Hierarchical Segregation: Issues and Analysis

Transcription

1 Hierarchical Segregation: Issues and Analysis Dina Shatnawi Naval Postgraduate School Ronald L. Oaxaca University of Arizona and IZA Michael R Ransom Brigham Young University and IZA November 3, 010

2 Anatomy of a Wage Decomposition Endowment e ects Pure wage discrimination Job/occupational segregation Example of a conventional model K 1 Ȳ j = X j ˆβ j + Ō jk ˆ jk, j = m, f k = Ō jk is the sample proportion of j workers in the kth job title. Without loss of generality, the left-out reference job is occupation 1. 3 X j includes the constant term.

3 Conventional Decomposition Ȳ m Ȳ f = ( X m X f ) ˆβ m + K k= X f ˆβ m K ˆβ f + Ō fk ( ˆ mk ˆ fk ) k= Unexplained/discrimination e ect = X f ˆβ m K ˆβ f + Ō fk ( ˆ mk ˆ fk ) k= Total endowment e ect = ( X m X f ) ˆβ m + K k= K k= (Ō mk Ō fk ) ˆ mk + (Ō mk Ō fk ) ˆ mk, where (Ō mk Ō fk ) ˆ mk is a measure of the segregation e ect on the wage gap.

4 Conventional Decomposition Some problems 1 When the occupational/job title category is discrete, often the occupational aggregation is arbitrary e.g. digit level, 3 digit level, etc. Depending on the degree of job title aggregation, there may be a lack of overlap in the occupational support, i.e. some occupational categories may contain no observations for one group. 3 What constitutes the e ect of pure wage discrimination? K K X f ˆβ m ˆβ f + Ō fk ( ˆ mk ˆ fk )? Ō fk ( ˆ mk ˆ fk )? k = k = Ideally, one would like a measure of gender wage gaps within jobs after conditioning on personal productivity measures. The K problem with Ō fk ( ˆ mk ˆ fk ) is that this measure is not k = invariant with respect to the omitted occupation reference group (Oaxaca and Ransom, 1999).

5 Hierarchical Segregation Baldwin et. al (001) Occupational segregation arises from (male) worker distaste for supervision by women The proportion of women in a given job title relative to the proportion of men in a given job title declines exponentially as one ascends the job ladder. The model dispenses with the need to commit to any degree of job title aggregation. With continuous wage distributions assumed, in e ect every wage constitutes a job.

6 Hierarchical Segregation Model For convenience let us assume that wages within a rm follow a lognormal distribution. Assume a wage distribution for males given by f m (w) = 1 wσ p π exp (`n(w) µ) σ (1) Identifying assumption: the female wage density comes from the same class of distributions as the male wage density. Female wage density = male wage density x sorting function The sorting function is the ratio of the proportion of females in a given job level to the ratio of males in the same job level In the limit each wage constitutes a job level.

7 Hierarchical Segregation Model For the log normal distribution, which is a special case of the class of exponential distributions, the sorting function turns out to be 8 >< g(w) = exp >: ( 1)`n(w) σ 6 4`n 0 exp w µ >= C7 A5 >; ()

8 Hierarchical Segregation Model The female wage density is given by f f (w) = w σ 1 pπ exp 6 4 `n(w) σ where 1 is the job segregation parameter. In the absence of job segregation, = 1. µ (3)

9 Hierarchical Segregation Model Worker heterogeneity within job titles can be incorporated into the wage density functions: and f m (w) = f f (z) = z 1 wσ p π exp σ 1 pπ exp 6 4 (`n(w) µm ) σ 3 (`n(z) µ f ) σ 7 5 where z represents the female wage in the presence of worker heterogeneity. (4) (5)

10 Hierarchical Segregation Model Conditioning on worker characteristics (x) introduces additional parameters (α): µ m = x m α m, µ f = x f αf, and αf = α f. Worker heterogeneity and wage discrimination imply 1 E [`n (z)] = [x f α m `n (φ)]. (6) where φ is a measure of wage discrimination. The female wage density (5) implies E [`n (z)] = µ f (7) = x f α f = x f α f.

11 Hierarchical Segregation Model Upon equating (6) to (7), one can solve for φ from `n(φ) = x f (α m α f ): φ = exp [x f (α m α f )]. (8) Pure wage discrimination (in logs): `n(φ) = x f (α m α f ) Clearly in the absence of wage discrimination, α m α f = 0 ) φ = 1.

