Evaluating the E®ectiveness of Washington State Repeated
Job Search Services on the Employment Rate of Prime-age
Female Welfare Recipients¤

July 29th, 2003

1

Abstract
This paper uses an unbalanced panel dataset to evaluate how repeated job search services and personal characteristics a®ect the employment rate of the prime-age female welfare
recipients in the State of Washington. We propose a transition probability model to takeaccount issues of sample attrition, sample refreshment and duration dependence. We also
propose a nonlinear two-stage least square estimator to allow for selection due to unobservables and generalize Honor¶e and Kyriazidou's (2000) conditional maximum likelihood
estimator to allow for state-dependent individual speci¯c e®ects and slope coe±cients. We
then provide a conditional nonlinear two-stage least square estimator that allows for both
unobserved individual heterogeneity and selection on unobservables. The speci¯cation tests
indicate that the assumptions of no selection due to observables or no unobserved individual
speci¯c e®ects are not violated. Our ¯ndings indicate that job search services do have positive and signi¯cant impacts on the employment rate of those who are initially unemployed,
but their impacts are insigni¯cant for those who are employed. Furthermore, there are
signi¯cant experience enhancing e®ects. These ¯ndings suggest that providing job searchservices to unemployed individuals to help them ¯nd jobs quickly may have a lasting impact
on raising their employment rate.
¤

This research was supported in part by Employment Security Department, State of Washington, but does
not necessarily represent an o±cial position of Employment Security Department. We thank M. Kaiser andGeert Ridder for very helpful comments and suggestions. After the completion of the preliminary version of thispaper, we have come across a paper by D'Addio and Honor¶e (2002) that proposes a similar conditional maximum
likelihood estimator for a state dependent coe±cients and individual e®ects logit model.

2

1

Introduction

The aim of this paper is to provide measurements of the e®ects of repeated job search services(JSS) on the employment rates of female welfare recipients who participated in the WorkFirst
program of the State of Washington. The WorkFirst program is the implementation of theFederal Temporary Assistance for Needy Family (TANF) program in the State of Washington.
Initiated in August 1997, its main goal is to help ¯nancially struggling families to ¯nd jobs,
keep themselves o® unemployment, and get better jobs. Emphasizing that getting a low-paying
job now is better than waiting for a high-paying job in the future, the WorkFirst program
has a process that focuses its main activities on job search services (JSS). Nevertheless, TANF
recipients have been returning to welfare in great numbers. Over 70 percent of the entrants towelfare were former TANF recipients during the sample period (1998.II - 2000.IV). To e±ciently
allocate the limited resources, policy makers are particularly interested in ¯nding out whether it
is e±cient to provide Job Search Services (JSS) to the same clients repeatedly.
In this study the population of interest are female TANF recipients between the age 25 35. However, our data set is not a balanced panel data as some of the early studies (e.g., Bassi(1984), Ashenfelter and Card (1985), Heckman and Hotz (1989)). A signi¯cant feature of ourdata is that clients entered and left the program at di®erent time periods. For instance, only
about 3.4 percent clients have the complete treatment history over the sample period 1998.II to
2000.IV. There is also the issue of right censoring because the data period ends at 2000.IV. If
we restrict our attention to the subsample of clients that entered and left a training programin the same time periods, it would greatly reduce available observations. Moreover, misleading
inference may occur if clients are not randomly selected from the whole population. If we choose3to start our sample later than 1998.II to keep as many balanced panel observations as possible,
participation histories would be incomplete for those who entered the program late, hence makes
the estimation of repeated treatment e®ects di±cult.
In this paper we propose a transition probability model of ¯nding employment or staying
on employment as a means to take account of issues arising from sample attrition, sample refreshment and duration dependence. Being state dependent, a transition probability model also
allows one to accommodate dynamics in a simple format. We ¯rst estimate such a model byassuming that (1) participations of JSS are not endogenous so that there is no bias stemmingfrom selection on unobservables; and (2) there are no unobserved individual speci¯c e®ects. To
check the validity of these assumptions, we also suggest three estimators that relax one or bothof the above two assumptions: a nonlinear two-stage least square estimator that deals with theendogeneity of participation decisions, a generalized conditional maximum likelihood estimatorthat allows for state-dependent ¯xed e®ects and slope coe±cients, and a conditional nonlinear
two-stage least square estimator that allows for both unobserved individual heterogeneity andendogeneity of the participation decision. The Hausman (1978) speci¯cation tests indicate thatthe above two assumptions are not contradicted by our sample information, therefore, we shalldiscuss our empirical ¯ndings from the model treating participation decisions as exogenous andwithout the presence of unobserved individual speci¯c e®ects.
Our ¯ndings show that the repeated job search services do have positive and signi¯cant e®ects
on the employment rate of initially unemployed clients, but their e®ects are not signi¯cant for
those who are already employed. Furthermore, the probability of employment is also in°uenced
by the duration in employment or unemployment, family factors, education level, geographic and4local labor market conditions as well as other welfare services.
Section 2 introduces the model and section 3 presents the estimation results. Diagnostic
checkings are discussed in section 4. Conclusions are in Section 5. Detailed descriptions of ourdata are available upon request.

2

The Model

2.1

A Transitional Probability Model for the Outcomes

In this section we propose a transitional probability framework to take account issues of sample
attrition, sample refreshment and duration dependence. Let yit be the binary indicator thattakes the value 1 if the ith client is employed and the value 0 if otherwise, t = ti , ti + 1, ...Ti ,where ti and Ti denote the ¯rst period and last period that client i is observed. We assume
that yit depends on the initial states, yi;t¡1 , previous JSS treatments, and strictly exogenous
socio-demographic variables, xit . We separate yit into two groups depending on the value ofyi;t¡1 being 0 or 1. Let yits be those yit where yi;t¡1 = s; s = 0; 1: Let yits¤ denote the potential
state given yi;t¡1 = s; s = 0; 1. We assume thatyits =

8
>
>
<
>
>
:

1; if yits¤ > 0;

(1)0; if otherwise.

