Detailed Wage Decompositions: Revisiting the Identification Problem

Transcription

1 Detailed Wage Decompositions: Revisiting the Identification Problem ChangHwan Kim August 1, 2012 Department of Sociology University of Kansas 1415 Jayhawk Blvd., Room 716 Lawrence, KS Tel: (785) Fax: (785) Comments from the editor of Sociological Methodology, five anonymous reviewers, Arthur Sakamoto, Daniel Powers, Yu Xie, and Myeong-Su Yun have greatly improved previous drafts. All remaining errors are the author s sole responsibility. i

2 Detailed Wage Decompositions: Revisiting the Identification Problem August 1, 2012 Abstract Yun (2005) suggested an averaging method to resolve the identification problem in detailed decompositions. Since then, detailed decomposition techniques have been widely discussed. This paper shows that the averaging method may not answer the identification problem. The method is built on unrealistic distributional constraints on a set of dummy variables and is sensitive to both the number of groups and the method of grouping. As an alternative, a weighted averaging method is suggested which is in line with Haisken-DeNew and Schmidt s (1997) two step re-normalization. This method gives a distinct meaning to the intercept term and makes detailed decomposition feasible using a reasonable assumption. However, this paper underscores that there are multiple solutions to the identification problem, and that other detailed decomposition methods may be acceptable depending on theoretical or practical considerations. Keywords: Blinder-Oaxaca Decomposition, Detailed Decomposition, Averaging Method, Grand-Mean Weighting Method 1

3 1 Introduction Blinder-Oaxaca (BO) wage decomposition methods have been extensively applied in sociology and economics (e.g., Dodoo 1991; Farkas and Vicknair 1996; Sakamoto et al. 2000; DeLeire 2001; Sayer 2004; Van Hook et al. 2004; Phillips and Sweeney 2006; Stearns et al. 2007; Berends and Penaloza 2008). Although several variants of wage decompositions have been developed since the initial work of Blinder (1973) and Oaxaca (1973), 1 all these techniques share the basic idea that the mean wage gap between two groups at a given time can be broken down into two components (Jones and Kelley 1984; Cotton 1985). The first component is the coefficient effect and the second component is the endowment effect. On top of the two components decomposition, researchers have often reported the contributions of individual variables or of combined sets of dummy variables after implementing a BO decomposition (e.g., Fields and Wolff 1995; DeLeire 2001; Bobbitt-Zeher 2007; citealtchangengland:2010). However, these detailed decompositions can be erroneous, because the contribution of each individual variable or set of dummy variables changes with the choice of the reference group. This is the identification problem which has been a much discussed (Jones 1983; Jones and Kelley 1984; Clogg and Eliason 1986; Oaxaca and Ransom 1999; Horrace and Oaxaca 2001; Gardeazabal and Ugidos 2004; Yun 2005) yet often ignored pitfall of detailed decomposition using BO techniques. To resolve the identification problem, Gardeazabal and Ugidos (2004) suggested normalizing the estimated coefficients, and Yun (2005) proposed an averaging method as a simple alternative to the unwieldy normalization. Since then, detailed BO-type decomposition techniques have been widely discussed in sociology and economics (e.g., Bhaumik et al. 2006; Powers and Yun 2009; Kim 2010; Fortin et al. 2010; Powers et al. forthcoming). The averaging method is, however, based on unrealistic distributional constraints on a set of dummy variables. In this paper, I suggest the application of an alternative method and consider its limitations. 1 Decomposition techniques were originally developed by sociologists and demographers (e.g., Kitagawa 1955). See Powers and Yun (2009: ) for a summary history. 2

