A Distributional Framework for Matched Employer Employee Data. Nov 2017

Transcription

1 A Distributional Framework for Matched Employer Employee Data Nov 2017

2 Introduction Many important labor questions rely on rich worker and firm heterogeneity - decomposing wage inequality, understanding earnings dynamics, mobility (individual and aggregate) - mobility between jobs, in and out of employment This heterogeneity might be unobserved - but we have repeated measures (matched data) - we can learn about latent types Economists have developed frameworks for two-sided heterogeneity, observed and unobserved

3 Two influential literatures for worker and firm heterogeneity Log linear fixed effect wages Abowd, Kramarz, and Margolis (1999); Card, Heining, and Kline (2013) y it = α i + ψ j (i,t) + ɛ it spurred both applied and theoretical literature pros: allows for 2-sided unobserved heterogeneity, tractable limitations: imposes additivity ( theory, Eeckhout and Kircher (2011)), suffers from limited mobility bias Equilibrium search structural models Burdett and Mortensen (1998); Shimer and Smith (2000); Postel-Vinay and Robin (2004); Hagedorn, Law, and Manovskii (2014) pros: allows for complex wage functions, can address efficiency/policy questions limitations: imposes strong structural assumptions (vacancy mechanism, wage setting, mobility decision...)

4 Two influential literatures for worker and firm heterogeneity Log linear fixed effect wages Abowd, Kramarz, and Margolis (1999); Card, Heining, and Kline (2013) y it = α i + ψ j (i,t) + ɛ it spurred both applied and theoretical literature pros: allows for 2-sided unobserved heterogeneity, tractable limitations: imposes additivity ( theory, Eeckhout and Kircher (2011)), suffers from limited mobility bias Equilibrium search structural models Burdett and Mortensen (1998); Shimer and Smith (2000); Postel-Vinay and Robin (2004); Hagedorn, Law, and Manovskii (2014) pros: allows for complex wage functions, can address efficiency/policy questions limitations: imposes strong structural assumptions (vacancy mechanism, wage setting, mobility decision...)

5 This paper: Proposes a distributional model of wages - assume discrete heterogeneity: firms (k) and workers (l) - non-parametric conditional wage distributions F kl (w) - unrestricted firm compositions π k (l) Non-parametric identification & estimation for 2 types of mobility assumptions: - 2 period static model ( AKM assumptions ) - 4 period dynamic model Applies method to Swedish matched employee-employer data Important properties: works with very short panels (2 to 4 periods) relax additivity and mobility provide a regularization testing framework: - compatible with many theoretical models: - informative about patterns without imposing full structure, - without further assumptions, can t address efficiency questions

6 This paper: Proposes a distributional model of wages - assume discrete heterogeneity: firms (k) and workers (l) - non-parametric conditional wage distributions F kl (w) - unrestricted firm compositions π k (l) Non-parametric identification & estimation for 2 types of mobility assumptions: - 2 period static model ( AKM assumptions ) - 4 period dynamic model Applies method to Swedish matched employee-employer data Important properties: works with very short panels (2 to 4 periods) relax additivity and mobility provide a regularization testing framework: - compatible with many theoretical models: - informative about patterns without imposing full structure, - without further assumptions, can t address efficiency questions

7 Plan of the talk 1 Framework & identification overview 2 Data and empirical results 3 Performance on a theoretical sorting model

8 Model and Indentification

9 Heterogeneity and wages Workers indexed by i with discrete types ω(i) {1,..., L} Firms indexed by j with discrete classes f (j ) {1,..., K }. Let j it denote the identifier of the firm where i works at t. The proportion of type-l workers in firm j is π f (j ) (l), where: Pr [ω(i) = l f (j i1 ) = k] = π k (l). The conditional cdf of log wages Y i1 is: Pr [Y i1 y ω(i) = l, f (j i1 ) = k] = F kl (y). Interactions between workers are ruled out. At this K and L are assumed known, which is an important restriction. In a different paper we are extending this. We also provide theorems of l continuous.

10 Heterogeneity and wages Workers indexed by i with discrete types ω(i) {1,..., L} Firms indexed by j with discrete classes f (j ) {1,..., K }. Let j it denote the identifier of the firm where i works at t. The proportion of type-l workers in firm j is π f (j ) (l), where: Pr [ω(i) = l f (j i1 ) = k] = π k (l). The conditional cdf of log wages Y i1 is: Pr [Y i1 y ω(i) = l, f (j i1 ) = k] = F kl (y). Interactions between workers are ruled out. At this K and L are assumed known, which is an important restriction. In a different paper we are extending this. We also provide theorems of l continuous.

