Full text of Working Papers (Federal Reserve Bank of Richmond) : Estimating Matching Efficiency with Variable Search Effort, Working Paper 16-13r

View original document

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Estimating Matching Efficiency with
Variable Search Effort

WP 16-13R

Andreas Hornstein
Federal Reserve Bank of Richmond
Marianna Kudlyak
Federal Reserve Bank of San
Francisco

Estimating Matching Eﬃciency with Variable Search
Eﬀort∗
Andreas Hornstein Marianna Kudlyak
FRB Richmond

FRB San Francisco

This Draft: December 6, 2016
First Draft: May 21, 2015
Working Paper No. 16-13 (revised)

Abstract

We introduce a simple representation of endogenous search eﬀort into the
standard matching function with job-seeker heterogeneity. Using the estimated
augmented matching function, we study the sources of changes in the average
employment transition rate. In the standard matching function, the contribution
of matching eﬃciency is decreasing in the matching function elasticity. In contrast, for our matching function with variable search eﬀort and small matching
elasticity, search eﬀort is procyclical, accounting for most of the transition rate
volatility; and the decline of the aggregate matching eﬃciency accounts for a
small part of the decline in the transition rate after 2007. For a large matching
elasticity, search eﬀort is countercyclical, and large movements in matching eﬃciency compensate for that; and the decline in the matching eﬃciency accounts
for a large part of the decline in the transition rate after 2007. The data on
employment transition rates provide evidence for endogenous search eﬀort but
do not separately identify cyclicality of search eﬀort and matching elasticity.
Key Words: Matching eﬃciency. Search eﬀort. Matching elasticity. Aggregate matching function. JEL Codes: E24, J63, J64.
∗

Email: andreas.hornstein@rich.frb.org, marianna.kudlyak@sf.frb.org. The authors thank Yongsung
Chang, Mark Watson and Eran Yashiv for helpful comments, and the participants at the 2016 NBER SI
Macro Perspectives group, European Economic Association Meetings in Geneve, Penn State, Erasmus University Rotterdam, Tinbergen Insitute, UC Santa Cruz, UC San Diego, UC Davis, UT Austin, U.S. Census,
the Fall 2015 Midwest Macro Meetings, University of Rochester, the 2015 Kiel-FRB Richmond conference
at CREI Barcelona, the 2015 FRB Richmond-UVA Jamboree at UVA, the 20th ENSAI Economics day "On
Labor Markets Instability," and the 2015 Macro Montreal "New Developments in Business Cycle Analysis."
The views expressed here are those of the authors and do not reﬂect those of the FRB Richmond, FRB San
Francisco or the Federal Reserve System.

Introduction

In Diamond-Mortensen-Pissarides matching models of the labor market, the matching function plays a role similar to the aggregate production function in macroeconomic models of
the goods market.1 The matching function is a reduced form representation of how in a
frictional labor market the combination of workers who are looking for employment and
vacant positions that need to be ﬁlled–the inputs to the matching function–results in
new matched workers and positions–the output of the matching function. In the standard
matching function, inputs are homogeneous and there is no utilization variation of inputs.
During and after the 2007-09 recession, the ability of the standard matching function to
account for the decline in the average employment transition rate through changes in its
inputs has deteriorated. Changes in the number of new matches that cannot be accounted
for by changes in inputs are attributed to the residual - matching eﬃciency. Similarly to the
measurement of total factor productivity in the growth accounting literature, measurement
of the matching eﬃciency depends, however, on the measurement of inputs.
There are substantial diﬀerences in the employment transition rates of diﬀerent groups of
job seekers (by duration of nonemployment, reason of nonemployment, etc.), and the 2007-09
recession and its aftermath are associated with large changes in the composition of the search
pool toward groups with typically low employment transition rates. Motivated by these
observations, we augment the standard matching function by allowing for heterogeneity in
search eﬀectiveness of diﬀerent groups of job seekers, which through the lens of the matching
function is a multiplicative shifter of the job seeker’s rate of ﬁnding a job. In contrast to
the existing literature, we not only allow for exogenous but also for endogenous changes
of eﬀectiveness over time.2 For this augmented matching function, the average employment
transition rate depends on the composition of the job seeker pool and job search eﬀectiveness,
in addition to the relative number of job seekers and vacancies, and matching eﬃciency.
We model search eﬀort of an observable type of job seekers as a constant elasticity function
of the aggregate matching rate, which itself is a function of the total job seeker input in the
economy. We allow for diﬀerences in the search eﬀort elasticity across types to account
for cyclical changes in relative transition rates. Our model is motivated by the standard
1

Petrongolo and Pissarides (2001) in their survey of the matching function provide an in-depth description
of the role the matching function plays in macroeconomics.
2
Existing approaches are limited to the analysis of composition eﬀects arising from exogenous heterogeneity. For example, Veracierto (2011), Barnichon and Figura (2016), Sahin Song, Topa, and Violante
¸
(2012), Elsby, Michaels, and Ratner (2016), Kroft, Lange, Notowidigdo and Katz (2016), and Hall and
Schulhofer-Wohl (2015).

search and matching model (Pissarides, 2000); however, in the estimation we do not restrict
search eﬀort to be procyclical as implied by the standard model. For this speciﬁcation of
endogenous search eﬀort we show, ﬁrst, that from the data on employment transition rates
the group-speciﬁc search eﬀort elasticities and matching function elasticity are not separately
identiﬁed.3 That is, the data are consistent with low matching function elasticity with respect
to vacancies () and procyclical search eﬀort as well as high  and countercyclical search
eﬀort. Second, the aggregate matching eﬃciency is identiﬁed up to a positive scalar, 1.
Third, the matching elasticity from the standard matching function, which ignores variable
search eﬀort, is identiﬁed but is not equal to the underlying ‘true’ matching elasticity.
Our model of variable search eﬀort thus deﬁnes the observed group-speciﬁc employment
transition rates as nonlinear functions of the observables (market tightness and the composition of search pool) and the unobserved state (aggregate matching eﬃciency and groupspeciﬁc exogenous eﬀects, i.e., group-speciﬁc matching eﬃciencies). We use an extended
Kalman ﬁlter to estimate the parameters and infer the unobserved state. In our benchmark
model, we estimate a ﬂexible random walk process for the aggregate matching eﬃciency
and constant group-speciﬁc exogenous eﬀects. We then extend the framework and allow for
stochastic time-varying group-speciﬁc exogenous eﬀects.
We apply our estimation procedure to three alternative groupings of the pool of nonemployed individuals: unemployed job seekers by duration of unemployment, unemployed
job seekers by reason of unemployment, and nonemployed job seekers by labor force status (unemployed and out of the labor force) and gender.4 The groupings are motivated by
an attempt to maximize the possible impact of ‘unobserved quality change’ on the average
employment transition rate: we want to have a decomposition of the search pool such that
groups have large and persistent diﬀerences in transition rates, and we want to see large
compositional changes of the search pool.
First, we ﬁnd that the decline of the identiﬁed aggregate matching eﬃciency in the Great
Recession was large, even after controlling for composition eﬀects and variable search eﬀort.
This conclusion is robust to alternative sample periods, measures of vacancies, or structural
breaks in the parameters. Second, the data reject the hypothesis under which our model is
isomorphic to a model with constant search eﬀort, that is, the data provide evidence in favor
of group-speciﬁc variable search eﬀort. We ﬁnd that a model with group-speciﬁc variable
3

Throughout the paper, we use terms "type" and "group" interchangeably.
In the estimation, we focus on the nonemployed job seekers. However, the framework can be easily
extended to take into account employed job seekers.
4

search and only one exogenous shock — aggregate matching eﬃciency (as opposed to a number
of type-speciﬁc exogenous shocks) — is suﬃcient to capture both comovement and relative
changes in group-speciﬁc transition rates. Little is gained from adding time-varying groupspeciﬁc exogenous eﬀects. If we instead shut down the group-speciﬁc variable search eﬀort,
as is typical in the literature, the data call for large movements in exogenous group-speciﬁc
eﬃciencies to capture relative movements in the group-speciﬁc transition rates.
Finally, using the augmented matching function, we account for the relative contributions of labor market tightness, aggregate matching eﬃciency, and variable search eﬀort to
changes in the average employment transition rate contingent on the elasticity of the matching function. As we just noted, in our framework the cyclicality of search eﬀort and the
magnitude of changes in matching eﬃciency depend on the value of the matching elasticity.
For small values of the matching elasticity, search eﬀort is procyclical and accounts for most
of the procyclical transition rate. In particular, the large decline of the average transition
rate in the Great Recession is mainly attributed to reduced search eﬀort, and the matching
eﬃciency does not decline much. On the other hand, when the matching elasticity is large,
search eﬀort is strongly countercyclical and large changes in matching eﬃciency are required
to maintain a procyclical transition rate. This is the opposite of what one obtains for the
standard matching function approach with constant search eﬀort for which a larger matching
elasticity implies a smaller contribution coming from the aggregate matching eﬃciency.
Independent evidence on the cyclicality of search eﬀort is limited and mixed. Shimer
(2004), Mukoyama, Patterson, and Sahin (2014), and Faberman and Kudlyak (2014) provide evidence for countercyclical search eﬀort, whereas DeLoach and Kurt (2013) argue for
acyclical search eﬀort, and Gomme and Lkhagvasuren (2015) argue for procyclical eﬀort.
Our reading of the literature is that it favors the presence of countercyclical search eﬀort.
The evidence for endogenous and countercyclical search eﬀort poses a hurdle for the standard
search and matching model, both in terms of its technical capacity to accommodate countercyclical search eﬀort and in terms of its ability to generate ﬂuctuations of the employment
transition rate of empirical magnitudes.
To our knowledge, ours is the ﬁrst study that allows for endogenous search eﬀort in the
matching function accounting framework. A number of papers have studied how accounting
for search heterogeneity might aﬀect estimates of matching eﬃciency.5 Almost all of these
5

Like all of this literature, we focus on the measurement of search eﬀectiveness for job seekers, taking the
recruiting eﬀectiveness for reported vacancies as given. In seminal work, Davis, Faberman and Haltiwanger
(2013) study recruiting eﬀectiveness for vacancies.

studies assume, as we do, that after correcting for quality diﬀerences across types, the types
remain perfect substitutes in the matching function. Davis (2011) and Kroft, Lange, Notowidigdo, and Katz (2016) study heterogeneity among the unemployed deﬁned by duration
of unemployment, and they construct correction factors for search eﬀort that recover the
eﬀective input of workers to the matching function. Sahin, Song, Topa, and Violante (2014)
show how the potential misallocation of unemployed workers across disaggregated labor markets aﬀects measured matching eﬃciency in the reduced form aggregate matching function.
Barnichon and Figura (2015) build on these contributions and study a decomposition of the
unemployed by type and market segment and allow for diﬀerences in ﬁxed type-speciﬁc efﬁciencies, but no aggregate matching eﬃciency. They argue that relative to a homogeneous
search pool model, composition eﬀects account for most of the observed decline in matching
eﬃciency. Veracierto (2011) extends the deﬁnition of the search pool beyond the unemployed to include those out of the labor force (OLF) and allows for time-varying relative
type-speciﬁc eﬃciencies. Sedlacek (2014) uses techniques similar to ours to study the role
of time-varying relative type-speciﬁc eﬃciencies when the search pool includes unemployed,
OLF, and employed. For the same broadly deﬁned search pool as in Sedlacek (2014) but
more demographic detail, Hall and Schulhofer-Wohl (2016) study the role of type-speciﬁc
matching eﬃciencies when diﬀerent observed search types remain imperfect substitutes in
the matching function.6 On the theory side, the cyclicality of search eﬀort is potentially
relevant for the ampliﬁcation of the volatility of vacancies and unemployment in search and
matching models (for example, Costain and Reiter (2008) or Gomme and Lkhagvasuren
(2015)).
The remainder of the paper is structured as follows. Section 2 describes a model of the
matching function with heterogeneity and endogenous search eﬀort and derives identiﬁcation results. Section 3 describes the estimation procedure. Section 4 describes the data
and basic facts that motivate variable search eﬀort. Section 5 presents our estimation results on the identiﬁed aggregate matching eﬃciency with variable search eﬀort. Section 6
presents the decomposition of the average employment transition rate conditional on proand countercyclical search eﬀort. Section 7 concludes.
6

Similar to type-speciﬁc variable search eﬀort in our model, Hall and Schulhofer-Wohl’s imperfect substitutability across types captures cyclical changes in relative transition rates. Compared with our approach
there are two conceptual diﬀerences to Hall and Schulhofer-Wohl. First, unlike our approach there is no
obvious interpretation for what imperfect substitutability across types means. Second, Hall and SchulhoferWohl exclude the possibility of changes in aggregate matching eﬃciency and limit themselves to changes in
type-speciﬁc matching eﬃciencies.

