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STATE DEPENDENCE IN MARKOV LABOR FORCE MODELS
by Steven Tynan
offered as partial fulfillment
of the requirements
for a Bachelor of Arts degree
in MMSS
1984
1 . Introduction
The use of Markov models in economics and other social sciences is
well established. Such models have been used for some time as models
of labor force behavior. Recent work in developing models based on an
assumed Poisson-type job offer generating function has been done by
Flinn and Heckman (1982a,b). Recently, Burdett, et al. (1981), Plinn
and Heckman (1983), Tuma and Robins (1980), and many others have used
Markov models to analyze specific features of the labor market.
The independence of the probability of exiting one's current labor
state with respect to both work history and time already spent in the
state is associated with the Poisson assumption of Markov models.
These assumptions contradict recent theories about the job search
process. For example, theory suggests that a worker's past is very
important in determining his ability to get a job. Theory also
suggests that the length of a worker's current labor force spell
affects his probability of remaining there.
Heckman and Borjas (1980) examine the possibility of allowing three
types of non-Markov state dependence to enter into models that are
similar in structure to the Markov models. The Heckman and Borjas
paper presents and adapts a two-state model with Markov origins. The
empirical work done at the end of the paper is called "tentative" by
the authors. It is based on a study of 122 young men and serves
primarily as an illustration of techniques rather than a test of the
validity of state dependence.
Originally this paper was intended to review the work done in this
area and related concepts, such as heterogeneity, and then to retest
for state dependence in labor force state participation using
Denver/Seattle income maintenance experiments (DIME/SIME) data. This
data source shows shows labor force state participation histories for
about 4800 families and has been used by Tuma and Robins (1980),
Burdett, et al. (1981), and Weiner (1982), among others.
Unfortunately, computer problems prevented the use of the DIME/SIME
data, so this paper is largely a review of the work done in the area
of state dependence.
The model presented in Heckman and Borjas (1980) uses two states,
employment and unemployment, thereby grouping the states of
unemployment and out of the labor force into a single non-employment
state. Plinn and Heckman (1983) have shown that there is reason to
believe that the two states are significantly different from each
other. This paper, therefore, utilizes a three-state model with
features that appear in the model in Burdett, et al. (1981), while
maintaining the important features of the Heckman and Borjas model.
Those features and some others are discussed below. It is
important to understand the underlying assumptions leading to the
Poisson-like feature in Markov economic models, so this paper outlines
the development of a single state job offer generating model, through
a two-state model, to a three-state model of labor force state choice
based on stationary and stochastic characteristics.
Current labor theories are a fundamental source of controversy
about the applicability of Markov models of labor. A section after
the models makes a quick, but necessary overview of some of these
theories. The next section defines and discusses four types of state
dependence, including three not allowed in Markov models, and examines
ways to modify models to accommodate for them.
In the next two sections, the problems of heterogeneity in
variables and censoring, or truncation of data, are described and
discussed. The section after that discusses the nature of the
DIME/SIME data that was to be used. In a final section, problems that
would have been encountered in trying to use the Heckman and Borjas
methods with the DIME/SIME data are discussed.
2. A one-state model of the search
To motivate and facilitate more complicated labor force models
later it will be beneficial to examine a model of job search in which
an unemployed worker is viewed as receiving job offers generated by a
Poisson process. A model of this type is presented in Lippman and
McCall (1976) and summarized in Flinn and Heckman (1982b). We will
use that summarized model as a starting point. The one-state model is
presented here roughly following the structure of Flinn and
Heckman (1982b).
A Poisson process
In this and subsequent models, workers (or agents) are assumed to
maximize income. Job search is examined from the viewpoint of an
unemployed worker who incurs an instantaneous cost c of job search.
The agent is assumed to receive job offers generated by a Poisson
process with parameter a independent of the level of c (c > 0). The
probability that the agent gets a job offer in the marginal time
interval At is afit + o(At), where o(At) has the property
lim o(^t} _—> 0. (2.1)
It^o At
Property (2.1) says that the probability of two or more job offers
in the time interval At is negligible. Consider that the job offer
generating Poisson process has a sample space Q on which there is a
probability measure f. There is a non-negative integer ff+(ir) for each
realization w e Q at any time after the start of the process
(i.e. % > Q) . Here ^ . will represent the number of arrivals of job
offers in the interval [0,t].
As an assumed Poisson process, the arrival process in this model
exhibits two familiar properties. First, the number of arrivals
t+tit ~ "t ^n ^ e interval (t,t + fit] depends only on j\t and not on t.
Second, the number of arrivals during (tjt + 2Js1s] (i.e. Jfft.4-) is
independent of , the number of arrivals during [0,t]. Alternatively
stated, the job search process is assumed to. be a continuous
time-parameter process with stationary and independent increments.
The assumptions of stationarity and independence lie at the heart of
the question of the effects of employment history on future
employment. One justification for the Poisson wage arrival assumption
can be found in Burdett and Mortensen (1978).
Using the terminology and reasoning of Cinlar (1975) we can restate
the probability of a job offer in the marginal increment fit as
follows,
lim .— = a (2.2) Z|t-»0 £t
where f{S^ = 1 } is the probability that one job offer will arrive in
marginal time interval £t and a is the Poisson parameter defined
above.
In these terms, the assumption that no more than one offer arrives
in £ft is stated as,
P{N.t > 2} lim <LL- = Q> ( 2 # 3 )
A W O fit
Proofs for (2.2) and (2.3) and additional characteristics of a
Poisson models can be found in Cinlar (1975)•
The search
Each job offer has an associated wage offer. In this model wage
offers are seen as independent realizations of F(x), a wage
distribution common to all agents. F(x) is a known absolutely
continuous distribution with a finite mean.
Wage offers are only available once, so that a worker can not
choose to accept an offer after he has turned it down. Jobs and
workers last forever and employed workers do not participate in the
job search process.
Letting V be the value to an unemployed worker of continuing his
job search, we can show that V consists of three parts plus a
negligible component o(£lt) .
V = _ c 4 t + i l z ^ l v + a ^ E max[x/r;V] + o(4t) , for V>0, 1+rfit 1+rflt 1+rflt
(2.4) = 0 otherwise.
