Discouragement and Job Search Dynamics during the Great Recession * Tristan Potter † Boston College October 9, 2015 Abstract During the Great Recession, the amount of time individuals devoted to job search fell sharply over the unemployment spell (Krueger and Mueller, 2011). I show that this decline is explained by variation in individuals’ experiences looking for work since job loss: search falls the more time an individual has spent unsuccessfully searching, and the decline is mitigated after receiv- ing job offers. Existing models of job search cannot account for these facts. I develop a tractable model of sequential search in which job seekers are uncertain about the existence of work. Non- monotonic search dynamics arise as search decisions interact with endogenously-evolving beliefs over the spell of unemployment. I structurally estimate the model and show that the mechanism accounts for non-linear profiles of search time, offer arrival probabilities and offer acceptance probabilities over the first two years of unemployment. The model outperforms a flexible alter- native and provides a novel explanation for the hump in search effort over the unemployment spell observed prior to the Great Recession. Keywords: unemployment; search theory JEL Classification: J64, E24, D83 * Special thanks to my advisors Sanjay Chugh, Ryan Chahrour and Peter Ireland for their continual guidance and support. Many thanks to Kit Baum and Fabio Schiantarelli for numerous enlightening conversations. Thanks also to Ana Lariau Bolentini, Donald Cox, Mario Crucini, Michael Grubb, John Lindner, Claudia Olivetti, Joseph Quinn, David Schenck, Utku Unver, Mathis Wagner, Zhijie Xiao, and seminar participants at Boston College for valuable conversations and suggestions. This draft is preliminary and incomplete. † Email address: [email protected]
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Discouragement and Job Search Dynamics during the Great
Recession∗
Tristan Potter†
Boston College
October 9, 2015
Abstract
During the Great Recession, the amount of time individuals devoted to job search fell sharply
over the unemployment spell (Krueger and Mueller, 2011). I show that this decline is explained
by variation in individuals’ experiences looking for work since job loss: search falls the more
time an individual has spent unsuccessfully searching, and the decline is mitigated after receiv-
ing job offers. Existing models of job search cannot account for these facts. I develop a tractable
model of sequential search in which job seekers are uncertain about the existence of work. Non-
monotonic search dynamics arise as search decisions interact with endogenously-evolving beliefs
over the spell of unemployment. I structurally estimate the model and show that the mechanism
accounts for non-linear profiles of search time, offer arrival probabilities and offer acceptance
probabilities over the first two years of unemployment. The model outperforms a flexible alter-
native and provides a novel explanation for the hump in search effort over the unemployment
spell observed prior to the Great Recession.
Keywords: unemployment; search theory
JEL Classification: J64, E24, D83
∗Special thanks to my advisors Sanjay Chugh, Ryan Chahrour and Peter Ireland for their continual guidance andsupport. Many thanks to Kit Baum and Fabio Schiantarelli for numerous enlightening conversations. Thanks also toAna Lariau Bolentini, Donald Cox, Mario Crucini, Michael Grubb, John Lindner, Claudia Olivetti, Joseph Quinn,David Schenck, Utku Unver, Mathis Wagner, Zhijie Xiao, and seminar participants at Boston College for valuableconversations and suggestions. This draft is preliminary and incomplete.†Email address: [email protected]
The Survey of Unemployed Workers in New Jersey (SUWNJ) is a weekly longitudinal survey of
unemployment insurance (UI) benefit recipients in New Jersey. The study was conducted by the
Princeton University Survey Research Center starting in the fall of 2009 and lasting for up to
24 weeks. Sampled individuals were asked to participate in a weekly online survey lasting for a
minimum of 12 weeks, and up to 24 weeks for the long-term unemployed. The weekly survey
consisted of questions pertaining to job search activity, time use, job offers, and consumption. See
Appendix A for a more complete description of the survey, and Krueger and Mueller (2011) for a
comprehensive description of methodology.
1Figure 1 plots average search time over the unemployment spell after removing individual-specific means usingthe Survey of Unemployed Workers in New Jersey.
3
2.2 Evidence from Job Offers
In the standard theory of sequential search, job offers arrive at predictable intervals.2 Such pre-
dictability implies that job offers per se should have no bearing on behavior.3
If, however, job seekers lack information about the availability or existence of work, then an offer
itself may have informational value, independent of its nominal value. Furthermore, if this in-
formation affects job seekers’ perceptions of the return to searching for work, then behavior may
be expected to change after the arrival of an offer. The detailed information on job offers and
individuals’ search decisions contained in the SUWNJ allow this simple hypothesis to be tested.
Specifically, I compare the change in search effort the week before and after an offer is received and
rejected with the average change in search effort over the unemployment spell, excluding the week
of the offer. Table 1 reports the results for both the unrestricted sample and the prime age sample
used in Section 2.3.
Table 1: Search Time Before and After Job Offers
Prime Age Full Sample
Time Diary∆offeri -0.03 (0.14) -0.06 (0.20)
∆avgi -0.32 (0.08) -0.40 (0.12)
Observations 211 139
Weekly Recall∆offeri -0.76 (0.52) -0.65 (0.71)
∆avgi -1.68 (0.42) -2.32 (0.63)
Observations 172 109
Standard errors in parentheses.
