A Distributional Framework for Matched Employer Employee Data Nov - - PowerPoint PPT Presentation

A Distributional Framework for Matched Employer Employee Data

Nov 2017

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

Two influential literatures for worker and firm heterogeneity

Log linear fixed effect wages

Abowd, Kramarz, and Margolis (1999); Card, Heining, and Kline (2013)

• yit = αi + ψj(i,t) + ǫit
• 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 ...)

Two influential literatures for worker and firm heterogeneity

Log linear fixed effect wages

Abowd, Kramarz, and Margolis (1999); Card, Heining, and Kline (2013)

• yit = αi + ψj(i,t) + ǫit
• 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 ...)

This paper:

• Proposes a distributional model of wages
• assume discrete heterogeneity: firms (k) and workers (ℓ)
• non-parametric conditional wage distributions Fkℓ(w)
• unrestricted firm compositions πk(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

This paper:

• Proposes a distributional model of wages
• assume discrete heterogeneity: firms (k) and workers (ℓ)
• non-parametric conditional wage distributions Fkℓ(w)
• unrestricted firm compositions πk(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

Plan of the talk

1 Framework & identification overview 2 Data and empirical results 3 Performance on a theoretical sorting model

Model and Indentification

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 jit denote the identifier of the firm where i works at t.
• The proportion of type-l workers in firm j is πf (j)(l), where:

Pr [ω(i) = ℓ | f (ji1) = k] = πk(ℓ).

• The conditional cdf of log wages Yi1 is:

Pr [Yi1 ≤ y | ω(i) = ℓ, f (ji1) = k] = Fkℓ(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.

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 jit denote the identifier of the firm where i works at t.
• The proportion of type-l workers in firm j is πf (j)(l), where:

Pr [ω(i) = ℓ | f (ji1) = k] = πk(ℓ).

• The conditional cdf of log wages Yi1 is:

Pr [Yi1 ≤ y | ω(i) = ℓ, f (ji1) = k] = Fkℓ(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.

Job mobility

static model: 2 periods

Yi1 Yi2 k k′ move

• Consider a worker of type ℓ in firm k in period 1
• He gets a wage Yi1 drawn from Fkℓ(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 Yi2 from a distribution Gk′ℓ(y′)

that depends on ℓ and k′, not on (k, Y i1).

Job mobility

dynamic model: 4 periods

Yi1 Yi2 Yi3 Yi4 k k′ move

• Consider a worker of type ℓ in firm k at t = 2
• Wages (Yi1, Yi2) 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 Yi2, not on Y i1.

• At t = 3, If he moves, the worker draws a wage Yi3 from a

distribution that depends on ℓ, k′, k, Yi2, not on Y i1.

• At t = 4, the worker draws a wage Yi4 that depends on ℓ, k′,

Yi3, not on (k, Y i2, Y i1).

Link to theoretical models

• 2-periods model:
• Example: Shimer and Smith (2000), without or with
• n-the-job search (workers’ threat points being the value of

unemployment).

• No role for match-specific draws, unless independent over time
• r measurement error. No sequential auctions.
• 4-periods model:
• All models where state variables (ℓ, kt, Yt) 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 (ℓt), no

permanent+transitory within-job wage dynamics (example: random walk+i.i.d. shock).

Plan of attack

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

Main restrictions

Static model

• Under the assumptions of the static model, we have,
• For movers from firm k to firm k′ we have:

Pr

• Yi1≤y, Yi2≤y′|k, k′

=

K

• ℓ=1

pkk′ (ℓ) Fkℓ(y)Fk′ℓ(y′),

• For the cross-section in k we have

Pr [Yi1 ≤ y| k] =

K

• ℓ=1

πk (ℓ) Fkℓ(y).

Main restrictions

Dynamic model

• Using mobility assumptions of the dynamic model
• conditioning on Y2 = y2Y3 = y3, we get:

Pr

• Yi1≤y, Yi4≤y′|y2, y3, k, k′

=

K

• ℓ=1

pkk′y2y3 (ℓ) Fkℓ(y|y2)Gk′ℓ(y′|y3)

• Similar structure as in static model:
• use 4 period of data
• replace Fkℓ(y) with Fkℓ(y|y′)
• replace pkk′ with pkk′y2y3

Identification Wage Functions

Large firms

• Consider two larger firms k and k′ and joint Y1, Y2 wages for

movers k → k′ Ak,k′(y1, y2) =

• ℓ

Fkℓ(y1)pkk′(ℓ)Fk′ℓ(y2)

• Discretize wage (nw) support and write in Matrix form:

A(k, k′)

nw×nw

= F(k)

nw×nℓ

P(k, k′)

nℓ×nℓ diag.