11 Job mobility static model: 2 periods move k k Y i1 Y i2 Consider a worker of type l in firm k in period 1 He gets a wage Y i1 drawn from F kl (y). The worker moves to a class-k firm with a probability that depends on k and l, not on Y i1. In period 2 he draws a wage Y i2 from a distribution G k l(y ) that depends on l and k, not on (k, Y i1 ).

12 Job mobility dynamic model: 4 periods move k k Y i1 Y i2 Y i3 Y i4 Consider a worker of type l in firm k at t = 2 Wages (Y i1, Y i2 ) are drawn from a bivariate distribution that depends on (k, l). At t = 2, the worker moves to a type-k firm with a probability that depends on l, k and Y i2, not on Y i1. At t = 3, If he moves, the worker draws a wage Y i3 from a distribution that depends on l, k, k, Y i2, not on Y i1. At t = 4, the worker draws a wage Y i4 that depends on l, k, Y i3, not on (k, Y i2, Y i1 ).

13 Link to theoretical models 2-periods model: - Example: Shimer and Smith (2000), without or with on-the-job search (workers threat points being the value of unemployment). - No role for match-specific draws, unless independent over time or measurement error. No sequential auctions. 4-periods model: - All models where state variables (l, k t, Y t ) are first-order Markov. - Examples: wage posting, sequential auctions (Lamadon, Lise, Meghir and Robin 2015), with aggregate shocks (Lise and Robin 2014). more - No latent human capital accumulation (l t ), no permanent+transitory within-job wage dynamics (example: random walk+i.i.d. shock).

14 Plan of attack 1 Identification with large firms 2 Empirical content of means & event study 3 Grouping firms in discrete types

15 Main restrictions Static model Under the assumptions of the static model, we have, For movers from firm k to firm k we have: Pr [ Y i1 y, Y i2 y k, k ] K = p kk (l) F kl (y)f k l(y ), l=1 For the cross-section in k we have K Pr [Y i1 y k] = π k (l) F kl (y). l=1

16 Main restrictions Dynamic model Using mobility assumptions of the dynamic model conditioning on Y 2 = y 2 Y 3 = y 3, we get: Pr [ Y i1 y, Y i4 y y 2, y 3, k, k ] = K p kk y 2 y 3 (l) F kl (y y 2 )G k l(y y 3 ) l=1 Similar structure as in static model: - use 4 period of data - replace F kl (y) with F kl (y y ) - replace p kk with p kk y 2y 3

17 Identification Wage Functions Large firms Consider two larger firms k and k and joint Y 1, Y 2 wages for movers k k A k,k (y 1, y 2 ) = l F kl (y 1 )p kk (l)f k l(y 2 ) Discretize wage (n w ) support and write in Matrix form: A(k, k ) = F (k) P(k, k ) }{{}}{{}}{{} n w n w n w n l n l n l diag. F (k ) Consider case where n w =n l, and both k k and k k: A(k, k )A 1 (k, k) = F (k)p(k, k )P 1 (k, k)f 1 (k) Which is an eigen value decomposition.

18 Identification Wage function In general, the identification relies on a joint diagonalization of all A(k, k ). A(k, k ) = F (k)p(k, k )F (k ) It is sufficient (but not necessary, see paper) for identification of F kl that: - p kk (l) 0 for l = 1,..., L. - p kk (l p k k (l), k = 1,..., L, are distinct. - The columns of F (k) (the F kl ) are linearly independent. once F kl is known, go to cross-section to get π k (l) In the 4 period model, replace Y 1, Y 2 with Y 1 Y 2, Y 4 Y 3 and do everything conditional on k, k, Y 2, Y 3.

19 Empirical content of wage means intro In the linear framework (AKM) where Y it = α i + ψ j (i,t) + ɛ it one can focus on movers to get: E(Y it+1 Y it m = 1) = ψ j (i,t+1) ψ j (i,t), (1) which can be recovered with OLS. Now consider an interacted model at the class level: Y it = a(k it ) + b(k it )α i + ɛ it with E[ɛ it α i, k i1, k i2, m i1 ] = 0. what can we do?

20 Empirical content of wage means intro In the linear framework (AKM) where Y it = α i + ψ j (i,t) + ɛ it one can focus on movers to get: E(Y it+1 Y it m = 1) = ψ j (i,t+1) ψ j (i,t), (1) which can be recovered with OLS. Now consider an interacted model at the class level: Y it = a(k it ) + b(k it )α i + ɛ it with E[ɛ it α i, k i1, k i2, m i1 ] = 0. what can we do?