The three non-negligible components of the value of search equation
are: 1) the discounted cost of search in interval At, 2) the product
of the probability of not receiving a job offer (1 - afct) times the
discounted value of search at end of time interval At, and J>) the
product of the probability of being offered a job (a$t) times the
discounted expected value of the more valuable of two options. Those
options are to accept a job offer with present value x/r or to reject
the offer and continue the search with value V.
The first, second, and third non-negligible components correspond
to the first, second, and third terms, respectively, on the right of
equation (2.4) above. The second value of search equation above (V=0
otherwise) indicates that there can he no negative value for search.
An agent for whom V=0 is defined as out of the labor force. In this
one-state model, stationarity demands that an agent may not re-enter
the labor force once he has left.
Robbins (1970) shows that E(|x|)<oo is a sufficient condition for
the existence of an optimal reservation wage policy. That reservation
wage is represented by rV, which is implicitly determined in the
equation
o
c + rV = (a/r) $ (x-rV) dF(x) , for V>0. (2.5) rV
Equation (2.5) [Lippman and McCall (1976)] is obtained by collecting
the terms in (2.4) and passing to the limit. The agent accepts any
offered wage x>rV, so the probability that a wage offer is refused is
P( rV).
An unemployment spell $ a ¥ m exceed length ta contingent upon two
events. The first prerequisite event is that the agent receives j
offers in time interval tu. This event is assumed to happen with
probability
P[j offers | tu] = (atu);]'e-atu/j! , with a>0. (2.6)
The second prerequisite event is that all of the j offers are turned
down by the agent. The discussion of equation (2.5) showed that this
probability is [P(rV)]J.
With independent arrival times and wage offers, the probability
that unemployment spell fu is longer than tu is
P[Tu>ta] = $ ((atu)J/j!)e-atu'[P(rV)]^ = e-a( 1-E( rV)) V ( 2 > ? )
0
This is the product of the two previous probabilities summed over j.
Equation (2.7) is called the "survivor function" in Elinn
and Heckman (1982b). The density of t is clearly
f(tu) = a(l-P(rV))ea^1-F^rV^ta.
To find the density of accepted wage offers (ie. transfers out of
unemployment) it is important to note that this density is the density
of wage offers given that those offers are greater than rV. The
associated random variable is the truncated wage offer random variable
with rV as the lowest point. This density is therefore
f(x|x>rV) = -1^1— , x>rV. (2.8) 1-P(rV)
Because wages offers assumed to be distributed independently of
their arrival times, the product of the densities of spell durations
tj. and of accepted wages x gives the joint density of the two random
variables. That joint density is
g(ta,x) = |a(l-F(rV))expl-a(l-F(rV))ta}} f U )
1-F( rV) (2.9)
= (a-exp{-a(l-F(rV))ta})f(x) , x>rV.
A hazard rate is defined to be the conditional density of exit
times from a state given the amount of time already spent in the
state. The assumptions of this stationary search model generate a
duration model with constant hazard rate
hu(tu) = h(ta)= -d In p[Tu>tu]/dtu = aO-F(rV)). (2.10)
High hazard rates are accompanied by rapid movement from the
unemployment state. Cases of constant hazard rates are very important
within the scope of this paper. Constant hazard rates [ dh( t )/dt =0]
signal the absence of duration dependence.
3. Increasing the number of states
A natural next step given the assumptions and structure of the
Poisson process model of job search above is the presentation of a
Markov model of labor state participation. Models of employment,
unemployment, and labor force non-participation utilizing Markov
assumptions have been widely presented. [see eg. Burdett, et
al. (1981), Plinn and Heckman (1982a,b,1983), Heckman and Borjas
(1980), Tuma and Robins (1980)] Plinn and Heckman (1982b) develop
two-state and multi-state models of sequential job search which serve
as intermediary models below. Those models help to develop the
theoretical link between the Poisson model above and the Markov choice
model to be presented here.
The essential extension made in the two-state model upon the
one-state model above is the expansion of value function (2.4) above
to reflect the agent's two options of employment and unemployment.
The inclusion into the model of the choice between employment and
unemployment is reflected in the addition of the subscripts a or e on
the value variable V. Hence, the value to an agent of a job with a
flow rate of output gn and a termination rate § is defined as V-»(x) .
It is assumed here that one-half of the flow rate of output (x) goes
to the employer while the other half (again x) goes to the worker.
The value to the worker of remaining unemployed and continuing the job
search is now called V ¥u
The value of employment, Ve(x), is the sum of three components plus
negligible component o(&t) . The components are the flow of x over
interval Z\t, the value of a job times the probability of keeping it,
and the value times the probability of unemployment. All three
components are discounted at rate r. The equation for the value of
employment is therefore,
1 Ve(x) = _{x6t + 0-sfit)V (x) + s£tVu} + o(4t). (3-1)
1 +r£t
This becomes V (x)=(x + sVn)/(r+s) when the terms are collected and
At is passed to 0.
The agent is assumed to be indifferent between a job with value
Ve(x*)=Vu and unemployment. Therefore x* is the reservation wage and
has the value
Ve(x*) = (x* + sVa)/(r+s) = Va, (3-2) so
x- = rVu.
The equation for the value of unemployment is similar to value
equation (2.4-) above, with a few changes. With the introduction into
the model of an employment state, equation (2.4) becomes
1+rflt 1+rflt 1+rflt
The three non-negligible terms on the right of (3«3) above again
represent the discounted worth to the agent of 1) the cost of job
search, 2) the value times the probability of continued unemployment,
and 3) the probability of a potential job match times the expected
maximum of two values. Here the options in three are taking a job
with value Ve(x) or remaining unemployed with value Vu.
Note that a no longer represents the Poisson parameter of a job
offer generating function, but is now the rate of arrival of
encounters between agents and potential employers. This rate is
assumed to be a differentiable function of L, the number of agents.
It is assumed that a rises with L, so a=g(L), g(0)=0, and g'(l)>0 for
all L20.