A client is considered to have taken JSS if records show that she has taken at least one JSS
within one quarter. Since over 95 percent participants took no more than 3 JSS over the periodof 1998.II - 2000.IV, we will focus on the evaluation of the e®ects of repeated JSS to be at most5three. We consider the e®ects of previous period JSS treatment on current yit since one JSS canlast up to 12 continuous weeks. A client taking at most 3 JSS before period t is in one of thefour possible potential states: (1) she has not taken any JSS; (2) she has taken one JSS; (3) she
has taken two JSS; and (4) she has taken three JSS. Let d~mit be mutually exclusive dummies such
that

d~m
it =

8
>
>
<
>
>
:

1, if exactly m JSS has(have) been taken before period t;(2)0, otherwise,m = 0; 1; 2; 3:

We assume that JSS participations in°uence the probability of employment through its impacts¤on potential outcomes. Depending on the realization of d~m
it ; yit can take one of the four possible

forms, yitsm¤ , m = 0; 1; 2; 3, where yitsm¤ denotes the the corresponding potential outcome when
s¤
sm¤
d~m
has the following relation,it = 1; m = 0; 1; 2; 3: At time t, yit and yitionhyits¤ = d~1it yits1¤ + (1 ¡ d~1it ) d~2it yits2¤ + (1 ¡ d~2it ) d~3it yits3¤ + (1 ¡ d~3it )yits0¤~2 ~ s2 + d~3 °~ s3 ;= yits0¤ + d~1it °~ s1
it + dit °
it
it it

(3)

sm¤
where °~ sm¡ yits0¤ ; m = 1; 2; 3; measures the cumulative e®ect of m JSS over no JSS for
it = yit

the ith individual at the tth time period, when the no-JSS state is treated as the base state, i:e:;d~0it = 1 when d~1it = d~2it = d~3it = 0, and °~ s0
it = 0.
There is a one-to-one relation between the cumulative e®ect and marginal e®ect. Rewrite

6equation (3) asyits¤ = yits0¤ + Dit ° sit ;(4)

where Dit = [d1it ; d2it ; d3it ]; with Dit = [0; 0; 0] when d~0it = 1; Dit = [1; 0; 0] when d~1it = 1; Dit =
s2
s3 0
sm
[1; 1; 0] when d~2it = 1, and Dit = [1; 1; 1] when d~3it = 1, and ° sit = (° s1
it ; ° it ; ° it ) with ° it =s(m¡1)~ it°~ sm
it ¡ °s(m¡1)¤= yitsm¤ ¡ yit; m = 1; 2; 3:

The value of ° sm
it measures the marginal impact of the mth job search service on the ith
individual at time t conditional on her last period employment status being s. The averagetreatment e®ect for the mth job search service conditional on last period employment status
being s is E (° sm
it ).
However, yitsm¤ is not observable. We shall therefore de¯ne the treatment e®ects of the mth
JSS on client i in terms of the changes in the probability of employment conditional on lastperiod's employment status being s. For simplicity, we assume that yitsm¤ can be decomposed as
the sum of the e®ects of observables xit ; g sm (xit ) ; and the e®ects of unobservables, usm
it ;yitsm¤ = g sm (xit ) + usm
it , m = 0; 1; 2; 3:(5)Then Pr (yit = 1jyi;t¡1 ; xit ) = Pr (yits¤ > 0jyi;t¡1 ; xit ) ; and the treatment e®ect of the mth JSSs(m¡1)¤

is de¯ned as ¢sm= Pr (yitsm¤ > 0jyi;t¡1 = s; xit ) ¡ Pr(yititm = 1; 2; 3: The average treatment e®ect (ATE)> 0jyi;t¡1 = s; xit ); s = 0; 1;of the mth JSS conditional on last pe-smsm¤
riod's employment status s is de¯ned as ¢sm> 0jyi;t¡1 = s; xit )¡AT E = E [¢i ] = E[Pr(yits(m¡1)¤

Pr(yit> 0j yi;t¡1 = s; xit )]. The treatment of the treated (TT) of the mth JSS conditional

7

sm m
on last period's employment status s is de¯ned as ¢sm
T T = E (¢it jdit = 1) :ATE of the mth JSS is the mean impact of the mth JSS if clients are randomly assigned to
JSS. It is of interest if one is interested in estimating the impact of the mth JSS on a randomly
selected clients. TT of the mth JSS is the mean impact of the mth JSS on those clients who
actually have taken the mth JSS. It is of interest if the same selection rule for treatment applies
in the future.
s(m¡1)¤

Because it is impossible to simultaneously observe yitsm¤ and yit; m = 1; 2; 3; we cannotsmdirectly estimate ¢sm
AT E and ¢T T . In essence, to evaluate treatment e®ect is to deal with a

missing data problem. If we approximate the no-JSS outcomes of the treated group by observed
outcomes from the control group and calculate the treatment e®ect by
X
1
1 X
yjt ¡
yjt ; m = 1; 2; 3
Nm j2©
Nm¡1 j2©
m

(6)

m¡1

where Nm ; Nm¡1 is the number of clients who have taken the mth and the (m ¡ 1)th treatment
respectively, and ©m ; ©m¡1 is the set that includes the corresponding client respectively, thenequation (6) approachesm
E (yt = 1jyt¡1 = s; dm
t = 1) ¡ E (yt = 1jyt¡1 = s; dt = 0)ns(m¡1)¤= E Pr (ytsm¤ > 0jyt¡1 = s; dm> 0jyt¡1 = s; dm
t = 1) ¡ Pr(ytt = 1)hios(m¡1)¤s(m¡1)¤mm+ Pr(yt> 0jyt¡1 = s; dt = 1) ¡ Pr(yt> 0jyt¡1 = s; dt = 0)

sm
= ¢sm
T T + Bt ;s(m¡1)¤

where Btsm = E([(Pr yts(m¡1)¤> 0jyt¡1 = s; dm
t = 1) ¡ Pr(yt8> 0jyt¡1 = s; dm
t = 0)]) is thebias for estimating the e®ect of the mth treatment conditional on employment status s resulted
from using control group to approximate the no-JSS state of the treated group. If Btsm = 0, then(6) provides a consistent estimate of ¢sm
TT .
When

m
usm
¹it ;
it ? dit jyi;t¡1 ; x(7)where x¹0it = (x0it ; x0i;t¡1 ; :::x0iti ); the treatment assignment is free from the in°uence of unobserved
factors a®ecting yitsm¤ conditional on (yi;t¡1 ; x¹it ). Condition (7) is called Conditional Independence Assumption (CI) by Rosenbaum and Rubin (1983) or Ignorable Treatment Assignment
Assumption by Heckman and Robb (1985) and Holland (1986). Thens(m¡1)¤
Pr(yits(m¡1)¤ > 0jyi;t¡1 = s; xit ; dm
> 0jyi;t¡1 = s; xit ; dm
it = 1) = Pr(yitit = 0):In other words, under CI, conditional on x; ATE(x) =TT(x).
For the unconditional bias Btsm to equal to zero we also need

xit ? dm
it jyi;t¡1 :(8)

