07350015%2E2012%2E698142

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Job Durations With Worker- and Firm-Specific
Effects: MCMC Estimation With Longitudinal
Employer–Employee Data
Guillaume Horny , Rute Mendes & Gerard J. van den Berg
To cite this article: Guillaume Horny , Rute Mendes & Gerard J. van den Berg (2012) Job
Durations With Worker- and Firm-Specific Effects: MCMC Estimation With Longitudinal
Employer–Employee Data, Journal of Business & Economic Statistics, 30:3, 468-480, DOI:
10.1080/07350015.2012.698142
To link to this article: http://dx.doi.org/10.1080/07350015.2012.698142

Published online: 20 Jul 2012.

Submit your article to this journal

Article views: 306

View related articles

Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]

Date: 11 January 2016, At: 22:46

Job Durations With Worker- and Firm-Specific
Effects: MCMC Estimation With Longitudinal
Employer–Employee Data
Guillaume HORNY
Bank of France, DGEI-DEMS-SAMIC, 75 049 Paris Cedex 01, France, and Department of Economics, Université
Catholique de Louvain (guillaume.horny@banque-france.fr)

Rute MENDES
International Training Centre of the ILO, Turin, Italy (mendesrutept@gmail.com)

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

Gerard J. VAN DEN BERG

Department of Economics, University of Mannheim; Institute for Evaluation of Labour Market and Education Policy,
Uppsala, Sweden; Department of Economics, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands;
Institute for the Study of Labor, Bonn, Germany (givdberg@uni-mannheim.de)
We study job durations using a multivariate hazard model allowing for worker-specific and firm-specific
unobserved determinants. The latter are captured by unobserved heterogeneity terms or random effects,
one at the firm level and another at the worker level. This enables us to decompose the variation in job
durations into the relative contribution of the worker and the firm. We also allow the unobserved terms to
be correlated in a model that is primarily relevant for markets with small firms. For the empirical analysis,
we use a Portuguese longitudinal matched employer–employee dataset. The model is estimated with a
Bayesian Markov chain Monte Carlo (MCMC) estimation method. The results imply that unobserved firm
characteristics explain almost 40% of the systematic variation in log job durations. In addition, we find a
positive correlation between unobserved worker and firm characteristics.
KEY WORDS: Assortative matching; Dynamic models; Frailties; Gibbs sampling; Job transitions;
Matched employer–employee data.

1.

INTRODUCTION

The basic stylized facts regarding job durations are well established. For example, the survey by Farber (1999) provides

abundant evidence that in OECD (Organisation for Economic
Co-operation and Development) countries, long-term employment relationships are common, most new jobs end early, and
the probability of a job ending declines with tenure. Unobserved heterogeneity in the probabilities of job exit can largely
account for these stylized facts. If workers are heterogeneous in
terms of mobility propensities, then the observed job exit rate
at any point in time depends on the proportions of those types.
Higher-mobility workers experience several short spells, while
lower-mobility workers engage in fewer but longer employment
relationships. The fact that most new jobs end early is explained
by a sufficiently large proportion of high-mobility workers. The
fact that the probability of job ending is observed to decline with
tenure is then explained by dynamic selection on unobservable
characteristics of the workers.
Since job exit is a decision that involves both the worker
and the firm, it is plausible that exit rates are affected simultaneously by characteristics of workers and by characteristics
of firms. Whereas the relevance of worker heterogeneity in job
durations is well established (see, e.g., Farber 1999; Bellmann,
Bender, and Hornsteiner 2000; and Del Boca and Sauer 2009),
the empirical evidence on the importance of firm heterogeneity is much more limited. Abowd, Kramarz, and Roux (2006)

included unobserved worker and firm heterogeneity in a model
for wages and job mobility and concluded that there is a large
amount of heterogeneity among firms and their tenure profiles.
Frederiksen, Honoré, and Hu (2007) analyzed job durations in
a framework with unobserved firm heterogeneity only, and they
showed that the latter is an important determinant of the variation in job durations. Mumford and Smith (2004) estimated
linear models for tenure allowing for individual and workplace
effects. The results emphasize the importance of allowing for
workplace effects.
For a number of reasons, it is relevant to know the relative
contributions of worker and firm characteristics as determinants
of job durations. First, notice that inequality in society depends
on the variation in the characteristics of the jobs that employed
individuals have. If the variation in job durations is primarily
driven by worker characteristics, then the ensuing inequality
will be more persistent. Conversely, if the variation is primarily driven by firm characteristics, then the restructuring of the
market form in a sector can have large effects on inequality in
society. Second, the results of the analysis are of importance
for the econometric analysis of job durations. If unobserved
firm heterogeneity is important, then the inclusion of very large

468

© 2012 American Statistical Association
Journal of Business & Economic Statistics
July 2012, Vol. 30, No. 3
DOI: 10.1080/07350015.2012.698142

