Presented at the NBER Labor Statistics Program Meeting and the UC-Berkley Labor Economics Seminar of the Committee on National Statistics.AbowdSchmutte_EndMob_Presentation.pdf: 142 downloads, before Oct. 1, 2020

Abowd, John

Schmutte, Ian M.

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Presentation: Endogenous Mobility

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2-24-2012 
Presentation: Endogenous Mobility 
John Abowd 
Cornell University, John.Abowd@cornell.edu 
Ian M. Schmutte 
University of Georgia 
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Presentation: Endogenous Mobility 
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Suggested Citation 
Abowd, J., & Schmutte, I.M. (2012, February). Presentation: Endogenous mobility. Presented at the NBER 
Labor Statistics Program Meeting and the UC-Berkley Labor Economics Seminar of the Committee on 
National Statistics. 
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Copyright held by authors. 
This article is available at DigitalCommons@ILR: https://digitalcommons.ilr.cornell.edu/ldi/8 
Endogenous Mobility
John M. Abowd1,2 Ian M. Schmutte3
1Cornell University
2Center for Economic Studies, U.S. Census Bureau
3University of Georgia
National Bureau of Economic Research
Labor Studies Program Meeting
February 24, 2012
Acknowledgements and Disclaimer
• This research was begun while Abowd was Distinguished Senior
Research Fellow and Schmutte was RDC Administrator at the U.S.
Census Bureau. Any opinions and conclusions expressed herein are
those of the authors and do not necessarily represent the views of the
U.S. Census Bureau. All results have been reviewed to ensure that no
confidential information is disclosed.
• This research uses data from the Census Bureau’s Longitudinal
Employer-Household Dynamics Program, which was partially supported
by the following National Science Foundation Grants: SES-9978093,
SES-0339191 and ITR-0427889; National Institute on Aging Grant
AG018854; and grants from the Alfred P. Sloan Foundation. Abowd also
acknowledges direct support from NSF Grants SES-0339191,
CNS-0627680, SES-0922005, TC-1012593, and SES-1131848.
AKM in the Presence of Endogenous Mobility
ln yit = Xitβ + θi + ψJ(i,t) + it
• The Goal: “Rehabilitate” the Abowd-Kramarz-Margolis
decomposition
• The Problem: Structural interpretations rely on the
assumption that job mobility is exogenous
• The Approach: Use the realized mobility network for
structural estimation
Realized Employment Network, t = 1
A
B
C
D
1
2
3
4
5
6
7
8
Realized Employment Network, t = 2
A
B
C
D
1
2
3
4
5
6
7
8
Realized Employment Network, t = 3
A
B
C
D
1
2
3
4
5
6
7
8
Realized Mobility Network
A
B
C
D
1
2
3
4
5
6
7
8
Estimating Individual and Employer Wage Effects
• The AKM (1999) specification for the wage determination
equation with individual and employer heterogeneity
y = Xβ + Dθ + Fψ + ε
• where y is the [N × 1] stacked vector of log wage
outcomes yit , now sorted by t , then i .
• X is the [N × k ] design matrix of observable individual and
employer time-varying characteristics (the intercept is
normally suppressed, with y and X measured as
deviations from overall means).
• D is the [N × I] design matrix for the individual effects.
• F is the [N × J − 1] design matrix for the employer effects
(non-employment is suppressed).
•  is the [N × 1] vector of statistical errors, whose properties
will be elaborated below.
Estimating Individual and Employer Wage Effects II
• [ β′ θ′ ψ′ ]′ are the unknown effects [k × 1] , [I × 1] ,
and [J − 1× 1] , resp., associated with each of the design
matrices.
Moment Equation Framework
• Solving the fixed-effects moment equations X ′X X ′D X ′FD′X D′D D′F
F ′X F ′D F ′F
 βθ
ψ
 =
 X ′yD′y
F ′y