Matching with endogenous search eﬀort

The aggregate search and matching function in macro-labor models describes the ‘production’
of hires as a function of the stocks of job seekers and vacancies, and an exogenous shift
term denoting the aggregate eﬃciency of the matching process. The standard approach
for the search and matching function assumes that the inputs are homogenous and that
search eﬀort does not vary endogenously with the state of the labor market. We augment
the standard search and matching function by allowing for time-varying heterogeneity in
search eﬀectiveness across observed groups of job seekers. Search eﬀectiveness may vary
for exogenous reasons or it may change in response to the aggregate matching rate. We
refer to the endogenous component of group-speciﬁc search eﬀectiveness as search eﬀort and
introduce a simple model in which search eﬀort is a function of the aggregate matching rate.

2.1

A model of heterogeneous search eﬀectiveness

In this section, we describe a simple extension of the aggregate matching function approach
that allows for heterogeneity in search eﬀectiveness of job seekers.
Consider an economy with a ﬁnite number of search types,  ∈ . Time is continuous.

At any point in time,  job seekers of type  engage in search and the types diﬀer in their
search eﬀectiveness,  , to be described in more detail below. Total eﬀective search input,
P
∗ ≡    , and vacancies,  , are inputs to a Cobb-Douglas matching function that

generates hires,  ,


 = exp ( )  (∗ )1− 


(1)

for a given aggregate matching eﬃciency,  , and matching elasticity, 0    1. In the
standard matching function, the search types are homogeneous,  = 1, and aggregate
P
search input is simply the sum of all job seekers,  ≡   .
The aggregate matching rate per search unit,  ≡  ∗ , is

 = exp ( )  − 
 ¯
¯
where  ≡   is the standard aggregate labor market tightness, and  ≡

(2)
P



   is

average search eﬀectiveness, with   =   being the share of type  in the search pool.
The transition rate to employment for a type  searcher is then the product of the type-

speciﬁc search eﬀectiveness and the per search unit matching rate,  =   . The average

P
¯
aggregate transition rate,  ≡     , is

¯
¯
 = exp ( ) 1−  


(3)

For the analysis of the average transition rate, the standard matching function approach
ignores changes in average search eﬀectiveness, whether due to changes in the composition
of the search pool or due to variations in type-speciﬁc search eﬀectiveness, and sets  = 1.
¯
In other words, with heterogeneous search eﬀectiveness, the standard approach conﬂates
changes in matching eﬃciency with changes in composition and search eﬀectiveness.

2.2

Modeling endogenous search eﬀectiveness

Let the type-speciﬁc search eﬀectiveness  be the product of a type-speciﬁc exogenous
component (exogenous type-speciﬁc matching eﬃciency),  , and a type-speciﬁc endogenous
component (search eﬀort),  ,
 =  exp ( ) 

(4)

We model type ’s search eﬀort as a constant elasticity function of the aggregate matching
rate,


 =   

(5)

where  is the type-speciﬁc elasticity of search eﬀort with respect to the aggregate matching
rate.
Our approach is motivated by the basic search and matching model, e.g. Pissarides (2000)
or Gomme and Lkhagvasuren (2015), for which the employment transition rate of searchers
is the product of search eﬀort and the aggregate matching rate per unit of search eﬀort, and
searchers face an increasing and convex cost from search eﬀort. For this model, search eﬀort
increases if the expected gains from search increase, in particular, if the aggregate matching
rate increases.7 In our set-up, the search elasticity  reﬂects the endogenous response of
search eﬀort to changes in the aggregate matching rate, assuming that more eﬀort makes a
searcher more eﬀective.
Combining equations (4) and (5) yields the following expression for type-speciﬁc search
eﬀectiveness:
ln  =  +  ln  
7

(6)

See Appendix A for the ﬁrst order condition from the basic search and matching model with endogenous
search eﬀort.

This speciﬁcation nests two special cases of time-varying type-speciﬁc search eﬀectiveness.
When  is constant, all time-variation is due to the type-speciﬁc search eﬀort. When   = 0,
all time-variation is due to the variation in the type-speciﬁc exogenous component.
We will call search eﬀort procyclical if   0 and countercyclical if    0. We do not
restrict search eﬀort to be procyclical as implied by the basic search and matching model, but
we impose a lower bound on the search elasticity,   ≥ −1, such that a type’s employment

transition rate is always procyclical,

ln  =  + (1 +  ) ln  for  ∈ .

(7)

Our model of heterogenous search eﬀectiveness thus deﬁnes observed employment transition
rates of groups as a function of the unobserved aggregate matching rate and exogenous groupspeciﬁc eﬀects, that may be ﬁxed or time-varying. Combining equations (2) and (7) then
yields the aggregate matching rate as a nonlinear function of the search pool composition,
aggregate matching eﬃciency, and exogenous type-speciﬁc eﬀects
ln  =  +  ln  −  ln

 exp ( +   ln  ) 

(8)



o
o
n
n


is observable,  =   ( )
de  ( )=1  (  )=1
=1
o
n
are parameters. Our goal is to estimate the
scribes the unobserved state, and  ( )
=1

In equations (7) and (8),  =

parameters and infer the unobserved state conditional on observable variables.

2.3

Identiﬁcation

We ﬁrst show that, conditional on using only observations on group-speciﬁc transition rates,
the parameters of our model are not uniquely identiﬁed. In particular, the sign of the search
eﬀort elasticity, that is, the cyclicality of search eﬀort, is not identiﬁed.
Proposition 1 Conditional on observations { }, the matching elasticity and search eﬀort
elasticities are identiﬁed only up to the restriction


(1 +  ) =  ≥ 0 for  ∈ 
1−

(9)

and the aggregate matching eﬃciency and matching rate are identiﬁed up to the restriction
 ≡
ˆ


1−
ˆ
and ln  ≡
ln  .


8

(10)

We obtain this proposition by applying the deﬁnition of  from (9) together with the
ˆ
transformation of variables  and  , to our model for employment transition rates (7) and
ˆ
(8) rewriting it as follows
b
ln  =  +  ln  

b
ln  =  + ln  − ln
b

X


´
³
b
 exp  +  ln  

(11)
(12)

³
´
ˆ ˆ
From these equations one can see that given observable  and a solution        , any

other solution (        ) that satisﬁes the constraints (9) and (10) is observationally
equivalent. Note that  is eﬀectively the identiﬁed transition elasticity in (11). It then
follows from equation (9) that for each type there exists a critical value of the matching
elasticity that depends on the type’s identiﬁed transition elasticity such that search eﬀort is
procyclical (countercyclical) if the matching elasticity is below (above) the critical value. In
other words, working with observations on group-speciﬁc transition rates only, the observations are consistent with either pro- or countercyclical search eﬀort.
Finally note that if the identiﬁed transition elasticities are the same for all types,  = 
∀ ∈ , that is,


(1 +  ) =  ∀ ∈ 
1−

(13)

our model is equivalent to one with common search eﬀort across types (  = ). Then by
setting  = 0 and  = 1 (1 + ) in equation (9), it encompasses a special case of constant
search eﬀort. Alternatively, if the identiﬁed transition elasticities diﬀer across types, we can
argue for the presence of endogenous search eﬀort. It is an empirical question whether the
identiﬁed transition elasticities diﬀer across types, and we address this issue in the estimation
section.
We get a better understanding of the model’s inability to identify the cyclicality of search
eﬀort if we ignore heterogeneity and consider the case of a representative searcher. For this
case, the closed form solution of the aggregate matching rate,  , and the employment
transition rate, 1 , in terms of the underlying parameters, matching elasticity and search
eﬀort elasticity, is

(ln  +  ) and
ˆ
1 + 1
1 + 1
= 
(ln  +  )
ˆ
1 +  1
| {z }

ln  =
ln 1

≡ˆ


where  is the eﬀective matching elasticity that relates observed transition rates to market
ˆ
tightness.8 Let us start with a matching function elasticity  and constant search eﬀort,
 = 0. Then a 1 percentage point increase of market tightness results in an  ppt increase
of the matching rate  and the transition rate 1 . Suppose that search eﬀort is procyclical,
1  0, then the higher matching rate leads to an increase in search eﬀort, 1 , and a decline
in eﬀective tightness,  1 , equation (2). Thus the matching rate increases less than 
ppts. The direct eﬀect of increased search eﬀort on the transition rate, however, more than
compensates for the decline in the matching rate, and the transition rate increases by more
than  ppts. The converse holds when search eﬀort is countercyclical, 1  0. In this case, a
1 ppt increase of market tightness results in a more than  ppt increase in the matching rate
because search eﬀort declines, and the eﬀective market tightness increases by more than 1
ppt. Again, the direct eﬀect of reduced search eﬀort more than compensates for the increase
in the matching rate, and the transition rate increases by less than  ppts. Thus the eﬀective
matching elasticity is increasing in the matching elasticity and the search eﬀort elasticity.
Any given eﬀective matching elasticity  can then be accounted for by the same matching
ˆ
elasticity,  = , and a-cyclical search eﬀort, 1 = 0, or by a larger (smaller) matching
ˆ
elasticity,    (  ) with a countercyclical (procyclical) search eﬀort, 1  0 ( 1  0).
ˆ
ˆ
Even though the matching elasticity and search eﬀort elasticities are not separately identiﬁed, for a log-linear approximation of the average employment transition rate as a function
of market tightness, the coeﬃcient on market tightness is uniquely determined.
Proposition 2 The ´
log-linear approximation of the average employment transition rate at
³
ˆ ˆ
a point 0  0  0   0 is
¯
∆ ln  =  (∆ ln  + ∆ˆ ) + (1 − ) ∆¯ + ∆¯, with
¯

¯ 

¯

 =
¯
¯
1+

(14)
(15)

¯
and , ∆¯, and ∆¯ are weighted averages of type-speciﬁc elasticities, eﬃciencies and mea

surement errors.
Proof. See Appendix.
The expression for the log-linear approximation of the average employment transition
rate is analogous to the standard homogeneous search and matching model, but with the
8

Here we ignore exogenous type-speciﬁc eﬀects since there is only one type.

reduced form matching elasticity  being a function of the weighted average of the identiﬁed
¯
transition elasticities  . Therefore, the ‘reduced form’ matching function elasticity  is
¯
identiﬁed but is not equal to .9
Finally, we note that the overall level of the aggregate matching eﬃciency and the typespeciﬁc eﬀects is not identiﬁed. Using equations (11) and (12). it is straightforward to show
the following proposition.10
o
n
is identiﬁed up to an additive shift
Proposition 3 The level of the state  =   ( )
ˆ
=1

term.