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

Horny, Mendes, and van den Berg: Job Durations With Worker- and Firm-Specific Effects

numbers of worker characteristics to a job duration model does
not remove the need to deal with unobserved heterogeneity.
Third, the results simply help in improving our understanding
of the determinants of job durations and job mobility. (In the following, we also discuss the relevance of assortative job matching
for the study.)
In this article, we estimate multivariate hazard models for
job exits (or, equivalently, job durations), allowing for workerspecific and firm-specific unobserved determinants. Workerspecific determinants encompass the propensity of the worker to
leave or lose a job, while firm-specific determinants can reflect

the firm’s preference to employ a stable workforce. Furthermore,
we allow the unobserved effects of matched firms and workers
to be correlated. To our knowledge, this is the first study that
allows for such a flexible modeling of job mobility. Obviously,
the econometric analysis requires observation of multiple job
spells per worker and/or multiple job spells per firm. A firm
is cross-sectionally and longitudinally connected to multiple
workers, whereas a worker is longitudinally connected to multiple firms. We use a matched employer–employee dataset in
which both workers and firms are longitudinally followed. The
data are from Portugal and are exhaustive for the private sector.
The estimates of the correlation between the worker-specific
and the firm-specific unobserved heterogeneity term are informative on the extent to which specific types of firms match with
specific types of workers. To see this, consider a firm where
the job durations are typically short. Is this only because of
high job exit rates at the firm, or is it also because the firm
attracts workers who have high job exit rates anyway, that is,
who would also have high job exit rates if employed at firms
where job spells are typically long? The former reflects firm
heterogeneity, whereas the latter leads to a positive correlation
estimate. Our article is therefore connected to the expanding

literature on assortative matching of workers and firms. Recent
advances in this literature focus on assortative matching in terms
of (worker-specific and firm-specific) productivity (see Lopes de
Melo (2008) and Mendes, van den Berg, and Lindeboom (2010),
for empirical analyses based on matched employer–employee
data). See also Abowd et al. (2004), Andrews et al. (2008), and
Woodcock (2008) for fixed-effects panel data analyses of linear
wage equations with fixed effects for workers as well as firms.
In the econometric analysis, we treat the unobserved heterogeneity terms as random effects, one at the firm level and
another at the worker level. This is in line with econometric duration analysis with unobserved heterogeneity (see van den Berg
2001). Due to right-censoring, fixed-effect panel data methods
are not feasible. More to the point, we are interested in the relative contributions of workers’ and firms’ characteristics in the
variation of job durations, and the estimation of the distribution
of the random effects enables such a decomposition.
The model structure is such that the unobserved worker and
firm effects are neither nested nor independent. Our model
thus relates to the literature on multiple nonnested random
effects. As long as the effects are mutually independent, crossed
random-effects models are extensions of nested random-effects
models, and they can be fitted using the same procedures.

However, with general dependence, the implementation is computationally more demanding since the data cannot be grouped
in independent blocks (Tibaldi 2004). Applications of crossed

469

random-effects models in a generalized linear model framework
include Raudenbush (1993), Van den Noortgate, De Boeck, and
Meulders (2003). Because of the complex nonlinear connections
between the unobserved individual-specific and firm-specific
effects and the observed outcomes, we estimate the models using a Bayesian Markov chain Monte Carlo (MCMC) approach,
based on the Gibbs Sampler and in line with Manda and Meyer
(2005). Robert and Casella (1999) provided a survey of MCMC.
Dependence between the worker and firm random effect in a
given job creates an additional complication. If the correlation
between worker and firm effects is sufficiently high, then this
can entail that the random effects of different workers at a given
firm are correlated, and/or that the random effects of different
firms employing a given worker over time are correlated. As
we will show, this is an implication of the required positive
semidefiniteness of the correlation matrix of the random terms

of firms and workers. A dependence across workers at a given
firm and across firms having employed a given worker implies
that many observed job spells of different workers and firms
are statistically dependent. Indeed, if many workers and firms
are connected, because firms have many employees and workers
move between many different firms, then most firms would have
jointly dependent random effects, and most workers would also
have jointly dependent random effects. Estimation of such a
model would involve an insurmountable computational burden.
In some particular cases, we consider restrictions on the correlation parameters of the worker and firm random effects that
ensure that the correlation matrix of the worker and firm random effects has the following set of properties: it is positive
semidefinite while at the same time the worker random effects
are iid across workers and the firm random effects are iid across
firms. We conclude that this holds true if the correlation between
the worker and firm random effects is restricted to an interval
strictly smaller than [−1, 1] and if firms are small (and if in
the observation window, workers are not highly mobile across
many firms). Notice that positive assortative matching may rule
out any mobility of workers across firms with widely different
productivities. In that case, the correlation may be large within

small labor market segments, whereas it is absent for connections across segments.
As we discuss in the following, the Portuguese labor market
displays low mobility and a high degree of stratification. Moreover, it has relatively many small firms. Hence, the empirical
analysis is unlikely to be affected in a major way by the above
complication. However, this does not imply that our modeling
approach is suitable for any type of labor market, particularly
if its workers make frequent job transitions and if many of its
firms are large. Positive-semidefiniteness of the correlation matrix may then be incompatible with iid worker random effects
and iid firm random effects.
We should point out that our article has a connection to some
recent studies in the literature on job seniority and wages. Specifically, the influential article by Dostie (2005) estimates a model
in which job durations and wages are simultaneously modeled
in a random-effects setting. His approach has been adopted by,
for example, Bagger (2007) and Battisti (2009). These studies
are primarily concerned with changes within job durations, and
less with the relative contribution of firm-specific and workerspecific determinants of job durations. Accordingly, the models