Collecting the terms of equation (3.3) and passing to the limit
yields
rVu+c = a 5 (Ve(x)-Vu)dF(x) rV a °°
r+s rV (x-rVu)dF(x) , for Vu>0. (3-4)
If &s§(i) is n0"t sufficiently large to insure that Yu>0, fie., if
a<c(r+s)/ j xdF(x)] job search does not take place. 0
As in the one-state model, the hazard rate for exit from
unemployment is fca(tu) = a(l-F(rVu). Of course, the hazard rate for
employment, fre(te), is the termination rate s.
The next step from this model to a three-state Markov model of
labor state participation is straight forward. Appendix B of Flinn
and Heckman (1982b) shows the expansion of the one- and two-state
models to include a third state of out of the labor force.
4. 3-state Markov model
Having assumed, for the sake of argument, the underlying one-state
Poisson model of job search, it is straight forward to model labor
force participation state choice as a Markov decisions process. For
the structure of the three-state labor state participation model
below, I am indebted to both a two-state Markov model appearing in
Heckman and Borjas (1980) and a three-state model appearing in
Burdett, et al. (1981) .
Consider an agent who is either employed, unemployed or out of the
labor force at time %. Initially, we will assign fn(t) , where
nteN={e, o, or u} , as the respective probabilities of the agent being
in a given state at time t. As in the models above, a central feature
of this model is the Poisson assumption that the probability of
changing state more than once in a marginal time interval At is
negligible.
Burdett, et al. formalize the idea that underlying the agents'
observed choices of labor state are stationary as well as stochastic
time varying characteristics associated with individual agents. Each
agent is identified by a stationary characteristic vector x (§ X.
[Note that this vector x is not to be confused with the wage offer,
scalar x, in the one-state model. The vector might, however, include
the agents wage or expected wage as a stationary variable, for
example, as ^ in x= (x1 ,x2, . . . ,xk) ] If characteristics are not
stationary as assumed or if differences in these characteristics
between agents are overlooked, the problem of heterogeneity arises.
This problem will be discussed below.
Also associated with each agent at any given time t is vector
y(t) €t T of stochastic time varying characteristics, labeled
"disturbances" by Burdett, et al. The disturbances are paired, many
to one, with one of the time varying state choices. That is, each
y(t) is only one pair (y(t),n(t)), while each of the three n(t)'s
might be in many. X and T are real vector spaces.
Choice of a state at time t is determined by the current values of
x and y(t). Since x(t) is assumed to be stationary, changes in n(t)
result only from changes in y(t). Since this is a Markov model, the
probability of such a change is assumed to be conditional on the
current state and the stationary vector, but it is thought to be
independent of the current y(t). The probability that an agent
transfers from one state to another in interval lt is therefore equal
to the conditional probability that the agent' s disturbance will be in
the subset of disturbances for which the second state is preferred at
t + At, given n(t) and x.
The probability that the disturbance of a worker with vector x in
state n at time t will be in a subset A at t + fit (ie.
P[y(t + /it) = Aj n(t) = n and x]) can be called ^n(A|x,£.t), stressing
the link between state choice and disturbance. Assuming that this
probability divided by At has a limit as 4t—>0, that limit can be
written to show that it is the instantaneous probability of a change
in disturbance times the distribution of new disturbances given a
change.
fVA|x,dt) iim _ = ft ( x ) U (dy;x) (4 .1 ) £t->0 fit
/I
The expected time to the first change is 1/h (x) and P (dy;*) is the
new disturbance distribution.
Given this instantaneous probability of a change to state n
associated with disturbance subset A, it is necessary to restrict A
such that it corresponds with the optimal state choice m(t) € N.
Am(x) = {y€Y|Vm(x,y)=max Vn(x,y)! , mCN, (4-2)
where Vn(x,y) , n£N, is a stationary characteristic contingent utility
indicator.
The event that an agent changes from any optimal state n at t to
any new optimal state m at t + fit requires that his disturbance
changes to an element of A^, given x and n. The probability of this
event is therefore
Pm(^tfx) = P{n(t + £t)=m| n(t) = n and x}
= P{y(t + £t)4Am(x)| n(t) = n and x} (4.3)
= ^ m(Aj x,6t) , m^n.
Dividing this probability by ^t and passing to the limit gives the
state to state transition rates
a ( x ) = l i r a ™ 4 W 0 4 t _ (4.4)
= f\ n^x) J X U y j x )
for all n and m Q N optimal at times t and t + 4t, respectively, where
ri^m. The transition rate is shown as the product of the incremental
probability of changes in y(t) (h n(x)H) times the probability that
the resulting y makes the state m optimal.
The transition rate ^ ( x ) above should be interpreted such that
fj^jC^t^) = a^Cx^t + o(t) where o(t) has property (2.1) from the
Poisson model above. In the context of testing for state dependence,
"the a (x)!s are determined by firm hiring and firing practices
relative to worker characteristics. With the Markov assumption still
intact, the rates depend solely on the states in question.
The probability of finding a worker in a given state m at time
t + £t is the sum over all n of the products of switching from the
original state to m in fit times the probability of being in the
original state at t. For example, the probability of being employed
after the time increment is
Pe(t+4t,x)=(aee(x)^t)Pe(t,x) + (aae(x)^t)Pu(t,x) + (aoe(x)4t)P0(t,x)
+ o(t). (4-5)
Note in equation (4-5) that the probability of remaining in the
same state from time ^ "to $ ± £% is (ann(x)(fit + l) +o(t)). This
property is important when explaining pure Markov state dependence,
the idea that those already in a state are the ones most likely to be
there at a later date if the time increment is short enough.
Continuing on with the example of employment above, we see that the
change in the probability of employment over <Qt is
Pe(t+6t,x)-Pe(t)=(aee(x)£t)Pe(t,x) + (aue(x)/St)Pu(t,x)
+ (aoe(x)/H)P0(t,x)+o(t) . (4-6)
Dividing by $t and passing to the limit gives
dPe(t,x)/dt = aee(x)Pe(t,x) + aue(x)Pu(t,x) + aoe(x)PQ( t ,x) . (4-7)
In general, the probability of being in state n after interval$t and
the first derivative of the probability are
Pn(t+fct,x) = (ann(x)At + l)Pn(t,x) + (amn(x)5t)Pm(t,x)
+ (aln(x)4t)P1(t,x) + o(t) (4.8) and
P;(t,x) = ann(x)pn(t,x) + amn(x)Pm(t,x) + aln( x)Px( t ,x) (4-9)
for distinct 1, m, and n in N.