When condition (8) does not hold, we say that the selection bias is due to observables or selection
on observables. In other words, for Btsm = 0 unconditionally, we need that there is neither
selection on observables nor unobservables.
Usually, the xit that a®ects the potential outcome may also a®ect the participation decision.

9

It is hard to assume that (xit ? dm
it jyi;t¡1 ). Therefore, we shall analyze the treatment e®ects bysimultaneously controlling the impacts of xit and JSS. The nonparametric approach that takes
account both the timing of treatments and interval between treatments involves many possible
regimes. It is very complicated to analyze and requires huge number of observations (e.g., Gill
and Robins (2001), Lechner (2001)). To extract information from ¯nite sample, in this paper weshall take a parametric approach. Since our sample only covers a relatively short period from1998 - 2000, for simplicity, we assume a linear structure for yits¤ ,yits¤ = x0it ¯ s +Dit ° s + usit :(9)

We shall derive our model speci¯cation and inference under the CI assumption in this section.
We then consider methods of testing CI assumption and methods of estimating ATE or TT
conditioning on x
¹it when CI is violated in section 5.
Let Pisk = Pr (yit = kjyi;t¡1 = s) ; s; k = 0; 1; be the transition probability that the ith individual is in state j in period t ¡ 1 and k in period t,

Pis1 = Pr (yit = 1jyi;t¡1 = s; xi )(10)

= Pr (xit ¯ s + Dit ° s + usit > 0)
= F (xit ¯s + Dit ° s ) = Fits ; s = 0; 1;

and F (¢) denotes a certain cumulative distribution function. In this study, we assume F takes

10the logit form,

exp (xit ¯ s + Dit ° s ); s = 0; 1; t = ti + 1; :::; Ti :1 + exp (xit ¯s + Dit ° s )Fits =(11)

Since the lagged values of the initial employment status are not observed, the initial state yiti isapproximated by an unconditional speci¯cation,

yiti = Q (¹xi ) + "iti ; i = 1; :::; N;

where x¹i = (Ti ¡t1i +1)

PTi
ti

(12)

xit ; and Q denotes a monotonic function. The means of the explanatory

variables are used instead of xi = (x0iti ; :::x0iTi ) as it usually yields better ¯nite sample results
when the variation of xit over time is limited (Hsiao (2003)).
¡¢
Let yi = yiti ; yi;(ti +1) ; :::yiTi and piti = Pr (yiti = 1j¹
xi ). Under the assumption that u1it and u0it

are independent across individuals, the likelihood function for all N individuals takes the form

L=

Ti
N
Y
Y

(1¡yit ) yi;t¡1

yit
[Pi11
Pi10

i=1 t=ti +1](1¡yit ) (1¡yi;t¡1 ) yiti]
piti (1 ¡ piti )(1¡yiti ) :

yit
[Pi01
Pi00

(13)

Equation (13) is similar to the likelihood functions of the binary qualitative response models.
Because the log likelihood function is the sum of the log likelihoods for the job-holder group (¡¢¡¢yi;t¡1 = 1), for the job-seeker group ( yi;t¡1 = 0) and for the initial states, ¯1 ; ° 1 and ¯0 ; ° 0

can be estimated separately using the data points from the job-holder group and those of thejob-seeker group respectively.
0

Let µs = (¯ s0 ; ° s0 ) be an m £ 1 vector of unknown parameters, and µs 2 £s . Let wit =11

(x0it ; D0it )0 . The MLE of model (13) is consistent and asymptotically normally distributed under
the assumptions that
A1 The parameter space £s is an open bounded subset of the Euclidean m-space.
A2 fwit g is uniformly bounded in i and t and limn!1 n1

P P
i

0
t wit wit is a ¯nite nonsingularmatrix, where n denotes the total number of observations over i and t. Furthermore, theempirical distribution of fwit g converges to a distribution function.

2.2

Evaluation of The Conditional and Unconditional Impacts

We are interested in two questions. First, how do repeated JSS and personal characteristics a®ect
the employment rate of the job-seekers and the job-holders, respectively? Second, what are theunconditional impacts of JSS and other characteristics on the employment rate regardless of anindividual's previous period employment status?
We can calcuate the conditional impact of the mth JSS for job seekers and job holders fromm
E (yt = 1jyt¡1 = s; dm
t = 1; xt ) ¡ E (yt = 1jyt¡1 = s; dt = 0; xt ) ; s = 0; 1; m = 1; 2; 3: If xt is¢¡
randomly drawn, E yt = 1jyt¡1 = s; dmt¡1 can be approximated by the sample average of thepredicted probabilities of those who have dm
it = d, d = 0; 1. We can also evaluate the impact

of one-unit change of xij on Pis1 by @Pis1 =@xij if xij is continuous. The population impact of a
one-unit change of xj is

R @Pis1

can be approximated by N1@xij

dF (xi ). Assuming that xij is randomly distributed, this impact

P @Pis1

i @xij .

The transitional probability framework also allows one to trace out an individual's dynamic
path of Pist from its initial state. However, for simplicity we shall only provide the unconditional impacts of JSS and other socio-demographic variables regardless of an individual's em12

ployment status. The equilibrium or marginal probability of employment is computed from
¼ 1i = Fi0 = (1 ¡ Fi1 + Fi0 ), where Fi0 is de¯ned in (10) that are are evaluated at x
¹i = N1

3

P

i xi .