21 Empirical content of wage means interactions Interacted model Y it = a(k it ) + b(k it )α i + ɛ it with 2 firms: wage k=2 a(2) + b(2) ᾱ 21 = ȳ 21 (2) k=2 ȳ 12 (2) = a(2) + b(2) ᾱ a(1) + b(1) ᾱ 12 = ȳ 12 (1) k=1 ȳ 21 (1) = a(1) + b(1) ᾱ 21 k=1 time Comparing changes: = (ᾱ 12 ᾱ 21 )(b(2) b(1)) - 0 if no interactions b(2) = b(1) - also 0 when composition is identical ᾱ 12 = ᾱ 21

22 Empirical content of wage means interactions Consider the following differences: ȳ 21 (2) ȳ 12 (2) = b(2)(ᾱ 21 ᾱ 12 ) ȳ 21 (1) ȳ 12 (1) = b(1)(ᾱ 21 ᾱ 12 ) Taking ration whenever ᾱ 21 ᾱ 12 : b(2) b(1) = ȳ21(1) ȳ 12 (2) ȳ 21 (2) ȳ 12 (1) this recovers the interaction term.

23 Empirical content of wage means Event Study in wages from Shimer Smith Wages in Shimer Smith Event study firm class model log wages (PAM) firm class quantile log wages (PAM) model log wages (NAM) quantile log wages (NAM)

24 Firm types Intro Identification relies on large flows of workers between k and k, However mobility is low at the firm pair. We propose to discretized firm types: assume K discrete type in the population k drives the unconditional firm wage distribution: H k (y) = l π k (l)f kl (y) Recover types using cross-section, then treat groups as large firms. Requires that H k (y) are separable. This first stage classification achieves a double purpose: - reduces the problem of limited mobility. - it breaks the complicated dependence structure between firms.

25 Firm types Grouping in practice Under the assumption that in the population i) firms are clustered in K groups and ii) the H k are separated, then: Then firms partition f (j ) 1..K is identified, and can recovered by k-mean on firms wage distributions. min f (1),...,f (J ),H 1,...,H K J j =1 n j D d=1 ) 2 ( F j (y d ) H f (j ) (y d ), when K is known and H k (y) are separable, this classification is super-consistent in firm size.

26 Firm types Distribution of wages firm class model log wages (PAM) firm class quantile log wages (PAM) firm class model log wages (NAM) firm class quantile log wages (NAM) Notes: Model based on Shimer and Smith (2000) with on-the-job search.

27 Recap of the full method 1 get Firm classes membership f (j ) [1..K ]. - group firms based on wage distributions - in practice we use k-means - cluster based on cross-section (or combine with movers) more 2 get p kk (l), and F kl or F kl (y 1 y 2 ) - use movers, treat worker type as random - non-parametric identification in the paper - in practice we use the EM algorithm 3 get firm compositions π k (l) - using stayers (cross-section or 4 periods) - another EM

28 Applying framework to Swedish data

29 Sample description We use matched employer-employee data from Sweden between 1997 and We select full-year employed males in 2002 (period 1) and in 2004 (period 2): 1, 000, 000 workers and 60, 000 firms. From this we define movers as workers whose firm IDs are different in 2002 and We focus on continuing firms and get 20, 000 job changers, with 13, 000 firms. more We use log pre-tax annual earnings, net of time dummies (interacted with education*cohort).

30 Estimated firm classes We estimate firm classes on the 2002 cross-section using a weighted k-means algorithm (empirical cdfs with 40 points, starting values). We allow for K = 10 classes. Wage variation across firms is captured well: the between-class variance of log wages is 90% of the between-firm variance. Note: the ordering of firm classes (by mean log wages) is arbitrary. Differences between classes in terms of worker composition (education, age), but also log value added per worker.

31 Descriptive statistics on estimated firm classes firm cluster: all number of workers 21,662 62, , , ,080 78, ,971 85,806 58,728 27, ,152 number of firms 6,487 7,972 7,804 6,494 4,663 3,748 4,209 3,984 3,157 2,812 51,330 % HS dropout % HS grade % some college % workers younger than % workers between 31 and % workers older than mean log wages variance of log wages between firm variance of log wages mean of log value added per worker variance of log value added per worker median number of workers per firm Notes: Sample 1 in All workers are males, employed during the full year HS is high school.