12 Hierarchical Segregation Identi cation Issues Ideally, one would want to jointly estimate α f,, x f, and α m from the likelihood function for the combined sample of males and females. The estimated value of φ is backed out from ( 8) using the MLE estimates of the parameters and the conditioning values of x f and x m. Alternatively, one could separately estimate α m and α f, In this case additional identifying restrictions are necessary to recover the parameter vector α f and the segregation parameter. Again, the estimated value of φ is backed out from ( 8).

13 Hierarchical Segregation Identi cation Issues Identi cation strategy adopted in Shatnawi et al. (010a) in the case of worker heterogeneity and pure wage discrimination. Set σ = σ m and σ = σ m = σ f and separately estimate the log wage equations for males and females The segregation parameter is recovered from = σ m σ f

14 Decompositions Absence of pure wage discrimination Without covariates (homogenous workers) the log wage gap is entirely the result of job segregation: 1 E [`n(w m )] E [`n(w f )] = µ Decomposition of wage levels E (w m ) E (w f ) = exp µ + σ µ exp + σ. Empirically, the decomposition of wage levels is given by w m w f = ˆµ exp ˆθ m exp( ˆµ ˆσ ˆσ ) exp ˆ ˆ ˆµ ˆσ h +exp ˆ ˆ exp(bθ m ) exp(bθ f )i.

15 Decompositions Absence of pure wage discrimination bθ m and bθ f are remainder terms that equate the means of the predicted wages for males and females to their sample means, i.e. bθ m = `n(w m ) ( ˆµ ˆσ ) bθ f = `n(w f ) ˆµ ˆ ˆσ ˆ

16 Decompositions Worker heterogeneity Conditioning on sample mean characteristics ( x), the expected log wage decomposition can be written as E [`n(w m jx m )] E [`n(w F jx f )] = (x m x f ) α m + x f (α m α f ) 1 +x f α f. Endowments = (x m x f ) α m Pure wage discrimination = x f (α m α f ) Segregation = x f α 1 f

17 Decompositions Worker heterogeneity Note that x f α 1 f re ects both discrimination and segregation when α f 6= α m. A better measure of segregation (and pure wage discrimination) would include an interaction term 1 1 x f α f = x f α m + x f (α f α m ) 1. 1 x f α 1 m measures segregation e ects on wages assuming no pure wage discrimination. x f (α f α m ) 1 measures the interaction e ect of pure wage discrimination and segregation.

18 Decompositions Worker heterogeneity In wage levels (conditioning on mean characteristics) the wage gap is given by E (w m jx m ) E (w f jx f ) = exp x m α m + σ xf α f exp + σ. The decomposition of the wage level gap can be expressed as E (w m jx m ) E (w f jx f ) = σ exp [exp ( x m α m ) exp ( x f α m )] σ + exp [exp ( x f α m ) exp ( x f α f )] +exp x f α f + σ x f α f exp + σ

19 Decompositions Worker heterogeneity [exp ( x m α m ) exp ( x f α m )] σ Endowments = exp Pure wage discrimination σ = exp [exp ( x f α m ) exp ( x f α f )] Segregation = exp x f α f + σ x f α f exp + σ

20 Decompositions Worker heterogeneity The empirical wage level decomposition can be expressed as w m w f = exp(0.5 ˆσ m + bθ m ) N 1 f " N 1 m # N f exp(x ˆα m ) +N 1 f exp(0.5 ˆσ m + bθ m ) +N 1 f exp(bθ f ) +N 1 f N m exp(x mi ˆα m ) ( ) Nf [exp(x ˆα m ) exp(x ˆα f )] ( Nf exp(x ˆα f ˆσ m) x bα f exp ˆ " Nf exp(x ˆα f ˆσ m)# h exp(bθ m ) exp(bθ f ) i ˆσ f )

21 Decompositions Worker heterogeneity Again, bθ m and bθ f are remainder terms that equate the means of the predicted wages for males and females to their sample means. Endowments = " exp(0.5 ˆσ m +bθ m ) N 1 m Pure wage discrimination = N 1 f exp(0.5 ˆσ m + bθ m ) Segregation = ( Nf Statistical adjustment N 1 f exp(bθ f ) = N 1 f N m exp(x mi ˆα m ) Nf 1 # N f exp(x ˆα m ) ) ( Nf [exp(x ˆα m ) exp(x ˆα f )] h exp(x ˆα f ˆσ m) exp x bα f ˆ " Nf h exp(x ˆα f ˆσ m)# exp(bθ m ) exp(bθ f ) ˆσ f i i )

22 Decompositions Worker heterogeneity Numerical example from Shatnawi, et. al (010a) No wage discrimination because of union contract.