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

470

Journal of Business & Economic Statistics, July 2012

are richer than ours in terms of the role of wages, but they do
not allow for a firm-specific effect. The models do allow the job
exit hazard to depend on a worker-specific random effect and
a match-specific random effect, where the latter is drawn anew
for each separate match between a worker and a firm, but these
worker- and match-specific effects are assumed to be independent across workers in a firm and from each other. Note that in
this approach, the match-specific effect is effectively nested in
the likelihood contribution of the worker, and all this facilitates
the empirical analysis.
The article is organized in six sections. The Portuguese
matched employer–employee data are described in Section 2.
These data have been used before in a number of studies. See,
for example, Vieira, Cardoso, and Portela (2005); Cardoso and
Portela (2005); and Mendes, van den Berg, and Lindeboom
(2010) for descriptions and analyses of the data and for summaries of the Portuguese labor market. Section 3 presents the
duration models that we estimate. Here we also provide some
economic intuition for the existence of assortative matching in
job durations. In Section 4, we discuss the estimation method
and the choice of the prior distributions. The results are discussed in Section 5. Section 6 concludes.
2.

DATA

The study is based on Quadros de Pessoal, a longitudinal matched employer–employee dataset gathered by the Portuguese Ministry of Labor and Solidarity. The data are collected
through a report that all firms with registered employees are
legally obliged to provide every year. All workers employed by
a firm in the month of October are in the data. Coverage is low
for the agricultural sector and nonexistent for public administration and domestic services, whereas the manufacturing and
private services sectors are almost fully covered.
An identifier is assigned to every firm when it enters the
dataset for the first time, while the identifier of the worker is
a transformation of his social security number. Based on these
identifiers, one can match workers and firms, and follow both
over time to identify job spell durations. Unlike some labor
force surveys, the Quadros de Pessoal do not have details on
a worker’s labor market events between consecutive years, nor
on the exact timing a job exit took place. We can thus identify
transitions of workers between firms occurring in time intervals
of 1 year, but not the occurrence of other short spells (job, unemployment, or nonparticipation spells) within that time interval.
As of 1994, the Ministry has implemented elaborate checks
on the accuracy of the firm identification code. Therefore we
use the data covering the period 1994–2000. The dataset comprises nearly 4 million workers and 385,000 firms. We apply
a few conditions on workers, firms, and spells: (a) we discard
firms that leave, temporarily or permanently, the market; (b) we
exclude workers who are observed in a nonpaid job or in selfemployment at some point; (c) we exclude spells with entry
before 1994, to avoid initial condition problems; and (d) workers older than 55 are discarded. This results in what we call the
“full data” covering around 338,000 workers and 55,000 firms.
To reduce the computational burden of the estimation procedure, we draw a 3% sample of workers that is as close as
possible to the full data in terms of worker and firm observables

and job durations. In particular, within each group formed by
the combination of all values of the observables, a 3% random
sample is drawn, thus maintaining the proportion of each group.
Descriptives of the full data and the 3% sample are presented
in Appendices A and B. In both, about 91% of the workers
experience only one new job spell in the observation window;
about 8% experience two spells, and about 1% experience three
spells.
In our model of job transitions, we use the following observed
characteristics of the worker: age, gender, and education. Age
may capture life-cycle effects, as “job shopping” tends to take
place mainly at an early age, when the worker is not aware of
his own abilities or of the characteristics of the labor market
(Johnson 1978). Age is grouped into the following categories:
16–25, 26–35, and 36–55 years old. Education is stratified into
primary school, lower secondary, upper secondary, and higher
education. We also observe whether the job is a part-time job
or not, and the corresponding dummy variable is included as
firms facing negative demand shocks may tend to first terminate
part-time jobs to minimize the loss of specific human capital.
Control variables at the firm level include industry, region, and
an indicator for multiple plants. Descriptives of firms’ and workers’ characteristics are presented in Appendix B. Finally, we use
calendar time as an additional explanatory variable.
3.