When we estimate our model we therefore normalize one of the states.

Estimation

Our model for the evolution of the type-speciﬁc employment transition rates has a straightforward state-space representation, albeit with a nonlinear measurement equation. Conditional
on the parameters, we are using an extended Kalman-ﬁlter approach to infer the state of the
system from observations on the type transition rates. We obtain parameter estimates by
maximizing the likelihood function.
Given the unobserved state  = (ˆ   { } ), equation (11) for the measured type

transition rate  with measurement noise  ,


ˆ
ln  =  +  ln  +  for  = 1  


(16)

with  ∼  (0 Σ ), and equation (12) for the identiﬁed matching rate deﬁne the measurement equations of the state-space model.

We use three diﬀerent speciﬁcations for the evolution of the unobserved state  . The
baseline speciﬁcation has time-varying aggregate matching eﬃciency only, and the two other
alternatives use either time-varying type-speciﬁc eﬃciencies only or aggregate and timevarying type-speciﬁc eﬃciencies jointly.
9

In the standard matching function approach that ignores heterogeneity and endogenous search eﬀort, the
matching function elasticity () is typically estimated from the relationship between the average transition
¯
rate and labor market tightness, i.e., ln  =  +  ln  , where  denotes the estimated aggregate matching
eﬃciency from the standard matching function approach.
10
The proof is in Appendix C.

Model 1. Time-varying aggregate matching eﬃciency and ﬁxed type-speciﬁc matching
eﬃciencies,
¡
¢
 = b−1 +   ,   ∼  0 2
b



 =  ∀ ∈ 

We use a random walk to capture any trend in the aggregate matching eﬃciency and also any
potentially substantial drop of the matching eﬃciency after 2007. We allow for permanent
diﬀerences across types through ﬁxed eﬀects, and time-varying diﬀerences across types are
captured through diﬀerences in the identiﬁed transition elasticities. In view of proposition 3
we normalize 1 = 0.
Model 2. Time-varying type-speciﬁc matching eﬃciencies and ﬁxed aggregate matching
eﬃciency,
 = 0
b

´
³
 = −1 +   ,   ∼  0  2 ∀ ∈ 

Similar to Model 1, we allow for ﬂexible random walk processes for type-speciﬁc matching
eﬃciencies. We normalize  = 0.
b

Model 3. Time-varying aggregate matching eﬃciency and time-varying type-speciﬁc match-

ing eﬃciencies,
¡
¢
 = −1 +   ,   ∼  0  2 
b
b


 − 

´
³
2
=   (−1 −  ) +   ,   ∼  0  ∀ ∈ 

In Model 3, we specify the aggregate matching eﬃciency as a random walk and type-speciﬁc
matching eﬃciencies as stationary (1)-processes. We normalize 1 = 0. We choose this
speciﬁcation for two reasons. First, anticipating our estimation results for Model 2, we see
substantial comovement of type-speciﬁc eﬃciencies at low frequencies, which suggests a common trend. We identify this common trend with the aggregate matching eﬃciency. Second,
with this speciﬁcation the stochastic processes for  and { } are separately identiﬁed to a
b

ﬁrst order approximation.11
11

Even though in Model 3 we have only  observations on type employment transition rates, but  + 1

Data and basic facts

We now construct alternative measures of the search pool using data from the Current Population Survey (CPS) and characterize these measures from the perspective of our matching
framework with heterogeneity and variable search eﬀort.

4.1

Deﬁnition of job seeker groups

In the matching function framework, we attribute diﬀerences in transition rates across job
seekers to diﬀerences in search eﬀectiveness. Therefore, we want to decompose the pool of job
seekers into observable groups that are characterized by large and persistent diﬀerences in
transition rates to employment. Speciﬁcally, we consider three alternative characterizations
of the pool of job seekers using the CPS.
The ﬁrst two characterizations deﬁne the search pool narrowly as the unemployed, that
is, those reporting to be actively engaged in search. We consider two decompositions of the
unemployed, by duration and by reason of unemployment. The ﬁrst classiﬁcation groups
the unemployed into those that report unemployment of less than 5 weeks, 5-26 weeks, or
more than 26 weeks. The second classiﬁcation consists of the four groups that report being
unemployed because they are on temporary layoﬀ, on permanent layoﬀ, have quit a job, or
have previously been out of the labor force.
The third characterization of the search pool includes all nonemployed, that is the unemployed and those that are out of the labor force (OLF). Although those that are OLF
do not report to be actively engaged in search, in any month a substantial number of them
do make the transition to employment. Thus a clear cut distinction between those that
are unemployed and those that are OLF may not be appropriate for a matching framework
that expressly allows for diﬀerences in search eﬀectiveness.12 We decompose this broader
deﬁnition of the search pool into four groups of nonemployed job seekers characterized by
their labor market status (unemployed or OLF) and gender (male or female).
unobserved states at any point in time, for the linear Kalman-ﬁlter the state is identiﬁed, if the state-space
system is ‘observable.’ See, for example, Harvey (1989). Formally, the -dimensional state of a system is
said to be observable, if the last  past observations are suﬃcient to determine the entire history of states
for the last  periods. If the measurement equations for our model were linear, we could easily verify that
the observability condition is satisﬁed for at least one type-speciﬁc eﬃciency being stationary. Since our
measurement equations are nonlinear, we are using an extended Kalman-ﬁlter and no general conditions for
identiﬁcation are available. In our estimation we check that the observability condition is satisﬁed for the
extended Kalman-ﬁlter along the inferred path of the state.
12
For these reasons Veracierto (2011), Barnichon and Figura (2015), and Hall and Schulhofer-Wohl (2015)
have all used a broader concept of the search pool in the matching function.

For our characterizations of the search pool we take a job seeker’s membership in a group
at the beginning of a period as given.13 We are agnostic of whether the groups within each
decomposition reﬂect some ex-ante (inherent) heterogeneity among job seekers in terms of
their transition rates, or whether diﬀerent transition rates are a result of being/becoming a
member of a particular group.

4.2

Data sources and construction of the series

For each deﬁnition of the search pool we construct the employment transition rates and the
search pool shares of the diﬀerent groups of job seekers.
We construct the employment transition rates for the diﬀerent groups of nonemployed job
seekers using the micro data from the Current Population Survey (CPS) basic monthly ﬁles,
from January 1976 to December 2015. We follow Madrian and Lefgren (1999) and Shimer
(2012) and match individuals from month to month using information on race, age and sex
besides individual and household’s identiﬁcation number. In the analysis, we weight each
individual by the average of the individual’s CPS sampling weights from adjacent months.
The transition probability of a group is the fraction of individuals that transition between
labor market states in two adjacent months.14 We transform the month-to-month transition
probabilities to monthly continuous time transition rates. For the search pool deﬁnitions
that cover the unemployed we use the exit probabilities to employment (E) and OLF (I), and
assume that job seekers who exit do not return to unemployment in the same month. This
deﬁnes a relation between the discrete time transition probabilities  and the continuous time
employment transition rates  , which we can solve for the transition rate from unemployment
to employment
 = −

log (1 −   −  )

1 +   

When the exit probability to OLF is small relative to the exit probability to employment,
the employment transition rate is approximately − log (1 −   ). For most of our samples

this is not a good approximation. We therefore prefer to use information on exit rates to all
states when calculating the employment transition rate. For the search pool deﬁnition that
covers all nonemployed, we use the exit probabilities of unemployed (OLF) to employment
13

The same approach is taken in other recent work that accounts for heterogeneity in matching eﬃciency,
for example, Barnichon and Figura (2015) or Hall and Schulhofer-Wohl (2015).
14
In the analysis, we follow the BLS approach and treat the reported labor force status as a true status.
Frazis, Robinson, Evans, and Duﬀ (2005) describes that the main reason for why the BLS does not correct
responses for a potential error is a lack of methodology or the data that would guide the correction.

and OLF (unemployment). Otherwise we proceed the same way to calculate the monthly
employment transition rates.15
We employ two alternative aggregate vacancy series: (1) the Help Wanted Index (HWI)
from the Conference Board, which is available since 1951, and (2) the Job Openings and
Labor Turnover Survey (JOLTS) program of the BLS, which is available starting in January
2001.16
For the baseline analysis, we use data from the CPS and the HWI covering the period
from January 1994 to December 2015. We start the baseline sample in 1994 because of the
structural breaks in the search pool shares and transition rates associated with the 1994
CPS redesign. We use the adjustment factors from Polivka and Miller (1998) to adjust the
search pool share series prior to 1994. Since no comparable adjustment factors are available
for our constructed exit probabilities, we introduce an additive adjustment factor in our
measurement equations for the group-speciﬁc employment transition rates prior to 1994.
All monthly series are seasonally adjusted using the Watson (1996) implementation of the
X-11 procedure. Since the monthly series remain highly volatile, even after this adjustment,
we estimate our models on quarterly averages of the seasonally adjusted monthly data.

4.3

Basic facts

Accounting for heterogeneity in the matching function framework matters for the measurement of matching eﬃciency if there are large and persistent diﬀerences in employment transition rates across diﬀerent groups of job seekers and large and systematic changes in the
composition of the search pool over time. We now show that this is indeed the case for our
two decompositions of the pool of unemployed. In particular, in recessions the search pool
composition shifts toward those with relatively low employment transition rates, that is,
the average search eﬀectiveness declines. Not accounting for this change in average eﬀectiveness is likely to bias one’s estimate of the aggregate matching eﬃciency downward. For
our broader deﬁnition of the search pool, which includes unemployed and OLF, this kind of
cyclical bias is not as pronounced.
First, consider the decomposition of the search pool of unemployed by duration of unemployment. Large diﬀerences in transition rates among the unemployed by duration are
immediately apparent: short-term unemployed are three times as likely to transition to em15

See Appendix D for the derivations.
Given the shift in job advertising from print media to web-based means the HWI may not be consistent
over time. Barnichon (2010) corrects for structural changes in the HWI series and we use his adjusted series.
16

ployment than long-term unemployed (Figure 1, Panel A.1). Furthermore, the diﬀerences in
the transition rates among these groups persist over time, keeping the ranking of transition
rates unchanged. But even though there is substantial comovement among the transition
rates, the relative transition rates do change (Figure 1, Panel A.2). For example, in the
2007-09 recession the transition rates of the medium- and long-term unemployed decline
more than those of the short-term unemployed. We will attribute changes in relative transition rates to time-varying diﬀerences in search eﬀectiveness. Finally, the composition of the
pool of the unemployed by duration changes systematically and substantially over time: the
share of long-term unemployed increases in recessions, e.g., following the 2007-09 recession
the share of long-term unemployed more than doubled (Figure 1, Panel A.3).
The alternative decomposition of the search pool of unemployed by reason has the same
features as the decomposition by duration (Figure 1.B). Relative to those who give a temporary layoﬀ as the reason for unemployment, the unemployed on permanent layoﬀ tend to
be half as likely to ﬁnd employment, they suﬀer a relatively larger decline of transition rates
in recessions, and they make up a larger share of the search pool in recessions.17
For the broader deﬁnition of the search pool that includes all of the nonemployed workingage population the pattern that emerges for average search eﬀectiveness is more ambiguous
(Figure 1.C). The unemployed are relatively more eﬀective at search than those who are OLF,
their employment transition rates are about ﬁve to 10 times higher than the OLF transition
rates. But the transition rates of the unemployed are also much more cyclical such that the
relative transition rates of those who are OLF are increasing in recessions. At the same time,
the search pool share of those who are OLF tends to decline in recessions. The cyclical bias
of composition changes for this search pool deﬁnition is therefore not obvious.

