aaaabaGaam4AaiabgIGiolaadwfadaWgaaadbaGaamiAaaqabaaale qaniabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaa igdaaaGcdaaeqbqaaiaahIhadaWgaaWcbaGaam4AaaqabaGccaWG5b WaaSbaaSqaaiaadUgaaeqaaaqaaiaadUgacqGHiiIZcaWGvbWaaSba aWqaaiaadIgaaeqaaaWcbeqdcqGHris5aOGaaGOlaaaa@5F2F@

They can be estimated from the sample by

β ^ h = ( k S h d k x k x k ) 1 k S h d k x k y k , ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaSbaaSqaaiaadIgaaeqaaOGaaGypamaabmaabaWaaabuaeaacaWG KbWaaSbaaSqaaiaadUgaaeqaaOGaaCiEamaaBaaaleaacaWGRbaabe aakiaahIhadaqhaaWcbaGaam4AaaqaamXvP5wqSX2qVrwzqf2zLnha ryqqYLwySbsvUL2yVrwzG00uaGqbaiaa=jrmaaaabaGaam4AaiabgI GiolaadofadaWgaaadbaGaamiAaaqabaaaleqaniabggHiLdaakiaa wIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaaeqbqaai aadsgadaWgaaWcbaGaam4AaaqabaGccaWH4bWaaSbaaSqaaiaadUga aeqaaOGaamyEamaaBaaaleaacaWGRbaabeaaaeaacaWGRbGaeyicI4 Saam4uamaaBaaameaacaWGObaabeaaaSqab0GaeyyeIuoakiaaiYca caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlai aaigdacaGGPaaaaa@6EA0@

where d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@3800@ are the sampling weights. The regression coefficients β ^ h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaSbaaSqaaiaadIgaaeqaaaaa@3862@ are called the group wage structure or the returns on characteristics and they represent the contribution of each characteristic to the wage.

Result 1 A sufficient condition to obtain the following equalities

Y ¯ h = X ¯ h β h and Y ¯ ^ h = X ¯ ^ h β ^ h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaaiaadIgaaeqaaOGaaGypaiqahIfagaqeamaaDaaaleaa caWGObaabaWexLMBbXgBd9gzLbvyNv2CaeHbbjxAHXgiv5wAJ9gzLb sttbacfaGaa8NeXaaakiaahk7adaWgaaWcbaGaamiAaaqabaGccaaM e8UaaGjbVlaabggacaqGUbGaaeizaiaaysW7caaMe8Uabmywayaary aajaWaaSbaaSqaaiaadIgaaeqaaOGaaGypaiqahIfagaqegaqcamaa DaaaleaacaWGObaabaGaa8NeXaaakiqahk7agaqcamaaBaaaleaaca WGObaabeaaaaa@5ACB@

is that there exists a vector ς p , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOWdiabgI Gioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xh Hi1aaWbaaSqabeaacaWGWbaaaOGaaiilaaaa@4561@  such that ς x k = 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOWdmaaCa aaleqabaWexLMBbXgBd9gzLbvyNv2CaeHbbjxAHXgiv5wAJ9gzLbst tbacfaGaa8NeXaaakiaahIhadaWgaaWcbaGaam4AaaqabaGccaaI9a GaaGymaiaacYcaaaa@4816@  for all k U h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadwfadaWgaaWcbaGaamiAaaqabaGccaaIUaaaaa@3B24@

Since it is assumed that x k 1 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbGaaGymaaqabaGccaaI9aGaaGymaaaa@3A5B@ for all k U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadwfacaGGSaaaaa@39F9@ with ς = ( 1 0 0 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOWdmaaCa aaleqabaWexLMBbXgBd9gzLbvyNv2CaeHbbjxAHXgiv5wAJ9gzLbst tbacfaGaa8NeXaaakiaai2dadaqadaqaaiaaigdacaaMe8UaaGimai ablAciljaaicdaaiaawIcacaGLPaaacaGGSaaaaa@4B9B@ the equality is always fulfilled. The proof of Result 1 can be found in Appendix A. Putting together the result above, equations (2.1) and (3.1), the average difference between the wages of two groups can be written as