Source: Survey of Unemployed Workers in New Jersey
Notes: The samples are restricted to those receiving andrejecting offers. See Appendix A for details.
∆offeri ≡ (spost
i − sprei )/2
∆avgi ≡ 1
T
∑t ∆sit
For both measures of search time, the decrease in search after an offer is received, ∆offer, is signif-
icantly attenuated relative to the typical decline, ∆avg. This result holds for both the prime age
sample as well as the full sample of job seekers in the SUWNJ. In light of the preceding discussion,
these results suggest that the simple informational structure of standard search models may be
missing an important component.
2If arrivals are stochastic, then the arrival time is constant in expectation.3Even when the parameters of the offer distribution are unobserved, the offer itself has no effect on behavior
independent of its value. See Burdett and Vishwanath (1988).
4
2.3 The Negative Effect of Past Search
If information about the outcomes of searching for work—job offers—affect subsequent search
decisions, then so too should information about the effort put into searching for work. In this
section I study how time spent searching for work since job loss affects subsequent search decisions.
2.3.1 Empirical strategy
Consider expressing time devoted to job search as a function of unemployment duration, time- and
individual-fixed effects, observed shocks to search time and—reflecting the preceding intuition—the
total time spent looking for work since job loss:
sit = α+ γdit + β
t−1∑k=0
sik + τt +[δ1 . . . δ10
]·
e1it...
e10it
+ (ηi + εit). (1)
For individual i in interview week t, sit denotes search time, dit denotes unemployment duration, τt
is an aggregate time effect, e1it,...,e
10it are indicators for reported shocks to search time, and ηi is the
individual effect.4 The coefficients of interest are γ and β, which determine the roles of duration
and the stock of past search, respectively, in driving time spent looking for work.
Equation (1) cannot be estimated directly from the SUWNJ data for two reasons. First, the
individual effect ηi is unobserved. Second, no individuals in the sample are observed from the
beginning of the unemployment spell, so the stock variable of interest is only partially observed.
However, both of these problems can be dealt with by taking first-differences of (1):
∆sit = γ∆dit + βsit−1 + ∆τt +[δ1 . . . δ10
]·
∆e1
it...
∆e10it
+ ∆εit. (2)
The presence of the lagged-dependent variable on the right-hand side of (2) now gives rise to an
endogeneity problem common to dynamic panel models: E [sit−1∆εit] 6= 0. Following Anderson and
Hsiao (1982), I address the endogeneity of sit−1 by instrumenting with its first lag, sit−2. Under
the assumption that εit is serially uncorrelated, ∆εit is an MA(1) process, and thus sit−2 is a valid
instrument for sit−1. The Arellano-Bond test for serial correlation confirms that sit−2 is indeed a
valid instrument. I refrain from including further lags of sit−1 because doing so entails considerable
loss of data, given that the average individual is observed for fewer than six weeks.5
4The indicators for reported shocks to search time are based on responses to the following question from theSUWNJ: “In the last 7 days, did anything happen that made you spend more time or less time looking for work thanusual? Please select all that apply.” Respondents were given 10 options from which to select, including, for example“I was sick/I was caring for a sick person in my family.”
5In Appendix A, I estimate the model using the GMM estimator developed in Arellano and Bond (1991) in order
5
2.3.2 Results
Table 2 reports results from the baseline specification described above.6 I report results for search
effort as measured by (i) time diary data documenting time spent looking for work in the day prior
to the interview; and (ii) weekly recall data documenting total time spent looking for work in the
week prior to the interview.7 For each measure, I report results from a specification that does not
include as a regressor the stock of past search (Static), and results from an identical regression
augmented with the stock of past search time (Dynamic).
Source: Survey of Unemployed Workers in New Jersey
Notes: All regressions use survey weights. The sample consists of respondentsages 20-65 who have not received a job offer and who left their previous jobinvoluntarily and do not expect to return. All first-differenced regressionsexclude constant terms.∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
Two principal conclusions emerge from Table 2. First, the coefficient on the stock of past search
is highly-significant and negative for both measures of search effort. Moreover, the dynamic model
provides a much better fit for the data as measured by the adjusted R2. Second, when the stock
of past search is included as a regressor, unemployment duration ceases to enter the model with a
significant negative coefficient. Put differently, the observed decline in effort over the unemployment
spell documented by Krueger and Mueller (2011) can be attributed to variation in past search.
An interesting implication of the foregoing analysis is that time devoted to job search follows an
AR(1) process. To see this, abstracting from exogenous search shocks and time effects, observe
that equation (2) may be rearranged as follows:
to exploit additional available moment conditions while mitigating the data loss associated with differencing. Theresults are consistent with the results from the more parsimonious instrumenting strategy discussed above. I alsoestimate the model using the within estimator, though the within transformation is known to induce bias in pointestimates for small T panels.
6For brevity, I exclude estimated coefficients associated with time effects and the exogenous search shocks. Fullresults are reported in Appendix A.4.