Matching eﬃciency with variable search eﬀort

In this section, we present estimates of the identiﬁed transition elasticities ( ) and the
identiﬁed aggregate matching eﬃciency ( ). We turn to the interpretation of these results
b

conditional on the aggregate matching elasticity  in the next section.

We start with estimates of the baseline model with type-speciﬁc variable search eﬀort and

time-varying aggregate matching eﬃciency only (Model 1) for the search pool decomposition
by unemployment duration. For this case, we ﬁnd evidence for an unprecedented decline of
17

The sharp jump of the transition rate in 1994 for those who report being on temporary layoﬀ likely
reﬂects the structural break associated with the CPS redesign.

the identiﬁed matching eﬃciency following the Great Recession and the presence of variable
search eﬀort. For a restricted version of the baseline model with constant search eﬀort, that
is, a model with a composition eﬀect only, we ﬁnd a smaller decline in identiﬁed matching
eﬃciency. The results are robust with respect to unobserved heterogeneity in search eﬀort
and the use of the HWI index for vacancies.
We then compare the results from this baseline model with the two alternative models
that allow for time-varying type-speciﬁc matching eﬃciencies (Models 2 and 3) and ﬁnd that
relative to these alternatives the baseline model with variable search eﬀort captures changes
in relative transition rates well. Furthermore, the identiﬁed aggregate matching eﬃciency
in the baseline model appears to capture a common trend in the time-varying type-speciﬁc
matching eﬃciencies in Model 2.
We then shut down the time-varying search eﬀort channel and force the model to capture
time-varying heterogeneity via type-speciﬁc exogenous shocks. We ﬁnd that the restricted
model requires substantial volatility in type-speciﬁc exogenous shocks to capture relative
movements in type-speciﬁc transition rates.
Finally, we show that estimates of the baseline model for our two alternative search
pool deﬁnitions, unemployment by reason and labor force status, yield qualitatively similar
paths for the identiﬁed matching eﬃciency. In particular, we ﬁnd comparable declines of the
identiﬁed aggregate matching eﬃciency following the Great Recession. For ease of exposition,
in this section we will frequently drop the qualiﬁer ‘identiﬁed’ for the transition elasticities
and aggregate matching elasticity when no confusion can arise.

5.1

Aggregate matching eﬃciency with variable search eﬀort

The baseline model with endogenous search eﬀort and aggregate matching eﬃciency captures
well the comovement of type-speciﬁc employment transition rates and changes in relative
transition rates.
5.1.1

Transition elasticities and matching eﬃciency in a baseline model

We begin with the baseline model with aggregate matching eﬃciency when heterogeneity in
the search pool of unemployed workers is deﬁned by duration of unemployment, Model 1 of
Section 3. For diﬀerent sub-samples and speciﬁcations, Table 1 displays the parameter estimates of the model and Figure 2 displays the smoothed posterior of the identiﬁed aggregate
matching eﬃciency.

Identiﬁed transition elasticities The baseline speciﬁcation covers the years 1994-2015
and uses the HWI for the vacancy measure, Table 1, Column (1). The type-speciﬁc ﬁxed
eﬀects,  , decline with unemployment duration, capturing the persistently lower employment
transition rates of the unemployed with longer durations.18
The identiﬁed transition elasticities,  , are monotonically increasing with the duration
of unemployment, that is, the employment transition rates of those unemployed with longer
durations are more cyclically sensitive than the transition rates of those with shorter durations.
The estimates of the type-speciﬁc transition elasticities are suﬃciently precise to reject
the hypothesis that they are the same. As noted in the discussion of Proposition 1, if the
identiﬁed transition elasticities are the same for all types, then the model is consistent with
constant search eﬀort for a particular choice of the aggregate matching elasticity. Since the
transition elasticities are signiﬁcantly diﬀerent from each other, we reject the hypothesis of
no variable search eﬀort.
Identiﬁed aggregate matching eﬃciency For the baseline speciﬁcation, the estimate
of the identiﬁed aggregate matching eﬃciency declined dramatically following the Great Recession, and it has only partially recovered over the last years; the solid black line (1) in
Figure 2. Since the aggregate matching eﬃciency is identiﬁed only up to a positive scalar,
namely the matching elasticity, we will defer discussion of the quantitative magnitude of
changes in the matching eﬃciency to the next section. The behavior of the identiﬁed matching eﬃciency is, however, informative about the magnitude of movements in the aggregate
matching eﬃciency during the Great Recession relative to the overall sample period.
To gain more perspective on the magnitude of the decline in the matching eﬃciency, we
re-estimate the model for the extended sample period 1976-2015. As noted in section 4.2,
we allow for a structural break in the employment transition rates prior to 1994 to account
for the 1994 CPS data revision. For the years after 1994, the estimated aggregate matching
eﬃciency for the extended sample and the baseline sample follow each other closely, lines
(1a) and (2a) in Figure 2. In particular the matching eﬃciency declines dramatically in
the Great Recession, and that decline remains exceptional even for the extended sample
period. The estimated transition elasticities are somewhat lower in absolute value for the
18
The ﬁxed eﬀect for short duration unemployment is normalized at one. For the Kalman-ﬁlter, we take
the initial prior of the identiﬁed aggregate matching eﬃciency as given, and this prior becomes a parameter
of the likelihood function reported in the ﬁrst row of Table 1.

extended sample period, but the diﬀerences between them remain signiﬁcant and the cyclical
sensitivity continues to increase with the duration of unemployment, columns (1a) and (2a)
in Table 1.
5.1.2

Robustness

Constant search eﬀort We evaluate the contribution of endogenous search eﬀort by
comparing our baseline model with a restricted version of the model where the transition
elasticities are the same for all types. As we have argued in the discussion of Proposition
1 in Section 2.3, imposing this restriction makes the model equivalent to one with constant
search eﬀort for a particular value of the matching elasticity.19 The restricted model can then
be reinterpreted as having ﬁxed type-speciﬁc search eﬃciencies, i.e., only ﬁxed heterogeneity
similar to Barnichon and Figura (2015).
The results of estimating the restricted model are presented in Table 1, column (1), and
the aggregate matching eﬃciency is the black o-line (1) in Figure 2. The restricted model has
no time-varying heterogeneity, only type-speciﬁc ﬁxed eﬀects, and the restricted transition
elasticity  is an average of the unrestricted type-speciﬁc transition elasticities. A priori
it is not obvious whether the restriction of constant search eﬀort will increase or decrease
estimated matching eﬃciency. It turns out that the estimated aggregate matching eﬃciency
of the restricted model adjusts to compensate for the model’s lack of ability to accommodate
relative movements in type-speciﬁc transition rates. Speciﬁcally, to compensate for the
decline in relative transition rates of long duration unemployment after 2007, the restricted
model requires a somewhat higher aggregate matching eﬃciency than the unrestricted model
with type-speciﬁc time-varying search eﬀort, Figure 2, lines (1) and (1). In other words,
allowing for variable search eﬀort makes the estimated matching eﬃciency somewhat more
volatile relative to a constant search eﬀort restriction.
Given the estimated transition elasticity from the restricted model, we infer that search
eﬀort is constant if the matching elasticity is  = 038. Thus, our estimate of the identiﬁed
aggregate matching eﬃciency  from the restricted model with ﬁxed heterogeneity, line (1)
b

in Figure 2, is equivalent to the scaled aggregate matching eﬃciency   from the standard
matching function that allows for ﬁxed heterogeneity and no time-varying changes in relative
search eﬃciencies with  = 038.
For this same matching elasticity, we can construct the aggregate matching eﬃciency
19

In Appendix E we provide a more detailed illustration of this case.

using the standard assumption of a homogeneous search pool using the expression for the
average transition rate, equation (3) with a constant match quality. The scaled matching
eﬃciency from this homogeneous case,  , is displayed as line (1) in Figure 2. It is
comparable to our identiﬁed matching eﬃciencies. As we have discussed previously in Section
4.3, for the decomposition of the unemployed search pool by duration of unemployment,
the composition of the search pool shifts toward groups with relatively lower employment
transition rates when unemployment is high and the average transition rate is low, that is,
average search quality declines. Moving from lines (1) to (1) demonstrates how accounting
for compositional changes alone reduces the volatility of the estimated aggregate matching
eﬃciency, that is, line (1) shows the estimate of the aggregate matching eﬃciency purged
of this procyclical composition eﬀect.
Unobserved heterogeneity If one believes that changes in the duration distribution
of unemployment mainly reﬂect unobserved heterogeneity within the three groups of the
unemployed by duration, then it is possible that the unobserved composition of our search
pool groups is systematically changing over time with aggregate labor market conditions.20
The time-varying employment transition rates of a group may therefore reﬂect not only the
actual variation of the transition rate of the job seekers in the group, but also the unobserved
compositional shifts within the group. Such time-varying unobserved heterogeneity within
our search pool groups would be reﬂected in the time-varying endogenous as well as exogenous
components of group-speciﬁc matching eﬃciencies.21
If unobserved heterogeneity is quantitatively important, the identiﬁed transition elasticities,  , should vary systematically with the aggregate state of the economy. To check
for this possibility, we allow for a structural break in the identiﬁed transition elasticities
that depends on the level of the unemployment rate. In particular, the identiﬁed transition
elasticities are allowed to be diﬀerent for periods when the unemployment rate exceeds the
average of the minimum and maximum unemployment rate. For this exercise we use the
extended sample period starting in 1974 and, to allow for the possibility of low frequency
changes in the unemployment rate, we deﬁne separate minima and maxima for the subsamples 1974-1987, 1987-1997, 1997-2005, and 2005-2012. Column (2) of Table 1 shows
the estimates from this speciﬁcation. The estimated breaks for the transition elasticities
20
For recent contributions see Hornstein (2012), Ahn and Hamilton (2015), and Alvarez, Borovickova, and
Shimer (2016).
21
See Appendix F for an illustration.

are quite small. Furthermore, allowing for these breaks does not much aﬀect the estimates
for the aggregate matching eﬃciency, the dashed-x red line (2) in Figure 2. Below we will
estimate the model with aggregate and time-varying type-speciﬁc matching eﬃciencies, and
the latter will reﬂect possible time-varying within-group heterogeneity.
JOLTS vacancies Finally, we consider how an alternative measure for vacancies, namely
the vacancy posting series from JOLTS, aﬀects our estimate of matching eﬃciency. Since
JOLTS is available only from 2001 we re-estimate our model with the HWI vacancy measure
for the shorter sample period, line (3) in Figure 2, and also estimate our model with the
JOLTS vacancy measure, line (3) in Figure 2. The decline in matching eﬃciency starting in
2008 is comparable for the two vacancy measures, but the JOLTS-based measure does not
replicate the pre-2008 decline that is quite prominent for the HWI measure. The estimates
for the identiﬁed transition elasticities for the shorter sample using either the HWI or JOLTS
vacancy measure are quite close to their respective estimates for the baseline sample, but
with larger standard errors, Columns (1a) and (3a), respectively, of Table 1.