Findings

In this section we provide estimates of model (10) that considers the following social-demographic
factors as explanatory variables: (i) participation of the WorkFirst program such as JSS, alternative services (AS) ( for clients who could not participate JSS directly due to problems like drug
abusement and family violence), and post-employment services (PS) (for clients who have got atleast part time jobs) dummies; (ii) duration dependence such as number of quarters employedor unemployed; (iii) welfare history; (iv) family information such as number of adults, numberof children, age of the youngest child, marital status; (v) race and ethnicity such as dummies forwhites, blacks and Hispanic; (vi) language and education such as English speaking and grade 12
and above dummies; (vii) local economy such as local unemployment rate; and (viii) geographic
and time dummies. A full description of these variables are presented in Table 1.
Table 2 and Table 3 columns 3 and 4 provide the estimates of the impacts of JSS and othersocio-demographic factors on the probability of being employed for job-seekers and job-holders,
respectively. For job-seekers we note that ¯rst, job search services have signi¯cant impacts on
the probability of employment (at 5% level), with the ¯rst JSS having the biggest impact (0.32),
and the second JSS and the third JSS having less but still statistically signi¯cant impacts (0.10
and 0.12 respectively); and second, the longer an individual stays unemployed, the less likely she
will ¯nd a job (the estimated coe±cient for duration of unemployment is -0.1240). These two
results put together make a strong case for the state to provide job search services to unemployed13

individuals quickly to get them stay out of unemployment.
For job-holders it shows that quite a few variables that are signi¯cant for the job seekers
turn out to be insigni¯cant. None of coe±cients of the job search services are statisticallysigni¯cant. The estimated coe±cients for the ¯rst, the second and the third JSS are 0.04, -0.01
and 0.06, respectively, and the corresponding standard errors are 0.04, 0.06 and 0.08, respectively.
Total number of previous participations in post employment services (PS) also has no signi¯cantimpacts on employment rate, neither is marital status nor the race or ethnic dummy for black
or for Hispanic. In addition, unemployment duration is a negative but insigni¯cant factor in
determining the employment rate of the job-holders. On the other hand, the longer an individualis employed, the higher the probability that she will stay employed next period. This seems tosuggest that once a client is employed, what matters is not unemployment history, but her
employment history.
The impacts of one-unit change in explanatory variables to the probability of being employed
for job seekers and for job holders are reported in column 3 and 4 of Table 4, respectively. This
table shows that for job seekers, the ¯rst JSS increases the probability of being employed by 8.45
percent on the treated group, the second JSS increases it by a further 2.15 percent, and thethird JSS increases it by an additional 0.8 percent respectively. On the other hand, none of thethree JSS have statistically signi¯cant impacts on job holders.
Column 5 of Table 4 presents the equilibrium impacts of the repeated JSS participations aswell as other characteristics regardless of an individual's initial state. It shows that in the longrun, the ¯rst job search service increases the probability of being employed by 3.7 percent, thesecond job search service increases it by a further 0.3 percent, and the third job search service

14increases it by an additional 0.7 percent.
Columns 3, 4 and 5 of Table 4 show that the longer one stays unemployed, the less chance
one has for employment. On the other hand, the longer one stays employed, the higher the
chance that she stays employed in the future. This information together with the ¯nding that
unconditionally, JSS have positive e®ects on the probability of employment may shed light on
the debate between the education-¯rst or the employment-¯rst strategy.

4

Diagnostic Checking

The inference reported on section 3 requires the validity of the conditional independence assumption. It is also assumed that there is no unobserved individual speci¯c e®ects that could becorrelated with wit . If the above two assumptions do not hold, then our maximum likelihood
estimates are biased. In this section we propose methods to relax the above assumptions andthen check whether these assumptions are valid. We start with the possibility that there areunobserved characteristics that can a®ect the selection decision, that is, we allow for the possibility that selection is due to unobservables. Then we generalize Honor¶e and Kyriazidou's (2000)
conditional maximum likelihood estimator to allow for the presence of unobserved individual speci¯c e®ects in the main equation assuming no selection on unobservables conditional on wit andthe individual e®ects. Finally, a conditional nonlinear two-stage least square estimator is proposed to allow for both selection due to unobservables and unobserved individual heterogeneity.
Hausman (1978) statistics are constructed to test for the validities of the CI assumption and thenonexistence of unobserved individual speci¯c e®ects. Our sequential (or conditional) procedures
allow for the use of more sample observations conditional on other assumptions being valid or15

relax the need for good instruments because consistent estimation methods that simultaneously
relax both assumptions impose severe restrictions on the data that can lead to signi¯cant loss ofsample information. Moreover, the conditional testing procedures are more powerful to detect
the alternative if the conditional event is true. Pedagogically, it is also much simpler to show thevalidity of proposed procedures before presenting a simultaneous test of CI and the presence ofunobserved individual speci¯c e®ects.

4.1

Controlling for the Selection on Unobservables

In this subsection we consider the situation that the CI assumption does not hold. Instead of
specifying a complete model of the interactions between potential or actual outcomes with sequential participation decisions that may depend on interim outcomes in a dynamic optimization
framework, as long as the employment status is given by (9), we can use a limited information
framework to take account of the issue that after controlling for the observed characteristics thetreatment decision can still be correlated with the unobserved personal characteristics in thepotential or actual state equation. We note thatyits = Fits + ´ sit :(14)

Then with probability Fits , yits = 1 and with probability (1 ¡ Fits ) ; yits = 0; it follows that E (´ sit ) =Fits (1 ¡ Fits )+(1 ¡ Fits ) (¡Fits ) = 0. Therefore, if we can ¯nd instruments zit such that Ezit ´sit = 0
and Ezit wit0 has full column rank, where wit0 = (x0it ; D0it ) then we can apply Amemiya's (1974)
nonlinear two stage least squares estimator to obtain a consistent estimator of µs in a limited
¡
¢
0 0
s0
s0 0
information framework. Let Y = (y10 ; :::; yN) , yi0 = yiti ; yi;(ti +1) ; :::yiTi ; F s = (Fs01 ; F2 ; :::FN ) ,
16

0
s
s
s
Fs0
i = (Fiti ; :::FiTi ) , Z = (z1 ; :::zN ) and zi = (ziti ; :::; ziTi ). The NL2SLS estimator of µ isde¯ned as the solution that minimizes