7The time diary measure of search time is likely more robust to concerns of reporting bias, as having to accountfor each hour of the previous day reduces the scope for manipulation.
6
sit = γ + (1 + β)sit−1 + ∆εit. (3)
Interpreted in this way, the results imply that progressive withdrawal from the labor force—
declining time devoted to job search—is an intrinsic feature of joblessness in the data from the Great
Recession. Put differently, negative duration dependence obtains absent any exogenous changes in
the availability of work over the course of the unemployment spell.8 To be concrete, the parameter
estimates in Table 2 suggest that on average during the Great Recession, the half-life of search
effort was just six weeks of unemployment as measured by the time diary data, and just over eight
weeks of unemployment as measured by the weekly recall data.
2.3.3 Robustness
I consider several modifications to the baseline model described above to ensure that the results
presented in Table 2 are a robust feature of the data. Specifically, I allow for search to depend
non-linearly on the duration of unemployment and account for forward-looking search behavior
through inclusion of leads of search shock indicators. I also confirm that the results are robust
to the use of various sample selection criteria, alternative observation weighting schemes and hold
when the sample is restricted to individuals spending a strictly positive amount of time searching.
These supplementary results, cataloged in Appendix A.5, provide robust support for the results in
Table 2.
2.3.4 Stock-flow matching
One plausible explanation for the results in Table 2 is the presence of stock-flow matching (Coles
and Smith, 1998; Ebrahimy and Shimer, 2006; Coles and Petrongolo, 2008). Specifically, suppose
that upon job loss, individuals observe a stock of relevant vacancies, and search time is devoted to
applying to those jobs. Once that stock has been exhausted, subsequent search is limited by the
flow of newly-posted vacancies. In this environment, the time devoted to search corresponds to the
rate at which the initial stock is drawn down, and thus individuals who devote more time to search
early in the unemployment spell may more rapidly reduce their search as they are forced to wait
for the arrival of new vacancies.
As a simple test of whether stock-flow matching is driving the results in Table 2, I replace the stock
of past search time on the right-hand side of equation (1) with the stock of past applications. If
search time is constrained by the availability of vacancies, as predicted by a stock-flow model, then
the total number of applications submitted since job loss should predict the amount of time devoted
to job search. The results, found in Appendix A, indicate that the stock of past applications is
8As noted above, there is considerable evidence that job-finding prospects do change over the unemployment spell.The point here is simply that progressive labor force detachment exists even in the absence of such effects.
7
not a significant determinant of time spent looking for work, as would be expected in a model of
stock-flow matching.9
2.3.5 A structural interpretation
An alternative explanation for the results in Table 2 is that job seekers are uncertain about their
job-finding prospects. If this is the case, and job seekers infer from their past failures to find
work that their job-finding prospects are relatively poor, then those who have spent more time
unsuccessfully looking for work in the past may become discouraged and substitute away from job
search.
To capture this intuition, I reinterpret (1) as the reduced-form of an underlying structural model in
which past failures to find work induce pessimism about job-finding prospects, which in turn causes
a substitution away from search effort. Because data on individuals’ perceptions about the job-
finding process are not available at a sufficiently high frequency, I use self-reported life satisfaction
as a proxy.10 Denoting by pit the SUWNJ measure of self-reported life satisfaction, consider such
a model:11
pit = φ0 + φ1dit + φ2
t−1∑k=0
sik + τt + νi + ωit (4)
sit = θ0 + θ1dit + θ2pit + τt +[δ1 . . . δ10
]·
e1it...
e10it
+ ηi + εit. (5)
Note that substituting (4) into (5) implies the following structural interpretation of the coefficients
in equation (1): γ = θ1 + θ2φ1 and β = θ2φ2. Duration can affect search decisions directly (θ1) and
indirectly through its effect on self-reported life satisfaction (θ2φ2). The effect of past search is the
composite of its effect on self-reported life satisfaction, and the effect of life satisfaction on search
effort.
The econometric issues discussed in the preceding section largely carry over to the system described
by (4) and (5). The methodology is discussed at length in Appendix A.8. The results in Table
A.14 suggest that, to a significant extent, total search time since job loss is affecting search deci-
sions via discouragement: increased time spent looking for work since job loss increases reported
dissatisfaction, which in turn reduces time spent looking for work.
9Faberman and Kudlyak (2014) corroborate this result using a new data set on internet job search. They documentthat applications to newly-posted vacancies account for only 17% of total applications in the sixth month of search,suggesting that stock-flow matching is of limited relevance to search dynamics.
10Insofar as reductions in the perceived probability of exiting unemployment reduce expected permanent income,they should be highly-correlated with reported quality of life.
11The variable pit is an ordinal response to the first question in the SUWNJ: “Taking all things together, howsatisfied are you with your life as a whole these days?” The available responses are: “Very satisfied” (1); “Satisfied”(2); “Not satisfied” (3); “Not at all satisfied” (4). The specification in equation (5) implicitly treats pit as a cardinalvariable to facilitate estimation in the presence of fixed effects. See Riedl and Geishecker (2014) for discussion ofprecedent.