5.2

Alternative models of matching eﬃciency

In the benchmark model the comovement of the type-speciﬁc employment transition rates is
captured by the aggregate matching eﬃciency, and relative changes of type-speciﬁc transition
rates are captured by diﬀerential responses of search eﬀort to the aggregate transition rate
while exogenous diﬀerences in type-speciﬁc eﬃciencies remain ﬁxed. In this section, we
present results for models that also allow for time-varying type-speciﬁc matching eﬃciencies,
Models 2 and 3 of Section 3. A comparison across the models is informative regarding
the extent to which type-speciﬁc variable search eﬀort with a common aggregate shock
(as opposed to a number of type-speciﬁc exogenous shocks) is suﬃcient to capture both
comovement and relative changes in type-speciﬁc transition rates.
In Table 2.A and Figure 3 we display the results from estimating Models 1, 2, and 3 for
the sample period from 1994-2015 when the search pool of unemployed is diﬀerentiated by
unemployment duration.22 The top panel of Figure 3 displays the smoothed posterior for
the aggregate matching eﬃciency of Models 1 and 3, and the lower three panels display the
smoothed posteriors for the type-speciﬁc matching eﬃciencies of Models 2 and 3.
22

The results for Model 1 represent speciﬁcation (2a) in Table 1 and Figure 2. Hornstein and Kudlyak
(2016) contains a complete listing of the results for Models 2 and 3 similar to Table 1 and Figure 2 for Model
1.

For the model with time-varying type-speciﬁc matching eﬃciencies and ﬁxed aggregate
matching eﬃciency, Model 2, we ﬁnd that the transition elasticities are more similar across
groups than in the baseline Model 1 with ﬁxed type-speciﬁc eﬀects, Table 2.A, Columns
(1) and (2). This means that Model 2 captures changes in relative type-speciﬁc transition
rates not so much through diﬀerences in the types’ responsiveness to the aggregate transition
rate, but through diﬀerential changes in type-speciﬁc matching eﬃciencies. For example, the
matching eﬃciency of the long-duration unemployment group is relatively more volatile and
displays a larger decline in the post-2008 period than the matching eﬃciencies for the other
two groups, lower three panels of Figure 3. Nevertheless, there is a substantial comovement between the type-speciﬁc matching eﬃciencies of the three groups, which suggests the
presence of a common component.
Comparing the model with time-varying aggregate and type-speciﬁc matching eﬃciencies,
Model 3, to the baseline Model 1, we ﬁnd only small diﬀerences for the estimated transition
elasticities and the estimated time paths for aggregate matching eﬃciency, Columns (1) and
(3) of Table 2.A, and the top panel of Figure 3. Relatively small movements in type-speciﬁc
matching eﬃciencies achieve some marginal improvement over the model with an aggregate
matching eﬃciency only, but otherwise diﬀerences in search eﬀort elasticities across types
are enough to capture changes in relative transition rates.
Comparing the estimates across the three alternative models of aggregate and typespeciﬁc matching eﬃciencies with variable search eﬀort, we argue that the variation of the
type-speciﬁc transition rates is well captured by movements in aggregate matching eﬃciency
together with diﬀerential endogenous type-speciﬁc responses to movements in the aggregate
transition rate. Through the lens of the aggregate matching function with heterogeneity
and endogenous search eﬀort, it means that there is little exogenous movement in the typespeciﬁc transition rates beyond what is captured by the type-speciﬁc search eﬀort and an
aggregate exogenous shock.
5.2.1

Exogenous time-varying heterogeneity with constant eﬀort

We have just argued that with variable search eﬀort allowing for type-speciﬁc exogenous
eﬃciencies in addition to the aggregate matching eﬃciency does not aﬀect much our estimates
and yields small estimated variation in the type-speciﬁc eﬃciencies. Once we shut down
variable search eﬀort, however, much more variation in type-speciﬁc eﬃciencies is required
to describe relative movements in type-speciﬁc transition rates.
Consider our Model 3 estimated under restriction (13), i.e, with constant search eﬀort.
22

Given the estimate of , the restricted model is akin to the model with constant search eﬀort
and  = 036. In Figure 4 we display the identiﬁed aggregate matching eﬃciency and the
type-speciﬁc matching eﬃciencies for the unrestricted model as the crossed red lines and
for the restricted model as the circled blue lines. The restricted model captures all relative
movements in the type-speciﬁc transition rates via exogenous eﬀects. It delivers similar
aggregate matching eﬃciency as the unrestricted model but requires much more volatile typespeciﬁc matching eﬃciencies to capture all relative movements in the type-speciﬁc transition
rates.

5.3

Alternative search pool deﬁnitions

We now estimate models of heterogeneous employment transition rates with variable search
eﬀort for our two alternative deﬁnitions of the search pool, unemployed by reason of unemployment and nonemployed by labor force status and gender. Estimated transition elasticities
are displayed in Table 2, panels B and C, for all three model speciﬁcations of the aggregate
and type-speciﬁc matching eﬃciencies. The smoothed posteriors of the aggregate matching
eﬃciencies for Model 1 are displayed in Figure 5.23 For both deﬁnitions of the search pool,
we continue to ﬁnd evidence in favor of diﬀerent transition elasticities across types, that is,
variable search eﬀort. We also ﬁnd declines of the aggregate matching eﬃciency in the years
following the Great Recession that are comparable to what we ﬁnd for the baseline search
pool of unemployed by duration.
Comparing the decomposition of unemployment by reason with the previous decomposition by duration, we ﬁnd that the characteristics of those who claim to have been laid oﬀ
temporarily are similar to those who report a short unemployment duration: their employment transition rates are higher and less cyclically sensitive than for the other groups. For
Model 1 with aggregate matching elasticity only, this is reﬂected in their transition elasticities being lower than for the other groups, Table 2.B, Column (1). Similar to the baseline
search pool deﬁnition, the diﬀerences between estimated transition elasticities of diﬀerent
groups are less pronounced for Model 2 with type-speciﬁc matching eﬃciencies only, Table
2.B, Column (1).
The second alternative deﬁnition of the search pool includes all nonemployed and diﬀerentiates between those who are actively engaged in search and those who are not. For this
23

Hornstein and Kudlyak (2016) contains a complete listing of the results for all search pool deﬁnitions
and model speciﬁcations.

broader deﬁnition of the search pool, the groups with the higher employment transition rates
tend to be more cyclically sensitive. This is the opposite of what we see for the two previous search pool deﬁnitions, which cover only the unemployed. The diﬀerences in cyclical
sensitivity of the transition rates are so large across groups that they show up as diﬀerences in estimated transition elasticities for all speciﬁcations of aggregate and type-speciﬁc
eﬃciencies, Table 2.C.
The qualitative features of the estimated aggregate matching eﬃciency are very similar
for the diﬀerent search pool deﬁnitions, Figure 5. All of them are characterized by a decline of
aggregate matching eﬃciency in the years following the Great Recession that is exceptional
relative to the full sample. Compared with the baseline search pool of unemployed by
duration, the two alternative search pool deﬁnitions indicate more volatility of the aggregate
matching eﬃciency.

Aggregate matching eﬃciency and the cyclicality of
search eﬀort

In our framework, changes in the observed average employment transition rate can be decomposed into contributions coming from aggregate labor market tightness, average search
eﬀectiveness, and aggregate matching eﬃciency, equation (3),
¯
¯
ln  =  ln  + (1 − ) ln  + 

(17)

With a heterogeneous search pool, changes in average search eﬀectiveness in turn depend
on changes in the composition of the search pool and changes of type-speciﬁc search eﬃciency/eﬀort,
ln  = ln
¯

   + ln
¯



   +  
¯

(18)



Our framework only allows us to infer whether search eﬀort responds to changes in the aggregate transition rate, but not whether search eﬀort is pro- or countercyclical. Nevertheless,
as discussed in Section 2.3, we can characterize the cyclicality of search eﬀort conditional
on the aggregate matching elasticity. In particular, for a suﬃciently small (large) matching
elasticity, search eﬀort for each type will be procyclical (countercyclical).
We now show how the relative contributions of the components of the average employment transition rate depend on the assumed aggregate matching elasticity for the case with
24

variable search eﬀort, time-varying aggregate matching eﬃciency, and no exogenous changes
for type-speciﬁc eﬃciencies, Model 1 from Section 3. We conduct our analysis for the unemployment search pool by duration and the nonemployment search pool by labor force status
for the period 1994-2015. Figure 6 illustrates the dependence of search eﬀort elasticities on
matching elasticity for this case. For 0    028, search eﬀort is procyclical for all three
unemployment duration groups, and for   042, search eﬀort is countercyclical for all three
unemployment duration groups.24
Figure 7 displays the decomposition of the average employment transition rate for the
unemployment search pool diﬀerentiated by unemployment duration. For the decomposition
in the top (bottom) panel of Figure 7 we choose an aggregate matching elasticity of  = 02
( = 05) such that search eﬀort is procyclical (countercyclical) for all types. Our identiﬁed
measure of aggregate matching eﬃciency,  , is a scaled version of the actual matching
ˆ
eﬃciency, , with the matching elasticity being the scale factor,  =  . We can then
ˆ
rewrite the expression (3) for the average employment transition rate as
b
¯
ln  =  (ln  +  ) + (1 − ) ln  

and it is immediate that the contributions of market tightness and aggregate matching efﬁciency to changes in the average employment transition rate increase with the matching
elasticity. This is unlike the standard matching function with homogeneous search where a
higher matching elasticity will reduce the measured contribution of matching eﬃciency. In
our setup the diﬀerence is made up for by movements in average search eﬀectiveness. For
low values of , procyclical average search quality more than compensates for the small impact of market tightness, and the estimated model requires a small contribution of matching
eﬃciency. Conversely, for high values of , countercyclical average search quality calls for
a larger contribution from matching eﬃciency than is suggested only by the contribution
from market tightness. For the two cases we study, implied matching eﬃciency with countercyclical search eﬀort is more than twice as volatile than with procyclical search eﬀort.
For example, following the Great Recession, the implied matching eﬃciency with  = 02
declines by about 15 percent, whereas it declines by more than 30 percent for  = 05.
Consider ﬁrst the case with a relatively low aggregate matching elasticity and procyclical
search eﬀort, the top panel of Figure 7. We start with the average employment transition
rate, the solid black line, and subtract the contribution coming from changes in market
24

Appendix E provides a more detailed explanation of Figure 6.