Since Pe(t,x) + Pu(t,x) + P0(t,x) = 1, Pg(t,x) + P^t.x) +
P (t,x) = 0. In other words, because the entire population is
accounted for by the three states, any change in the probability of
inhabiting one state is made up for by an equal and opposite change
divided among the other two probabilities. Therefore (temporarily
ignoring x)
(aee+aeu+aeo)pe^t) + aue+auu+auo)pu^t)+(aoe+aou+aoo)po(t) = 0- (4-10)
Prom the discussion above, it can be seen that for non-degenerate
states, iLnn(x)<0 and anm(x)>0, n^m. Therefore, ann(x) + a ^ x ) +
an-j_(x) = 0 for all possible combinations of distinct n, m, and 1 in N.
The hazard rate associated with any state is the conditional density
of exit times given the amount of time already spent in the state. In
this Markov model the hazard rate associated with any state is the
constant escape rate,
hn(T'|x) = an(x) = ? anm(x) , n,m£N. (4-11)
The resulting duration in n is tn, a random variable with the
conditional distribution function
P{tn<Tn|x} = 1 - exp{-Tnan(x)}. (4-12)
The unconditional probability distribution at t is the solution to the
differential equation system given in (4.9).
5• Theories of state dependence
Fundamental to the notion of state dependence in labor market
models is an economic theory generally known as scar theory. In order
to discuss state dependence below, it is essential to quickly focus
some general ideas about scar theory. Ellwood (1982) and Heckman and
Borjas (1980) present some very general and short ideas about theories
that suggest state dependence. The ideas behind the theories,
nevertheless, are very important.
Ellwood's discussion of scar theory is centered on the loss of work
experience during teenage unemployment. The ideas are easily related,
however, to general questions of work experience.
Human capital theory assumes that workers, especially early in
their careers, acquire experience or education. This acquisition is
viewed as an investment in "human" capital. This early investment
leads to concave aggregate age-earnings profiles. Loss of early
employment opportunities shifts back human capital investment and
delays increased earnings. These delayed earnings must be discounted
and total lifetime earninge might be truncated by a short career.
Unemployment later in life also involves some loss of investment.
The greatest amount of investment is undertaken by the young, however,
and later unemployment is not assumed to he as costly.
Dual lahor market theory also suggests some degree of state
dependence. Its theorists hypothesize that the main signals of
ability available to employers are education and work record. Agents
who are not in school and who show long or frequent spells out of a
job are assumed to have poor labor force attachment. These people are
poor risks.
The mirror image of scar theory which suggests the presence of
state dependence in employment is brought about by job- or
firm-specific human capital investment. Over their tenures at a
certain job, workers are assumed to develop skills which are valuable
to their employers. This suggests that the longer an agent is
employed, the more likely he is to remain in the same job. Unemployed
workers who have experience are also more attractive to employers with
similar jobs available. This suggests that their chances of being
hired are dependent upon previous occurrences of hiring.
6. Four types of state dependence
The models above exhibit two familiar key features that should be
stressed again. First, the length of time already spent in a given
spell has no effect on the rate of exit from that spell. This feature
is due to the exponential parameterization of the model and is, in
fact, unique to exponential models. Second, the labor market history
of an individual is inconsequential in determining his rates of exit
from spells. Effectively, these features preclude any structural
dependence on the state occupied, except that the model allows an
agent to remain in a state more easily than he can switch states in a
small interval.
Heckman and Borjas (1980) discuss and label four types of
structural state dependence that might exist in sequential labor state
participation models. The four types are pure Markov state
dependence, which is allowed in the Markov model above, and duration,
occurrence, and lagged-duration dependence, which are not. This
section will discuss these four types and some alterations which
Heckman and Borjas suggest to accommodate for the final three types.
Pure Markov dependence
Section 4 above has already alluded to the idea of pure Markov
state dependence. Consequently, and because it does not create major
theoretical difficulties, this discussion will be very short.
Continuous-time discrete state Markov models allow the probability
of entering a state n to differ from the probability of remaining in
that state n. Taken to the extreme, this suggests that for very small
fit's, the probability of changing states goes to zero. Prom the
probabilities given above it is clear that
lim a (x)(dt > 0 and lim (a ( x ) 4 t + 1) > 1 (6 .1 ) Awo™ d w o
for all distinct n and m in IT. Markov state dependence also allows
the probability of entering one state, say e, from another, say u, to
differ from the probability of entering first state from a third, say
o.
Duration dependence
The idea of duration dependence captures the notion that the length
of time spent in a state during a given spell will affect transition
from that spell. In the topic of labor state participation, this
encompasses two ideas. First, each unit of time spent in an
employment spell might diminish the chances of being fired. Second,
for either non-employment state, continued lack of a job might affect
transition rates either positively, due to decreasing reservation
wage, or negatively, due to decreasing attractiveness of the worker to
employers.
The exclusion of duration dependence is a result of the exponential
functional form assigned to exit time distributions in the model
above. Exponential distributions uniquely ignore time spent in the
spell in determining exit times. The model can be adjusted to allow
or to test for duration dependence by replacing equation (4-12) above
with some conditional distribution function f (T|X) where T is the
time in the state before termination. Pn(T|x) has conditional density
fn(T|x), is not affected by previous spells, and cannot be
exponential.
Recalling the concept of a hazard rate function, we see that the
hazard rate is
fn(T|x) h (T|x) = — (6.2)
1-Fn(T|x)
With these hazard functions for states, it is possible to derive the
distribution functions and densities which are
"T
Fn(Tlx) = 1-exp{- Jh (u|x)du} (6.3) and
fn(T|x) = hn(T|x)'exp-{ hn(u|x)du} , n£ N. (6.4)
c
If 4hn(Tlx)/dT>0, rf=N, the hazard rate increases with T, so we say
positive duration dependence exists. If the derivative is negative
(or 0), negative (or no) duration dependence exists.