S s = (Y ¡ Fs )0 Z(Z 0 Z)¡1 Z (Y ¡ Fs ) :(15)

Theorem 1 Under A1, A2 and the assumptions that
A3 If µs 6= µ ¤s , Pr [Pr (y s = 1jw; µs ) 6= Pr (y s = 1jw; µ¤s )] > 0;A4 lim n¡1
A5 lim n¡1

P P
P P

@F s
t zit @µ s0i

A6 plimn¡1
A7 lim n¡10t zit ziti exists and is nonsingular;iconverges in probability uniformly in £s ;P P

@F s st zit @µ s jµ is full rank;

i

P P
i

@2F s
s0
t zit @µ s0
j @µ

converges in probability to a ¯nite matrix uniformly in µs 2 £s ,where µsj is the j-th element of the vector µs ;sthe values of ^µ that minimizes(15) is consistent with asymptotic covariance matrix´³^NL2SLS = [G0 Z (Z 0 Z)¡1 Z 0 G]¡1 £ [G0 Z(Z 0 Z)¡1 Z 0 V Z (Z 0 Z)¡1 Z 0 G]
Asy Cov µ£[G0 Z (Z 0 Z)

¡1

Z 0 G]¡1 ;where G is the stacked matrix of @F s =@µs0 , and V is a diagonal matrix with diagonal elements
equal to Fits (1 ¡ Fits ).
We use sociodemographic variables that are excluded from xit as instruments. Among them
are regional dummies and number of children in the household. Using the MLE of µ as initial
17

values, we apply quadratic hill climbing procedure (Quandt (1983)) to the method of score toiterate until convergence. The convergence criteria is 0.001. Column 5 and column 6 of Table
2 and Table 3 present the NL2SLS estimates and the corresponding standard errors for initiallyunemployed clients and for those who already have jobs, respectively. These results show that
the MLE and the NL2SLS are quite similar. Because the NL2SLS is consistent regardless of thevalidity of the CI assumption while the MLE is only consistent when the CI assumption holds,
and also because the MLE is e±cient under the CI assumption, a Hausman (1978) test statistic
can be constructed to test the validity of the CI assumption. The calculated Hausman statistic
is 2.66 for those who are initially unemployed and 0.54 for those who already have jobs. They
are not signi¯cant at 5% level for a Chi square distribution with 16 degrees of freecom. In otherwords, the information of the data does not appear to contradict the CI assumption.

4.2

Controlling for Individual speci¯c E®ects

In this subsection we propose to generalize Honor¶e and Kyriazidou's (2000) conditional maximum
likelihood estimator to allow for the presence of state-dependent individual speci¯c e®ects as well
as slope coe±cients. We assume that the error term can be decomposed into two parts,

usit = ®si + "sit ; s = 0; 1;(16)where ®si denotes the unobserved individual speci¯c e®ects given yi;t¡1 = s, and "sit is the error
term that has zero mean and variance ¾ 2" . Assuming (16) holds, we combine yits¤ , s = 0,1; into
one equation:

18

¡
¢
yit¤ = ®0i (1 + ±i yi;t¡1 ) + xit (¯ 0 + byi;t¡1 ) + Dit ° 0 + gyi;t¡1 + "it ;(17)where "it = "1it yi;t¡1 + "0it (1 ¡ yi;t¡1 ) ; ®1i = ®0i (1 + ± i ) ; ¯1 = (¯ 0 + b); and ° 1 = ° 0 + g.
If ®1i and ®0i are treated as randomly distributed, one can obtain the MLE provided theirconditional distributions given xi can be speci¯ed. However, the consistency of the estimatedparameters depends on whether the conditional distributions of ®1i and ®0i are correctly speci¯ed.
Moreover, even if the distribution assumptions of ®1i and ®0i are correctly speci¯ed, the estimation
can be quite involved due to multiple integrations of ®1i and ®0i over (Ti ¡ti ) period. On the other
hand, if ®1i ; ®0i are treated as ¯xed, there is no need to specify their distributions conditional on
xi a priori. Therefore, we focus on ¯xed e®ect models. Assuming that conditional on xit ; "1itand "0it follow a standard type I extreme value distribution, then¡¢Pr yit = 1jwit ; ®0i ; ± i ; yiti ; :::yi;t¡1 =exp [®0i (1 + ± i yi;t¡1 ) + wit (µ0 + cyi;t¡1 )];1 + exp [®0i (1 + ± i yi;t¡1 ) + wit (µ0 + cyi;t¡1 )](18)

where wit0 = [x0it ,D0it ]; µ0 = (¯ 00 ; ° 00 )0 and c = (b0 ; g0 )0 . When ± i = 0 for all i, and c = (°; 0; :::; 0),equation (18) becomes¡¢Pr yit = 1jwit ; ®0i ; ± i ; yiti ; :::yi;t¡1 =exp [®0i + wit µ0 + °yi;t¡1 ];1 + exp [®0i + wit µ 0 + °yi;t¡1 ](19)which is of the same form as equation (6) in Honor¶e and Kyriazidou (2000). Therefore, the model

19

considered in Honor¶e and Kyriazidou (2000) can be treated as a special case of this model (17) .
For ease of exposition, we shall only present the case that Ti ¡ ti = 3 and consider two events

A = fyiti ; yi(ti +1) = 0; yi;(ti +2) = 1; yi;(ti +3) g;
B = fyiti ; yi(ti +1) = 1; yi;(ti +2) = 0; yi;(ti +3) g:

To simplify notations, we denote yi(ti +m) = yim , xi(ti +m) = xim ; m = 0; 1; 2; 3 in the remaining of
this section. Under the assumptions that wi2 = wi3 and yi0 = yi3 ,¡¢P AjA [ B; wit ; ®0i ; ± i =1:
1 + exp [(wi1 ¡ wi2 ) µ0 + (wi1 yi0 ¡ wi3 yi3 ) c]

(20)

The conditional probability no longer depends on ®0i and ± i when wi2 = wi3 and yi0 = yi3 hold.
Therefore, we propose to estimate µ0 and c by maximizing the objective functionNX
i=1

1 (yi1 + yi2 = 1) ¢ 1(yi0 ¡ yi3 = 0) ¢ 1 (Di2 ¡ Di3 = 0) ¢ K(

£ ln

Ã

~0 + (wi1 yi0 ¡ wi3 yi3 ) ~exp[(wi1 ¡ wi2 ) µc]yi1~0 + (wi1 yi0 ¡ wi3 yi3 ) ~1 + exp[(wi1 ¡ wi2 ) µc]!

xi2 ¡ xi3
)
¾n

(21)

~0 and ~with respect to µc over the parameter space, where 1 (A) denotes the indicator function,i3) denotes a kernel density function that gives more weight to those observations whoseK( xi2¾¡xnxi2 is closer to xi3 , and ¾ n is a bandwidth that shrinks toward 0 as n increases.