MODEL

3.1 Discrete-Time Job Duration Model
Since we only observe job entry and job exit on an annual basis, we specify the job duration models in discrete time. Specifically, we use the time-aggregated mixed proportional hazard
(MPH) model for the discrete-time hazard function or conditional job exit probability.
At any point in time, an existing firm has one or more workers, and these workers change between firms when they move to
another job. A firm is thus cross-sectionally and longitudinally
connected to multiple workers, whereas a worker is longitudinally connected to multiple firms. This means that there is no
hierarchical structure between workers and firms.
We denote by i = 1, . . . , I the firm index and by j =
1, . . . , J the worker index. We observe all existing worker–firm
combinations at one point in time for every calender year in the
observation window. As a result, job durations are realized in
discrete time k = 1, . . . , K.
The hazard function corresponding to individual j in firm i
at elapsed job duration k is the conditional probability of a job
termination at this duration k. It depends on (1) the elapsed duration k, on (2) worker- and firm-specific observed explanatory
variables xkij that are potentially time varying, on (3) unobserved
characteristics of firm i, and on (4) unobserved characteristics
of the worker j. We summarize (3) and (4) by their hazard effects vi and wj , respectively. The firm effect vi is invariant
across job spells at the firm. It may arise for several reasons, including cross-firm variation in labor contracts, tenure profiles,
workforce management policies, and product market conditions
or monopsony power. The worker effect wj is invariant across
different jobs occupied by the worker. As noted in the introduction, the existence of such an effect has been demonstrated in

Horny, Mendes, and van den Berg: Job Durations With Worker- and Firm-Specific Effects

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

many empirical studies. Both effects are time-invariant. In line
with survival analysis or biostatistics, they may be referred to
as “unobserved heterogeneity” terms or “frailties.”
In the empirical analysis, we estimate a range of model specifications. The simplest model accounts for observed heterogeneity only. The second specification introduces a worker-specific
effect in this simplest model. The third specification allows for
worker- and firm-specific effects that are independently distributed of each other. Finally, the most general specification
allows the two effects to be correlated within a pair of a worker
and a firm.
Following Kalbfleisch and Prentice (1980), the discrete-time
MPH model without unobserved heterogeneity is defined as







(1)
λ k|xkij , β0k , β1 = 1 − exp − exp β0k + xkij β1 ,
With worker-specific unobserved heterogeneity, we obtain







λ k|xkij , β0k , β1 , wj = 1 − exp − exp β0k + xkij β1 + wj .

(2)

A discrete-time model with two frailties is defined as






λ k|xikj , β0k , β1 , vi , wj = 1 − exp − exp β0k + xkij β1

+ vi + wj .
(3)

Let us denote by λij k the value of the hazard function at k. The
likelihood contributions of job spell durations are based on the
probability (density) of a possibly right-censored duration conditional on the observed and unobserved explanatory variables.
With censoring at k, this equals
k

(1 − λij s ),

(4)

s=1

whereas in case of a spell completion at duration k,
λij k

k−1

s=1

(1 − λij s ).

(5)

The “full likelihood” is the product of the above expressions for
all observed job spells of the workers and firms in the sample.
Notice that this full likelihood is defined as a function of the
parameter vectors β0 , β1 and the unobserved worker and firm
effects.
The vector of unobserved worker and firm effects has a distribution in the population. As we sample workers upon inflow into
jobs, and we follow workers over time after that, we effectively
take this inflow of workers to be the underlying population.
Hence, the sample constitutes a random sample of workers from
this population, and it is appropriate to argue that the worker
effects wj are marginally iid. If we regard firms as infinitely
exchangeable, then by De Finetti’s theorem, the firm effects vi
can be modeled as iid conditional on a latent variable. This does
not automatically imply that they are marginally iid. To proceed,
we make the further assumption that the firm effects are iid as
well. In addition, for the remainder of this section, we assume
that the firm effects vi are independent of the worker effects wj .
In Section 3.3, we incorporate dependence between wj and vi .
Notice that the “random effects assumption” states that the
frailties are independent of the observed explanatory variables

471

in the population. This is clearly restrictive. Decisions of individuals and firms may be reflected by the values of the frailties,
and these decisions may depend on the relevant values of x.
However, contrary to linear models and some nonlinear models
for panel data, the identification of duration models with unobserved heterogeneity typically warrants an assumption that x is
independent of unobserved heterogeneity. A detailed survey of
the identification of such models was provided by van den Berg
(2001). In case the data contain multiple spells sharing the same
unobserved determinants, the independence assumption can be
discarded. However, in our case, this would amount to observing
multiple spells of the same worker in the same firm, which is
very unlikely to occur. It is conceivable that such multiple-spell
data requirements are unnecessarily strong to be able to relax the
assumption that x is independent of the frailties. We address this
issue in a rather modest way, by reestimating the model with
samples in which multiple spells per worker and/or per firm
are over-represented. Given our limited observation window,
this entails an over-representation of short spells. Admittedly,
it is also conceivable that allowing for dependence between the
worker frailty and the firm frailty will make it more difficult to
relax the assumption that both frailties are independent of x.
Clearly, the identifiability of duration models with nonnested
dependent frailties is an important topic for further research.
Notice that we implicitly assumed that for a given worker,
the firm effects in consecutive jobs are drawn from an identical distribution. Economic models of search on the job predict
that workers move to better jobs as they become older. In that
case, the job exit rate decreases with each successive job. This
suggests that the firm effects are systematically decreasing with
each successive job. We do not allow for this because it complicates the analysis dramatically. First, the ensuing joint distribution of job durations would need to satisfy specific features (see,
e.g., van den Berg and Van Vuuren 2003). Second, inference on
the correlation between worker and firm effects would be more
difficult (see Section 3.3). Third, we would face an unsolvable
initial conditions problem, as our data do not provide job spells
dating back to the labor market entry of the workers. The assumptions that we make instead are in line with the literature in
which matched employer–employee data are analyzed to study
wage determinants. Typically, it is assumed that a job-to-job decision consists of two decisions that are made sequentially: a job
separation and a job acceptance, where the characteristics of the
new job are unrelated to those of the previous job, conditional on
the observed and fixed unobserved characteristics of the worker.
This keeps the empirical inference manageable, and, at least, it
constitutes a transparent condition for model identification.
3.2