4 2 Identification Problem and Averaging Method Using ordinary least square (OLS) regression, we can estimate the log wage, y, as follows: y = a + K J j b jk x jk + e (1) j=1 k=1 where j = 1, 2, 3,..., J and k = 1, 2, 3,..., K j ; j refer to the jth factor, and k represents the kth level of each of these J factors. In equation 1, each factor includes a complete set of dummies, including the typically left out reference group. The reference group E[b jk x jk x j1 = 1] takes a value of zero, which implies the identification constraint of b j1 = 0. For simplicity, I restrict J = 1 in this paper, so that equation 1 becomes y = a+ K k=1 b kx k +e. Consider a comparison of white workers (group W) and black workers (group B). Estimating equation 1 separately for two groups, the mean wage gap between groups W and B (ȳ W ȳ B ) can be partitioned into several components as follows: ȳ W ȳ B = ( â W â B) K + (ˆb W k }{{} k=1 D1A ˆb B k ) xb k } {{ } D1B } {{ } D1 + K ( x W k k=1 xb k )ˆb W k } {{ } D2 (2) where D1 is the sum of D1A, which denotes the intercept component, and D1B, which denotes the coefficient component. D1 represents the total coefficient effect. D2 refers to the total endowment effect. By definition in OLS, the mean residual difference between ē W and ē B is zero. In a regression model, an estimated intercept varies with the choice of reference group, and that is the case with the intercept component (D1A) in equation 2. As the reference group changes in a regression model, the estimated coefficient b k changes accordingly. Therefore, estimates of the extent to which individual factors and factor levels contribute to the mean wage gap between groups W and B also vary with the choice of reference group. This illustrates the identification problem in detailed decompositions. 3

5 To resolve the identification problem, Gardeazabal and Ugidos (2004) suggest normalizing the coefficients of the dummy variables by imposing a restriction of b k = 0. As a simple way to impose such restriction, Yun (2005) proposes the following averaging method: y = ( a + b ) K + (b k b)x k + e k=1 K = a + b k x k + e k=1 (3) where a refers to a + b and b refers to (b k b). A b is computed as the sum of b k divided K k=1 by the number of factor levels; that is, b k K. For the reference group of equation 1, b 1 is zero by definition, thus the transformed coefficient b 1 b is equal to b. This transformation of b k causes K k=1 b k to become zero. In ANOVA, the restriction of b k = 0 is referred to as a sigma constraint. It is one of a number of possible identifying normalizations (Fox 2008:145). Under this constraint, the constant or intercept term is a generalized grand mean (i.e., the mean of the means), and the effects b k are deviations from this. As a result of this normalization, both the new coefficients for the independent variables, (b k b), and for the new intercept, a + b, do not depend on the choice of reference group. As the coefficient b 1 for the reference group becomes b, no group is omitted. Because equation 3 has exactly the same structure as equation 1, the BO decomposition technique for equation 2 can also be applied to equation 3. On the surface, the averaging method appears to solve the identification problem. In fact, the averaging method does not resolve it but merely conceals it. In equation 3, the intercept a is the expected wage given x k = 1/K for k = 1,..., K. That is, E[y (x k = 1/K)] = a. The difference between the intercepts of the two groups, a W a B, is the expected wage difference between hypothetical groups W and B, assuming the means of all x s equal 1/K. The normalized coefficients b k represent the expected differences of y for a group whose x k = 1 for a specific k compared to the hypothetical reference point (or group), of which the mean of x k is equal to 1/K for all k s. That is, b k = E[y (x k = 1)] a. 4