32 Parametrization of the 2-periods model We specify a model with Gaussian error: - Y it N (µ tkl, σ tkl ) - π k (l) and p kk (l) left unrestricted With K = 10, L = 6 we get 900 parameters. - We have also estimated a mixture of mixture models. We also have an interacted model - Y it = a(k) + b(k)α + ɛ it

33 Estimated mean log wages and type proportions (2-periods model) Mean log wages Proportions of worker types log earnings type proportions firm class k firm class k Notes: The left graph plots the mean of F kl. The L = 10 firm classes (on the x-axis) are ordered by mean log wage. The K = 6 worker types correspond to the 6 different colors. 95% confidence intervals based on the parametric bootstrap (200 replications). The right graph plots type proportions π k (l).

34 Variance Decomposition and mean effects (2-periods model) Var(α) Var(α+ψ) 80.3 (.8) Var(ψ) Var(α+ψ) 3.4 (.2) Variance decomposition ( 100) 2Cov(α,ψ) Var(α+ψ) Corr(α, ψ) R (.6) 49.1 (.9) 74.8 (.6) Reallocation exercise ( 100) Mean Median 10%-quantile 90%-quantile Variance.5 (.09).6 (.10) 2.7 (.20) 1.2 (.30) 1.1 (.11)

35 Simulations and decompositions We simulate the model based on the estimated parameters, conditional on the job moves in the data. We simulate entire employment spells, using the spell lengths in the data. We run linear regressions of the form: Y i1 = α(ω 0 i ) + ψ(f 0 j i1 ) + ε i1 We compare our results with AKM on real and simulated data.

36 Variance decompositions (2-periods model) Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ) Corr(α, ψ) Data (K = 10) estimate 79.6% 4.3% 16.1% Monte-Carlo (K = 10, 100 reps) mean 80.3% 3.4% 16.3% quantile 79% 3% 15% quantile 82% 3.9% 17.4% Varying the number of classes K = % 3.6% 14.9% K = 15 79% 4.5% 16.5% K = % 4.7% 16.2% 0.42 Mixture model estimate 77% 5.2% 17.8% Monte-Carlo (K = 10, 100 reps) mean 77.8% 4.4% 17.8% quantile 76.3% 4% 16.8% quantile 79.1% 5% 18.8% 0.494

37 Fixed effect, limited mobility bias min spell rep Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ) Corr(α, ψ) Data This paper Fixed-effects Simulated from the model This paper Fixed-effects Simulated from the model without limited mobility Fixed-effects Fixed-effects

38 Estimation of the 4-periods model We add two employment periods: 2001 and There are 12,519 workers moving in 2003 employed in the five years. We write a model for Pr[Y 1, Y 2, Y 3, Y 4 l, k, k ] : Y i1 = ρ 1 2 Y i2 + a 1 (k) + b(k)α+ ε i1 Y i2 = a 2 (k) + b(k)α+ ξ 2 (k )+ ε i2 Y i3 = a 3 (k ) + b(k )α+ ξ 3 (k)+ ε i3 Y i4 = ρ 4 3 Y i3 + a 4 (k ) + b(k )α+ ε i4 with ε s covariance matrix has to respect the Markovian property. we leave E(α k, k ) and Var(α k, k ) unrestricted

39 Estimated mean log wages and type proportions (4-periods model) Mean log-earnings Proportions of worker types log earnings 10.5 type proportions firm class k firm class k

40 Variance decompositions on Swedish data (4-periods model) Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ) Corr(α, ψ) ρ 1 2 ρ 4 3 Data (K = 10) estimate 79.9% 5.4% 14.8% Monte-Carlo (K = 10, 100 reps) mean 77.6% 5.9% 16.5% quantile 69.9% 4% 13.6% quantile 81.8% 9.4% 21% Varying the number of classes K = % 3.9% 14.3% K = % 6.4% 17.3% K = % 7.3% 17.9%

41 Dynamic results (4-periods model

42 Dynamic results (4-periods model)

43 Dimensionality of firm heterogeneity (4-periods model)

44 Performance on sorting models

45 Theoretical search-matching model: wage distributions A simple extension of Shimer and Smith (2000) worker x and firm y, on-the-job search (λ 0, λ 1 ) production function f (x, y) = a + (νx ρ + (1 ν)y ρ ) 1/ρ wages are continuously bargained ( outside option is unemployment) eqs consider PAM (ρ = 3) and NAM (ρ = 4)

46 Theoretical search-matching model: wage distributions firm class model log wages (PAM) firm class quantile log wages (PAM) firm class model log wages (NAM) firm class quantile log wages (NAM) Notes: Model based on Shimer and Smith (2000) with on-the-job search.