Δ = Y ¯ ^ M Y ¯ ^ F = ( X ¯ ^ M X ¯ ^ F ) β ^ F + X ¯ ^ M ( β ^ M β ^ F ) . ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaaG ypaiqadMfagaqegaqcamaaBaaaleaacaWGnbaabeaakiabgkHiTiqa dMfagaqegaqcamaaBaaaleaacaWGgbaabeaakiaai2dadaqadaqaai qahIfagaqegaqcamaaBaaaleaacaWGnbaabeaakiabgkHiTiqahIfa gaqegaqcamaaBaaaleaacaWGgbaabeaaaOGaayjkaiaawMcaamaaCa aaleqabaWexLMBbXgBd9gzLbvyNv2CaeHbbjxAHXgiv5wAJ9gzLbst tbacfaGaa8NeXaaakiqahk7agaqcamaaBaaaleaacaWGgbaabeaaki abgUcaRiqahIfagaqegaqcamaaDaaaleaacaWGnbaabaGaa8NeXaaa kmaabmaabaGabCOSdyaajaWaaSbaaSqaaiaad2eaaeqaaOGaeyOeI0 IabCOSdyaajaWaaSbaaSqaaiaadAeaaeqaaaGccaGLOaGaayzkaaGa aGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiodaca GGUaGaaGOmaiaacMcaaaa@6A14@

The difference between average wages of the groups contains two elements: an explained part, also called the composition effect ( X ¯ ^ M X ¯ ^ F ) β ^ F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdHqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWabeaace WHybGbaeHbaKaadaWgaaWcbaGaamytaaqabaGccqGHsislceWHybGb aeHbaKaadaWgaaWcbaGaamOraaqabaaakiaawIcacaGLPaaadaahaa WcbeqaamXvP5wqSX2qVrwzqf2zLnharyqqYLwySbsvUL2yVrwzG00u aGqbaiaa=jrmaaGcceWHYoGbaKaadaWgaaWcbaGaamOraaqabaaaaa@4B43@ and an unexplained part, or the structure effect X ¯ ^ M ( β ^ M β ^ F ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdHqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaary aajaWaa0baaSqaaiaad2eaaeaatCvAUfeBSn0BKvguHDwzZbqegeKC PfgBGuLBPn2BKvginnfaiuaacaWFsedaaOWaaeWabeaaceWHYoGbaK aadaWgaaWcbaGaamytaaqabaGccqGHsislceWHYoGbaKaadaWgaaWc baGaamOraaqabaaakiaawIcacaGLPaaacaGGUaaaaa@4C16@ The former encompasses differences in characteristics between the two groups. The latter is the difference in the returns on characteristics between the two groups, the part that is not attributable to objective factors (Oaxaca, 1973; Blinder, 1973). It is obtained using characteristics as a proxy for productivity. The estimation of the structure effect is the central element of this paper. Equation (3.2) has the same elements as the one proposed by Oaxaca (1973) and Blinder (1973). The methodology applied to obtain the estimated average values and coefficients differs from the traditional regression technique. The BO method uses the estimated regression coefficients obtained through ordinary least squares (OLS) and the vectors of average values of the observed explanatory variables. The proposed approach takes into account the survey weights. However, the weighted BO method is the same as the original BO method if the sampling weights are all equal to 1.

3.2  A note on the structure effect

The two elements in equation 3.2 have different names across the literature. The first one, whose denomination we retained as composition effect is also termed endowments effect. The second one, which we call structure effect is also found in the literature as unexplained residual, price effect, sex effect, calculated effect or unequal treatment (Weichselbaumer and Winter-Ebmer, 2006). Using the BO method, the structure effect is an estimation of the discrimination level. However, discrimination is an intricate phenomenon that might not be always fully observed. Unobserved variables, selection bias or some mechanisms on the labour market can help to increase the explained part of the wage difference. Moreover, Weichselbaumer and Winter-Ebmer (2005) note two potential issues regarding the chosen model. First, if the characteristics chosen in the linear model are themselves subject to discrimination, then the resulting structure effect will be over-estimated. Second, if the characteristics are not a proper measure of the productivity, then again, the structure effect might be under- or over-estimated. Weichselbaumer and Winter-Ebmer (2006) warn about the legitimacy of the characteristics as productivity indicators, since “wages may also be determined by bargaining power, compensating differentials or efficiency wages”. However, for simplicity, in what follows, we will assume that there are no such issues and that the estimated structure effect is the result of discrimination on the labor market. Moreover, we do not examine sample selection bias or other mechanisms underlying the distribution of men and women in certain jobs.

