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) • y it = α i + ψ j ( i , t ) + ǫ 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 ...)

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 ) + ǫ 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 ...)

This paper: • Proposes a distributional model of wages - assume discrete heterogeneity: firms ( k ) and workers ( ℓ ) - non-parametric conditional wage distributions F k ℓ ( 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

This paper: • Proposes a distributional model of wages - assume discrete heterogeneity: firms ( k ) and workers ( ℓ ) - non-parametric conditional wage distributions F k ℓ ( 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

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 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 i 1 ) = k ] = π k ( ℓ ) . • The conditional cdf of log wages Y i 1 is: Pr [ Y i 1 ≤ y | ω ( i ) = ℓ, f ( j i 1 ) = k ] = F k ℓ ( 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 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 i 1 ) = k ] = π k ( ℓ ) . • The conditional cdf of log wages Y i 1 is: Pr [ Y i 1 ≤ y | ω ( i ) = ℓ, f ( j i 1 ) = k ] = F k ℓ ( 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 move k ′ k Y i 1 Y i 2 • Consider a worker of type ℓ in firm k in period 1 • He gets a wage Y i 1 drawn from F k ℓ ( y ) . • The worker moves to a class- k ′ firm with a probability that depends on k and ℓ , not on Y i 1 . • In period 2 he draws a wage Y i 2 from a distribution G k ′ ℓ ( y ′ ) that depends on ℓ and k ′ , not on ( k , Y i 1 ) .

Job mobility dynamic model: 4 periods move k ′ k Y i 1 Y i 2 Y i 3 Y i 4 • Consider a worker of type ℓ in firm k at t = 2 • Wages ( Y i 1 , Y i 2 ) 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 i 2 , not on Y i 1 . • At t = 3 , If he moves, the worker draws a wage Y i 3 from a distribution that depends on ℓ , k ′ , k , Y i 2 , not on Y i 1 . • At t = 4 , the worker draws a wage Y i 4 that depends on ℓ , k ′ , Y i 3 , not on ( k , Y i 2 , Y i 1 ) .

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).

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: � K � Y i 1 ≤ y , Y i 2 ≤ y ′ | k , k ′ � p kk ′ ( ℓ ) F k ℓ ( y ) F k ′ ℓ ( y ′ ) , Pr = ℓ =1 • For the cross-section in k we have � K Pr [ Y i 1 ≤ y | k ] = π k ( ℓ ) F k ℓ ( y ) . ℓ =1

Main restrictions Dynamic model • Using mobility assumptions of the dynamic model • conditioning on Y 2 = y 2 Y 3 = y 3 , we get: K � � Y i 1 ≤ y , Y i 4 ≤ y ′ | y 2 , y 3 , k , k ′ � p kk ′ y 2 y 3 ( ℓ ) F k ℓ ( y | y 2 ) G k ′ ℓ ( y ′ | y 3 ) Pr = ℓ =1 • Similar structure as in static model: - use 4 period of data - replace F k ℓ ( y ) with F k ℓ ( y | y ′ ) - replace p kk ′ with p kk ′ y 2 y 3

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 ) = F k ℓ ( y 1 ) p kk ′ ( ℓ ) F k ′ ℓ ( y 2 ) ℓ • Discretize wage ( n w ) support and write in Matrix form: A ( k , k ′ ) P ( k , k ′ ) F ( k ′ ) ⊺ = F ( k ) � �� � � �� � � �� � n w × n w n w × n ℓ n ℓ × n ℓ diag. • 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.

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 k ℓ 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 k ℓ ) are linearly independent. • once F k ℓ 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 .

Empirical content of wage means intro • In the linear framework (AKM) where Y it = α i + ψ j ( i , t ) + ǫ 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 + ǫ it with E [ ǫ it | α i , k i 1 , k i 2 , m i 1 ] = 0 . • what can we do?