6 The identification problem occurs because the estimates of the intercept and those of all other coefficients change with the arbitrary change of reference group. If the intercept of one possible reference group (e.g, E[y x 1 = 1 & x 2,3,4,...,K = 0] when x 1 is the reference group) and that of the other possible reference group (e.g, E[y x 2 = 1 & x 1,3,4,...,K = 0] when x 2 is the reference group) are arbitrary, then the intercept of the averaging method (i.e., E[y x 1,2,3,...,K = 1/K] when a hypothetical point of which the means of all xs equal 1/K is the reference) is also arbitrary in essence. Some will argue that the averaging method nonetheless offers a normalized, standardized, and detailed decomposition. They will say that it is better to use 1/K than to cherrypick the reference group. However, the averaging method is sensitive to the number of k. This sensitivity leads to another kind of identification problem. Suppose a researcher uses the four factor levels for education. One possible four-factor grouping would be LTHS (less than high school), HSG (high school graduate), SC (some college), and BA+ (bachelor degree or higher), in which case the intercept calculated using the averaging method is E(y x LT HS = x HSG = x SC = x BA+ =.25). If the researcher changes her education categories into five, dividing BA+ into BA and Grad (graduate degrees), the intercept of the averaging method is changed to E(y x LT HS = x HSG = x SC = x BA = x Grad =.20). In the former case, BA+ would include a 25% share of workers, whereas in the second grouping it would include a 40% share. As the size of K changes, so does the intercept estimate, as well as all the other coefficients. The averaging method is sensitive not only to the number of groups, but also to the method of grouping. Suppose a research uses another way of four-factor grouping, which combines LTHS and HSG into <HSG and divides BA+ into BA and Grad, so that four groups are <HSG, SC, BA, and Grad. In the previous four-factor grouping, BA+ would include a 25% share of workers, whereas in the new grouping it would include a 50% share. As the hypothetical distribution of x for the intercept differs as a function of grouping method, almost all estimated effects including the intercept effect (D1A), the sum of coefficient effects (sum of D1B), coefficient effects (D1B) of factor level and endowment effects (D2) of 5

7 factor level differ as well. Only the sum of D1 and the sum of D2 are consistently estimated free from the variation of model specifications. Note that the original BO decomposition also yields the same results regarding the sums of D1 and D2. There is also an identification problem with dichotomous variables (Yun 2005). Which values should be coded as 1 and 0 is an arbitrary decision made by the researcher. With the averaging method, the intercept for the dichotomous variables is the value expected when the proportions of x = 1 and x = 0 are equal (i.e.,.50). This simply is not the case for many variables of interest, such as union membership. In sum, despite contentions to the contrary, the averaging method suffers from the same identification problem as the original BO decomposition. Different numbers of groups and different methods of grouping can lead to substantially divergent results. 3 A Suggested Alternative Approach: The Grand-Mean Weighting Method If neither the current BO decomposition nor the averaging method can solve the identification problem, are detailed decompositions feasible at all? They are said to be infeasible because the intercept term in regression models depends on the choice of reference group. In fact, any normalization with the linear restriction of the parameters K k=1 b k w k = 0 will yield model-free decomposition estimates. There are an infinite number of possible restrictions. The averaging method (where the weighting factors w k are equal to 1/K) is a special cases of these restrictions. Even the original BO decomposition can be considered a special case of the restriction, with w 1 = 1 for k = 1 (factor level 1 is the reference group) and w k = 0 for all other k. Given the restriction of K k=1 b k w k = 0, all detailed decompositions are not mathematically wrong. The feasibility of detailed decomposition does not hinge on mathematical tweaks, but rather on acceptable restrictions involving the weighting factors w k. As an alternative approach, I suggest to apply a weighted averaging method in which the 6

8 weighting factors are grand-means. The weighted normalization is in line with the two-step re-normalization proposed by Haisken-DeNew and Schmidt (1997) and the restricted least squares method discussed by Greene and Seaks (1991). I will call this the grand-mean (GM) weighting method. In the following section, I first discuss the mathematical advantage of the GM weighting method and then turn to its theoretical implications. To carry out a detailed decomposition using the GM weighting method, it is necessary to transform the estimated regression coefficients of equation 1, b k, as follows: y = ( a + b ) K + (b k b )x k + e k=1 K = a + b k x k + e k=1 where b = K b k x k. k=1 (4) Yun (2005) used a simple arithmetic mean to compute b in equation 3. The normalization of the averaging method by K k=1 b k = 0 is equivalent to the normalization based on K k=1 b k (1/K) = 0. The GM weighting method, instead, normalizes the estimated coefficients by imposing K k=1 b k x k = 0, where x refers to the grand mean for both group W and group B. That is, b is treated as the grand-mean-weighted sum of b k in equation 4. After these transformations, the usual BO decomposition techniques can be applied. The insensitivity of a to the choice of reference group can easily be proved. For simplicity, let s assume there is only one dummy variable on the right side of equation. When x 0 is a reference group, the regression model looks like equation 5a, and as we change the reference group to x 1, the estimated regression model becomes equation 5b. y = a + bx 1 + e (5a) y = (a + b) + ( b)x 0 + e (5b) 7