Occurrence dependence
The idea behind what Heckman and Borjas call occurrence dependence
is that each spell spent in a given state by an agent and the total
number of those spells might affect the agent' s future probability of
re-entering the state once he has left it. The theoretical basis for
such state dependence in labor markets is found in scar theory as
discussed above and in the concept of human capital.
Since allowing occurrence without duration dependence removes only
the restriction against work history affecting exit times, time
already spent in the current spell is still assumed not to affect exit
times. Exponential distribution functions can still be used when
allowing for occurrence dependence alone.
To modify the model above to allow occurrence dependence, it is
necessary only to distinguish between the escape rates a^(x) for each
spell based upon the number h of previous spells. This can be
accomplished in either of two ways depending upon the underlying
assumptions. If only spells in state a are thought to affect a (x),
the hazard rate for an agent with h previous spells of state m can be
designated &!}(x). If previous spells in any state in N can affect the
&n(x)'s, designate the hazard rates as a^1*I(x) , where h, i, and j are
the number of spells in 1, m, and n. For simplicity we will assume
the first case. If i£(x) a** (x) for h^h1 , occurrence dependence is
said to exist. If ar(x) > a**~'(x), for all h, increased incidence of
a state decreases the expected waiting time of returning to the state.
Lagged duration dependence
If the concepts of occurrence dependence and duration dependence
are combined, they suggest that the total time previously spent in
certain spells affects the current probability of exiting a state.
This is called lagged duration dependence. To introduce lagged
duration dependence into the model it is necessary to restate the
hazard function as a function of past and present state durations. If
there is no current state duration dependence, the derivative for time
spent in the present spell is zero. The lagged duration hazard rate
is
hn(Tjjij|x) = gn(T^,...,T
1e,Ti,...,T
1a,TJ,...,T
10!x), n£N (6.5)
If duration, occurrence, and lagged duration dependence are allowed in
model, the hazard function above can be written as
h h i j ( T h i j J x ) . g h i j ( . } > n 4 N ( 6 . 6 )
7- Heterogeneity
The problem of heterogeneity has been widely discussed in the
literature cited for this paper. [see Ellwood (1982), Plinn and
Heckman (1982a,b), Heckman and Borjas (1980), Salant (1977)] Broadly
interpreted, the term "heterogeneity" refers to any differences in
observed or unobserved variables, either among members of a population
or for an individual over time. These differences in variables can be
further classified as "pure heterogeneity" or "state dependent
heterogeneity" as is done in Heckman and Borjas (1980). In its more
precise usage, the term "heterogeneity" is commonly used to refer to
pure heterogeneity in unobserved components.
Pure vs. State Dependent Heterogeneity
The major problem due to heterogeneity in a sample population is
caused by the inability to always distinguish between pure
heterogeneity and state dependent heterogeneity. The essential
distinction between the definitions of these two terms lies in the
cause of the changes in observed or unobserved variables.
Differences in variables are termed as "pure heterogeneity" if
changes in those variables are not determined by the outcomes of
participation in any specific state of the labor pool (ie.
employment, unemployment or out of the labor force). While purely
heterogeneous characteristics may change over time, their changes are
not a function of state participation. As exogenous variables to the
state selection process, these components affect the process, but they
are not affected by it.
Pure heterogeneity, as defined in the previous paragraph, creates a
problem that has been called "the Mover-Stayer problem". Basically,
the problem is that pre-existing differences among agents might
falsely indicate that a worker's participation in a state affects his
escape rate from that state. These differences must therefore be
controlled for. A major conceptual problem arises in trying to
control for this heterogeneity. Salant (1977) discusses this problem.
Theories of job search predict a constant or decreasing reservation
wage for individual job seekers. The escape rates for these agents
should therefore remain constant or rise. This seemingly plausible
assumption seems, however to contradict empirical evidence of a
declining aggregate escape rate.
The apparent contradiction above can be explained by assuming the
presence of pure heterogeneity. In fact, differences among agents in
the determinants of a reservation wage are necessary to prevent the
distribution of wages from collapsing to a single wage.
The assumption of pure heterogeneity among workers explains the
contradictions discussed above. Search theory predicts a constant or
rising escape rate among individuals, but pure heterogeneity allows
groups of individuals to have different constant rates. In any state
there will be agents with high escape rates and agents with low escape
rates. Those with high rates are termed "Movers" while those with low
rates are termed "Stayers". As time passes, the Movers will, on
average, leave more quickly while Stayers will generally remain in the
state. Thus, while each individual might have a constant escape rate,
the aggregate escape rate falls over time. Failure to account for
pure heterogeneity in unobserved variables across agents will lead to
an over-estimation of duration dependence.
Heterogeneity in unobserved variables that is caused by or affected
by the state selection process (ie. state dependent heterogeneity)
can cause problems in the detection of duration dependence as well.
Spells in a given state may affect different agents differently. For
example, unemployment might inflict different costs on one worker than
it does on another. In such a case, it is difficult to measure
comparative losses. Such changes in variables, however, are affected
by the state selection process, and in turn, affect it. Unlike
variables exhibiting pure heterogeneity, variables with state
dependent heterogeneity have joint distributions which are partially
determined by the history of the process.
Even if there is no state dependence in observed variables, it is
possible to find it in unobserved variables. In any event, failure to
correctly control for heterogeneity in dynamic models can seriously
bias the estimation of state dependence.
Functional forms of job search model parameters
The theoretical solution to the problem of heterogeneity in the
estimation of state dependence in dynamic job search models involves
assigning functional forms to the parameters of the model. In
principle, the parameters of the search model are functions of
observed or unobserved variables. Parameters associated with
observables can be estimated. Unobservables must be assigned
distributions with parameters that can be estimated and then
integrated out.
In practice, the problem of functional form selection arises when
trying to use the method outlined in the previous paragraph. Economic
theory commonly does not suggest functional forms that might be
applicable to a given parameter. Aside from certain qualitative
guidelines, the selection of functional forms for unobservables seems
to be quite arbitrary.