23 constant age age squared tenure ten σ Table 6 Heterogenous Case No Discrimina8on Lognormal Results and Decomposi8on Pooled Men and Women N 1976 based off of joint es8ma8on of the male and female likelihood func8ons Decomposi8on Wage Difference Percent Difference Endowment Discrimina8on Segrega8on Non linear Total

24 Decompositions Conventional wage decomps Conventional log wage model with occupational controls: `n(w j ) = X j ˆβ j + K k= Ō jk ˆ jk, j = m, f Conventional type wage level decomposition with occupational controls: w m w f Endowments + Segregation = exp(0.5 σ m + θ m )[N 1 m N 1 f N f exp(x ˆβ m + O ˆ m )] N m exp(x mi ˆβ m + O mi ˆ m ) Note that the endowment and segregation e ects cannot be separately identi ed in wage levels from a log wage model.

25 Decompositions Conventional wage decomps Pure ( wage discrimination = Nf 1 exp(0.5 σ m + Nf θ m ) exp(x ˆβ m + O β m ) exp(x ˆβ f + O ˆ f ) ) # Statistical adjustment = N 1 f exp(0.5 σ m + θ m ) " Nf exp(x ˆβ f + O ˆ f ) exp(0.5 σ f + θ f )

26 Fixed E ects Models Shatnawi, et. al (010b) Balanced design `n (w mit ) = x mit β m + α mi ᾱ m + ε mit, t = 1,..., T `n (w t ) = x t β f + α ᾱ f + ε t, t = 1,..., T β f = β f, α = α Segregation parameter: = σ mε ᾱ j = N j σ f ε N j α ji N j, j = m, f from which the normalization (α ji ᾱ j ) = 0 yields a constant term of ᾱ j at the overall sample mean.

27 Fixed E ects Models Decompositions Expected log wage decomp at overall sample mean: E `n(w m jx m ) E `n(w F jx f ) = x m x f βm +[x f (β m β f ) + (α m α f )] 1 +x f β f Overall sample mean: x = n T t=1 nt x it

28 Fixed E ects Models Decompositions Expected wage level gap E (w m jx m ) E (w f jx f ) = exp x m β m + α m + σ εm x f β exp f + α f + σ εf.

29 Fixed E ects Models Empirical Wage Level Decompositions w m w f = exp + exp bσ εm + bθ m bσ εm + bθ m + exp(bθ f ) 4exp + " exp! h exp x m bβ m + α m! h exp x f bβ m + α m x f bβ f + α f + bσ εm x f bβ f + α f + bσ εm! i exp x f bβ m + α m i exp x f bβ f + α f 0 x f b β f + α f!# h i exp(bθ m ) exp(bθ f ) + bσ εm b 13 A5

30 Fixed E ects Models Empirical Wage Level Decompositions Unbalanced Design Endowments = exp(0.5 ˆσ εm + bθ m ) (T f n f ) 1 n f T if t=1 " T im n m (T m n m ) 1 # exp(x t bβ m + bα mi ) t=1 exp(x mit bβ m + bα mi ) Pure wage discrimination = (T f n f ) 1 exp(0.5 ˆσ εm + bθ m ) h i exp(x t bβ m + bα mi ) exp(x t bβ f + bα ) n f T if t=1 Segregation = (T f n f ) 1 exp(bθ f )f exp x t bβ f + bα ˆ n f T if t= ˆσ εm b [exp(x t bβ f + bα ˆσ εm)! ]g

31 Fixed E ects Models Empirical Wage Level Decompositions Unbalanced Design Statistical adjustment = (T f n f ) 1 " nf T if t=1 h exp(bθ m ) exp(bθ f ) exp(x t bβ f + bα ˆσ εm) i. #

32 Random E ects Models Shatnawi et.al (010b) `n (w mit ) = x mit β m + v mit, t = 1,..., T `n (w t ) = x t β f + v t, t = 1,..., T vit m = ui m + ε m it vit f = ui f + ε f it E [v it] = σ u + σ ε = σ v Segregation parameter: = σ vm σ vf = σ um + σ εm σ εm + σ εm

33 Random E ects Models Decompositions Note ln w = xbβ + u, where u = ln w 6= xbβ. n u i n Log wage decompositions ln w m ln w f = x m x f βm + x f (β m β f ) 1 +x f β f + (u m u f ) The empirical wage level decompositions are identical in form to the FE decompositions with the RE constant term appearing in the place of the individual xed e ects.