8
3 Sequential Search with Endogenous Discouragement
This section develops a simple theory of unemployment in which job seekers are uncertain about
the effectiveness of their search. I make two modifications to an otherwise standard McCall (1970)-
style model of sequential search: (i) job seekers choose how much time to spend looking for work
each period; and (ii) job seekers do not observe the rate at which job offers arrive per unit of time
devoted to search. Because the arrival rate is unobserved, job seekers are endowed with beliefs that
evolve endogenously in response to the arrival of new information. Search time and reservation
wage dynamics over the unemployment spell are thus driven by the evolution of beliefs, which in
turn respond to the idiosyncratic outcomes of search.
3.1 Environment
3.1.1 Timing
Unemployment duration is discrete and measured in weeks. Unemployed job seekers are assumed to
maximize the present discounted value of income net of search costs: E0∑∞
t=0 δt(yt − ηst). Search
costs may be thought of as monetary costs or as forgone home production.
At the beginning of each week t, job seekers choose to devote fraction st of their week to searching
for work. While searching, job offers arrive according to a Poisson process with true average rate
parameter λT .12 Letting τt denote the stochastic arrival time of the first offer, the true probability
of a job offer arriving before search ends is given by
Pr(τt ≤ st) ≡ F (st;λT ) = 1− e−λT st . (6)
If a job offer arrives before search ends (τt < st), the job seeker updates her estimate of λT , and
then decides whether or not to accept the offer as in a standard McCall-style search framework.
Offers are drawn from a fixed non-degenerate distribution Φ(ω) with density φ(ω). If the offer is
accepted, the job seeker receives ω each period until she is separated from the job, at which point
a new unemployment spell begins. Separations occur at the beginning of each week at constant
rate ρ. If the offer is rejected, the job seeker receives flow value of unemployment b and continues
searching next period.
If no offer arrives before search ends (τt ≥ st), the job seeker receives flow value of unemployment b
and updates her estimate of λT to reflect the fact that searching for fraction st of the week yielded
no offers.
12Unemployment duration is discrete, but offers arrive continuously within periods. This setup allows the model tomap directly into the weekly time use data from the SUWNJ without needing to appeal more generally to continuoustime. When an offer arrives, I assume that agents must stop searching for the balance of the period to update beliefsand evaluate the offer, so agents never receive more than one offer per week. This assumption could be relaxed byassuming that the number of offers arriving each period follows a Poisson distribution.
9
Figure 2 depicts the timing of the model.
Figure 2: Timing of Events
3.1.2 Beliefs
I assume that job seekers do not know their true job offer arrival rates λT . Instead, they form beliefs
over the value of λT , which take the form of a Gamma distribution, parameterized by αt and βt. The
assumptions that observed arrival times follow a (right-censored) exponential distribution and that
beliefs follow a Gamma distribution together imply that beliefs are time-invariant up to parameters
αt and βt, a result which affords the model considerable tractability. See Appendix B for proof of
this claim.
The density of beliefs in week t is thus given by
Pr(λ = λ) ≡ γ(λ;αt, βt) =βαtt λ
αt−1e−βtλ
Γ(αt). (7)
The mean and variance of the distribution of beliefs in week t are
Et(λ) =αtβt
V art(λ) =αtβ2t
. (8)
The parameters of the belief distribution αt and βt evolve endogenously over the unemployment
spell according to the following laws of motion:
αt+1 =
αt + 1 if τt < st (offer)
αt if τt ≥ st (no offer)(9)
βt+1 =
βt + τt if τt < st (offer)
βt + st if τt ≥ st (no offer).(10)
10
Note that αt counts the number of job offers received since job loss and βt measures accumulated
search time since job loss. The endogeneity of beliefs arises from the presence of st in (9) and (10).
Figure 3: Endogenous Beliefs
Arrival Rate (λ)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Pr(λ
=λ)
0
0.2
0.4
0.6
0.8
1
1.2
Pessimistic BeliefsOptimistic Beliefs
Figure 3 depicts two belief distributions associated with different values of αt and βt. As more job
offers arrive, job seekers become more optimistic, and the belief distribution shifts outward. Con-
versely, as more time is spent searching without receiving an offer, job seekers become pessimistic,
and the belief distribution shifts inward.13
In keeping with much of the macroeconomic literature on learning, I assume that job seekers opti-
mize within an anticipated utility framework.14 This assumption serves to simplify the exposition
of the model, and provides a significant reduction in the computational burden associated with
estimating the model in Section 4. In Appendix B I numerically solve the model under rational
expectations and demonstrate that search decisions are not significantly altered when individuals
anticipate the evolution of their own beliefs. I therefore restrict attention to anticipated utility
throughout the remainder of the paper.
13Note that, conditional on not receiving an offer, arrival time τt is not observed. This is not a problem for theanalysis because the Gamma distribution also serves as the conjugate prior for the relevant right-censored Exponentialdistribution, according to which observed arrival times are distributed. See Appendix B.
14Kreps (1998).