(then, imposing identification) yields estimates of the
components of heterogeneity that can be used as the
basis for consistent estimation of functions of the individual
and employer effects.
• Next, we show how this specification relates to network
models.
Labor Markets as Bipartite Graphs
• Imagine a set of I individuals, A (t), and a set of J
employers, E (t) arranged in a bipartite graph where A (t)
and E (t) are the two (disjoint) vertex (or node) sets
• There is a link between i ∈ A (t) and j ∈ E (t) if and only if i
is employed by j at date t
• The totality of these links active at date t can be
represented by the I × J adjacency matrix B (t)
• Assuming that we are modeling primary job holders, this
adjacency matrix has a special form that will be critical in
the modeling
Labor Markets as Bipartite Graphs
• The labor market bipartite graph summarized by B (t)
evolves over time
• Since the employment relations between firms and
workers can change at any time, it is reasonable to think of
t as a continuous variable, sampled at intervals reflected in
the data
• These considerations motivate adopting the dynamic
network modeling tools to try to address the endogenous
mobility issues
Individual Degree Distribution
• We distinguish primary employment from other forms of
employment
• The primary employer at time t is the current employer if
there is only one
• Otherwise, the primary employer is the one to whom the
individual supplies the most labor market time
• This assumption puts constraints on the row degree
distribution of B (t)
Individual Degree Distribution
• Specifically, assume that j = 0 refers to the
non-employment state
• Including the column j = 0 ensures that every individual in
the population at date t has exactly one “employer”
although the (shadow) log wage outcome will be
unobserved for individuals who are not employed at t
• Hence, B (t)eJ+1 = eJ+1, where eJ+1 is the (J + 1)× 1
column vector of 1s
Employer Degree Distribution
• Given this setup the column degree distribution, e′IB (t) , is
the size distribution of employers (technically only the
columns 1 to J are included in this distribution)
• The employer size distribution (including not-employed) is
also the column degree distribution of this bipartite graph
• We note that the (very hard) problem of entry and exit of
individuals and employers can be included in this
formalism by including columns in B for potential and
defunct employers and allowing for birth and death of
individuals. For the moment, we are not going to worry
about this complication
The Evolution of the Labor Market
• The existing data are snapshots of the labor market at
points in time, B(t1), ...,B(tT ), where T is the total number
of available time periods
• These adjacency matrices describe outcomes sampled at
discrete points in time from the I × (J + 1) potential
outcomes at each moment of time
• The objective is to use these snapshots of the labor market
to test various assumptions about how the labor market
evolves over time
Restating in Terms of the Adjacency Matrix Sequence
• Note that when the sort order is t then i we have:
F =

B (1)
B (2)
...
B (T )

where B (t) is the adjacency matrix from the bipartite labor
market graph
• A direct strategy modeling endogenous mobility is to model
the evolution of B (t)
Latent Heterogeneity Wage Decomposition and
Mobility
• Workers, firms, and matches belong to latent heterogeneity
classes
• ai is the ability class of worker i ∈ {1, . . . , I}.
• bj is the productivity class of employer j ∈ {1, . . . , J}.
• kij is the quality of the match between i and j
• Match quality depends on ability and productivity
• Earnings and mobility both depend on all three
components
• Wages:
wijt = α + aiθ + bjψ + kijµ+ εijt
• Mobility: Probability of separation and transition depends
on a,b and k .
Observed Data, Latent Data and Parameters
• Observed Data
yit =
[
wi J(i,t)t , sit , i , J (i , t)
]
for i = 1, ..., I and t = 1, ...T .
• Latent Data Vector:
Z = [a1, . . . ,aI ,b1, . . . ,bJ , k11, k12, . . . , k1J , k21, . . . , kIJ ]
• Parameter Vector:
ρT =
[
α, θT , ψT , µT , σ, γ, δ, pia, pib, pik |ab
]
, ρ ∈ Θ
Complete Data Likelihood Function
L (ρ|Y , Z ) ∝
I∏
i=1

T∏
t=1
1√
2piσ2
exp
− (wi J(i,t)t−α−aiθ−bJ(i,t)ψ−ki J(i,t)µ)2
2σ2

×
T−1∏
t=1
[
1− γ〈ai 〉
〈
bJ(i,t)
〉〈
ki J(i,t)
〉]1−sit [γ〈ai 〉〈bJ(i,t)〉〈ki J(i,t)〉
]sit
×
T−1∏
t=1
[
δ〈
bJ(i,t+1)
〉
|〈ai 〉
〈
bJ(i,t)
〉〈
ki J(i,t)
〉]sit

×
I∏
i=1
J∏
j=1
 L∏
`=1
M∏
m=1
Q∏
q=1
(pia`)
ai` (pibm)
bjm
(
piq|`m
)kijq
Gibbs Sampler Estimation of Posterior Distributions
We use the Gibbs sampler to draw from P(ρ,Z |Y )
σ
(1) ∼ p
(
σ|α(0), θ(0)T , ψ(0)T , µ(0)T , γ(0), δ(0), pi(0)a , pi(0)b , pi
(0)
k|ab, Z
(0)
, Y
)