F(k′)⊺

• Consider case where nw=nℓ, 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.

Identification Wage function

• In general, the identification relies on a joint diagonalization
• f all A(k, k′).

A(k, k′) = F(k)P(k, k′)F(k′)⊺

• It is sufficient (but not necessary, see paper) for identification
• f Fkℓ that:
• pkk ′(ℓ) = 0 for ℓ = 1, ..., L.
• pkk′(ℓ

pk′k(ℓ), k = 1, ..., L, are distinct.

• The columns of F(k) (the Fkℓ) are linearly independent.
• once Fkℓ is known, go to cross-section to get πk(ℓ)
• In the 4 period model, replace Y1, Y2 with Y1|Y2, Y4|Y3 and

do everything conditional on k, k′, Y2, Y3.

Empirical content of wage means

intro

• In the linear framework (AKM) where Yit = αi + ψj(i,t) + ǫit
• ne can focus on movers to get:

E(Yit+1 − Yit|m = 1) = ψj(i,t+1) − ψj(i,t), (1) which can be recovered with OLS.

• Now consider an interacted model at the class level:

Yit = a(kit) + b(kit)αi + ǫit with E[ǫit|αi, ki1, ki2, mi1] = 0.

• what can we do?

Empirical content of wage means

intro

• In the linear framework (AKM) where Yit = αi + ψj(i,t) + ǫit
• ne can focus on movers to get:

E(Yit+1 − Yit|m = 1) = ψj(i,t+1) − ψj(i,t), (1) which can be recovered with OLS.

• Now consider an interacted model at the class level:

Yit = a(kit) + b(kit)αi + ǫit with E[ǫit|αi, ki1, ki2, mi1] = 0.

• what can we do?

Empirical content of wage means

interactions

• Interacted model Yit = a(kit) + b(kit)αi + ǫit with 2 firms:

¯ y21(2) ¯ y21(1) ¯ y12(1) ¯ y12(2) a(2) + b(2) · ¯ α21 = a(1) + b(1) · ¯ α12 = = a(1) + b(1) · ¯ α21 = a(2) + b(2) · ¯ α12 k=2 k=2 k=1 k=1 ∆21 ∆12 wage time

• Comparing changes: ∆21 + ∆12 = (¯

α12 − ¯ α21)(b(2) − b(1))

• 0 if no interactions b(2) = b(1)
• also 0 when composition is identical ¯

α12 = ¯ α21

Empirical content of wage means

interactions

• Consider the following differences:

¯ y21(2) − ¯ y12(2) = b(2)(¯ α21 − ¯ α12) ¯ y21(1) − ¯ y12(1) = b(1)(¯ α21 − ¯ α12)

• Taking ration whenever ¯

α21 = ¯ α12: b(2) b(1) = ¯ y21(1) − ¯ y12(2) ¯ y21(2) − ¯ y12(1)

• this recovers the interaction term.

Empirical content of wage means

Event Study in wages from Shimer Smith

Wages in Shimer Smith Event study

0.90 0.95 1.00 1.05 2.5 5.0 7.5 10.0

firm class

model log wages (PAM)

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:

Hk(y) =

• ℓ

πk(ℓ)Fkℓ(y)

• Recover types using cross-section, then treat groups as large
• firms. Requires that Hk(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.

Firm types

Grouping in practice

• Under the assumption that in the population i) firms are

clustered in K groups and ii) the Hk 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),H1,...,HK J

• j=1

nj

D

• d=1
• Fj (yd) − H f (j) (yd)

2 ,

• when K is known and Hk(y) are separable, this classification

is super-consistent in firm size.