tightness, the dashed red line. This gets us the standard measure of matching eﬃciency from
the homogeneous search model, the dashed blue line. Since the average transition rate and
market tightness move together, subtracting the matching elasticity-weighted contribution of
market tightness from the average search eﬀort yields a measure of matching eﬃciency that
becomes less volatile for higher values of  for the standard approach. For the assumed low
value of the matching elasticity, measured matching eﬃciency for the homogeneous search
model is quite volatile and declines substantially in the Great Recession. For the second
step, we subtract the contribution coming from changes in average match quality holding
type-speciﬁc search eﬀort ﬁxed, the dash-dot green line. This gets the matching eﬃciency
after accounting for changes in the composition of the search pool as in Barnichon and
Figura (2015), the dash-dot blue line. Accounting for the composition eﬀect reduces the
volatility of measured matching eﬃciency since the composition shifts toward low transition
rate components–from short-duration to long-duration unemployment–as market tightness
and the average employment transition rate decline. For the third step we take into account
that for the assumed low value of matching elasticity, search eﬀort is procyclical, that is,
it declines as market tightness declines. Therefore movements in search eﬀort amplify the
decline in average search eﬀectiveness, the solid green line, and reduce the volatility of
measured matching eﬃciency even more, the solid blue line.
Now consider the case when the matching elasticity is relatively large,  = 05, and search
eﬀort is countercyclical, the bottom panel of Figure 7. Given the larger matching elasticity, the contribution of market tightness increases, the dashed red line, and the measured
matching eﬃciency from the homogeneous search model is less volatile, the dashed blue line.
Again, accounting for changes in the composition of the search pool reduces the volatility of
measured matching eﬃciency even more, the dash-dotted green and blue lines. For a large
matching elasticity, search eﬀort is, however, countercyclical, and more than compensates for
the procyclical composition eﬀect, the solid green line. Accounting for the large increase in
search eﬀort following the decline in market tightness then recovers the much larger decline
in matching eﬃciency, the solid blue line, than what is implied by the standard homogeneous
search model.
We now brieﬂy discuss the role of search eﬀort for a decomposition of the average employment transition rate when the search pool is characterized according to labor force status,
Figure 8. This case is of interest, since for this deﬁnition of the search pool, composition
eﬀects increase average search eﬃciency when market tightness and the average employment
transition rate decline: the share of unemployed with relatively high exit rates increases.
26

Now countercyclical search eﬀort is reinforcing the composition eﬀect, whereas procyclical
search eﬀort is dampening the composition eﬀect, but in any case the impact of cyclical
search eﬀort dominates the impact of composition eﬀects.25

6.1

Existing empirical evidence on the cyclicality of search eﬀort

We show above that data on type-speciﬁc transition rates and the vacancy-unemployment
ratio cannot separately identify  from search eﬀort elasticities,   . To separately identify
, additional data are required. Such data might come from the evidence on the cyclical
behavior of search eﬀort.
While evidence on the cyclicality of search eﬀort is relatively scarce, the existing studies suggest that search eﬀort is countercyclical. Speciﬁcally, using the CPS data, Shimer
(2004) ﬁnds that the number of search methods used by the unemployed increases during
the 2001 recession. Using the CPS and the data from the Annual Time Use Survey (ATUS),
Mukoyama, Patterson and Sahin (2014) conclude that the time spent on search is countercyclical. Faberman and Kudlyak (2014) ﬁnd that the number of applications sent by a
job seeker per week on an online job board is signiﬁcantly higher in metropolitan areas with
more slack labor markets. Using ATUS, DeLoach and Kurt (2013), however, ﬁnd that search
appears to be acyclical. Speciﬁcally, they argue that workers reduce their search in response
to deteriorating labor market conditions, but these eﬀects are oﬀset by the increase in search
eﬀort due to declines in household wealth. Gomme and Lkhagvasuren (2015) argue that
search eﬀort of an individual worker is procyclical and that the measured countercyclical
average search eﬀort is due to a composition eﬀect.
Taking as given the evidence that search eﬀort is countercyclical implies that the matching
function elasticity is closer to 1 than to 0. That is, vacancies play an important role in
the production of new hires in the matching function framework and the countercyclical
search eﬀort exacerbates changes in the job seeker input. Consequently, large changes in the
aggregate matching eﬃciency are required to describe changes in the average transition rate.
25

See Hornstein and Kudlyak (2016) for a complete listing of the results on the cyclicality of search eﬀort
for all search pool deﬁnitions.

Conclusion

Modeling search eﬀort as a constant elasticity function of the aggregate transition rate,
we ﬁnd a substantial decline of the aggregate matching eﬃciency after 2007, even after
accounting for endogenous search eﬀort. Endogenous search eﬀort accounts well for variation
in relative transition rates of diﬀerent groups of job seekers. The data are consistent with
both pro- and countercyclical search eﬀort. Without additional data, we can only make a
statement about cyclicality conditional on the elasticity of the matching function. We ﬁnd
countercyclical eﬀort for a wide range of the elasticity of the matching function with respect
to vacancies, (13 1). Countercyclical eﬀort dampens transition rate volatility, and larger
volatility in aggregate matching eﬃciency is required to compensate for that, in contrast
with the standard model that ignores endogenous search eﬀort.

References
[1] Ahn, Hie Joo , and James Hamilton. 2015. "Heterogeneity and Unemployment Dynamics,” Working Paper, University of San Diego.
[2] Alvarez, Fernando, Katarina Borovickova, and Robert Shimer. 2016. Decomposing Duration Dependence in a Stopping Time Model, mimeo.
[3] Barnichon, Regis. 2010. Building a composite help-wanted index. Economics Letters
109, 175-178.
[4] Barnichon, Regis, and Andrew Figura. 2015. Labor Market Heterogeneity and the Aggregate Matching Function. American Economic Journal: Macroeconomics, Vol. 7(4):
222-49.
[5] Baxter, Marianne, and Robert G. King. 1999. Measuring Business Cycles: Approximate
Band-Pass Filters for Economic Time Series. Review Of Economics And Statistics. Vol.
81 (4): 575-593.
[6] Christensen, Bent Jesper, Rasmus Lentz, Dale T. Mortensen, George R. Neumann, and
Axel Werwatz. 2005. On-the-job Search and the Wage Distribution. Journal of Labor
Economics, Vol. 23 (1): 31-58.
[7] Costain, James S., and Reiter, Michael. 2008. Business Cycles, Unemployment Insurance, and the Calibration of Matching Models. Journal of Economic Dynamics and
Control, Vol. 32(4): 1120-1155.
[8] Davis, Steven J. 2011. Comment on “Job Search, Emotional Well-Being, and Job Finding in a Period of Mass Unemployment,” by Alan Krueger and Andreas Mueller, Brookings Papers on Economic Activity: 58—81.
[9] Davis, Steven J., R. Jason Faberman, and John C. Haltiwanger. 2013. The
Establishment-Level Behavior of Vacancies and Hiring. Quarterly Journal of Economics
128 (2): 581-622.
[10] DeLoach, Steve, and Mark R. Kurt. 2013. Discouraging Workers: Estimating the Impact
of Macroeconomic Shocks on Search Intensity of the Unemployed. Journal of Labor
Research, Vol. 34(4): 433-454.

[11] Elsby, Michael W.I., Ryan Michaels, and David Ratner. 2015. The Beveridge Curve: A
Survey. Journal of Economic Literature: Vol. 53 (3): 571-630.
[12] Faberman, Jason R., and Marianna Kudlyak. 2014. The Intensity of Job Search and
Search Duration. Richmond FED WP No. 14-12.
[13] Frazis, Harley J., Edwin L. Robinson, Thomas D. Evans, and Martha A. Duﬀ. 2005.
Estimating Gross Flows Consistent with Stocks in the CPS. Monthly Labor Review,
Vol. 128 (9): 3-9.
[14] Fujita, Shigeru, and Giuseppe Moscarini. 2013. “Recall and Unemployment,” NBER
Working Paper No. 19640.
[15] Gomme, Paul, and Damba Lkhagvasuren. 2015. Worker Search Eﬀort as an Ampliﬁcation Mechanism. Journal of Monetray Economics, Vol. 75 (October): 106—122.
[16] Hall, Robert E., and Sam Schulhofer-Wohl. 2015. Measuring Matching Eﬃciency with
Heterogeneous Jobseekers. NBER Working Paper No. 20939.
[17] Harvey, Andrew C. 1989. Forecasting, Structural Time Series Models and the Kalman
Filter. Cambridge University Press.
[18] Hornstein, Andreas. 2012. Accounting for Unemployment: The Long and Short of It.
FRB Richmond WP No. 12-07.
[19] Hornstein, Andreas, and Marianna Kudlyak. 2016. Estimating Matching Eﬃciency
with Variable Search Eﬀort: Tables and Figures. Available at
https://www.richmondfed.org/publications/research/working_papers/2016/wp_16-13r
[20] Kroft, Kory, Fabian Lange, Matthew J. Notowidigdo, and Lawrence Katz. 2015. LongTerm Unemployment and the Great Recession: The Role of Composition, Duration
Dependence, and Non-Participation. Journal of Labor Economics, Vol. 34 (S1): S7-S54.
[21] Krueger, Alan, and Andreas Mueller. 2011. Job Search, Emotional Well-Being, and Job
Finding in a Period of Mass Unemployment, Brookings Papers on Economic Activity,
Vol. 42(1): 1-81.
[22] Lise, Jeremy. 2013. On-the-job Search and Precautionary Savings. Review of Economic
Studies (80): 1086—1113.

[23] Madrian, Brigitte C., and Lars John Lefgren. 1999. A Note on Longitudinally Matching
Current Population Survey (CPS) Respondents. NBER Working Paper No. t0247.
[24] Mukoyama, Toshihiko, Christina Patterson, and Aysegul Sahin. 2014. Job Search Behavior over the Business Cycle. Federal Reserve Bank of New York Staﬀ Report No.
689.
[25] Petrongolo, Barbara, and Christopher A. Pissarides. 2001. Looking into the Black Box:
A Survey of the Matching Function. Journal of Economic Literature 39, 390-431.
[26] Pissarides, Christopher A.. 2000. Equilibrium Unemployment Theory. The MIT Press.
[27] Polivka, Anne E., and Stephen M. Miller. 1998. The CPS after the Redesign: Refocusing the Economic Lens. In "Labor Statistics Measurement Issues, " John Haltiwanger,
Marilyn E. Manser and Robert Topel, editors. University of Chicago Press.
[28] Sahin, Ay¸egül, Joseph Song, Giorgio Topa, and Givanni L. Violante. 2014. Mismatch
¸
s
Unemployment. American Economic Review, Vol. 104(11): 3529-64.
[29] Sedlacek, Petr. 2016. The Aggregate Matching Function and Job Search from Employment and Out of the Labor Force. Review of Economic Dynamics, Vol. 21(7): 16-28.
[30] Shimer,

Robert.

2004.

Intensity.

Mimeo,

downloadable

from

https://sites.google.com/site/robertshimer/research/workingpapers.
[31] Shimer, Robert. 2012. Reassessing the Ins and Outs of Unemployment. Review of Economic Dynamics 15, 127-148.
[32] Veracierto, Marcelo. 2011. Worker Flows and Matching Eﬃciency. Federal Reserve Bank
of Chicago Economic Perspectives v.35 4th Quarter, 147-169.
[33] Watson, Mark M. 1996. Comment on “Is Seasonal Adjustment a Linear or Nonlinear
Data Filtering Process,” Journal of Business and Economic Statistics, Vol. 14, No. 3.