20

Theorem 2 Let qi = [wi1 ¡ wi2 wi1 yi0 ¡ wi2 yi3 ]; Ã =(µ 00 ; c0 ),¶exp(qi Ã)yi1
hi (Ã) = 1 (yi1 + yi2 = 1) ¢ 1(yi0 ¡ yi3 = 0) ¢ 1 (Di2 ¡ Di3 = 0) £ ln:
1 + exp(qi Ã)µ(22)

Let the following assumptions hold:C1 f(yi0 ; yi1 ; yi2 ; yi3 ; wi1 ; wi2 ; wi3 ); i = 1; :::; N g is a random sample of N observations from a
distribution that satis¯es (18).
C2 The true value of the parameters of interest, Ã 0 ; is in the parameter space ª, which is acompact subset of the Euclidean K-space(RK ), where K = k + q + 1.
C3 (i) The random vector xi2 ¡ xi3 conditional on Di2 = Di3 is absolutely continuously distributed with density function f (¢). f (¢) is bounded from above, strictly positive and has
support in the neighborhood of zero. (ii) Pr (Di2 = Di3 ) > 0:
C4 E [jjwi1 ¡ wi2 jj2 jA] and E [jjwi1 yi0 ¡ wi3 yi3 jj2 jA] are bounded on their supports, whereA = [(xi2 ¡ xi3 ) = 0; Di2 = Di3 ] for assumptions (C4) - (C6):C5 The function E (h (Ã) jA) is continuous in a neighborhood of zero for all Ã2ª::
C6 The functions E [(wi1 ¡ wi2 )0 (wi1 ¡ wi2 ) jA] and E [(wi1 yi0 ¡ wi2 yi3 )0 (wi1 yi0 ¡ wi2 yi3 ) jA]
have full column rank in the neighborhood of zero.
C7 K : Rk ! R is a function of bounded variation that satis¯es:(i) supv2R jK (v) j < 1; (ii)R

jK (v)j dv < 1; and (iii)R

K (v) dv = 1:C8 ¾ n is a sequence of positive numbers that satis¯es: ¾ n ! 0 as n ! 1:21

^ be the solution to the problem
Let Ã

max
Ã2ª

N
X
i=1

K(

xi2 ¡ xi3
)hi (Ã) ;
¾n

(23)

p

^ ! Ã0:then Ã
Assumptions (C2) to (C5) are the regularity conditions required for the objective function toconverge to a nonstochastic limit which is uniquely maximized at Ã 0 by a law of large numbers.
Assumption (C6) is required for the identi¯cation of µ0 and c. Assumptions (C7) and (C8)
are standard for kernel density estimation. The above assumptions are quite similar to those
imposed in Honor¶e and Kyriazidou (2000), except that we separate the continuous explanatory
variables and discrete variables in moment conditions and in kernels. Similar to that of Honor¶e¡¢1=2
and Kyriazidou (2000), the convergence rate is much slower than root-n. It is at rate n¾ kn.

The assumptions for asymptotic normality are similar to those imposed in Honor¶e and Kyr-iazidou (2000) except that in addition to conditioning on xi2 = xi3 ; we also need to condition
on Di2 = Di3 . The proof of consistency and asymptotic normality follows straightforwardly tothose in Honor¶e and Kyriazidou (2000).
Whether individual speci¯c e®ects need to be controlled is critical for the adequacy of themodel presented in section 2. The conditional MLE remains consistent when individual speci¯ce®ects are not present. Signi¯cant information loss, however, occurs as the conditional MLE
greatly restricts data points: only about 10 percent of the observations used in MLE satis¯es the
key condition that xi;s+1 ¼ xi;t+1 for the consistency of the conditional MLE. Furthermore, very
few clients have taken the third JSS. The conditional MLE fails to converge when the third JSS
dummy is included.
22

Table 5 presents the estimated coe±cients for the ¯rst two JSS from MLE (model 1) andconditional MLE (model 2). The estimated coe±cients and standard errors are in columns 3
and 4 for model 1, and in columns 5 and 6 for model 2. For initially unemployed clients, theestimated coe±cients of the ¯rst JSS are both signi¯cant at 5% level; further, they are very close(0.32 versus 0.346). The second JSS has signi¯cant impact in model 1 but not so in model 2,
probably due to the reduction of available observations that satisfy the consistency conditions of
the conditional MLE. For those who are already employed, the estimated impacts of the ¯rst two
JSS are insigni¯cant both in model 1 and in model 2. The Hausman statistic for misspeci¯cationis merely 0.18 for the job-seeker group, which is not signi¯cant at a chi-square distribution
with two degrees of freedom (5.99 at 5% level). These results appear to suggest that there isno evidence of the presence of signi¯cant unobserved individual heterogeneity conditioning on
observed clients characteristics. They appear to further con¯rm that using model 1 is adequate
to evaluate the e®ectiveness of repeated JSS.