Firm types

Distribution of wages

0.90 0.95 1.00 1.05 2.5 5.0 7.5 10.0

firm class

model log wages (PAM)

• 0.4

0.5 0.6 0.7 2.5 5.0 7.5 10.0

firm class

quantile log wages (PAM)

0.96 1.00 1.04 1.08 1.12 2.5 5.0 7.5 10.0

firm class

model log wages (NAM)

• 0.50

0.55 0.60 0.65 0.70 2.5 5.0 7.5 10.0

firm class

quantile log wages (NAM)

Notes: Model based on Shimer and Smith (2000) with on-the-job search.

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 pkk′(ℓ), and Fkℓ or Fkℓ(y1|y2)

• use movers, treat worker type as random
• non-parametric identification in the paper
• in practice we use the EM algorithm

3 get firm compositions πk(ℓ)

• using stayers (cross-section or 4 periods)
• another EM

Applying framework to Swedish data

Sample description

• We use matched employer-employee data from Sweden

between 1997 and 2006.

• 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 2004. 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).

Estimated firm classes

• We estimate firm classes on the 2002 cross-section using a

weighted k-means algorithm (empirical cdfs with 40 points, 10000 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.

Descriptive statistics on estimated firm classes

firm cluster: 1 2 3 4 5 6 7 8 9 10 all number of workers 21,662 62,929 110,792 114,324 100,080 78,837 137,971 85,806 58,728 27,023 798,152 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 28.9 28 26.6 26.9 23.7 21.1 18.9 12.2 5.31 3 20.7 % HS grade 59.7 62.5 62.6 62.5 61.7 57.8 58.6 47.2 32.9 23.9 56.1 % some college 11.4 9.42 10.7 10.7 14.6 21.2 22.5 40.5 61.8 73.1 23.3 % workers younger than 30 25 20.9 20.5 18.4 16.4 18.4 14.3 15.3 16 14.8 17.5 % workers between 31 and 50 52.7 52.2 53.2 54.1 55.4 54.6 56 57 59.1 63.6 55.3 % workers older than 51 22.3 26.9 26.2 27.5 28.2 27.1 29.7 27.7 24.9 21.6 27.2 mean log wages 9.6 9.87 9.99 10.1 10.1 10.1 10.2 10.4 10.5 10.8 10.2 variance of log wages 0.15 0.0841 0.0934 0.0732 0.0699 0.141 0.0918 0.114 0.116 0.177 0.148 between firm variance of log wages 0.0576 0.00614 0.0039 0.00185 0.0016 0.0056 0.00184 0.00367 0.00456 0.039 0.0544 mean of log value added per worker 12.4 12.5 12.7 12.7 12.8 12.8 12.9 13 13 13.2 12.7 variance of log value added per worker 0.202 0.163 0.155 0.141 0.175 0.255 0.249 0.304 0.398 0.594 0.52 median number of workers per firm 2 4 6 5 5 6 5 5 4 3 4

Notes: Sample 1 in 2002. All workers are males, employed during the full year

• 2002. “HS”is high school.

Parametrization of the 2-periods model

• We specify a model with Gaussian error:
• Yit ∼ N(µtkℓ, σtkl)
• πk(ℓ) and pkk ′(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
• Yit = a(k) + b(k)α + ǫit

Estimated mean log wages and type proportions (2-periods model)

Mean log wages Proportions of worker types

9.5 10.0 10.5 11.0 1 2 3 4 5 6 7 8 9 10

firm class k log−earnings

0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 6 7 8 9 10

firm class k type proportions

Notes: The left graph plots the mean of Fkℓ. 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 πk(ℓ).

Variance Decomposition and mean effects (2-periods model)

Variance decomposition (×100)

Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ)

Corr(α, ψ) R2 80.3

(.8)

3.4

(.2)

16.3

(.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)

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:

Yi1 = α(ω0

i ) + ψ(f 0 ji1) + εi1

• We compare our results with AKM on real and simulated data.