47 Theoretical search-matching model: simulation results dim %bw %wwbf %wwwf Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ) Corr(α, ψ) PAM model BLM NAM model BLM PAM model BLM NAM model BLM Notes: Model based on Shimer and Smith (2000) with on-the-job search.

48 Theoretical search-matching model: wage distributions

49 Conclusion We introduce a new framework for wages in matched data: - unrestricted interactions, short panels, robust to low mobility - compatible with many structural models (micro and macro) - the method is important for many applications: teachers value added, sorting among cities, intergenerational mobility,... Important lessons for model of the labor market: - serial correlation for movers is large and first order - prelimanary results suggest large firm effects - endogenous mobility is empiricaly important (effect of l on Y 2 and of l on Y 3 ) Clustering on distribution: - Important insight for structural estimation, - We are currently working on statistical properties when heterogene- ity may not be grouped in the population, and clustering provides an approximation to the structure of heterogeneity

50 Wages of job movers E l1 l 2 (Y i2 Y i1 ) (x-axis) vs E l2 l 1 (Y i1 Y i2 ) (y-axis), l 1 < l E[y2 y1 l1,l2] wage gain when moving from l1 to l2 E[y1 y2 l2,l1] wage loss when moving from l2 to l1 N

51 Estimated standard deviations of log wages by worker type and firm class sd log wage factor(k) firm cluster (ordered my mean wage)

52 Cluster to cluster transitions for each type posterior transitions

53 Fit of log wage densities density lw Notes: Marginal densities of log wages for each x cell (in rows) and firm class (in columns). Sample 1, The red line is the model, the shaded area is from the data. back

54 Fit of log wage correlations 1.0 model N data Notes: Log wage correlations Corr(Y 1, Y 2 l 1, l 2), for job movers, by pairs of firm classes. Sample 2. In the data (x-axis) and in the simulated data (y-axis). back

55 Theoretical search-matching model: setup worker x and firm y, on-the-job search (λ 0, λ 1 ) back firm post vacancies production function f (x, y) = a + (νx ρ + (1 ν)y ρ ) 1/ρ Surplus equation is given by: (r + δ)s(x, y) = (1 + r) (f (x, y) δ(b(x) c(y))) r(1 δ)(π 0(y) + W 0(x)) + (1 δ)λ 1 P(x, y, y )(αs(x, y ) S(x, y))v 1/2 (y )dy. Wage equation is given by: (1 + r)w(x, y) = (r + δ)αs(x, y) + (1 δ)rw 0(x) (1 δ)λ 1 P(x, y, y )(αs(x, y ) αs(x, y))v 1/2 (y )dy.

56 Theoretical search-matching model: plots Production PAM Surplus PAM Allocation PAM y x y x y x Production NAM Surplus NAM Allocation NAM y x y x y x Notes: Model based on Shimer and Smith (2000) with on-the-job search.

57 Descriptive statistics and data selection earnings from 2002 to 2003 earnings from 2001 to 2005 all empl. either empl. both cont. firms empl. either empl. both cont. firms firms in ,753 53,610 46,597 43,884 53,159 43,670 40,987 firms in ,623 54,674 47,553 43,845 54,218 44,760 40,958 firms in ,374 54,867 46,450 43,887 54,427 43,605 40,986 workers in ,091, , , , , , ,489 workers in ,082, , , , , , ,681 workers in ,073, , , , , , ,489 mean reported firm size in median reported firm size in movers between 2002 and , ,090 54,968 19, ,389 50,629 17,504 % movers employed 12 months in co-movers 90 percentile co-movers 99 percentile co-movers 100 percentile 2,458 2,439 2, ,432 2, quaterly j2j probability quaterly e2u probability quaterly u2e probability Description of the data in the different samples. back

58 Wages of job movers E[y1+y2 l1,l2] E[y1+y2 l2,l1] N E l1 l 2 (Y i1 + Y i2 ) (x-axis) vs E l2 l 1 (Y i1 + Y i2 ) (y-axis), l 1 < l 2 back