Behavioral Explanations for Correlated Worker and
Firm Effects

In the most extensive version of the empirical model, we
allow the worker-specific effect wj on the job duration and
the firm-specific effect vi on the job duration to be correlated.
A nonzero correlation between wj and vi in jobs reveals that
assortative matching takes place between workers and firms.
Broadly speaking, there are two sets of possible explanations
for this. First, the assortative matching is driven by an inherent
preference for high or low job durations in both workers and

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

472

firms. Second, job stability is correlated to another variable that
triggers assortative matching. In the latter case, this may be a
more fundamental job characteristic, or it may be some other
omitted variable. We discuss these possible explanations in some
more detail.
Consider the first explanation. It is possible that job stability
as such is deemed valuable by both workers and firms, or at least
by a fraction of them. The job matching process may then lead to
an alignment of preferences. For example, some firms may have
a stable product market and may value long-run relationships
between sales employees and customers. It is also possible that
some jobs warrant high job-specific human capital investments.
At the same time, some workers may have a stronger preference
for stable jobs than others. For example, some workers may be
more risk averse than others. In these cases, workers and firms
with similar preferences may sort themselves into matches with
each other.
An example of the second explanation is inspired by the literature mentioned in Section 1 on assortative matching between
workers and firms. This literature focuses on the matching of
high-productivity workers with high-productivity firms. Such
matches are usually in their joint interest if the production function is supermodular. In a perfectly competitive labor market,
the equilibrium results in perfect sorting in terms of productivity, but this does not provide a reason for why the job durations
would be sorted as well. Now suppose that it takes a certain
amount of time for workers and firms to learn about each others’ productivity after the job has started. Then one would expect
that the matches where both the workers’ and firms’ productivities are relatively high will lead to stable jobs, whereas other
matches are sometimes insufficiently attractive for at least one
participant to be sustainable. Suppose, in addition, that idiosyncratic job separation rates are decreasing in the productivity of
the worker and/or firm. Then what we observe in terms of job
durations is that the workers and firms that have low job exit
rates often form matches. In sum, the worker and firm effects
wj and vi are positively correlated.
Another example of the second explanation relies on a more
mechanical omitted-variable argument. Suppose that each value
of the vector of observed explanatory variables x captures a separate labor market. It may be that within each of these markets
there are submarkets (e.g., local regions) with their own degree
of labor market volatility. In that case, workers with low job exit
rates are often employed at firms with low job exit rates, simply
because the job duration is a proxy for unobserved local market
characteristics.
3.3 Modeling of Correlated Worker and Firm Effects
We may capture the dependence between vi and wj by the
correlation matrix associated with the vector of all vi and wj
in the population. Recall that the vi are independent across
firms and the wj are independent across workers. In principle
one may consider flexible correlation patterns between worker
and firm effects, for example, by allowing some worker effects
to be more strongly correlated with a certain firm effect than
other worker effects. However, it is clear that parsimony is important to prevent the number of parameters from exploding.
Smith and Kohn (2002) and Liu, Daniels, and Marcus (2009)

Journal of Business & Economic Statistics, July 2012

developed parsimonious Bayesian approaches to estimate the
covariance matrix generated by multivariate Gaussian longitudinal processes. However, these studies do not examine matrices of dimension over (30 × 30), and it is unclear whether such
larger dimensions can be handled by these methods. Raknerud
(2009) considered a linear wage equation with worker-specific
and firm-specific random effects that may be correlated with
each other. He developed an estimation method based on a state
space representation, under the assumption that terms of order
equal to the square of the correlation coefficient can be ignored.
This method seems only applicable in a linear-equation setting.
To proceed, we assume equicorrelation, that is, corr(vi , wj ) =
corr(vi ′ , wj ′ ) = ρ, ∀(i, j ) = (i ′ , j ′ ). This reduces inference on
the correlation matrix to inference on a single parameter ρ.
Note that an n × n matrix R is a correlation matrix of n random
variables if and only if the following three conditions are satisfied: the matrix is symmetric, its diagonal elements are equal
to one, and the matrix is positive semidefinite. The latter property imposes restrictions on the range of the possible values of
the elements of the matrix. In the case of the equicorrelation
assumption, this amounts to a restriction on the range of values
of ρ. In particular, the condition of positive-semidefiniteness
rules out that one random variable is strongly correlated to other
random variables that are themselves uncorrelated to each other.
As an example, consider one firm with n workers (or, equivalently, one worker with n jobs). For n = 2, the candidate correlation matrix of the effects (v1 , w1 , w2 ) is defined as
⎛
⎞
1 ρ ρ
(6)
R = ⎝ρ 1 0 ⎠.
ρ 0 1
If v1 is strongly correlated with w1 and also with w2 , then
this is not compatible with w1 and w2 being independent. It is
not difficult to show that for a general number of employees n
of the firm, positive-semidefiniteness is equivalent to
1
|ρ| ≤ √ .
n