Appendix
A

Endogenous search eﬀort

A simple modiﬁcation of the basic matching model allows for variation of individual search
eﬀort that is related to the aggregate employment transition rate, e.g. Pissarides (2000)
or recently Gomme and Lkhagvasuren (2015). Let  and  denote the value of being
unemployed and employed, respectively. Then, the return on unemployment is
 =  −  () +  ( −  ) 
where  is the rate of time discount and  is the ﬂow return from unemployment. Devoting
eﬀort to search increases the rate at which the worker becomes employed but it comes at a
cost,  (). Determining the optimal choice of eﬀort is a well-deﬁned problem if the eﬀort
cost is an increasing convex function of eﬀort. For simplicity, assume that the cost function
is of the constant elasticity variety,
 () = 0  with   1.
The ﬁrst order condition yields the optimal search eﬀort as
 = 1(−1) [( − )  (0 )]1(−1) 

(19)

that is, search eﬀort is a constant elasticity function of the aggregate transition rate.
For the basic matching model, search eﬀort is an increasing function of the aggregate
transition rate: as the marginal beneﬁt from search increases, the worker will devote more
eﬀort to search, yielding procyclical search eﬀort. We propose to estimate a reduced form
expression that relates the search intensity for each type to the aggregate matching rate,
and a type-speciﬁc persistent component,  . The elasticity of search eﬀort with respect to
the aggregate transition rate is  . We do not impose any restrictions on search eﬀort to be
pro- or countercyclical, but we do impose the restriction that the type transition rate is a
nondecreasing function of the aggregate transition rate,  ≥ −1.

Proof of proposition 2

In this section, we obtain the log-linear approximation of the relation between the average
transition rate and the standard measure of market tightness. The ﬁrst order log-linear
approximation of the average transition rate is
¯
∆ ln  = ∆ ln  + ∆ ln  
¯
where ∆ ln  ≡ ln  − ln 0 .

Using the ﬁrst order log-linear approximation of the average search eﬀectiveness as a

function of the aggregate transition rate
 ¯
∆ ln  = ∆¯ + 0 ∆ ln  
¯
and the ﬁrst order log-linear approximation of the aggregate transition rate yields
1
( (∆ ln  − ∆¯ ) + ∆ ) 

1 + ¯0


∆ ln  =

where ∆  =



 0 ∆ ,  0 =
¯



 0  , with  0 =

0
exp(0 +  ln 0 )
0
0
 0 exp(0 +  ln 0 )



and



 0 = 1.

We then obtain the following relation between the average transition rate and the standard measure of market tightness
¶
¸
µ
∙
¯
¯
¯
¯  =  1 + 0 ∆ ln  +  1 + 0 ∆ + 1 −  1 + 0 ∆¯ 
∆ ln 

1 + ¯0

1 + ¯0 

1 + ¯0


(20)

Intuitively, in equation (20) the coeﬃcient on ln  captures a direct eﬀect of the labor
¯
market tightness on ln  as well as an indirect eﬀect of the labor market tightness via search
eﬀort. That is, the matching function elasticity in the standard matching function is
=
¯

¯
1 + 0
¯
0
≡
¯ 
1 + ¯0

1 + 0

P
¯
where 0 =    0 .

Proof of proposition 3

Consider the following shift of the state vector (∆0  ∆  ∆)
b
0 =  + ∆0
b

0
 =  + ∆

b0
b
ln  = ln  + ∆

such that

∆ = − ∆ and ∆0 = ∆
Such a shift does not have any impact on the system (11) and (12), i.e.,
b0
ln  = 0 +  ln 

³
´
b+∆
= ( + ∆ ) +  ln 

b
=  +  ln 

and
0

b
ln 

= 0 + ln  − ln
b

"
X 

#
³
´
0
b
exp 0 +  ln 


#
"
³
³
´´
³
´
X 
b
b
exp  + ∆ +  ln  + ∆
ln  + ∆ = (b + ∆0 ) + ln  − ln



"
#
´
³
X 
b
b
ln  =  + ln  − ln
b
exp  +  ln 




remain satisﬁed.

Continuous time transition rates from discrete time
transition probabilities

First, consider the deﬁnition of the unemployed search pool by (1) duration or (2) reason.
Suppose we have  groups of unemployed. For the continuous time transition rates, only
consider the exit from unemployment and ignore the possibility that an individual ﬂows
back into unemployment. Then the transition probabilities are related to the continuous
34

time transition rates as follows

 =


− }
| {z

¡
¢
 −   = 
|
{z
}

0 no exit to  by  transition to  at 


 + 

=
 + 
=


Solve for 

−( + ) 

£
¤
1 − −( + )

£
¤
1 − −( + ) 


=


 +  = 1 − −( + )

log (1 −  −  ) = − ( +  )
µ
¶

= − 1+


Thus,
 = −
 = −

log (1 −  −  )
³
´

1 + 
log (1 −  −  )
³
´

1 + 

= −

log (1 −  −  )
( +  )

and for  small relative to  we have
 ≈ − log (1 −  )
Suppose instead that the pool includes OLF and unemployed, then the above equation
becomes
 ≈ − log (1 − ( +  ))



 + 

Matching elasticity and cyclicality of search eﬀort

For estimated identiﬁed transition elasticities  , we can solve the expression for the transition
elasticities, equation (11), for the search eﬀort elasticities  conditional on the matching
elasticity ,
  = 

1−
− 1


Without any additional data, the model can accommodate procyclical (countercyclical)
search eﬀort with a low (high) matching elasticity,
 R 0 for  Q



1 + 

For the special case that the identiﬁed transition elasticities are the same for all types,
 = , the implied eﬀort elasticities are also the same,  = . Since the eﬀort elasticities
are not identiﬁed, constant search eﬀort,  = 0, is then a special case. In other words,
our model with variable search eﬀort is observationally equivalent to a model with constant
search eﬀort.
For given estimates of the identiﬁed transition rates, we can compute the range of 
for which the model requires pro(counter)cyclical search eﬀort. Figure A.1 below illustrates
the dependence of search eﬀort elasticities on matching elasticity using the estimates from
Model 1 for the three duration-contingent groups of the unemployed. For 0    028,
search eﬀort is procyclical for all three unemployment duration groups, and for   042,
search eﬀort is countercyclical for all three unemployment duration groups.
Since the estimated  increase with the groups’ unemployment duration, the search eﬀort
elasticity also increases with groups’ unemployment duration. In particular, for   028,
when the search eﬀort of all three groups is procyclical, search eﬀort of long-term unemployed increases more when the aggregate transition rate increases and declines more when
the aggregate transition rate declines as compared to the search eﬀort of medium-term unemployed or short-term unemployed. For   042, when the search eﬀort of all three groups
is countercyclical, search eﬀort of long-term unemployed declines less when the aggregate
transition rate increases and increases less when the aggregate transition rate declines as
compared to the search eﬀort of medium-term unemployed or short-term unemployed.

Unobserved heterogeneity within groups

In the analysis so far, we assume that job seekers within speciﬁed groups are homogenous.
However, it is possible that job seekers within the groupings are heterogeneous and the
composition of a group with respect to this heterogeneity is time-varying. In such a case,
the time-varying transition rate of a group might reﬂect not only the actual variation of the
transition rate of the job seekers in the group but also the compositional shifts within the
group.
To see this formally, suppose that job seekers within each group  are not homogeneous.
¡
¢
Let     represent the unobserved heterogeneity within group  and (  ) the distribuP
tion over types within the group,    = 1. The transition rates for the diﬀerent types 
are

¡
¢
ln  =  + 1 +   ln  

We observe only the ‘average’ transition rate for group . To keep things simple, suppose we
observe the geometric mean of the unobserved transition rates
ln  =

X


  ln  + 
¡
£
¢
¤
   + 1 +  ln  + 

=  + (1 +   ) ln  +  
where  =



   and   =



   .

That is, not only does unobserved heterogeneity show up in the estimated type-speciﬁc
matching eﬃciency, it also shows up in our estimated search eﬀort elasticities. Without
further disaggregation of our search groups, we could check if the search eﬀort elasticities
vary systematically with the cycle.

Time-varying heterogeneity with constant search effort and no aggregate matching eﬃciency

We now consider the eﬀect of imposing constant search eﬀort for Model 2 with no aggregate
matching eﬃciency and forcing the model to capture time-varying relative heterogeneity via
type-speciﬁc exogenous shocks in a model. For the duration-contingent search pool deﬁnition,

the type-speciﬁc matching eﬃciencies for the unrestricted version of Model 2 are represented
by the solid red lines in Figure A.1. When restriction (13) is imposed, the type-speciﬁc
matching eﬃciencies are represented by the dashed blue lines in Figure A.1. There is little
diﬀerence between the type-speciﬁc matching eﬃciencies from the restricted and unrestricted
model. This is so because in the unrestricted model, there is no aggregate shock, type-speciﬁc
variable search eﬀort does not diﬀer much across types ( '  ∀ ∈ ), and the type-speciﬁc

exogenous shocks essentially capture all movements in type-speciﬁc transition rates. Given

the estimate of , the restricted model is akin to the model with constant search eﬀort and
 = 038.
Importantly, the volatility of the type-speciﬁc exogenous shocks might diﬀer signiﬁcantly
from the unrestricted case if the restricted model is estimated under an assumption for a
value of  that diﬀers from the one implied by the estimate of . Speciﬁcally, when the
search eﬀort is constant, the identiﬁcation issue for  does not arise, it is then typical to use
 = 05 or another value from the range of admissible values. Setting  = 0 and  =  in
e
our model amounts to the following restriction

e
 =  ∀ ∈ 

e
(1 + e) and e = 0. For e = 0 and  = 05,  = 1. The estimated type-speciﬁc



e
e
matching eﬃciencies from Model 2 estimated under  = 1 are represented by the dotted green
e
where  =



1−


lines in Figure A.1. The estimated transition rate elasticities with respect to the aggregate

transition rate are set to 1 for all types, which substantially exceeds the estimated sensitivity

to the aggregate transition rate in the unrestricted model or even the model with restriction
 =  ∀ ∈ , where the estimate of  is 0557 with the standard error of 0049. Consequently,

in this restricted model, the estimated type-speciﬁc matching eﬃciencies compensate for the
e
imposed degree of comovement under  = 1 and diﬀer substantially from the unrestricted
model.

Type-Specific Effect z1 for 1-4 Weeks
0
-0.1
-0.2
1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

2010

2012

2014

2010

2012

2014

Type-Specific Effect z2 for 5-26 Weeks
0
-0.1
-0.2
1994

1996

1998

2000

2002

2004

2006

2008

Type-Specific Effect z3 for >=27 Weeks
0
-0.2
-0.4
1994

φi
φi=φ
φi=φ=1, ie α=0.5

1996

1998

2000

2002

2004

2006

2008

A.1 Variable vs Constant Search Eﬀort for Model 2

Note: Smoothed posterior estimates of the type-speciﬁc matching eﬃciencies from Model 2 deﬁned in the
text for search pool deﬁned as unemployed by duration. The solid red line denotes the unrestricted model,
the dashed blue line denotes the restricted model with equalized identiﬁed transition elasticities, and the
dotted green line denotes the model with identiﬁed transition elasticities restricted to one. The speciﬁcation
follows column (1a) in Table 1. All eﬃciencies are normalized to zero in 2001Q1.