11
3.2 Recursive Formulation
The value of entering week t unemployed with beliefs characterized by αt and βt may be written
recursively as
V Ut (αt, βt) = max
st
Eλt
[F (st;λ)Eωt
[V Ot (ω, αt+1, βt+1)
]+ (1− F (st;λ))[b+ δV U
t+1(αt+1, βt+1)]
]− ηst
(11)
where V Ot (ω, ·) denotes the value of having offer ω in hand and may be written as
V Ot (ω, αt+1, βt+1) = max
ω + δV E
t+1(ω, αt+1, βt+1), b+ δV Ut+1(αt+1, βt+1)
. (12)
The value of entering period t+ 1 employed at wage ω is given by
V Et+1(ω, αt+1, βt+1) = (1− ρ)
[ω + δV E
t+2(ω, αt+1, βt+1)]
+ ρV Ut+1(αt+1, βt+1). (13)
3.3 Solution
I solve the model in two stages. First, I characterize optimal behavior at the end of the period
for job seekers who have received offers. This turns out to take the form of a familiar reservation
wage policy. Second, I determine optimal search time at the start of the period conditional on the
reservation wage policy determined in the first stage.
3.3.1 Reservation wage
Consider first the problem of an unemployed job seeker with a known offer ω in hand. Assuming
that the wage rate during employment is expected to be constant and that no offers arrive during
employment, V E(·) is time-invariant which implies that (12) and (13) reduce to
V Ot (ω, αt+1, βt+1) = max
ω +
δ
1− δ(1− ρ)
[(1− ρ)ω + ρV U
t+1(αt+1, βt+1)
], b+ δV U
t+1(αt+1, βt+1)
.
(14)
Because the first argument in the max operator in equation (14) is strictly increasing in ω, while
the second is constant, the optimal choice between accepting and rejecting the offer may be char-
acterized by a standard reservation wage policy:
V Ot (ω, αt+1, βt+1) =
ω +δ
1− δ(1− ρ)
[(1− ρ)ω + ρV U
t+1(αt+1, βt+1)
]if ω >
¯wt
b+ δV Ut+1(αt+1, βt+1) if ω ≤
¯wt
(15)
and
¯wt = (1− δ(1− ρ))b+ (1− δ)(1− ρ)δV U
t+1(αt+1, βt+1). (16)
12
3.3.2 Search time
Consider next an unemployed job seeker at the beginning of week t who has not yet begun to search
for work. Imposing the restriction that job seekers are myopic with respect to their own beliefs as
discussed above, and making explicit the belief distribution, (11) may be written as
V Ut (αt, βt) = max
st
∞∫
0
[F (st;λ)Eωt
[V Ot (ω, αt, βt)
]+ (1− F (st;λ))[b+ δV U
t+1(αt, βt)]
]γ(λ;αt, βt)dλ− ηst
.
(17)
The first-order condition for the choice of st is given by
η =
∞∫0
f(st;λ)
[1
1− δ(1− ρ)
∫ B
¯wt
(ω −¯wt)φ(ω)dω
]γ(λ;αt, βt)dλ. (18)
3.4 Characterizing Search Dynamics
Elimination of the value function from (16), (17) and (18) yields the two key equations describing
search time and reservation wage dynamics in the model:
st = βt
( 1
η(1− δ(1− ρ))
∫ B
¯wt
(ω −¯wt)φ(ω)dω
(αtβt
)) 1αt+1
− 1
(19)
¯wt = b+
[1−
(βt
βt + st
)αt]( δ(1− ρ)
1− δ(1− ρ)
∫ B
¯wt
(ω −¯wt)φ(ω)dω
)− δ(1− ρ)ηst. (20)
3.4.1 Motivation vs. discouragement
Search dynamics in the model are driven by endogenously-varying beliefs regarding the unobserved
arrival rate of job offers. Specifically, a reduction in the perceived probability of finding work can
be decomposed into competing discouragement and motivation effects:
∂st∂βt
=
[βt + stαt + 1
]·[
αtβt− αt + 1
βt + st︸ ︷︷ ︸Discouragement Effect
− (1− Φ(¯wt))∫ B
¯wt
(ω −¯wt)φ(ω)dω
∂¯wt∂βt︸ ︷︷ ︸
Motivation Effect
]. (21)
A reduction in the perceived job finding rate increases search time through the motivation effect:
as the perceived likelihood of finding a job falls, so too does the option value of unemployment, thus
stimulating search to find an offer more quickly. By contrast, the same reduction in the perceived job
finding rate decreases search time through the discouragement effect: as the perceived likelihood of
13
finding a job falls, so too does the opportunity cost of home production or leisure, thereby inducing
a substitution away from search effort.
Because αt and βt in (21) are endogenous, the relative strength of these effects varies over the
unemployment spell. The model is therefore capable of generating non-monotonic dynamics in
time devoted to job search. To make more precise statements about the dynamic trajectory of
search time over the unemployment spell, it is necessary to first take a stance on the primitives of
the model and the distribution of beliefs at the time of job loss, as embodied in α0 and β0. I revisit
this question in the context of the Great Recession after estimating the structural parameters of
the model in Section 4. First, however, I explore some potential implications of the model.