(7)

These bounds do not depend on the functional form of the
joint distribution. Clearly, the assumption of mutually independent worker effects and mutually independent firm effects is
not compatible with a strong correlation between worker and
firm effects. However, this does not mean that any model with
ρ = 0 is misspecified. Consider a labor market where firms are
small and where workers are not mobile across different firms
in the course of their lifetime. In that case, the above restriction
may be satisfied. Similarly, if the labor market is segmented, for
example, due to positive assortative matching across segments,
and if the segments are small, then the restriction is weaker than
may be suggested by (7). With two equally large segments, the
correlation matrix (v1 , w1 , w2 , v2 , w3 , w4 ) can be expressed as



R 0
,
(8)
0 R
with
√ R as defined in (6). This is positive-semidefinite iff |ρ| ≤
1/ 2, which is the same condition as for (6), although the
number of workers and firms is now twice as large. In general,

Horny, Mendes, and van den Berg: Job Durations With Worker- and Firm-Specific Effects

with m equally large segments, inequality (7) is replaced by

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

|ρ| ≤ √

1
.
n/m

(9)

To avoid confusion, notice that the parameter ρ does not give
the overall correlation between worker and firm effects but the
within-segment correlation. For segments with small firms and
low-mobility workers, the boundary values of ρ are relatively
far away from zero.
With complex longitudinal patterns of worker-firm combinations, it is difficult to derive precise bounds for admissible values
of ρ, and therefore more difficult to assess how likely it is that
the model is misspecified. Notice that correlations of observed
outcomes across workers and firms may also be explained by
common observed determinants that are included as explanatory
variables in the model. In the end, it is an empirical question
whether, within a market, firms are sufficiently small and worker
mobility is sufficiently low, for the assumption of mutually independent worker effects and mutually independent firm effects to
be compatible with certain nonzero values of ρ. Cabral (2007),
in his study on the importance of small firms in Portugal, stated
that 86% of firms have fewer than 20 employees. In our own
data, this number is around 90%, and around 60% of the firms
have four employees or less. Of course, firms themselves may be
segmented in different parts that have no (or almost no) common
determinants of job spell durations, on top of those captured by
the observed covariates. Mendes, van den Berg, and Lindeboom
(2010) included a brief overview of the Portuguese labor market. They concluded that in the years covered by our data, it
has been much more stable than in other countries. Institutional
features and barriers make mobility costly and tend to hamper
worker mobility across jobs. In this sense, the Portuguese labor
market may be better suited for our modeling approach than the
labor markets in other countries.
We therefore include estimation results for the model with ρ
as an additional parameter. This approach can be further justified by the low numbers of job spells per worker in the data as
described in Section 2. Recall that the observation window per
worker consists of at most 7 years, and for 99.9% of all workers,
we observe less than four job spells. Obviously, with a limited
number of spells per worker and per firm, models with nonzero
ρ may not violate the positive-semidefiniteness of the correlation matrix for the firms and workers in the data, under the
maintained assumption of mutually independent worker effects
and mutually independent firm effects.
There is an apparent contradiction between, on the one hand,
the requirement that firms are sufficiently small and worker mobility is sufficiently low for the model with correlated worker
and firm effects to be coherent, and, on the other hand, the usefulness of a large firm size and worker mobility across firms for
the model to be identifiable. Concerning the latter aspect, notice that duration models with one-dimensional random effects
are identified in the absence of multiple spells per unit. This
compares favorably to regression models with one-dimensional
random effects and a single observation per unit. However, with
single-spell data, inference results are not robust to parametric
functional form assumptions in the model (see van den Berg
2001, for an overview). This can be remedied if the observed
explanatory variables are time-varying. Our data contain annual