3.3  The counterfactual wage distribution

In general, the counterfactual wage distribution is an artificial distribution obtained by using the characteristics of a group to estimate the wages of another group (see, for instance Bourguignon, Ferreira, and Leite, 2002). Examples of counterfactual distributions are found in DiNardo et al. (1996) or DiNardo (2002). The term X ¯ ^ M β ^ F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaary aajaWaaSbaaSqaaiaad2eaaeqaaOGabCOSdyaajaWaaSbaaSqaaiaa dAeaaeqaaaaa@3A50@ that appears in equation (3.2) is called the women’s counterfactual average wage. It is interpreted as the estimated average wage of women if they had the same average characteristics as men and if their return on characteristics remained unchanged. Women’s counterfactual wage distribution is obtained by using the characteristics of men ( X M ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WHybWaaSbaaSqaaiaad2eaaeqaaaGccaGLOaGaayzkaaaaaa@396D@ and the wage structure of women ( β F ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WHYoWaaSbaaSqaaiaadAeaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa @3A75@ In terms of interpretation, it is the wage distribution of women, if they had the same characteristics as men.

Using Result 1 from the previous section, women’s counterfactual mean wage equals

Y ¯ F | M = X ¯ M β F , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaamaaeiaabaGaamOraiaayIW7aiaawIa7aiaayIW7caWG nbaabeaakiaai2daceWHybGbaebadaqhaaWcbaGaamytaaqaamXvP5 wqSX2qVrwzqf2zLnharyqqYLwySbsvUL2yVrwzG00uaGqbaiaa=jrm aaGccaWHYoWaaSbaaSqaaiaadAeaaeqaaOGaaGilaaaa@4F76@

and is estimated from the sample by

Y ¯ ^ F | M = X ¯ ^ M β ^ F , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaary aajaWaaSbaaSqaamaaeiaabaGaamOraiaayIW7aiaawIa7aiaayIW7 caWGnbaabeaakiaai2daceWHybGbaeHbaKaadaqhaaWcbaGaamytaa qaamXvP5wqSX2qVrwzqf2zLnharyqqYLwySbsvUL2yVrwzG00uaGqb aiaa=jrmaaGcceWHYoGbaKaadaWgaaWcbaGaamOraaqabaGccaaISa aaaa@4FA4@

where X ¯ ^ M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaary aajaWaaSbaaSqaaiaad2eaaeqaaaaa@3801@ are estimated in equation (2.1) and β ^ F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaSbaaSqaaiaadAeaaeqaaaaa@3840@ are the coefficients estimated by means of equation (3.1). With this notation, the BO decomposition given in (3.2) is re-expressed as