3.4.2 Declining search time
A natural question is whether or not the model is capable of generating search dynamics consistent
with the regression results in Table 2. To answer this question, note that the variable βt ≡∑t−1
τ=0 sτ
in the model is precisely the stock variable on the right-hand side of the regressions in Table 2.
We are therefore simply interested in the conditions under which the partial derivative in (21) is
negative. The condition is stated as Proposition 1:
Proposition 1. For a given number of job offers αt, time devoted to search is decreasing in the
stock of past time spent searching iff
βtαt
> st + s2t
(αt + 1
αt
)δ(1− ρ)η
11−Φ(
¯wt)
∫ B¯wtωφ(ω)dω − c
. (22)
Proof. See Appendix B.
The left-hand side of the condition in Proposition 1 is the inverse of the expected value of the
unobserved arrival rate. In light of the interpretation of βt as the total stock of past search time,
and αt as the total number of job offers received, the left-hand side of the expression is just the
observed average search time per offer.
As a theoretical matter, Proposition 1 states that the model is capable of generating declining search
over the unemployment spell as individuals become discouraged. I directly address the empirical
relevance of this claim in Section 4.
3.4.3 Negative duration dependence
The foregoing discussion suggests that the model may also be capable of generating negative dura-
tion dependence in unemployment exit rates despite a constant underlying arrival rate of job offers
per unit of search time and a time-invariant wage offer distribution. Whether or not negative dura-
tion dependence obtains depends on whether the effect of potentially declining time devoted to job
14
search (which implies a falling exit probability) dominates the effect of a monotonically declining
reservation wage (which implies an increasing exit probability). The precise condition is given in
Proposition 2:
Proposition 2. For a given number of job offers αt, the probability of exiting unemployment is
declining in the stock of past time spent searching iff
f(st;λT )
F (st;λT )
(βt + stαt + 1
)[αtβt− αt + 1
βt + st−
δ(1−ρ)ηstβt
11−Φ(
¯wt)
∫B¯wtωφ(ω)dω−c+δ(1−ρ)ηst
]
<φ(
¯w′t)
1− Φ(¯w′t)
δ(1−ρ)ηs′t
βt+τt
1+(1−Φ(¯w′t))
δ(1−ρ)1−δ(1−ρ)
(1−(
βt+τtβt+τt+s
′t
)αt+1) . (23)
Proof. See Appendix B.
As discussed in Appendix B, the prime notation on the right-hand side of the inequality in Propo-
sition 2 results from the fact that in the model, job seekers update beliefs upon receiving an offer,
but before deciding whether or not to accept. This implies that the relevant reservation wage for
the accept-reject decision after receiving an offer is different from the reservation wage anticipated
at the beginning of the period when the time allocation decision is made.
As before, however, it is difficult to assess the empirical relevance of this condition without providing
empirical grounding for the initial values of αt and βt, the key endogenous variables in the model,
as well as other structural parameters of the model. I therefore turn to structurally estimating the
model in order to assess its ability to match key moments in the data.
4 Structural Estimation
This section structurally estimates the model developed in Section 3. I identify key structural
parameters—including those governing beliefs at the time of job loss—using the empirical profiles of
search time, job-finding probabilities and job-acceptance probabilities over the first two years of the
unemployment spell from the SUWNJ data. The estimated model can account for unemployment
dynamics along all three dimensions. By contrast, a flexibly-parameterized alternative model which
allows for dynamic selection on heterogeneous search costs, a cubic trend in the offer arrival rate,
and a cubic trend in the mean of the wage offer distribution—but no uncertainty—cannot.
15
4.1 Empirical Strategy
I use Indirect Inference to estimate the six key structural parameters of the model:15
Θ = [ α0, β0︸ ︷︷ ︸Beliefs
, λT , b, η, v︸ ︷︷ ︸Physical
]′. (24)
Estimation proceeds in three steps. First, I specify the auxiliary model. This is the lens through
which I compare the model with the data. Next, I estimate the parameters of the auxiliary model—
the auxiliary parameters—using the SUWNJ data. Finally, I choose the structural parameters Θ
so as to minimize the distance between the auxiliary parameters generated by the SUWNJ data
and the auxiliary parameters generated by simulating the model.16
4.1.1 Auxiliary model
Identifying the parameters governing perceptions about the job-finding process is nontrivial. In a
static setting, these parameters are not separately identified from standard structural parameters
such as the marginal cost of search η. However, if individuals are observed for sufficiently long
spells of unemployment and at a sufficiently high frequency, as in the SUWNJ, then identification
is possible simply because the learning process eventually comes to an end, after which point
the model is stationary. Initial beliefs are therefore identified from search dynamics early in the
unemployment spell, whereas the remaining structural parameters are identified once beliefs are
sufficiently focused and dynamics have died out.17
I implement the preceding identification strategy by specifying the auxiliary model as three linear
For individual i in interview week t, sit denotes the fraction of daily time devoted to job search, jit
is an indicator for whether or not a job offer was received, and ait is an indicator for whether or
not an offer was accepted conditional on having received an offer.18 The right-hand side variables
are indicators for unemployment duration, grouped into five-week bins: d1,it is an indicator for the
first five weeks of unemployment; d2,it for the second five weeks, so on and so forth through the
first two years of unemployment.