9 Applying equation 4, equations 5a and 5b are transformed to equations 6a and 6b respectively: y = (a + b x 1 ) + (0 b x 1 )x 0 + (b b x 1 )x 1 + e (6a) y = (a + b + [ b x 0 ]) + ( b [ b x 0 ])x 0 + (0 [ b x 0 ])x 1 + e (6b) The intercept of equation 6a is (a + b x 1 ) and that of equation 6b is (a + b + [ b x 0 ]). Because x 1 is a dichotomous variable, x 0 = 1 x 1 and x 0 = 1 x 1. If we replace x 0 with 1 x 1 for the intercept of 6b, it is reduced to (a + b x 1 ) which is identical with the intercept of 6a. This proves that the intercept of the GM weighting method is insensitive to the choice of reference group. The intercept effect (D1A) of the GM weighting method quantifies the extent to which members of the disadvantaged group are, on average, treated differently than members of the advantaged group in a society. Thus, this value can also be interpreted as the average extent of discrimination (assuming no unobserved heterogeneity). The estimated contribution of the individual factor level (D1B) of the GM weighting method indicates a deviation from the mean discrimination level. The sum of D1B of the GM weighting method is close to zero. 2 When there is group-based discrimination, all members of the disadvantaged group suffer to a similar degree. If the extent of discrimination varies greatly within a given minority group, it would be hard to label the discrimination as groupbased. Therefore, the small effects of D1B, along with the large intercept effect are what one would expect if there is group-based discrimination. The sums of D2 are identical between averaging method and the GM weighting method. Unlike the averaging method, however, the GM weighting method yields consistent estimates of the D2 for individual factor levels regardless of model specifications. 2 The sum of D1B of the GM weighting method will be meaningfully different from zero only if the value of x for each group differs from the grand mean. A similar limitation is inevitable to all detailed decomposition methods. For the averaging method, the sum of D1B can be substantively large only if the distribution of x differs from 1/K, and the sum of D1B becomes larger as a researcher arbitrarily applies a model specification that increases x k 1/K. For the original BO method, the sum of D1B will be near zero if the proportion of the reference group approaches to 1, and conversely it becomes larger as a researcher arbitrarily chooses a reference group of which the proportion (i.e., x k ) is smaller than other groups. 8

10 Table 1: Descriptive Statistics Total White Black Gap x x W x B x W x B Log Wage, ȳ Less Than High School (LTHS) High School Graduaate (HSG) Some College (SC) Bachelor Degree (BA) Graduate Degree (Grad) Never Married Currently Married Widow/Divorce/Seperated An Illustrative Example Using the 2009 Current Population Survey Monthly Outgoing Rotation Group (CPS-MORG), I decompose the log wage gap (.284 log dollars) between white male workers and black male workers. To examine the sensitivity of decomposition results to model specifications, I estimate three models, applying both the averaging method and the GM weighting method. Table 1 shows the grand means and two group means. Table 2 presents the decomposition results. 3 Model 1 decomposes the racial gap into five educational levels, three age groups, and three marital status. In Model 2, LTHS and HSG are collapsed to <HSG and marital status is divided into two categories. Model 3 has the same model specification with Model 2 except educational categories. BA and Grad are collapsed to BA+ and LTHS and HSG are separately identified. 3 The variance-covariance matrix of the averaging method is discussed in detail by Yun (2008). The same technique can be applied for the modified GMC method with slight modification. The variance-covariance matrix of the normalized regression coefficients of the averaging method is computed as Σ b = W Σ B 0W where W is a weight matrix and Σ B 0 is a reformatted variance-covariance matrix of the original regression coefficients (Σ B). For the GM weighting method, everything except the weighting matrix W is the same as the averaging method. The weighting matrix needs to be rebuilt by replacing a set of matrix of 1/K j for each factor with a set of matrix of weighting values using grand means. The new coefficient for the GM weighting method is obtained by taking diagonal of W B. The variance-covariance matrix of the new coefficients is computed as Σ b = W Σ B W. 9