α
θ
ψ
µ

(1)
∼ p


α
θ
ψ
µ
 |γ(0), δ(0), pi(0)a , pi(0)b , pi(0)k|ab, Z (0), σ(1), Y

γ
(1) ∼ p
(
γ|δ(0), pi(0)a , pi(0)b , pi
(0)
k|ab, Z
(0)
, α
(1)
, θ
(1)T
, ψ
(1)T
, µ
(1)T
, Y
)
.
.
.
k(1)IJ ∼ p
(
kIJ |ρ(1), a(1)1 , . . . , a
(1)
I , b
(1)
1 , . . . , b
(1)
J , k
(1)
11 , . . . , k
(1)
IJ−1, Y
)
Simulation Study: Correlations
AKM Gibbs True
y θ ψ µ ε θ ψ µ ε θ ψ µ ε
y 1
A
K
M
θ .879 1
ψ .393 -.069 1
µ .141 -.000 -.000 1
ε .024 0 0 0 1
G
ib
bs
θ .867 .985 -.066 .000 0 1
ψ .395 -.062 .990 -.000 0 -.065 1
µ -.097 -.292 .183 .607 0 -.404 .166 1
ε .025 .001 -.002 .017 .960 -.000 -.000 .001 1
Tr
ue
θ .867 .985 -.066 .000 0 1 -.065 -.404 -.000 1
ψ .395 -.062 .990 -.000 0 -.065 1 .166 -.000 -.065 1
µ -.097 -.292 .183 .607 0 -.404 .166 1 .001 -.404 .166 1
ε .051 .020 .016 .017 .960 .020 .018 -.005 .999 .020 .018 -.005 1
Distribution of Wage Parameters: Simulated Data
−4.885−4.884−4.883−4.882−4.881 −4.88
100
150
200
θ1
  ← −4.8826
3.12 3.121 3.122 3.123
150
200
250
300
θ2
  ← 3.1217
−2.205−2.204−2.203−2.202−2.201 −2.2
100
150
200
ψ1
  ← −2.2025
1.8 1.801 1.802 1.803 1.804
100
200
ψ2
  ← 1.8021
1.094 1.095 1.096 1.097 1.098 1.099
100
150
200
µ1
  ← 1.0968
−0.905 −0.904 −0.903 −0.902 −0.901
100
200
µ2
  ← −0.9032
1.079 1.08 1.081 1.082 1.083 1.084
100
200
α
  ← 1.0813
0.098 0.099 0.1 0.101
200
400
σ
  ← 0.0992
Data: Universe
• Matched employer-employee data from LEHD program
• All individuals employed in IL, IN, WI between 1999-2003.
• 16.9 million persons
• 719 thousand unique employers
• 39 million unique person-employer matches
• Summaries of AKM decomposition (as described in
Abowd, et al. [2003]) provide starting values and
benchmarks.
Data: Estimation Sample
• 0.25% simple random sample of individuals
• retain all matches and employers attached to those
individuals
• 42,228 persons
• 39,458 employers
• 97,455 matches (including non-employment spells)
• 211,140 person-year observations
Distribution of Wage Parameters: LEHD Data
−0.254 −0.252 −0.25 −0.248 −0.246
50
100
θ1
  ← −0.2497
0.605 0.61 0.615 0.62
40
60
80
θ2
  ← 0.6112
−1.1 −1.095 −1.09
50
100
ψ1
  ← −1.0961
1.22 1.225 1.23
20
40
60
80
ψ2
  ← 1.2256
−0.735 −0.73 −0.725 −0.72 −0.715
40
60
µ1
  ← −0.7243
1.045 1.05 1.055 1.06 1.065
20
40
µ2
  ← 1.0562
9.065 9.07 9.075 9.08 9.085 9.09
30
40
50
α
  ← 9.0777
0.43 0.432 0.434 0.436
50
100
150
σ
  ← 0.4330
Posterior Distribution of Wage Equation Parameters:
LEHD Data
Parameter Mean Std. Dev
θ1 -0.2497 0.0032
θ2 0.6112 0.0051
ψ1 -1.0961 0.0044
ψ2 1.2256 0.0055
µ1 -0.7243 0.0066
µ2 1.0562 0.0082
α 9.0777 0.0082
σ 0.4330 0.0024
Correlation Matrix of Wage Parameters: LEHD Data
AKM Gibbs
y θ ψ µ ε θ ψ µ ε
y 1
A
K
M
θ 0.5284 1
ψ 0.5683 0.0632 1