It is obvious that heterogeneity and the associated problem of
functional form selection are important problems in the study of state
dependence. In fact, Plinn and Heckman (1982b) report that an
unpublished paper by Heckman and Singer suggests that estimates of
parameters for unobservables are very sensitive to functional form
choice. The Heckman and Singer paper does, however, suggest a
non-parametric estimator of unobservable variable distribution
functions to partially diminish the arbitrariness of functional form
selection. Flinn and Heckman (1982b) demonstrate this method of
controlling for heterogeneity, first in observables and then in
unobservables, for the one-state search model above. Heckman and
Borjas (1980) use a simpler method of parameterizing the hazard
functions. Their analysis deals with log-normal and Weibull
distributions and suggests guidelines for more general
parameterizations.
For their analysis, Heckman and Borjas assumed that the explanatory
variables in vectors x and y are spell-specific, remaining constant
throughout a spell. This assumption helps make the DIME/SIME data
especially well suited to the Heckman and Borjas methods at least from
the standpoint of observahles. As Weiner (1980) points out, each
DIME/SIME spell has an associated vector of variables measured at the
beginning of the spell. The assumption might seem artificial, but for
many variables such as education or age, initial values are most
important. Especially for employment spells, most variables are most
important at the beginning of the spell for making hiring decisions.
The procedure for controlling for heterogeneity
Following the techniques used by Heckman and Borjas, each
stochastic state-determining vector y(t) explained above is separated
into an exogenous explanatory vector z(t) and an unmeasured
explanatory vector v(t). These vectors actually depend on the states
of origin ,n, and destination, m, and might depend on some or all
previous spells. Their true representations should therefore be
zniJ(Te'---'Tl'Tu'*--'Tu'To'---'To) a n d vnmJ(,)' b u t f o r s i mP l i c i ty we
will refer to them as z and v, with only those labels that are needed
for clarity. Stationary characteristics are still contained in vector
x, but extreme care must be taken in classifying a characteristic's
stationarity.
One example in Heckman and Borjas uses these vectors to
parameterize the hazard function assuming a Weibull distribution.
With this assumption the hazard is distributed as
hn(Tn|x) = a n y n ( T n ) ^ -1 - (7.1)
With this distribution,Y n>1 signifies positive duration dependence,
V n<1 signifies negative, and "7^=1 signifies the lack of duration
dependence with an exponential distribution as in Markov models.
Introducing the possibilities of occurrence and lagged-duration
dependence, the hazard is distributed as
^./JkllogT^+i 4 n(k)logii-kl. (7.2)
The < / (k), p&N, are parameters which equal 0 if waiting time
distributions for the current spell are unaffected by visits to state
P-
To parameterize the Weibull hazard rate to reflect heterogeneity in
z and v, write
ahi^ = exp-{BhiJzhiJ + vhiJ) (7-3) n r 'rn n n ; w y'
Where p is a parameter vector with the appropriate dimension. One
extreme treatment of unmeasured pure heterogeneity sets v as a vector
of constants for all combinations of spell counts hij and for all
states n. The other extreme treatment lets the v's be mutually
independent for each work history and in any current state. A
treatment that is more intuitively appealing sets vj| = crj1*' 'f where
c is a spell-specific factor loading coefficient vector and y is &
person-specific vector of constants.
Heckman and Borjas also show that in general, characteristics can
be allowed to change within a spell while still remaining stationary
for small, discrete time periods. To allow this non-stationarity
within spells, write the hazard for a spell starting at calendar time
X [h^CTJjiJ)] as
h S i J C T S- z S ( T S + ' t : } - T S " 1 - • • • » T i » T i » • • • » T l » T i i i . v S 1 J ^ S + " ^ ) • ( 7 - 4 )
The density function for exit times from the hth spell of n, Tz, given
the unobservables at all times within the spell fv •'(a +T )» a<I 1 is
h^^T^expf- ^ hhij(a)du|. ((7-5)
Breaking the spell down into periods in which characteristics are
constant, we can replace the integral with a finite sum to obtain
likelihoods conditional on unobservables. The unobservables can then
be integrated out.
8. Censoring
Miller (1981) discusses three classes of censored observations. A
censored observation is one which contains only partial information
about the random variable in question. In the context that is used
here, censoring results from a time truncation of the period over
which labor force spells are observed.
Miller's initial classification of censored spells determines
whether we observe only partial information due to a time limit or due
to a limit on the number of observations. Miller classifies these
causes as Type I and Type II, respectively. Miller's third class of
censoring, random censoring, contains the type of spells with which we
are concerned. Spells in this class might be truncated due not only
to a pre-set ending time for the experiment, but also due to agents
leaving the sample before the end of the experiment. In the DIME/SIME
data, some mid-experiment truncation resulted from interview refusals,
expulsion due to fraud, and relocation out of the Denver or Seattle
areas.
The distinction between right and left censoring is also discussed
briefly in Miller's book. Right censored spells are truncated such
that an ending time is not observed. Eor left censored spells, no
starting time is observed and the spell is joined in progress. It is
clear that the DIME and SIME experiments caught some spells already in
progress and left some spells still in progress. When selecting a
span of the experiment from which to obtain observations, however, it
is advisable to pick a period near the end of the experiment. In this
way, a labor market history of at least two or three years can be
associated with each spell. Since the spells in question took place
after the surveys had been conducted for a while, their starting times
can probably be observed. We need to be concerned, then, primarily
with random right censored observations.
Formally, S.. ,Sp, • • • ,S_ are the iid censoring times associated with
spell lengths f, ,T2,.-.,Tn (not necessarily iid). For each spell, we
see either the true or the censored length and we know if the spell is
censored or not. That is, we observe (S1 ,I1 ),...,(Dk,Ik) where
Di = minCT^Si) = T± A S±. (8.1)
I± = l(Tij<Si) = 0 if censored (ie., Ti>Si)
= 1 if not (ie., T^Sj) . (8.2)
With this type of random censoring it becomes crucial when
estimating parameters to make the assumption that §. and S are
independent of each other. For this to be so, attrition must be
random and not a result of the labor state selection process. This
assumption seems reasonable for all three causes of attrition;
expulsion, interview refusal, and relocation. It must be pointed out,
however, that there is a slight possibility that interview refusals
(and a very slight possibility that mobility) is related to the
agent's job status. Even if there is a relationship between state and
attrition, however, it is even less likely that the censoring is
related to the length of the spell.