473

observations of the explanatory variables, so they may vary over
time, and our models incorporate this. Recall also that although
the fraction of workers with multiple observed spells is small,
the size of the dataset is considerable.
If the resulting estimate of ρ is large in absolute value, then
this may indicate misspecification in the sense that the assumption of mutually independent worker effects and mutually independent firm effects is violated. This assumption is shared by
many empirical studies in which job duration models are estimated. In this sense, evidence for a large value of ρ may be of
importance for empirical studies of job durations with worker
data and the empirical analysis of job turnover with firm data,
as these types of analyses assume independent outcomes across
workers and firms, respectively. Obviously, since we also estimate models that impose ρ = 0, we can examine whether the
fit of the latter model is significantly worse.
Notice that in principle, the above concerns can be avoided
by estimating a model in which the assumptions of mutually
independent worker effects and mutually independent firm effects are relaxed. This means relaxing the almost universally
made econometric assumption that random effects are independently distributed across subjects. As noted in the beginning of
this section, in the most extreme case, if we do not make any
assumption on the correlation matrix of the joint distribution
of the random effects, the sample data simply provide a single
joint observation of a large number of correlated spells, and the
number of parameters in the correlation matrix exceeds the sample size. The more modest the relaxation of the assumption of
mutually independent worker effects and mutually independent
firm effects, the more parsimonious the model. As we have seen,
the development and statistical inference of models that are less
parsimonious than ours is still in its infancy. The computational
burden is substantial. Even with simulation methods one has to
draw from a joint normal distribution with a dimension equal
to the sum of the number of workers and the number of firms,
with a structured variance matrix of which each element has
to satisfy a number of constraints increasing with the matrix
dimension to ensure its positiveness. In a sensitivity analysis
in Section 5.4, we report some results based on a model that
allows worker effects to have a fixed correlation across workers
and firm effects to have a fixed correlation across firms. This
increases the range of admissible values of ρ. For example,
with one firm and three workers, the positive-semidefiniteness
√
of the correlation matrix implies that ρ ≤ (1 + 2α)/3, with
α > −1/2 being the correlation between worker fixed effects.
This is larger than the upper bound in Equation (7) for n = 3
for any α > 0. In another sensitivity analysis, we estimate the
model with samples that restrict the number of workers per firm.
A potentially promising alternative approach could be to define
clusters of workers and firms and to use fixed-effects estimation methods, see, for example, Sargent (1998) for a univariate
analysis.
With the above models, there are a number of reasons to
perform inference with Bayesian MCMC estimation methods.
First, these methods do not require large-sample asymptotics.
Second, these methods do not require numerical integration
over multidimensional random effects. Nevertheless, we know
that Bayesian approaches for nested random-effects Cox proportional hazard models are computationally intensive. With

474

Journal of Business & Economic Statistics, July 2012

massive data, the extension to nonnested effects is not trivial
(Tibaldi 2004); Horny et al. 2008). In the next section, we discuss our inferential approach in detail.

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

4.

BAYESIAN INFERENCE

Concerning the functional form for the unobserved heterogeneity distribution, it is useful to have a family of distributions
where the correlation or covariance of the dependence between
the worker and firm random effects is a separate parameter that
can attain positive as well as negative values. We adopt a normal
distribution for the random effects. We normalize the mean to
zero, and we denote the variances by σv2 for vi and by σw2 for wj .
For two values vi , wj in a newly formed job, this means that






vi
σv2 ρσv σw
∼ N 0,
.
wj
ρσv σw σw2
Recall that we estimate four model specifications. The simplest model accounts for observed heterogeneity only, so σv =
σw = 0. The second specification only imposes σv = 0. The
third specification allows for nonzero σv and σw = 0 but imposes ρ = 0. Finally, the most general specification does not
impose constraints on the parameters of the bivariate normal
distribution. The effect of the misspecification of the functional
form of the unobserved heterogeneity distribution has been investigated in detail for discrete duration models. Nicoletti and
Rondinelli (2010) showed that an incorrect normality assumption for the unobserved heterogeneity distribution does not bias
the baseline hazard and the covariate parameter estimates.
The Bayesian approach augments the assumed model with the
prior beliefs on the parameters. We choose proper but uninformative priors, allowing unlikely values of the parameters to not
necessarily have a zero posterior probability. Assigning a zero
prior probability to some parameter values would lead to a zero
posterior probability, even if the likelihood reaches its maximum
at these values. Manda and Meyer (2005) specified a baseline
hazard with steps related through a first-order autocorrelated
process, and Grilli (2005) used a polynomial specification. Because the time period under study consists of six time intervals,
we specify a piecewise constant baseline hazard with unrelated
coefficients β0k . (We also estimated models using a polynomial
specification in preliminary analysis and were led to a 6 degrees
polynomial.) The coefficients β1 associated with the x variables
are given independent Gaussian priors with means 0 and very
large variances 1000.
To complete the specification of the posterior distribution, we
must specify a prior distribution for the parameters of the unobserved heterogeneity distribution. The precision of each random
effect (i.e., σv−2 and σw−2 ) follows a gamma distribution, that is,
the conjugate prior of the Gaussian distribution, and we base our
prior elicitation on descriptive statistics. The annual transition
probability per worker is about 3.5% for the 5th percentile of
the duration distribution and 0.9% for the 95th percentile. So for
90% of the population, there is at most a four-fold variation between the odds of two workers. The corresponding confidence
interval on the transition probability is thus of width 3, which
suggests σw = 0.5. We set our prior for the precision σw−2 to
a gamma distribution with expectation 2 and variance 4. Similarly, the transition probability per firm is in a range from 1.3%

to 4% for 90% of the population, suggesting a gamma prior with
expectation 3 and variance 9 for σv−2 . Finally, for ρ, a noninformative prior, that is, a uniform distribution, is specified over its
range.
Our model is a Bayesian hierarchical model. We first specified
the duration model, then the prior distribution, and finally the
distributions on the parameters of the priors. Our posterior is not
a common distribution, even with all priors being independent.
However, we can construct a Markov chain with elements following the posterior distribution, and approximate the Bayesian
estimator using a Monte Carlo method. Notice that such inference is straightforward if the vi and wj were observed. We use
Gibbs sampling (Gelfand and Smith 1990), a data augmentation algorithm, to simulate unobserved heterogeneity terms for
all firms and workers. These are used to sample the parameters
from the posterior density. In the Technical Appendix, we explain the algorithm in detail. Notice that the approach places
distributional constraints on the vi and wj through their second
moments. As σv and σw approach zero, vi and wj get more concentrated around 0, indicating homogeneous firms and workers.
Conversely, large variances indicate heterogeneous firms and
workers. The model thus adjusts to the amount of unobserved
heterogeneity in the sample.
5.