11 In all three models, the sum of D1 (D1A + D1B) and the sum of D2 are identical between two decomposition methods. The sum of D2 for each factor (e.g., D2 of Edu Effect) is also identical across methods. All other estimated decomposition components, however, differ by decomposition methods. Importantly, the decomposition results of the GM weighting method are consistent across model specifications, while those of the averaging method are substantially altered by models. The intercept component (D1A) of the averaging method is.175 in Model 1, but it becomes.167 in Model 2 and.200 in Model 3. In contrast to the averaging method, the intercept components using the GM weighting method barely change across the models. The coefficient effects (D1) of individual factors and factor levels are not consistently estimated with the averaging method. For example, in Model 1 which uses five educational categories, the sum of D1 of education is.005. When the education factor levels are reduced to four categories in Model 2, the sum of D1 of education becomes.023. When I modify the classification of education factors again in Model 3, the effect now turns out to be These results imply that we cannot determine whether the coefficient effect of education contributes to the reduction of racial gap or to the increase of racial gap with the averaging method. For another example, the sum of D1 for two educational factor levels, LTHS and HSG, of the averaging method is.014 in Model 1, but it is.024 in Model 2 and.007 in Model 3. Unlike the averaging method, the GM weighting method yields consistent estimates of the effects of D1 for individual factors and factor levels under the different classifications of factor levels. The averaging method does not produce consistent estimates of the endowment effects (D2) for individual factor levels either. The estimated effect of D2 for BA and Grad combined in the averaging method is either.028,.039, or.049 depending on the model specifications, while that in the GM weighting method is almost identical across models. When a dichotomous variable is used, the averaging method reports even effects for two dichotomous categories by design. As a result, D2 s of married and not-married are equally.016 in Model 2. However, when three marital status factor levels are used in Model 1, D2 10

12 Table 2: Decomposition of the Mean Log Wage Gap between White and Black Men Using Averaging and GM weighting Methods Averaging GM weighting (Equation 3) (Equation 4) D1 D2 D1 D2 A. Model I LTHS HSG **.010*.016** SC BA ** ** Grad ** * [Σ Edu Effect] [.005] [.066]** [.007]** [.066]** [Σ Age Effect] [.000] [-.001]** [.001]* [-.001]** Never Married -.011*.012* -.015**.019** Currently Married.016**.020**.009**.012** Wid/Div/Sep [Σ Marriage Effect] [.005]* [.034]** [-.009]** [.034]** Intercept.175**.187** [Total] [.185]** [.099]** [.185]** [.099]** B. Model 2 <HSG.024**.031*.017**.024 SC BA Grad [Σ Edu Effect] [.023]** [.063]** [.007]** [.063]** [Σ Age Effect] [.000] [-.001]** [.001] [-.001]** Currently Not-married -.013** **.021 Currently Married.012** **.012 [Σ Marriage Effect] [-.001]** [.032]** [-.008]** [.032]** Intercept.167**.190** [Total] [.189]** [.095]** [.189]** [.095]** C. Model 3 LTHS HSG *.016 SC BA ** ** [Σ Edu Effect] [-.012] [.064]** [.005] [.064]** [Σ Age Effect] [.000] [-.001]** [.000] [-.001]** Currently Not-married -.014** **.021 Currently Married.013** **.012 [Σ Marriage Effect] [-.001]** [.033]** [-.009]** [.033]** Intercept.200**.191** [Total] [.188]** [.096]** [.188]** [.096]** 11