4.3

Controlling for both Selection Bias and Individual Speci¯c E®ects

We have presented the nonlinear two-stage least square estimator that allows for selection due
to unobservables but no unobserved individual heterogeneity, and the conditional maximum
likelihood estimator that controls for unobserved individual speci¯c e®ects but no selection due
to unobservables. If both selection due to unobservables and unobserved individual heterogeneity
are present, the above two estimators remain biased. In this subsection we propose a conditional
nonlinear two-stage least square estimator that allows for unobserved individual heterogeneity23

as well as selection on unobservables:

min S =NX
i=1

Ki Li (yi ¡ Fi ) zi0 (

N
X
i=1

Ki Li zi zi0 )¡1 zi (yi ¡ Fi )(24)

exp(qi Ã)
i3where Li = 1 (yi1 + yi2 = 1) ¢ 1(yi0 ¡ yi3 = 0) ¢ 1 (Di2 ¡ Di3 = 0), Ki = K( xi2¾¡x); Fi = 1+exp(qni Ã)as in equation (22), zi0 = (~zi0 ; q0i ) and z~i0 is vector of instruments for individual i.
Theorem 3 Under A1 - A7 and C1 - C8, the conditional nonlinear two-stage least squares
estimator is consistent.
The conditional nonlinear two-stage least square estimator not only greatly reduces the number of available observations, but also needs good instruments. Hence, it is unlikely to yieldaccurate estimates. The MLE estimates are e±cient but not consistent when both the no selection on unobservables assumption and the no inidividual heterogeneity assumption hold but
are inconsistent if either or both are violated. Again we use a Hausman test to check whether
the null hypothesis of no selection on unobservables assumption and no inidividual heterogeneity
hold. The resulting Hausman test statistic is merely of magnitude of 0.1, hence does not reject
the null.

5

Conclusion

In this study we have evaluated the e®ects of repeated job search services and individual characteristics on the employment rates of the prime-age female TANF recipients in Washington
State. We have suggested a transition probability framework to deal with the complicated issuesof sample attrition, sample refreshment and duration dependence. We estimated conditional
24

and unconditional impacts of the repeated job search services for job seekers and those whoare already employed. We also proposed a nonlinear two-stage least square method to allow
for selection due to unobservables, a generlized conditional maximum likelihood estimator to allow for state-dependent ¯xed e®ects and slope coe±cients, and conditional nonlinear Two-Stage
Least Square estimator to allow for both unobserved individual heterogeneity and selection due
to unobservables. The speci¯cation tests indicated that the CI assumption and the no individual speci¯c e®ects assumption conditional on included socio-demographic variables were notcontradicted by the information of the data.
Our ¯ndings show that Job Search Services do have positive and signi¯cant impacts on
the employment rates of those who are initially unemployed, with the ¯rst JSS increases theprobability of being employed by 8.45 percent, the second JSS increases it by a further 2.15
percent, and the third JSS increases it by an additional 0.8 percent for the treated group. But
the impacts of JSS are insigni¯cant for those who are already employed. Combining these ¯ndings
with the ¯nding that for each additional quarter that an individual stays unemployed, her chanceof being employed is decreased by 0.19 percent, it makes a strong case for the state to introduce
job search services quickly to those who are unemployed so as to mitigate the self-enhancing
e®ect of unemployment.

25

References[1] Amemiya, T. (1974), \The Nonlinear Two-Stage Least-Squares Estimator", Journal of
Econometrics, 2, pp. 105 - 110.
[2] Ashenfelter, O.C. and Card C.(1985), \Using the Longitudinal Structure on Earnings to
Estimate the E®ect of Training Programs", The Review of Economics and Statistics, 67(4),pp. 648 - 60.
[3] Bassi, L. J. (1984). \Estimating The E®ect of Training Programs with Non-Random Selection", The Review of Economic and Statistics 66(1): 36 - 43.
[4] Gill, R.D. and Robins, J.M. (2001). \Causal inference for complex longitudinal data: thecontinuous case", Annals of Statistics, Vol. 29, No. 6.
[5] Hausman, J. A., (1978) \Speci¯cation Tests in Econometrics", Econometrica, 6, 1251 - 1271.
[6] Heckman, J. J. and V. J. Hotz (1989). \Choosing Among Alternative Nonexperimental
Methods for Estimating the Impact of Social Programs: The Case of Manpower Training."
Journal of American Statistical Association 84(408): 862 - 874[7] Honor¶e, B. E., and Kyriazidou, E. (2000). \Panel Data Discrete Choice Models with Lagged
Dependent Variables", Econometrica, Vol. 68, No. 4, 839 - 874.
[8] Heckman, J., and Robb R. (1985). \Alternative methods for Evaluating the Impact of Intervention", Howard Wainer eds, Longitudinal Analysis of Labor Market Data Eds. Heckman
J. and Singer B.

26

[9] Holland, P. (1986). \Statistics and causal inference", Journal of the American Statistical
Association, Vol.
[10] Hsiao, C. (2002). Analysis of Panel Data, Cambridge: Cambridge University Press.
[11] Lechner, M., and Miquel, R. (2001), \A Potential Outcome Approach to Dynamic Programme Evaluation: Nonparametric Identi¯cation", memio, University of St. Gallen.
[12] Quandt, R. E., (1983) \ Computational Problems and Methods", Handbook of Econometrics,
Vol. I, Edited by Z. Griliches and M.D. Intriligator, North-Holland Publishing Company.
[13] Rosenbaum, P.R. and D. B. Rubin (1983). \The central role of propensity score in observational studies for causal e®ects", Biometrika, 70, 41 -55.

27

Table 1 Variable Definitions

Variable
Category

Variable Name

Definitions

Indicator for whether the first Job Search Services (JSS) had
been taken before period t.
LJSS2
Indicator for whether the second JSS had been taken before
period t.
LJSS3
Indicator for whether the third JSS had been taken before periodt.
Ltotal_AS
Total number of Alternative Services (AS) before period t.
Ltotal_PS
Total number of Post employment Services (PS) before period t.
Employment
lunemploycoun Total unemployed quarters before period t.
tHistory
lemploycountTotal employed quarters before period t.
Welfare history lafdcnow
Total quarters in AFDC and/or TANF before period t. (AFDC is
the predecessor of TANF).
Familynum_adlt
Number of Adults in the Assistance Unit.
num_chld
Number of Children in the Assistance Unit.
Age_youngest Age of the youngest child in the Assistance Unit. Calculated
based on the first quarter that WorkFirst began, 1997.IV.
MarriedMarital status. 1 indicates married.
RaceWhitesRace indicator. 1 indicates client is white.
BlacksRace indicator. 1 indicates client is black.
Hispanics
Race indicator. 1 indicates client is Hispanics.
Language andEnglishLanguage indicator. 1 indicates client can speak English.
Education
grade12
Education indicator. 1 indicates client’s highest grader higher
than 12.
Geographic
region1
Location indicator. 1 indicates client is from Region 1.
Information
region2
Location indicator. 1 indicates client is from Region 2.
region3
Location indicator. 1 indicates client is from Region 3.
Local economy Unemployrate The unemployment rate of the county that client is in.
Time
year98Year indicator. 1 indicates the record is in year 1998.
year99Year indicator. 1 indicates the record is in year 1999.
quarter1
Quarter indicator. 1 indicates the record is in quarter 1.
quarter2
Quarter indicator. 1 indicates the record is in quarter 2.
quarter3
Quarter indicator. 1 indicates the record is in quarter 3.
WorkFirst
Participation