Maximum likelihood estimation with random censoring
For reasons discussed in the final section of this paper, I feel
that the method of maximum likelihood is the best estimation method
for the problem at hand. Miller explains the use of maximum
likelihood estimators in his second chapter.
Each uncensored spell (ditl) has a likelihood function defined by
the density f(dA). The likelihood for a censored spell (d^O) is
defined by the survivor function
S(d.) = 1-FUi) = P{T>di}. (8.3)
The likelihood for a spell that might or might not be censored is
therefore
LU^i.) = f(d.)i S U . )1 " 1 , (8.4)
so the likelihood for the sample as a whole is
n
J^Kd^ii) =Tf f(di)"^S(di), (8.5)
where ' and " are products over uncensored and censored spells,
respectively. Recall that this likelihood function assumes the
independence of true and censored spell lengths.
If J| = (p. , . . . ,p )' is a vector of parameters, maximizing L(jB) over
all J3 is equivalent to solving for p in the likelihood equations
sp.iog Kp) = ^^ . log igCd^ii) = i | i o g fe(d i)4-nj1°« V d i ) = 0.
Solving generally requires iteration on a computer. Miller (1981)
gives several methods of solution.
9• DIME/SIME data
A test of state dependence using the methods of Heckman and Borjas
requires individual state duration data. A history of continuous
spells of employment, unemployment, and non-participation must he
acquired or constructed for each individual in the survey. The
National Longitudinal Survey of Young Men (NLS) used by Heckman and
Borjas provides such histories. Unfortunately, the sample size used
by Heckman and Borjas was necessarily very small to avoid
heterogeneity and spell censoring.
The Public Use Files of the Denver and Seattle Income Maintenance
Experiments (DIME/SIME) provide the information needed to construct
labor market histories for a much larger sample of workers. It is
likely that the increased reliability of estimates due to the greater
sample size would far outweigh the added problems brought about by
increased diversity in the sample. The addition of heterogeneity and
censoring must be controlled for in the manners discussed earlier.
It is also important not to over-emphasize the extent of the added
diversity in the new sample. It must be remembered that while the
young men in the NLS sample may seem similar to each other, they are
not totally homogeneous. Introducing and controlling for an element
of heterogeneity might, in fact, lead to more accurate estimates than
assuming that similar workers are homogeneous.
The primary purpose of the DIME and SIME surveys was to monitor
the effects of negative income tax on labor supply, participation in
transfer programs, and marital stability. Eour years of panel data
sets were compiled into the Public Use Piles. In Denver, about 2800
families were surveyed between January, 1971 and December, 1974* In
Seattle, about 2000 families were surveyed from January, 1971 through
December, 1973. Each of the families that remained in the program was
surveyed at regular intervals throughout the test periods.
At the outset of the experiments, participants had to meet a series
of requirements. Weiner (1982) says that these requirements were not
as strictly enforced after the experiment was underway. At the
outset, enrolled families had to have either two heads or one head
with at least one dependent. According to Weiner, this requirement
was later slackened. The heads of these families were required to be
between 18 and 58 and able to work.
Pre-experiment earnings were restricted to below $9000 for
single-head families (based on a family size of four) and to below
$11,000 for double-head families. Both Burdett, et al. (1981) and
Weiner (1982) note that these single year earnings restrictions might
not exclude as many higher-income families as might be expected.
Transitory downward fluctuations in income allowed some families with
normally higher incomes to participate. The exclusion of middle
income families resulting in estimates that show "higher rates of
leaving employment and lower rates of entering employment than the
population at large" noted in Tuma and Robins (1980) might not be as
serious as it seems
For each spell, the length of the spell and a vector of individual
characteristics measured at the beginning of the spell were recorded.
The characteristics were education, race, age, number of children,
marital status, wage, liquid assets, and skill level. Wages were
recorded only for spells of employment, but expected wages can be
estimated for the other two states by regressing against some of the
other variables. Since wages were reported for different years, they
must be deflated. Liquid assets included stocks, cash, checking
accounts, equity in homes or cars, and so on. Skill level was
classified as either skilled or unskilled and was determined by the
worker's anticipated ability to find work in a spot auction market.
Since transition between states undoubtedly depends not only upon
individual characteristics, but also on cyclical forces, an attempt
must be made to correct for any non-stationarity in the environment.
Denver experienced little fluctuation in their economy over the sample
period, so we need to be concerned only with corrections for Seattle.
Weiner (1982) suggests the use of two local cyclical measures: the
Conference Board's monthly (standardized months) help-wanted ad index
and the Employment Security Department of Washington's Seattle/Everett
SMSA unemployment rate.
Attrition from the sample resulted due to interview refusals,
expulsion due to fraud, or relocation. Depending upon the techniques
used to select spells for observation, it might be necessary to
correct for this censoring as well as the censoring of spells that
were still in progress at the end of the experiment.
Finally, since sixty percent of the families in these studies were
given negative income tax treatments, their histories are probably not
representative of society at large. These "test" families should be
excluded. Those families in the control groups still provide a large
sample of labor state histories. Even when the DIME/SIME data is
limited to that from the control families, it provides a larger sample
to test the methods and findings of Heckman and Borjas (1980) than the
NLS data.
10. Problems in matching the data to the methods
Undoubtedly, a number of problems would be encountered while trying
to use the methods of Heckman and Borjas to test for state dependence
in DIME/SIME spells. A number of these problems have already been
discussed in the contexts of previous sections. I am sure that I
haven't even thought of a good many others. Three conceptual problems
that I feel are very important, however, will he discussed here.
Those problems are the selection of the correct variable to measure,
selection of an assumed distribution for unobservables, and selection
of a method of estimation.
Selecting observation variables
The problem of variable selection can take two forms. First, there
is the problem of selecting a sufficient number of appropriate
explanatory variables. A second problem is the question of what
exactly to measure to determine the durations of past or current
spells. These measurements are then used to estimate the parameters
important in determining state dependence.
The first problem is a moot question with respect to the DIME/SIME
data. Explanatory variables have already been observed, so the only
question is whether to use all of them. Obviously, it is wise to use
as many as possible without over-complicating the analysis. The
important question, then, is what to use as an indicator of a person's
work history.