59 Variance of firm fixed-effects

60 Approximate clustering In a second paper we consider the case where Y it = η(i, t) + ɛ(i, t) where bot η(i, t) and ɛ(i, t) are unobserverd. Assuming that η(i, t) is low complexity d - meaning the number of ɛ balls to cover η is ɛ d - for example η(i, t) = φ(ξ i, t) with x i d dimensional We show that the convergence rate of the clustering as (K, N, T ) is: back O p ( log K T ) + O p( K N ) + O p(k 2 d )

61 stayers movers y in same firm m = 1 decision to move l realization y in new firm Pr[y k, l, y, Ω] 4 Pr[m k, l, y, Ω] Pr[l k, l, y, m = 1, Ω] Pr[y k, l, y, l, Ω] 2-period - 3 (k, l) (k, l) (k, l ) 4-period (k, l, y) (k, l, y) (k, l, y) (k, l, l, y) Shimer Smith (k, l) - 2 (k) (k, l ) Shimer Smith + OTJ 1 (k, l) (k, l) (k, l) (k, l ) Postel-Vinay Robin (k, l, y) (k, l) (k, l) (k, l, l ) Lamadon, Lise, Meghir, Robin 5 (k, l, y) (k, l) (k, l) (k, l, l ) Burdett Mortensen - wage posting 6 (k, l) (k, l) (k, l) (k, l) Burdett Coles - contract posting (k, l, y) (k, l, y) (k, l, l ) (k, l ) Lise Robin 2014 (z, k, l, y) (z, k, l) (z, k, l, l ) (z, k, l ) 4 Call Ω the information set that contains all the past. Each cell of the table shows what the subset of variable that are sufficient for the probability. 1 We consider a model where bargaining is with value of unemployment, not a sequential acution model. 2 Mobility here is only by exogenous match separation 3 We don t model stayers in this version 5 A model with sequential auction and sorting in equilibrium 6 Burdet Mortensen does not include heterogeneity directly, we refer to the wage posting mechanism. This also includes Shi and Delacroix paper. 7 z is the aggregate state. back

62 Re - classification step Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ) Corr(α, ψ) cor(k j, k j ) Data (K = 10) % 4.3% 16.1% Reclassifying the firms % 6.6% 19.1% % 7.3% 18% % 5.7% 17.8% % 5.4% 17.3% % 5.6% 17% % 6.6% 18.3% % 6.9% 18.5% % 6.7% 18.8% % 6.4% 18.1% % 6% 18.1%

63 Estimation of ρ rho 1 2 rho value of the objective value of rho

64 Mobility pattern probability of moving to higher firm classes worker type

65 Worker type composition drop out high school some college proportions firm cluster (ordered my mean wage)

66 Probability model static model Pr[Y 1, Y 2, k k, l] = Pr[Y 2 Y 1, k, k, l] Pr[k Y 1, k, l] Pr[Y 1 k, l] = Pr[Y 2 k, l] Pr[k k, l] Pr[Y 1 k, l] dynamic model Pr[Y 3, Y 4, k Y 1, Y 2, k, l] = Pr[Y 4 Y 1, Y 2, Y 3, k, k, l] Pr[Y 3 Y 1, Y 2, k, k, l] Pr[k Y 1, Y 2, k, l] Pr[Y 4 Y 3, k, k, l] Pr[Y 3 Y 2, k, k, l] Pr[k Y 2, k, l]

67 Table of content Main Supplements lit review Endogenous mobility contribution Proportions without x Fit marginal wages distr framework Fit cov(y 1, y 2 ) mobility wages sd identification mini model Search model desc general case Search model plots data data selection firm classes sec-approx-clustering conclusion wages Approximate clustering proportions p k (l, l ) plot *** 2 period var decomposition 4 period var decomposition

68 References Abowd, J. M., F. Kramarz, and D. N. Margolis (1999): High Wage Workers and High Wage Firms, Econometrica, 67(2), Burdett, K., and D. T. Mortensen (1998): Wage differentials, employer size, and unemployment, Int. Econ. Rev., pp Card, D., J. Heining, and P. Kline (2013): Workplace Heterogeneity and the Rise of West German Wage Inequality*, Q. J. Econ., 128(3), Eeckhout, J., and P. Kircher (2011): Identifying sorting in theory, Rev. Econ. Stud., 78(3), Hagedorn, M., T. H. Law, and I. Manovskii (2014): Identifying Equilibrium Models of Labor Market Sorting, Working Paper. Postel-Vinay, F., and J.-M. Robin (2004): To match or not to match?: Optimal wage policy with endogenous worker search intensity, Rev. Econ. Dyn., 7(2), Shimer, R., and L. Smith (2000): Assortative Matching and Search, Econometrica, 68(2),