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% 0.436

Monte-Carlo (K = 10, 100 reps)

mean 80.3% 3.4% 16.3% 0.489 0.025 quantile 79% 3% 15% 0.455 0.975 quantile 82% 3.9% 17.4% 0.526

Varying the number of classes

K = 5 81.5% 3.6% 14.9% 0.437 K = 15 79% 4.5% 16.5% 0.436 K = 20 79.2% 4.7% 16.2% 0.42

Mixture model

estimate 77% 5.2% 17.8% 0.443

Monte-Carlo (K = 10, 100 reps)

mean 77.8% 4.4% 17.8% 0.481 0.025 quantile 76.3% 4% 16.8% 0.465 0.975 quantile 79.1% 5% 18.8% 0.494

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 0.7766 0.0473 0.1762 0.4598 Fixed-effects 0.9813 0.3014

• 0.2826
• 0.2599

Simulated from the model

This paper 1 1 0.7669 0.0466 0.1866 0.4934 Fixed-effects 1 1 1.0879 0.3447

• 0.4326
• 0.3532

Simulated from the model without limited mobility

Fixed-effects 4 1 0.8948 0.1602

• 0.055
• 0.0727

Fixed-effects 4 10 0.7816 0.053 0.1654 0.4064

Estimation of the 4-periods model

• We add two employment periods: 2001 and 2005. There are

12,519 workers moving in 2003 employed in the five years.

• We write a model for Pr[Y1, Y2, Y3, Y4|ℓ, k, k′] :

Yi1 = ρ1|2Yi2+ a1(k) + b(k)α+ εi1 Yi2 = a2(k) + b(k)α+ ξ2(k′)+ εi2 Yi3 = a3(k′) + b(k′)α+ ξ3(k)+ εi3 Yi4 = ρ4|3Yi3+ a4(k′) + b(k′)α+ εi4 with ε’s covariance matrix has to respect the Markovian property.

• we leave E(α|k, k′) and Var(α|k, k′) unrestricted

Estimated mean log wages and type proportions

(4-periods model)

Mean log-earnings Proportions of worker types

9.5 10.0 10.5 11.0 1 2 3 4 5 6 7 8 9 10

firm class k log−earnings

0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 6 7 8 9 10

firm class k type proportions

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) ρ1|2 ρ4|3

Data (K = 10)

estimate 79.9% 5.4% 14.8% 0.357 0.2452 0.6603

Monte-Carlo (K = 10, 100 reps)

mean 77.6% 5.9% 16.5% 0.389 0.2138 0.6639 0.025 quantile 69.9% 4% 13.6% 0.33 0.205 0.6601 0.975 quantile 81.8% 9.4% 21% 0.441 0.2219 0.6675

Varying the number of classes

K = 5 81.8% 3.9% 14.3% 0.4 0.2674 0.6565 K = 15 76.3% 6.4% 17.3% 0.391 0.2469 0.6582 K = 20 74.8% 7.3% 17.9% 0.382 0.2476 0.6566

Dynamic results

(4-periods model

Dynamic results

(4-periods model)

Dimensionality of firm heterogeneity

(4-periods model)

Performance on sorting models

Theoretical search-matching model: wage distributions

• A simple extension of Shimer and Smith (2000)
• worker x and firm y, on-the-job search (λ0, λ1)
• production function f (x, y) = a + (νx ρ + (1 − ν)yρ)1/ρ
• wages are continuously bargained ( outside option is

unemployment)

eqs ⊲

• consider PAM (ρ = −3) and NAM (ρ = 4)

Theoretical search-matching model: wage distributions

0.90 0.95 1.00 1.05 2.5 5.0 7.5 10.0

firm class

model log wages (PAM)

• 0.4

0.5 0.6 0.7 2.5 5.0 7.5 10.0

firm class

quantile log wages (PAM)

0.96 1.00 1.04 1.08 1.12 2.5 5.0 7.5 10.0

firm class

model log wages (NAM)

• 0.50

0.55 0.60 0.65 0.70 2.5 5.0 7.5 10.0

firm class

quantile log wages (NAM)

Notes: Model based on Shimer and Smith (2000) with on-the-job search.

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 6 × 10 0.693 0.103 0.203 0.791 0.054 0.156 0.377 BLM 6 × 10 0.636 0.101 0.263 0.756 0.069 0.175 0.385 NAM model 6 × 10 0.661 0.136 0.203 1.082 0.125

• 0.206
• 0.281

BLM 6 × 10 0.625 0.114 0.262 1.049 0.099

• 0.148
• 0.23

PAM model 50 × 50 0.693 0.108 0.2 0.758 0.071 0.171 0.367 BLM 6 × 10 0.591 0.121 0.288 0.701 0.095 0.204 0.396 NAM model 50 × 50 0.685 0.115 0.201 1.079 0.107

• 0.186
• 0.273

BLM 6 × 10 0.668 0.044 0.288 1.009 0.041

• 0.05
• 0.122

Notes: Model based on Shimer and Smith (2000) with on-the-job search.