RESULTS

5.1 Unobserved Heterogeneity
The estimates of the unobserved heterogeneity distributions
for the three model specifications as defined in Section 4 are
reported in Table 1. The estimation results of parameters shared
by multiple model specifications are quite similar across the
three specifications, meaning that increasing the unobserved
heterogeneity complexity does not strongly affect the estimates
of those parameters. In the three specifications, σw is estimated
around 0.3 and is significant with posterior standard deviations
around 0.04. The estimates of σv range from 0.76 to 0.87, with
posterior standard deviations around 0.07. These results provide
strong evidence that both variances are effectively nonzero, with
the effects of firms’ unobserved characteristics being more dispersed than those of workers’ unobservables.
The correlation between the worker and firm effects is estimated to be positive and is highly significant at conventional
levels. Notice that the estimated value of 0.51 is reasonably high
Table 1. Estimates of the standard deviations and correlation of the
unobserved heterogeneity distributions
Type of heterogeneity

Parameter

Mean

2.5%

97.5%

Correlated frailties
Firm effect
Worker effect
Correlation

σv
σw
ρ

0.76
0.30
0.51

0.64
0.23
0.34

0.88
0.39
0.58

Independent frailties
Firm effect
Worker effect

σv
σw

0.87
0.26

0.76
0.20

0.98
0.33

Single frailty
Worker effect

σw

0.30

0.21

0.42

Horny, Mendes, and van den Berg: Job Durations With Worker- and Firm-Specific Effects

475

Table 2. Estimates of the β coefficients
Random effects
Variable

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:46 11 January 2016

Elapsed tenure
2 years
3 years
4 years
5 years and more
Worker’s characteristics
Female
Age:
16– 25
26 – 35
Education:
Primary school
Lower secondary
Part-time
Firm’s characteristics
Multiple plants
Region:
Center
Lisbon and Tagus Valley
Alentejo, Algarve and Islands
Sector:
Construction
Trade
Transports
Financial
Constant
Log-likelihood
Number of workers
Number of firms

None

Worker

Independent

Correlated

−0.60
−0.96
−1.36
−2.14

−0.58
−0.93
−1.33
−2.09

−0.27
−0.50
−0.80
−1.50

−0.40
−0.68
−1.00
−1.74

−0.25

−0.26

−0.34

−0.31

0.59
0.32

0.59
0.32

0.76
0.41

0.69
0.37

0.33
0.33
0.56

0.33
0.33
0.57

0.21
0.25
0.66

0.17
0.20
0.62

0.19

0.20

0.31

0.27

0.12
0.36
0.18

0.13
0.37
0.19

0.19
0.45
0.30

0.19
0.41
0.28

0.38
0.33
0.05
0.67
−2.48

0.39
0.34
0.05
0.68
−2.53

0.42
0.34
0.32
0.71
−3.08

0.37
0.30
0.32
0.60
−2.78

−7700
9222
6582

−7570
9222
6582

−6135
9222
6582

−6640
9222
6582

NOTES: Coefficients in bold are significant at 5% level. Omitted categories: elapsed tenure is 1 year, male, age 36–55, upper secondary or university education, full-time job, firm only
has one plant, northern region, mining, manufacturing, electricity/gas/water sectors. Year dummies are included but not reported since they are all insignificant at the 5% level.

in the light of the discussion in Section 3.3. The value does provide strong evidence for positive assortative matching in terms
of job exit determinants, but at the same time, it may suggest that
the maintained assumption of firm-specific (worker-specific) effects being independent across firms (workers) may be violated.
Robustness checks are provided in Section 5.4.
5.2 Observed Heterogeneity
The posterior means for the β coefficients together with information regarding their significance are reported in Table 2.
Negative duration dependence is found to be significant, with the
probability of separation declining monotonically with tenure
(i.e., with the elapsed duration). This suggests that the empirically observed inverse relationship between separation rates and
job tenure would not be fully explicable by pure heterogeneity
models that do not account for duration dependence. Notice that
the magnitude of the tenure effect reduces once we allow for
firm-specific random effects.
Regarding the controlled worker characteristics, we find that
women tend to move less. This result contradicts the findings of
many previous studies of job mobility. The main reason could
be the fact that the gender difference in terms of mobility rates is

changing over time. Light and Ureta (1992) found that women
belonging to early U.S. birth cohorts appeared to be more mobile
than men but this conclusion is reversed when more recent
cohorts are considered.
The results for age a