Figures and Tables

Figure 1: Transition Rates to Employment and Search Pool Composition
1

1.A Duration: Trans Rate

0.5

1990

2000

2010

1.B Duration: Relative Rate

0.5

0
1980

1990

2000

2010

2.B Reason: Relative Rate

0.5

0
1990

2000

2010

1.C Duration: Shares

0.5

1990

2000

2.C Reason: Shares

1990

2000

2010

1980

1990

2000

2010

3.C Gender and LFS: Shares
1

0.5

0
1980

2000

3.B Gender and LFS: Relative Rate
1

2010

0.5

1990

0
1980

1980

0.5

0
1980

3.A Gender and LFS: Trans Rate
1

0.5

0
1980

2.A Reason: Trans Rate

0.5

0
1980

1990

2000

2010

1980

1990

2000

2010

Note: The columns represent the three diﬀerent search pool deﬁnitions and are characterized
by the employment transition rates (ﬁrst row), the relative employment transition rates (second
row), and the search pool shares (third row). For the search pool deﬁned as unemployed by
duration, red denotes 1-4 weeks, blue denotes 5-26 weeks, and green denotes more than 26
weeks of unemployment. For the search pool deﬁned by reason of unemployment, red denotes
temporary layoﬀ, blue denotes permanent layoﬀ, green denotes job leavers, and purple denotes
new entrants or re-entrants. For the search pool deﬁned by LFS and gender, red denotes male
unemployed, blue denotes female unemployed, green denotes male OLF, and purple denotes
female OLF.

Figure 2: Identified Aggregate Matching Efficiency from the Model with
Aggregate Matching Efficiency and Variable Search Effort, Unemployment
by Duration

Aggregate Matching Efficiency, κ/α
0.1
0

Log Level, Mean of 2001=0

-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8

(1a) 1994-2015
(1b) 1994-2015, φ i =φ
(1c) 1994-2015, α= 0.38
(2a) 1976-2015
(2b) 1976-2015, Δ φ
(3a) 2001-2015, HWI
(3b) 2001-2015, JOLTS

-0.9
1980

1985

1990

1995

2000

2005

2010

2015

Note: Lines (1a)-(3b) show smoothed posterior estimate of b from a model with a random

walk in aggregate matching eﬃciency (Model 1 in the text). Lines (1a), (1b), (2a)-(3b) show
the estimates from the respective columns in Table 1, see note to Table 1 for details. Line
(1c) shows the constructed  for the standard matching function without heterogeneity and
b
 = 038; see Section 5.3.3 for details. All eﬃciencies are normalized to zero in 2001Q1.

Figure 3: Aggregate and Type-Specific Matching Efficiencies from Alternative Models of Stochastic State, Unemployment by Duration

Note: Smoothed posterior estimates of the aggregate and type-speciﬁc matching eﬃciencies
from Models 1-3 deﬁned in the text for search pool deﬁned as unemployed by duration. The
speciﬁcation follows column (1a) in Table 1. All eﬃciencies are normalized to zero in 2001Q1.

Figure 4: Variable vs Constant Search Effort for Model 3 with Duration
Contingent Search Pool
Aggregate Efficiency κ/α,
0
-0.2

φi

-0.4

φi=φ

-0.6
1994

1996

1998

2000
2002
2004
2006
2008
Type-Specific Effect z1 for 1-4 Weeks

2010

2012

2014

1994
1996
×10-6

1998

2000
2002
2004
2006
2008
Type-Specific Effect z2 for 5-26 Weeks

2010

2012

2014

1996

1998

2000
2002
2004
2006
2008
Type-Specific Effect z3 for >=27 Weeks

2010

2012

2014

1996

1998

2000

2010

2012

2014

0.1
0.05
0
-0.05

2
0
-2
1994
0.05
0
-0.05
-0.1
1994

2002

2004

2006

2008

Note: Smoothed posterior estimates of the aggregate and type-speciﬁc matching eﬃciencies
from Model 3 deﬁned in the text for search pool deﬁned as unemployed by duration. The red
crossed line imposes no restrictions on identiﬁed transition elasticities, and the blue circled line
imposes equality for the identiﬁed transition elasticities. All eﬃciencies are normalized to zero
in 2001Q1.

Figure 5: Aggregate Matching Efficiencies from Alternative Search Pool
Definitions

Aggregate Matching Efficiency, κ/α

Log Level, Mean of 2001=0

-0.2

-0.4

-0.6

-0.8

-1

Duration Contingent
Reason of Unemployment
Gender and LFS

1980

1985

1990

1995

2000

2005

2010

Note: Smoothed posterior estimates of the aggregate matching eﬃciency from a model with
a random walk in aggregate eﬃciency (Model 1 in the text) for search pool deﬁned as (1)
unemployed by duration, (2) unemployed by reason, or (3) LFS by gender. The speciﬁcation
follows column (2a) Table 1. All eﬃciencies are normalized to zero in 1994Q1.

Figure 6: Search Effort Elasticity and Matching Elasticity

6
1-4 weeks
5-26 weeks
>26 weeks

Search Effort Elasticity

-1
0.1

0.2

0.3

0.4

0.5

0.6

Matching Elasticity

0.7

0.8

0.9

Figure 7: Decomposition of the Average Transition Rate, Unemployment by
Duration
Average Transition Rate, α =0.20
0.4

Log Level, 1994: 1=0

0.2

-0.2

ln(E[u ρ λ ])
i i i

-0.4

α ln(θ)
ln(κ), No accounting for quality
(1-α) ln(E[uiρ i]), ρ i fixed
ln(κ), composition effect with ρ i fixed

-0.6

(1-α) ln(E[uiρ i])
ln(κ)

1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

2010

2012

2014

Average Transition Rate, α =0.50
0.4

Log Level, 1994: 1=0

0.2

-0.2

ln(E[u iρ iλi])

-0.4

α ln(θ)
ln(κ), No accounting for quality
(1-α) ln(E[uiρ i]), ρ i fixed
ln(κ), composition effect with ρ i fixed

-0.6

(1-α) ln(E[u ρ ])
i i

ln(κ)

1994

1996

1998

2000

2002

2004

2006

2008

Note: Components of the average employment transition rate from a model with a random
walk in aggregate eﬃciency (Model 1 in the text) for a search pool deﬁned by duration of
unemployment. The speciﬁcation follows column (1a) Table 1. All variables are normalized to
zero in 1994Q1.

Figure 8: Decomposition of the Average Transition Rate, LFS by Gender
Average Transition Rate, α=0.10

0.3

0.2

Log Level, 1994: 1=0

0.1

-0.1
ln(E[u iρiλi])
α ln(θ)
ln(κ), No accounting for quality
(1-α) ln(E[u iρi]), ρi fixed

-0.2

ln(κ), composition effect with ρi fixed

-0.3

(1-α) ln(E[u iρi])
ln(κ)

-0.4
1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

2010

2012

2014

Average Transition Rate, α=0.50

0.3

0.2

Log Level, 1994: 1=0

0.1

-0.1

-0.2

ln(E[u iρiλi])

-0.3

ln(κ), composition effect with ρi fixed

α ln(θ)
ln(κ), No accounting for quality
(1-α) ln(E[u iρi]), ρi fixed
(1-α) ln(E[u iρi])
ln(κ)

-0.4
1994

1996

1998

2000

2002

2004

2006

2008

Note: Components of the average employment transition rate from a model with a random
walk in aggregate eﬃciency (Model 1 in the text) for a search pool deﬁned by labor force status
and gender. The speciﬁcation follows column (1a) Table 1. All variables are normalized to zero
in 1994Q1.

Table 1: Estimates from the Model with Aggregate Matching Efficiency and
Variable Search Effort, Unemployment by Duration
(1a)
[ˆ 1 ] -2.401

(1.416)
2
-0.325
(0.245)
3
-0.587
(0.091)
1
0.405
(0.021)
2
0.562
(0.139)
3
0.717
(0.081)
 1
0.048
(0.009)
 2
0.038
(0.006)
 3
0.088
(0.017)

0.054
(0.017)
1
2
3
1
2
3
-

(1b)
-1.914
(0.809)
-0.572
(0.008)
-1.076
(0.014)
0.611
(0.073)
0.065
(0.006)
0.042
(0.004)
0.110
(0.009)
0.036
(0.012)
-

(2a)
-2.548
(0.912)
-0.349
(0.051)
-0.682
(0.058)
0.358
(0.024)
0.483
(0.047)
0.579
(0.029)
0.045
(0.003)
0.036
(0.003)
0.086
(0.005)
0.060
(0.012)
0.156
(0.025)
0.069
(0.019)
0.055
(0.031)
-

(2b)
-2.838
(0.218)
-0.314
(0.039)
-0.451
(0.171)
0.303
(0.036)
0.429
(0.072)
0.628
(0.087)
0.045
(0.003)
0.036
(0.003)
0.079
(0.006)
0.063
(0.013)
0.154
(0.027)
0.066
(0.031)
0.050
(0.050)
0.017
(0.009)
0.015
(0.012)
-0.025
(0.021)

(3a)
-2.749
(0.938)
-0.216
(0.417)
-0.423
(0.421)
0.355
(0.107)
0.543
(0.067)
0.702
(0.092)
0.049
(0.006)
0.039
(0.005)
0.085
(0.010)
0.069
(0.019)
-

(3b)
-2.881
(6.912)
-0.212
(0.415)
-0.461
(0.839)
0.377
(0.111)
0.579
(0.070)
0.730
(0.584)
0.050
(0.006)
0.038
(0.005)
0.091
(0.009)
0.038
(0.014)
-

Note: Col. (1a) - (3a) display parameter estimates of a model with a random walk for aggregate
matching eﬃciency using the HWI as a measure of vacancies. Col. (1a) displays estimates for
the sample period 1994-2015, and col. (1b) imposes the restriction that  is the same for
all types. Col. (2a) displays estimates for the sample period 1976-2015 with adjustment of
the search pool data prior to the 1994 CPS revision based on Polivka and Miller (1998) and a
ﬁxed estimated structural change in measured transition rates  , and col. (2b) introduces an
unemployment rate contingent structural break in the parameter  as explained in the text.
Col. (3a) re-estimates the model from col. (1a) for the sub-sample 2001-2015 with HWI for
vacancies, and col. (3b) does the same with vacancies from JOLTS.
49

Table 2: Identified Matching Elasticities
(1)
(2)
(3)
A. Unemployment by Duration
1-4 weeks
0.381
0.405
0.368
(0.037) (0.036) (0.174)
5-26 weeks
0.506
0.538
0.488
(0.037) (0.075) (0.294)
27 weeks and above
0.604
0.523
0.521
(0.038) (0.108) (0.138)
B. Unemployment by Reason
Laid oﬀ temporarily
0.116
0.587
0.498
(0.018) (0.061) (0.074)
Laid oﬀ permanently 0.563
0.719
0.634
(0.053) (0.103) (0.069)
Left job
0.571
0.668
0.584
(0.056) (0.069) (0.062)
LF entrant
0.516
0.461
0.451
(0.049) (0.070) (0.051)
C. Unemployment and OLF by Reason
Unemployed, male
0.686
0.731
0.588
(0.058) (0.064) (0.082)
Unemployed, female
0.686
0.588
0.589
(0.059) (0.058) (0.079)
OLF, male
0.261
0.218
0.195
(0.028) (0.043) (0.041)
OLF, female
0.215
0.193
0.189
(0.022) (0.042) (0.033)
Note: The columns display the estimated identiﬁed matching elasticities  of a model with
(1) a random walk for aggregate matching eﬃciency only, (2) random walks for type-speciﬁc
matching eﬃciencies, and (3) a random walk for aggregate matching eﬃciency and stationary
AR(1)’s for type-speciﬁc matching eﬃciencies. All models are estimated for speciﬁcation (2a)
in Table 1, that is, the 1976-2015 sample period with appropriate adjustment for the 1994 CPS
revision. See note to Table 1 for details.

Full text of Working Papers (Federal Reserve Bank of Richmond) : Estimating Matching Efficiency with Variable Search Effort, Working Paper 16-13r

FRASER