Theoretical search-matching model: wage distributions

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 Y2

and of ℓ on Y3)

• 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

Wages of job movers

Eℓ1ℓ2 (Yi2 − Yi1) (x-axis) vs Eℓ2ℓ1 (Yi1 − Yi2) (y-axis), ℓ1 < ℓ2

• 0.0

0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4

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

• 200

400 600

Estimated standard deviations of log wages by worker type and firm class

0.0 0.2 0.4 2.5 5.0 7.5 10.0

firm cluster (ordered my mean wage) sd log wage factor(k)

1 2 3 4 5 6

Notes: The graph plots the standard deviation of Fkℓ. The L = 10 firm classes

Cluster to cluster transitions for each type

posterior transitions

Fit of log wage densities

1 2 3 4 5 6 7 8 9 10 1 2 3 1 2 3 0.0 0.5 1.0 1.5 1 2 3 1 2 3 0.0 0.5 1.0 1.5 2.0 1 2 3 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 1 2 3 4 5 6 7 8 9 8 9 1011 8 9 101112 8 10 12 8 10 12 14 8 9 10111213 8 10 12 148 10 12 8 10 12 148 9 101112138 10 12

lw density

Notes: Marginal densities of log wages for each x cell (in rows) and firm class (in columns). Sample 1, 2002. The red line is the model, the shaded area is from the data.

back ⊲

Fit of log wage correlations

• ●
• ●
• ●
• ●
• ●
• −1.0

−0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

data model

N

• 200

400 600

Notes: Log wage correlations Corr(Y1, Y2|ℓ1, ℓ2), for job movers, by pairs of firm classes. Sample 2. In the data (x-axis) and in the simulated data (y-axis).

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Theoretical search-matching model: setup

• worker x and firm y, on-the-job search (λ0, λ1)

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• firm post vacancies
• production function f (x, y) = a + (νx ρ + (1 − ν)yρ)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)(Π0(y) + W0(x)) + (1 − δ)λ1

• P(x, y, y′)(αS(x, y′) − S(x, y))v1/2(y′)dy′.

Wage equation is given by:

(1 + r)w(x, y) = (r + δ)αS(x, y) + (1 − δ)rW0(x)− (1 − δ)λ1

• P(x, y, y′)(αS(x, y′) − αS(x, y))v1/2(y′)dy′.

Theoretical search-matching model: plots

Production PAM

x y

Surplus PAM

x y

Allocation PAM

x y

Production NAM

x y

Surplus NAM

x y

Allocation NAM

x y

Notes: Model based on Shimer and Smith (2000) with on-the-job search.

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 2002 54,753 53,610 46,597 43,884 53,159 43,670 40,987 firms in 2003 55,623 54,674 47,553 43,845 54,218 44,760 40,958 firms in 2004 56,374 54,867 46,450 43,887 54,427 43,605 40,986 workers in 2002 1,091,509 907,883 635,186 599,963 891,256 587,614 554,489 workers in 2003 1,082,028 910,454 635,135 598,834 893,392 587,573 553,681 workers in 2004 1,073,174 886,573 635,186 599,963 870,419 587,614 554,489 mean reported firm size in 2002 34.4 34.9 37.6 37.1 35.1 39.4 39 median reported firm size in 2002 10 10 10 10 10 11 11 movers between 2002 and 2004 142,580 121,090 54,968 19,745 117,389 50,629 17,504 % movers employed 12 months in 2003 0.755 0.868 0.952 0.968 0.873 0.953 0.969 co-movers 90 percentile 2 2 2 1 2 2 1 co-movers 99 percentile 13 15 22 7 15 22 7 co-movers 100 percentile 2,458 2,439 2,137 233 2,432 2,033 222 quaterly j2j probability 0.0196 0.0183 0.0149 0.00678 0.0181 0.0148 0.00652 quaterly e2u probability 0.0228 0.0133 0.00396 0.0127 0.00375 quaterly u2e probability 0.155 0.34 0.477 1 0.348 0.509 1

Description of the data in the different samples.