Possible indicators of an agent's work record are the number of
spells, the length of individual spells, or the total length of time
spent in a given state 1, m, or n. The number of previous visits to a
state can be misleading because it fails to say anything about time
spent in the spell. An agent might have very few spells of
unemployment and therefore seem to be able to hold a job, when in
actuality, he has a few very long unemployment spells.
Average lengths of individual spells, on the other hand, fail to
capture the number of transitions. A person with short unemployment
spells might be thought to be employed most of the time, when his
unemployment spells are really so short because he has so many. The
best duration variable to measure, then, is total time in a given
state.
Selecting distributions for unobservables
As has been said above, the problem of assigning distributions to
unobservables for the purpose of integration has very arbitrary
solutions. Nevertheless, a distribution must be assigned. The
solutions listed here are very tentative and undoubtedly will lead to
some error. Some structure is needed, however, and hopefully the
error will be minimized.
There is nothing to indicate that most unobservables will not be
distributed about a mean with decreasing frequency of observation with
increasing distance from the mean in either direction. That is to
say, it seems harmless to assume a distribution that at least looks
somewhat like a Normal distribution. For some values of a and , the
Weibull distribution demonstrates a bell-shaped curve of this type.
Considering the Markov-exponential history of the model (a special
case of the Weibull) it is appealing to think of the variables as
Weibull-distributed.
If evidence should exist that contradicts the assumption of a
Weibull distribution, it is reasonable to assume a Normal or
log-Normal distribution Procedures that can be used under either
assumption are well documented in the literature.
Selection of an estimation method
A final problem area deals with the choice of a method to estimate
parameters given the observations of an experiment like DIME/SIME.
Certain properties of the regression procedure make it unsuitable for
use with the DIME/SIME data. I would use the maximum likelihood
method mentioned above.
Regression procedures have two very serious limitations which are
noted in Heckman and Borjas (1980). First, regressions using censored
observations result in serious bias. Maximum likelihood estimators,
as it has been shown, are well suited for censored data. Second,
regression techniques make it difficult to introduce explanatory-
variables. Since the DIME/SIME data contains a significant number of
censored spells and explanatory variables must be introduced to
decrease the possibility of spurious, heterogeneity dependence,
maximum likelihood methods must be chosen over regression.
11. Conclusion
Heckman and Borjas (1980) develop some very important procedures to
test for state dependence in labor force transitions. They tested
these methods with a very small study of young men's labor force state
participation. Their test found no duration or occurrence dependence
after controlling for heterogeneity. The test was, however, really
just a demonstration of the procedures. A true test remains to be
done on a large sample such as the DIME/SIME data.
REFERENCES
Burdett, Kenneth, et al. A Markov Model of Employment, Unemployment, and Labor Force Participation; Estimates from the DIME Data, The Center Tor Mathematical ~STudies Irf Economics and Management Science, Northwestern University Discussion Paper No. 483, May 1981 .
Burdett, Kenneth and Dale Mortensen. "Labor Supply under Uncertainty," Research in Labor Economics, Vol. 2 (1978), pp. 109-157-
Cinlar, Erhan. Introduction to Stochastic Processes. Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1975-
Plinn, Christopher J. and James J. Heckman. "Models for the Analysis of Labor Force Dynamics," Advances in Econometrics, Vol. 1 (1982a), pp.35-95. See Erratum in Vol. 2.
Plinn, Christopher J. and James J. Heckman. "New Methods for Analyzing Structural Models of Labor Force Dynamics," Journal of Econometrics, Vol. 18, No. 1 (1982b), pp. 115-168.
Plinn, Christopher J. and James J. Heckman. "Are Unemployment and Out of the Labor Force Behaviorally Distinct States?" Journal of Labor Economics, Vol. 1, No. 1 (Jan. 1983), pp. 28-42.
Ellwood, David T. "Teenage Unemployment: Permanent Scars or Temporary Blemishes?" in Richard B. Freeman and David A. Wise, eds., The Youth Labor Market Problem. Chicago: The University of Chicago Press, 1982.
Lippman, S., and J. McCall. "The Economics of Job Search: A Survey, Part I," Economic Inquiry, Vol. 14 (June 1976), pp. 155—189 •
Heckman, James J. and George J. Borjas. "Does Unemployment Cause Future Unemployment? Definitions, Questions and Answers from a Continuous Time Model of Heterogeneity and State Dependence," Economica, Vol. 47, No. 187 (Aug. 1980), pp. 247-283-
Miller, Rupert G. Survival Analysis. New York: Wiley, 1981.
Robbins, H. "Optimal Stopping," American Mathematical Monthly, Vol. 77(1970), pp.333-343-
Salant, Steve. "Search Theory and Duration Data: A Theory of Sorts," Quarterly Journal of Economics, Vol. 91 , No. 1 (Feb. 1977), pp. 39-58.
Tuma, Nancy Brandon and Philip K. Robins. "A Dynamic Model of Employment Behavior: An Application to the Seattle and Denver Income Maintenance Experiments," Econometrica, Vol. 48, No. 4 (May 1980), pp. 1031-1052.
Weiner, Stuart. A Survival Analysis of Adult Black-White Unemployment Rate Differentials, Ph.D. dissertation, Northwestern University, 1982.
ACKNOWLEDGEMENTS
Although a paragraph such as this suggests that this paper is of a higher quality than it actually is, I owe and offer my gratitude to the following people. Any shortcomings of this paper are my own fault as all contrihutions by those listed below were purely positive. Dale Mortensen has served me as seminar co-leader, professor, advisor, reference source, bibliography, library, tutor, and department chairman throughout the year. Michael Dacey also served as seminar co-leader. Prior to and perhaps even more important than that, he attempted to teach my classmates and me the fundamentals of stochastic processes even though he was suffering from very poor health at the time. Erhan Cinlar, who wrote the text, continued our instruction in stochastic processes while Professor Dacey recuperated.
Although I have never met him, I would like to thank James J. Heckman of the University of Chicago. Much of my most important source material was written either by Professor Heckman with an associate or by professors in the basement of Andersen Hall or elsewhere on Northwestern's campus. Most of all, I thank everyone in the program for giving me a second chance after I messed up the first one.