15I describe the alternative model and its parameters in detail below.16This approach is simply a generalization of the Simulated Method of Moments (SMM). Indeed, the auxiliary
parameters described in the next section are conditional first moments.17Note that, in the model, beliefs need not converge to the true parameter value, though in some cases they will.18Search time is measured using the SUWNJ time diary data. Results are qualitatively similar if instead the weekly
recall data are used.
16
The associated auxiliary parameters, which can be estimated from either actual or simulated data,
Indeed, but for the initial upward bias in beliefs, search effort and thus job-finding rates would have
been considerably lower during the Great Recession than they were.
21
5 Search Dynamics Prior to the Great Recession
This section considers whether the model developed in Section 3 can help to understand search
dynamics prior to the Great Recession. Using CPS data from prior to the Great Recession, Shimer
(2004) documents a hump-shaped profile in search effort over the first year of unemployment. More
recently, Mukoyama et al. (2014) exploit overlap in the ATUS and CPS sample frames to construct
an improved measure of search effort, and also find a hump-shaped profile over the unemployment
spell.19 In this section I demonstrate that when the model is recalibrated to match the average
job-finding rate in the US prior to the Great Recession, a sufficient degree of dispersion in beliefs
yields precisely the results documented by Shimer (2004) and Mukoyama et al. (2014).
I recalibrate the model by choosing values for α0, β0 and λT in order to match the average pre-Great
Recession unemployment exit probability of the short-term unemployed in the United States (p0)
for various degrees of bias (B) and dispersion (V ) in the initial belief distribution.20 This allows
me to study the effect of belief dispersion and the aggregate state of the labor market on model
dynamics. The key assumption underlying the calibration strategy is that individuals’ beliefs at the
time of job loss are conditioned so as to be consistent with the aggregate short-term unemployment
exit probability in the data (allowing for possible bias in those beliefs), after which point beliefs
evolve endogenously in response to search outcomes. The remaining structural parameters are fixed
at their values from Section 4. See Appendix C for details on the calibration procedure.
The restrictions used to pin down α0, β0 and λT are thus:
Bias: B =
[α0
β0− λT
]/λT (33)
Dispersion: V =α0
β20
(34)
Unempl. exit probability: p0 =
s(α0,β0)∫0
[1− Φ
(¯w(α0 + 1, β0 + τ)
)]f(τ ;λT )dτ. (35)
Figures 6a and 6b plot the estimated profiles of search time over the first 100 weeks of unemployment
generated from simulated data. The estimation procedure is identical to that used for the auxiliary
model described in Section 4. Qualitatively, the similarity of the high-dispersion scenario in Figure
6 with Figure 6 in Shimer (2004) and Figure B2 in Mukoyama et al. (2014) is striking. The muted
dynamics in the medium- and low-dispersions scenarios fail to capture the peak in search after the
first year of unemployment, but nonetheless generate rising search throughout the first year of the
19The CPS asks respondents who are searching for work what they are doing to find a job. Shimer (2004) measuressearch effort as the number of reported methods among searching respondents. Mukoyama et al. (2014) exploit overlapbetween the CPS and the ATUS to construct time-intensity weights for each of the search methods considered in theCPS, and use the weights to impute search time for the full CPS sample.
20I use the short-term unemployment exit probability as an approximation of the unemployment exit probabilityin the first week of unemployment.
Table A.3 reports the tests for first- and second-order autocorrelation in the first-differenced resid-
uals developed in Arellano and Bond (1991).21 If the errors εit of equation (1) in levels are serially
uncorrelated, then we should expect to see no evidence of second-order autocorrelation in the dif-
ferenced residuals.22 Evidence of second-order autocorrelation suggests that the assumption of no
serial correlation in εit is invalid, which in turn implies that sit−2 is not a valid instrument for sit−1,
thus necessitating the use of further lags.
Table A.3: Tests for Serial Correlation
Time Diary Weekly Recall
Statistic p-value Statistic p-value
Arellano-Bond test for AR(1) z = −6.89 0.0000 z = −4.47 0.0000
Arellano-Bond test for AR(2) z = 1.40 0.1601 z = −0.30 0.7679
Source: Survey of Unemployed Workers in New Jersey
H0: No serial correlation.
The results in Table A.3 suggest that the disturbances εit are serially uncorrelated, and therefore
that sit−2 is a valid instrument for sit−1. In Appendix A.6, I consider an expanded set of internal
instruments in a GMM framework, the results of which are consistent with the more parsimonious
approach developed in the body of the paper.
21The test was developed in the context of a GMM framework, but is nonetheless applicable to the simple 2SLSprocedure used in the body of the paper.
22First-order autocorrelation in the first-differenced residuals results mechanically from the process of taking firstdifferences. In general, differencing an MA(n) process yields an MA(n+ 1) process.