µ 0.4236 0.0335 -.0182 1
ε 0.2345 -.0000 -.0000 0.0000 1
G
ib
bs
θ 0.3361 0.2401 0.1682 0.0816 -.0000 1
ψ 0.5486 0.2037 0.5599 0.1179 -.0000 0.0359 1
µ -.02219 0.0951 -.2577 0.1396 0.0000 -.1202 -.7236 1
ε 0.4989 0.2288 0.1498 0.2677 0.4703 -.0000 .0002 -.0000 1
Regression of Gibbs on AKM: LEHD Data
θGibbs ψGibbs µGibbs εGibbs
θAKM 0.151 0.317 0.154 0.175
ψAKM 0.1441 1.492 -.529 0.164
µAKM 0.072 0.332 0.266 0.307
εAKM 0.000 0.000 0.000 0.988
Constant 0.014 0.185 -.091 -.003
Separation Probabilities, γ: AKM and Structural
Estimates
θ Type ψ Type µ Type Separation Probability
AKM Gibbs
1 1 1 0.4127 0.6603
1 1 2 0.0530 0.4418
1 2 1 0.0169 0.2796
1 2 2 0.0055 0.2506
1 3 - 0.0552 0.3128
2 1 1 0.7109 0.3814
2 1 2 0.0448 0.1799
2 2 1 0.0103 0.1923
2 2 2 0.0123 0.2373
2 3 - 0.1095 0.3179
Destination Probabilities, δ: AKM and Structural
Estimates
Origin Destination Employer Type
θ Type ψ Type µ Type AKM Gibbs
1 2 – 1 2 –
1 1 1 0.4103 0.1440 0.4457 0.4546 0.1129 0.4324
1 1 2 0.4758 0.2057 0.3185 0.4929 0.2160 0.2911
1 2 1 0.2315 0.3990 0.3695 0.2486 0.4681 0.2834
1 2 2 0.2205 0.4791 0.3004 0.0910 0.2815 0.6275
1 3 - 0.6912 0.3088 – 0.7247 0.2753 –
2 1 1 0.5000 0.1988 0.3012 0.5079 0.0319 0.4602
2 1 2 0.4929 0.2622 0.2449 0.5226 0.2324 0.2450
2 2 1 0.2178 0.5139 0.2683 0.1120 0.6474 0.2406
2 2 2 0.1660 0.5727 0.2613 0.1089 0.6242 0.2671
2 3 - 0.6189 0.3811 – 0.7229 0.2770 –
Structural Markov Transition Matrix: LEHD Data
θ Type
ψ Type 1 1 2 2 3
µ Type 1 2 1 2 -
1 1 1 0.4127 0.2272 0.0708 0.0038 0.2855
1 1 2 0.0530 0.7230 0.0906 0.0048 0.1286
1 2 1 0.0169 0.0526 0.8447 0.0066 0.0792
1 2 2 0.0055 0.0173 0.0670 0.7530 0.1572
1 3 - 0.0552 0.1715 0.0818 0.0043 0.6872
2 1 1 0.7109 0.1014 0.0118 0.0004 0.1755
2 1 2 0.0448 0.8693 0.0405 0.0013 0.0441
2 2 1 0.0103 0.0113 0.9283 0.0038 0.0463
2 2 2 0.0123 0.0135 0.1436 0.7672 0.0634
2 3 - 0.1095 0.1202 0.0854 0.0027 0.6822
Mobility and Selection
.53
.71.68
.05
.06
.03
.02
.75
.82
.52
.70
ψ2
ψ1
θ1 θ2
Conclusions
• Showed that endogenous mobility affects the AKM
decomposition via the realized mobility network, which is
the tool used for identification in that model
• Developed a complete posterior predictive distribution for
incorporating endogenous mobility into the AKM wage
decomposition
• The Markov transition matrix that describes the evolution of
the network adjacency matrix reveals that the probability of
transitions into better matches do depend on the worker
type, firm type and match type in the current job
• Future work will refine the regression-based approach we
used here for estimating the expected structural effect
given the AKM wage components


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