Δ = Y ¯ ^ M Y ¯ ^ F = ( X ¯ ^ M X ¯ ^ F ) β ^ F + X ¯ ^ M ( β ^ M β ^ F ) = ( Y ¯ ^ F | M Y ¯ ^ F ) + ( Y ¯ ^ M Y ¯ ^ F | M ) . ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaaG ypaiqadMfagaqegaqcamaaBaaaleaacaWGnbaabeaakiabgkHiTiqa dMfagaqegaqcamaaBaaaleaacaWGgbaabeaakiaai2dadaqadaqaai qahIfagaqegaqcamaaBaaaleaacaWGnbaabeaakiabgkHiTiqahIfa gaqegaqcamaaBaaaleaacaWGgbaabeaaaOGaayjkaiaawMcaamaaCa aaleqabaWexLMBbXgBd9gzLbvyNv2CaeHbbjxAHXgiv5wAJ9gzLbst tbacfaGaa8NeXaaakiqahk7agaqcamaaBaaaleaacaWGgbaabeaaki abgUcaRiqahIfagaqegaqcamaaDaaaleaacaWGnbaabaGaa8NeXaaa kmaabmaabaGabCOSdyaajaWaaSbaaSqaaiaad2eaaeqaaOGaeyOeI0 IabCOSdyaajaWaaSbaaSqaaiaadAeaaeqaaaGccaGLOaGaayzkaaGa aGypamaabmaabaGabmywayaaryaajaWaaSbaaSqaamaaeiaabaGaam OraiaayIW7aiaawIa7aiaayIW7caWGnbaabeaakiabgkHiTiqadMfa gaqegaqcamaaBaaaleaacaWGgbaabeaaaOGaayjkaiaawMcaaiabgU caRmaabmaabaGabmywayaaryaajaWaaSbaaSqaaiaad2eaaeqaaOGa eyOeI0IabmywayaaryaajaWaaSbaaSqaamaaeiaabaGaamOraiaayI W7aiaawIa7aiaayIW7caWGnbaabeaaaOGaayjkaiaawMcaaiaai6ca caaMf8UaaiikaiaaiodacaGGUaGaaG4maiaacMcaaaa@7DA5@

3.4  Using the counterfactual distribution to estimate the composition and the structure effects

Building the counterfactual average wage allows for the estimation of the two effects that make up the wage difference at the average levels. From equation (3.3), the composition effect is equal to

( X ¯ ^ M X ¯ ^ F ) β ^ F = ( Y ¯ ^ F | M Y ¯ ^ F ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaace WHybGbaeHbaKaadaWgaaWcbaGaamytaaqabaGccqGHsislceWHybGb aeHbaKaadaWgaaWcbaGaamOraaqabaaakiaawIcacaGLPaaadaahaa WcbeqaamXvP5wqSX2qVrwzqf2zLnharyqqYLwySbsvUL2yVrwzG00u aGqbaiaa=jrmaaGcceWHYoGbaKaadaWgaaWcbaGaamOraaqabaGcca aI9aWaaeWaaeaaceWGzbGbaeHbaKaadaWgaaWcbaWaaqGaaeaacaWG gbGaaGjcVdGaayjcSdGaaGjcVlaad2eaaeqaaOGaeyOeI0Iabmyway aaryaajaWaaSbaaSqaaiaadAeaaeqaaaGccaGLOaGaayzkaaGaaGOl aaaa@58D7@

The composition effect can be interpreted as the difference between what women would earn on average if they had the characteristics of men and what they actually earn. Thus, it reflects the inequality due to the differences in characteristics. The structure effect in equation (3.3) is equal to

X ¯ ^ M ( β ^ M β ^ F ) = ( Y ¯ ^ M Y ¯ ^ F | M ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaary aajaWaa0baaSqaaiaad2eaaeaatCvAUfeBSn0BKvguHDwzZbqegeKC PfgBGuLBPn2BKvginnfaiuaacaWFsedaaOWaaeWaaeaaceWHYoGbaK aadaWgaaWcbaGaamytaaqabaGccqGHsislceWHYoGbaKaadaWgaaWc baGaamOraaqabaaakiaawIcacaGLPaaacqGH9aqpdaqadaqaaiqadM fagaqegaqcamaaBaaaleaacaWGnbaabeaakiabgkHiTiqadMfagaqe gaqcamaaBaaaleaadaabcaqaaiaadAeacaaMi8oacaGLiWoacaaMi8 UaamytaaqabaaakiaawIcacaGLPaaacaaIUaaaaa@5934@

The structure effect is the difference between the actual average wage of men and what women would earn if they had the average characteristics of men and their wage own structure. The equations above express the composition and structure effects at the average levels, since this is the limitation of the BO method. The next section presents a method that allows for the construction of the entire counterfactual distribution. This in turn results in the ability of estimating the composition and structure effects along the entire wage distribution.


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