LJSS1

28

Table 2 MLE and NL2S Estimations for Initially Unemployed Clients( yi , t −1 = 0 )Category

Variable

MLE Estimation

NL2S Estimation

CoefficientStandard Error

CoefficientStandard
Error-1.288755

0.0857847-1.288699

0.3209854

WorkFirst
LJSS1
Participation LJSS2

0.3208265

0.0336228

0.3210588

0.8014488

0.0974651

0.0460288

0.0971165

2.6166248

LJSS3

0.127637

0.0623509

0.1276976

6.2028023

Ltotal_AS-0.063875

0.0136483-0.064106

0.0418666

Ltotal_PS

0.1732402

0.054055

0.1733488

0.0783051

Employment Lunemploycount -0.121042
History
Lemploycount0.1768895

0.0100561-0.121247

0.0244949

0.0291352

0.1769557

0.076339

FamilyIntercept

Race

EducationTime

Num_adlt-0.184392

0.0407723-0.184336

0.0531489

Married-0.135805

0.0450123-0.135767

0.0479468

Whites-0.139019

0.0401963-0.138914

0.0682148

Blacks

0.0886408

0.0526743

0.0886086

0.0757805

Hispanics

0.1162661

0.04886660.1163025

0.0514206

English0.5122075

0.0576963

0.5122995

0.1662142

Grade120.2229153

0.0390711

0.2229834

0.0423111

Year980.2476005

0.0370489

0.2476764

0.1510935

Hausman Test Statistic = 5.476 < chi-2 (16)= 26.30 at 5% significance level. The Hausman test
therefore does not reject the MLE estimates.

29

Table 3 MLE and NL2S Estimations for Initially Employed Clients( yi , t −1 = 1 )Category

Variable

MLE Estimation

NL2S Estimation

CoefficientStandard
Error

CoefficientStandard Error

1.3670399

0.107082

1.3669012

0.3064295

WorkFirst
LJSS1
Participation LJSS2

0.0397998

0.0383007

0.0407879

1.515584

0.0124084

0.0592132

0.0126616

4.0119876

LJSS3

0.0677785

0.080673

0.0679399

4.9785368

Ltotal_AS-0.107349

0.0197488-0.107377

0.0206929

Ltotal_PS

0.034416

0.0271343

0.034242

0.0342981

Intercept

Employment LunemploycountHistory
Lemploycount-0.058854

0.0316035-0.058877

0.0406632

0.1060398

0.0126613

0.1062231

0.0169491

FamilyNum_adlt-0.107615

0.0472479-0.107726

0.0478171

Married

0.051051

0.0503581

0.0510632

0.0557685

Whites-0.079711

0.0483444-0.079871

0.057891

Blacks

0.0056359

0.0590434

0.0056498

0.089708

Hispanics

0.0175155

0.0542377

0.0173954

0.0719317

English-0.429121

0.0833468-0.429282

0.1318459

Grade120.1902404

0.0457895

0.1902577

0.0644864

Year980.3847283

0.0496395

0.3847448

0.0972272

Race

Education
Time

Hausman Test Statistic = 0.98, does not reject the MLE estimates.

30

Table 4 Mean Group Impacts and Equilibrium Impacts
Variable Category

WorkFirst
Participation

Employment
History
Earnings history
FamilyRace

Language and
Education
Geographic
Information
Local economy
Time

Parameter

LJSS1
LJSS2
LJSS3
Ltotal_AS
Ltotal_PS
lunemploycountlemploycountPre_earn
num_adlt
num_chld
Age_youngest
MarriedWhitesBlacks
Hispanics
Englishgrade12
Region1
Region2
Region3
Unemployrate
year98
year99quarter1
quarter2

Group Impacts
Job-Seeker
Group
0.08450.02150.008-0.0120.026-0.0200.029
0.00001-0.02800.002-0.069-0.02890.04650.03290.10550.04840.03650.02330.043900.0831
0.054
0.069
0

Job-holder
Group
0.0066*0.004*
0.002*-0.0175-0.0188*-0.02*
0.019
0.00001-0.0267
0.006
0.001
0.009*-0.0306
0.0057*0.004*-0.08150.02920.01850.033*
0.025*-0.0030.0616-0.01-0.0670.0327

Equilibrium
Impacts

0.037
0.003
0.007
-0.0270.024-0.00190.044
0.00001-0.0510.00560.0038-0.021-0.039-0.025
0.016
0.005
0.054
0.060
0.02440.0491-0.0030.08990.0130-0.084-0.022* Insignificant variables are considered as having coefficient zero in calculating the unconditional
equilibrium effects, and its long run impacts on job seekers (job holders) are considered as zero.

31

Table 5 With or Without Individual Heterogeneity *Without Individual
Heterogeneity
Probabilityof being
employed
Probabilityof staying
employedWith Individual Heterogeneity

Parameter

Chi-Square

Parameter

Standard
Error

LJSS1_0
LJSS2_0

0.32280.1053

91.40
5.19

0.35
0.20

0.176
0.24

LJSS1_1
LJSS2_1
LJSS1_1LJSS1_0
LJSS2_1LJSS2_0

0.044-0.016
.

1.128
0.066
.

.
.
0.14

.
.
0.25

.

.

-0.09

0.29

* LJSS1_0 and LJSS2_0 are the impacts of the first and the second JSS on the probability of
being employed respectively.
LJSS1_1 and LJSS2_1 are the impacts of the first and the second JSS on the probability ofstaying employed respectively.

32