Source: Survey of Unemployed Workers in New Jersey
Notes: Baseline regression augmented with various trends in unemployment durationas described in the body of the text.∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
Source: Survey of Unemployed Workers in New Jersey
Notes: Baseline regression augmented with one-week leads of reported shocksto time spent looking for work as described in the body of the text.∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
34
Table A.7: Alternative Weights
Time Diary Weekly Recall
Person-Week Person Unweighted Person-week Person Unweighted
Past Search -0.105∗∗∗ -0.132∗∗∗ -0.119∗∗∗ -0.0783∗∗∗ -0.0735∗∗∗ -0.0520∗∗∗
Source: Survey of Unemployed Workers in New Jersey
Notes: Two-step Difference GMM with Windmeijer-corrected standard errors.∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
Table A.10: Tests of serial correlation and over-identifying restrictions (differences)
Time Diary Weekly Recall
Statistic p-value Statistic p-value
Arellano-Bond test for AR(1) z = −8.31 0.000 z = −5.02 0.000
Arellano-Bond test for AR(2) z = 1.21 0.225 z = −0.36 0.721
Sargan test of over-ID restrictions χ223 = 240.11 0.000 χ2
23 = 372.84 0.000Hansen test of over-ID restrictions χ2
23 = 33.13 0.079 χ223 = 19.79 0.654
Source: Survey of Unemployed Workers in New Jersey
H0 (AB): No serial correlation; H0 (Sargan/Hansen): Instruments are jointly exogenous.
23Specifically, I focus on Difference GMM and Orthogonal Deviations GMM. A System GMM approach is ruledout due to the fact that, for most individuals, the stock variable of interest is itself partially unobserved.
36
A.6.2 Orthogonal deviations
In order to circumvent the data loss associated with differencing, I also estimate a version of the
model in which individual effects are purged by taking forward-orthogonal deviations.24 Table A.11
reports the results, and Table A.12 reports the associated tests of instrument validity.
Source: Survey of Unemployed Workers in New Jersey
Notes: Baseline regression with the stock of past search time replaced by the stock of pastapplications submitted. Applications is also included as a left-hand side variable.∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
loss does not significantly affect the amount of time individuals devote to job search, as would be
predicted by a model of stock-flow matching.
39
A.8 A structural interpretation
To estimate the system described by (4) and (5), I begin by taking first differences of both equations
to eliminate the individual effects and the unobserved component of total search time since job loss.
Estimation then proceeds in three steps: I first construct instruments for the endogenous variables
appearing in the equations in differences (sit−1 and ∆pit) by projecting these variables on all
included exogenous variables and sit−2. I then estimate (4) and (5) via 2SLS instrumenting with
predicted values from the first step. Finally, I use the residuals from these regressions to construct
a consistent estimate of the covariance matrix needed to implement generalized least squares.25
Table A.14 reports the parameter estimates from the structural model.
Source: Survey of Unemployed Workers in New Jersey
Notes: The sample consists of respondents ages 20-65 who have not received a joboffer and who left their previous job involuntarily and do not expect to return.The model is estimated via 3SLS as described in the text, with the second lag ofsearch time included as an instrument.∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
25This procedure amounts to estimating a seemingly-unrelated regression (SUR) model in which residuals areobtained in the first step using 2SLS.
40
B Model Solution and Proofs
B.1 Posterior distribution of beliefs
In this section I demonstrate that the Gamma distribution is the conjugate prior for the right-
censored exponential distribution, and derive the laws of motion for the parameters of the belief
distribution assuming Bayesian updating. Consider an individual who has been unemployed for n
weeks. For each week t = 1, ..., n of the unemployment spell, the individual has devoted st units of
time to searching for work. Define K ≡ t : τt ≤ st as the set of weeks in which an offer (below
the reservation wage) arrives before search ends, ns ≡ #K and nf ≡ n − ns. For weeks t ∈ K,
individuals observe the exact arrival time τt ≤ st. For the remaining weeks t 6∈ K, individuals only
observe that τt > st.
Because offers arrive according to a Poisson process with unobserved rate parameter λ, arrival
times are distributed according to a right-censored exponential distribution with distribution func-
tion F and density f . The corresponding likelihood function for λ is thus given by
L (λ) = L (λ|τtt∈K ; stt6∈K) =∏t∈K
f(τt|λ)∏t6∈K
(1− F (st|λ)) (B.1)
=∏t∈K
λe−λτt∏t6∈K
e−λst (B.2)
= λnse−λ(
∑t∈K τt+
∑t6∈K st) (B.3)
= λnse−λ(nsτ+nf s) (B.4)
where τ ≡ 1ns∑
t∈K τt and s ≡ 1nf
∑t6∈K st.
Suppose now that prior beliefs over λ follow a Gamma distribution with hyperparameters α0 and
β0, distribution function G(λ|α0, β0) and density g(λ|α0, β0). Applying Bayes’ rule and using the
expression for the likelihood function above, the posterior distribution of beliefs over λ is given by