A SIMPLE SOLUTION TO THE IDENTIFICATION PROBLEM IN DETAILED WAGE DECOMPOSITIONS

I. INTRODUCTION Oaxaca and Ransom (1999) show that the Oaxaca decomposition suffers from one nagging conceptual problem: The detailed Oaxaca decomposition of wage differentials is not invariant to the choice of reference group when dummy variables are used. (1) That is, if one uses dummy variable(s), then the detailed coefficients effect attributed to dummy variables is not invariant to the choice of the omitted group(s). (2) This invariance or identification problem is well-known to labor economists and has plagued decomposition and discrimination analysis for a long time. (3) This article proposes a simple and practical solution to this identification/invariance problem. The solution is based on the intuitive idea that if alternative reference groups yield different estimates of the characteristics and coefficients effects for each individual variable, then it is natural to obtain estimates of the two effects for every possible specification of the reference groups and take the average of the estimates of the two effects with various reference groups as the "true" contributions of individual variables to wage differentials. A close inspection shows that averaging the two effects in the decomposition equation with varying reference groups (the "averaging approach") is equivalent to finding average estimates of constant and dummy variables in wage equations with varying reference groups and then using the average estimates to calculate the Oaxaca decomposition equation. Researchers may react a bit negatively to this suggestion because it could be quite cumbersome and tedious to estimate wage equations permuting the choice of reference groups. This article shows that there is no need to estimate multiple wage equations. One set of regression estimates is sufficient to resolve the identification problem in the detailed decompositions. That is, the average of the estimates with varying reference groups can be easily calculated by using only "one" set of regression estimates with any specific reference group(s). The identification problem for the detailed decomposition disappears once the contribution of the dummy variables and the constant in the regression equations are identified. (4) II. IDENTIFICATION PROBLEM AND INVARIANT DECOMPOSITION: AN ILLUSTRATION To understand what the identification problem in the detailed wage decomposition is, I will first illustrate the problem with a simple example. Suppose that there are three categories, and the regression equation has only a constant and two dummy variables on the right-hand side where the reference group is the first category. (5) That is, (1) [y.sub.i] = [[alpha].sub.i] + [3.summation over (k=2)] [D.sub.ki] [[beta].sub.ki] + [e.sub.i], where y is log-wages and i = A or B for two comparison groups. Coefficients for two dummy variables taking category one as the reference group are estimated using ordinary least squares (OLS) regression and are reported in the second column of Table 1 for sample A and for sample B. Note that the coefficient of the reference group (category 1) is restricted to zero. Similarly, I may change my reference group specification and estimate the wage equation two more times. The estimates taking category 2 or 3 as the reference group are reported on the next two columns. Because I have three sets of estimates with varying reference groups, I may decompose the wage differentials (0.15 = 0.76 - 0.61) into characteristics and coefficients effects three times. The characteristics effect ([[DELTA].sub.X(1)]) and coefficients effects ([[DELTA].sub.[beta](1)] are specified as follows when category 1 is the reference group: [[DELTA].sub.X(1)] = [[summation].sup.3.sub.k=2] ([[bar.D].sub.kA] - [[bar.D].sub.kB]) [[beta].sub.kB], and [[DELTA].sub.[beta](1)] = [[alpha].sub.A] - [[alpha].sub.B] + [[summation].sup.3.sub.k=2] [[bar.D].sub.kA] ([[beta].sub.kA] - [[beta].sub.kB]). …