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Wages of job movers

• ●
• ●
• 19.5

20.0 20.5 21.0 21.5 19.6 20.0 20.4 20.8 21.2

E[y1+y2|l1,l2] E[y1+y2|l2,l1]

N

• 200

400 600

Eℓ1ℓ2 (Yi1 + Yi2) (x-axis) vs Eℓ2ℓ1 (Yi1 + Yi2) (y-axis), ℓ1 < ℓ2

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Variance of firm fixed-effects

Approximate clustering

• In a second paper we consider the case where

Yit = η(i, t) + ǫ(i, t) where bot η(i, t) and ǫ(i, t) are unobserverd.

• Assuming that η(i, t) is low complexity d
• meaning the number of ǫ−balls to cover η is ∝ ǫ−d
• for example η(i, t) = φ(ξi, t) with xi d−dimensional
• We show that the convergence rate of the clustering as

(K, N , T) → ∞ is: Op(log K T ) + Op(K N ) + Op(K − 2

d )

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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.

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Re - classification

step

Var(α) Var(α+ψ) Var(ψ) Var(α+ψ) 2Cov(α,ψ) Var(α+ψ)

Corr(α, ψ) cor(kj , ˜ kj )

Data (K = 10)

79.6% 4.3% 16.1% 0.436 1

Reclassifying the firms

1 74.2% 6.6% 19.1% 0.43 0.976 2 74.7% 7.3% 18% 0.383 0.966 3 76.5% 5.7% 17.8% 0.428 0.961 4 77.3% 5.4% 17.3% 0.424 0.957 5 77.3% 5.6% 17% 0.408 0.954 6 75.1% 6.6% 18.3% 0.412 0.951 7 74.6% 6.9% 18.5% 0.408 0.949 8 74.5% 6.7% 18.8% 0.421 0.945 9 75.4% 6.4% 18.1% 0.412 0.939 10 75.8% 6% 18.1% 0.424 0.934

Estimation of ρ

rho 1|2 rho 4|3 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 10 20 30 40

value of rho value of the objective

Mobility pattern

• 0.40

0.45 0.50 0.55 0.60 2 4 6

worker type probability of moving to higher firm classes

Worker type composition

drop out high school some college 0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

firm cluster (ordered my mean wage) proportions

Probability model

static model Pr[Y1, Y2, k′|k, ℓ] = Pr[Y2|Y1, k, k′, ℓ] · Pr[k′|Y1, k, ℓ] · Pr[Y1|k, ℓ] = Pr[Y2|k′, ℓ] · Pr[k′|k, ℓ] · Pr[Y1|k, ℓ] dynamic model Pr[Y3, Y4, k′|Y1, Y2, k, ℓ] = Pr[Y4|Y1, Y2, Y3, k′, k, ℓ] · Pr[Y3|Y1, Y2, k′, k, ℓ] · Pr[k′|Y1, Y2, k, ℓ] Pr[Y4|Y3, k′, k, ℓ] · Pr[Y3|Y2, k′, k, ℓ] · Pr[k′|Y2, k, ℓ]

Table of content

Main Supplements lit review Endogenous mobility contribution Proportions without x Fit marginal wages distr framework Fit cov(y1, y2) 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 pk(ℓ, ℓ′) plot *** 2 period var decomposition 4 period var decomposition

References

Abowd, J. M., F. Kramarz, and D. N. Margolis (1999): “High Wage Workers and High Wage Firms,”Econometrica, 67(2), 251–333. Burdett, K., and D. T. Mortensen (1998): “Wage differentials, employer size, and unemployment,”Int. Econ. Rev., pp. 257–273. Card, D., J. Heining, and P. Kline (2013): “Workplace Heterogeneity and the Rise of West German Wage Inequality*,”

• Q. J. Econ., 128(3), 967–1015.

Eeckhout, J., and P. Kircher (2011): “Identifying sorting in theory,”

• Rev. Econ. Stud., 78(3), 872–906.

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), 297–330. Shimer, R., and L. Smith (2000): “Assortative Matching and Search,”Econometrica, 68(2), 343–369.