13 of not married (i.e., the combination of never-married and widowed/divorced/separated) is smaller than that of married. Unlike the averaging method, the GM weighting method again produces consistent amounts of D2 for married regardless of models. Another noteworthy point is about the effects of age. Unlike other variables, the estimated effects D1 and D2 are near zeros in both methods. This is simply because the proportion of each age group happens to equal 1/K (i.e., 1/3) for both racial groups. In short, this illustration clearly shows that the detailed decomposition of the GM weighting methods are model-free and consistent, while the averaging method is sensitive to model specifications. 5 The Best Practice Even though I suggested to apply the GM weighted method for detailed decomposition, I hasten to add that there are no methods, including even the GM weighting method, that can ultimately solve the identification problem and thus be universally applied to all situations. Recall that given the restriction that K k=1 b k w k = 0, then all detailed decompositions are mathematically correct. Different restrictions lead to different interpretations of the estimates. Thus, the most essential question is what method is best and what principles should be applied when choosing a specific decomposition method. If the task is to compare groups within a nation on economic performance, I argue that the GM weighting method is generally preferable. According to this method, the currently observed distribution of x per se should be accepted as given. The intercept terms in equations 4 measure the expected wage when x is distributed as x. The reason why w k should be the grand-mean rather than a group-specific mean (or other weighting factor), is that the wage is determined by supply and demand in the whole labor force of a society, not a specific group. For example, the supply of highly educated workers can be measured best by x Grad, not by x W Grad or xb Grad. If the currently observed labor market reflects an equilibrium in employment, which in turn affects wages, the most reasonable and practical assumption 12

14 on the current status of the labor market is x. 4 The grand-means need not be of the two groups (W and B); their estimation can include other groups not of interest in the current study. Which distribution of x yields the most realistic estimate of actual wages depends upon the researcher s judgment. When the sample in a given dataset is representative, the grand means are unbiased and consistent estimates of E(x) for a given population. As far as E(x) is considered a reflection of the current social conditions in a given society, the GM weighting method accurately estimates the extent to which each factor and factor level contributes to the group differences under these social conditions. If there are theoretical or practical reasons to define a specific group as the reference group, the original BO decomposition method can be applied. For example, suppose a researcher conducts a detailed decomposition of a wage difference, with a particular focus on college premiums. College premiums are defined as the net difference in the results of HSG and BA. Therefore, HSG may be the natural choice for the reference group. Even in cases such as this, however, I recommend combining the original BO method with the GM weighting method. The dummy coding used in the original BO method need be applied only for the education factor, with the GM weighting method applied to all the others. This is because there are no strong theoretical or practical reasons to pick a certain factor level (e.g., the Pacific region) as the reference for computing college premiums. One caveat researchers should bear in mind in interpreting the college premium effect of the BO decomposition is that the college premiums are computed relative to the inner group counterparts. By setting HSG as a reference point, we implicitly assume that HSG does not contribute to the wage gap between two populations we are interested in. A positive college premium effect in account for the wage gap between group B (black workers) and group W (white workers) does not necessarily indicate that the wage of the college educated workers of group B exceeds that of group W. The higher college premium of group B can be a reflection of the excessive discrimination against the low educated workers of group B. 4 A similar logic was applied in a study of inter-industry wage differentials by Krueger and Summers (1988). 13

