Model specification

W e distinguish three different possible states in our model: unemployment, w age-w ork and self-employment. Self-em ploym ent is considered as an alternative to paid employment. M ovements from and to “out o f the labour force” are not considered here. An individual can move from any o f these states (source state, denoted by the first subscript) to the others (destination state, denoted by the second subscript) at any time. Six types of transition can then be defined as shown in Table 3.1 below.

This specification o f the model is coherent with on-the-job search theories. U nemployed individuals devote some o f their time searching for jobs. But, once they accept a job and start w orking (as paid w orkers or self-employed) they continue to

search for a better job. Tw o types o f “jo b s” are considered here: paid w ork and self- employment"^.

Transition intensities are defined as the probability o f departure from state k to state I in the short interval (r, t+dt) and are denoted as Qu (t\Z;^), w here Z is a set of observable and unobservable individual characteristics {X and v respectively) and p is a set of unknow n param eters to estimate, t, the elapsed duration, is m easured in m onths. In particular, the following functional form represents the transition intensities:

= e x p { g „(f) + z P j, + 8 j,v } (3.1)

where gu(t) is a function o f time spent in state k, before departure tow ards /. This specification allows for a flexible and non-m onotonic relation between elapsed duration and the hazard function. Its functional form will be discussed in Section 3.4.2. The set p, includes all param eters o f interest in g(.), and 0^/, for all possible (k,l). An unobservable individual fixed effect is denoted by v which w ould be correlated with the time spent in each state. It can reflect differences in tastes for working or starting up a business. The estimation o f param eters specific to every state allows state dependence along with duration dependence. Finally, X is a set of dem and conditions and demographic variables.

Table 3.1: Possible transition intensities.

Source State Self-Employment

Destination State

Em ployment Unemployment

Self-Employment e.,„(rlZ;P) _9,„('iz;P)

Employment e , . ( ' i z ; p )

Unemployment e . . ( d Z ; p ) e„,(tlZ;P)

Note: U denotes unemployment, E paid em ploym ent and SE self-em ploym ent.

Therefore the contribution to the likehhood function for each individual and com pleted spell is the probability o f surviving in state k until t (survival function) times the probability of m oving from k t o l i n t (transition intensity),

P„ (<IZ; P ) = e x p { - 0 , (fIZ; P )} 0 „ (0 Z ; P) (3.2)

f,

where 0^ is the corresponding integrated hazard function (0 ^ , = | Z ; p ) B 5 ) . For

0

each individual, the data consist o f one or more spells in every state. N ot every spell is com plete by the time o f the interview. Hence it is necessary to account for right censored spells. The contribution to the likelihood function o f an incomplete spell is the survivor function, that is

F ,(;IX ;P ) = e x p { - e ,(f lX ;p )} , (3.3)

Assuming that v equals zero for all individuals i. e. there is no unobserved heterogeneity, the likelihood function for an individual with a sequence o f spells {ti,

would be,

A C A A c,

. (3 4)

V c = l k l^k c=\ I

A(Pi'„ c,)= nnn^«ci^A P)''“ fin^iki^np)''

w here is an indicator variable which equals 1 if the individual exited state k tow ards state I in the cth spell; is a dummy which equals one if the cth spell is incomplete and the individual did not move from state k.

Taking logs and considering a sample o f N i.i.d. individuals the log-likelihood function is given by^

^ For a step-by-step derivation o f the likelihood function with and without unobserved heterogeneity, see Lancaster(1990)

N C,

>0gZ.

= XËZ

1=1 c = l k l^k

(3.5)

In the presence of unobservable heterogeneity among the individuals the model becom es m ore complicated. The individual fixed effect, denoted by v„ is an unobservable variable that varies over the population. Therefore, we cannot condition the individual probabilities on v, and use it as an additional explanatory variable. To get the unconditional probabilities it is necessary to integrate v, over all its possible values. In this case the individual likelihood takes the form

q

n

c=\ k l*k

(3.6)

w here h(vi) is the unknown distribution function o f the individual effect. The log- likelihood function for all individuals w ould then be.

logL = ^L,(ph.,,...,r.c^,X.)

/=]

The distribution of the unobserved heterogeneity could be fully specified and the previous equation estimated by maximum likelihood. H ow ever, H eckman and Singer (1984) pointed out that misleading results can be obtain by using these procedures when the chosen distribution for unobservables is not the right one. Therefore we alternatively use the N on-Param etrie M axim um Likelihood Estim ator (N PM LE) proposed by both authors which does not require any distributional assum ption. This procedure approxim ates the distribution function o f unobservables, /zfv), with a finite mixture distribution. The points o f support o f this finite distribution are the unknown values

V V m to w hich the M unknow n probabilities are attached. Then, the contribution to the likelihood o f an individual becomes:

A ( P ’ ^ ^ 1 » • • ’ ’ he ’ -^1 ) ~

M [ / C, Y Q , \

m = l 1 V c = l k l ^ k J \ c = l /

(3 ^0 being the log-likelihood function its summation over all individuals. The points of support as well as the probabilities assigned to each o f them are now param eters of interest to be estimated by the EM -algorithm (see Appendix B for description of im plem entation). The function is maximised at different num ber o f support points until the param eters o f the criterion function relatively stable^.