15 The example in the previous section illustrates how more generically analytic and substantive grounds may be invoked when choosing a specific decomposition method. In this spirit, I would recommend the averaging method only when it is reasonable to assume a uniform distribution across factor levels (although in the case of studies of labor market outcomes, such an assumption seems dubious). Other constraints of K k=1 b k w k = 0 might be possible, but there needs to be compelling reasons to bypass the grand means. The choice of weighting values becomes more subtle when detailed decomposition techniques are applied beyond the labor market. For group comparisons within a nation, such as racial differences in voting rates or gender differences in subjective well-being, the GM weighting method is still preferable in most cases. However, the GM weighting method is not always the best choice. In particular, grand means would not be appropriate weighting factors for international comparisons. Suppose a researcher is interested in the difference in mortality rates between the US and China. As the Chinese economy develops, the distributions of age, education-level, and other covariates in China would approach those in the US. If a researcher wants to perform a decomposition under the assumption that the distributions of the xs in China are equal to those in the US, the weighting factors used to normalize the coefficients should be x US, not the (weighted) means of groups means, x US and x CN. There are other studies for which the averaging method is most appropriate. An example is the total fertility rate (TFT), which is a hypothetical fertility rate when a woman experiences the current, age-specific population fertility rates throughout her life. A uniform distribution of age groups should be assumed for computing the TFT. 5 If researchers want to yield the intercept component (D1A) representing the TFT after controlling for the other covariates, they can apply the averaging method to the age variable and the modified GM weighting method to the covariates. 6 5 Yu Xie pointed out this in his comments on the previous draft of this paper at the quantitative methodology session of the 2011 American Sociological Association annual meeting. 6 Note that the identification problem and its solutions as discussed in this paper are relevant to the methods used for rate standardization and the decomposition of rate differences, matters that have long been discussed in the demography literature ( see, for example, Kitagawa (1955); Clogg and Eliason (1988); Liao (1989); 14

16 In short, there is no single best method of obtaining components for decomposition (Clogg et al. 1990:191). In principal, any detailed decomposition method is acceptable as long as there are theoretical or practical reasons to believe that the researcher s choice of reference group (or weighting factors) produces a meaningful decomposition result. This is why I refer to the GM weighting method A Suggested Alternative, not The Solution. 6 Conclusion Since the development of the BO decomposition techniques, detailed decompositions have often been reported despite the caveats about the identification problem that has been raised many scholars. To address this concern, Yun (2005) suggested the averaging method. Although that method is a notable advance and is undoubtedly applicable in some cases, it does not resolve the identification problem entirely. It is based on unrealistic distributional constraints on a set of dummy variables, and it is sensitive to the number of groups and the method of grouping. The legitimacy of any detailed decomposition depends on the acceptability of the assumption of how the independent variables are distributed for the purpose of computing the intercept. There are multiple solutions to the identification problem. Different model specifications for detailed decompositions are appropriate, depending on various theoretical and/or practical considerations. Unless such considerations are compelling, however, the GM weighting method is likely to be more generally preferred for studies of labor market outcomes. This conclusion is based on the following reasons: (1) a state of equilibrium (or current social conditions) is the most reasonable assumption to make for labor market phenomena; (2) estimates of the contributions of individual factors and factor levels from the GM weighting method are the least sensitive to model specifications, such as the choice of coding scheme; and (3) as a result, the GM weighting method provides clear substantive interpretations of the intercept Clogg et al. (1990)). 15

17 and coefficients components for individual factor levels. Detailed decomposition can be applied to various sociological issues. Given the rise in the number of highly educated workers, estimating the detail contributions of the differences in levels of education, fields of study, occupation, and other covariates in accounting for gender/race earnings gaps is especially promising. Wealth inequality is another area of interest, where a key issue is how to compute the extent to which each factor contributes to wealth accumulation (Spilerman 2000); detailed decomposition is an essential tool for such calculations (Scholz and Levine 2004). Application of the GM weighting method to non-linear models is warranted. Although there is no general agreement on how to decompose the results of quantile regressions, the GM weighting method can be easily applied to quantile regressions as long as the use of the Blinder-Oaxaca type decomposition implemented by García et al. (2001) is acceptable. 7 7 See Gardeazabal and Ugidos (2005) for further discussion. 16

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