# Model specification Directory UMM :Data Elmu:jurnal:L:Labour Economics:Vol7.Issue2.Mar2000:

In both of our samples we expect to find different exit patterns between persons with and without entitlement to benefits. Furthermore, the 1992 rule changes are expected to have no effect on unemployed individuals not receiving UB. The non-receivers therefore function as a control group. We now proceed by presenting our model specification in Section 2. In Section 3 we describe our data sets and discuss the definition of unemployment spells. Our results are presented and discussed in Section 4. It turns out that the analysis does not give empirical support to the notion that the 1992 reform produced significant changes in the behaviour around 80 weeks of unemployment duration. However, the reform appears to have had effects on the earlier stages of the search process, in that the first year’s exit rates for the second sample UB-receivers are lower than in the first sample. Section 5 contains a critical assessment of the findings.

## 2. Model specification

The hazard rate out of unemployment and into employment is defined as the limit of the conditional probability of a transition taking place in a small interval d t after time t if no transition occurred until t, when that interval approaches zero. Formally, Pr t F T - t q d t T G t Ž . r t s lim , 2.1 Ž . Ž . d t d t ™0 where it is understood that t is a realisation of a random variable, T, that measures unemployment duration. The conditional probability in the numerator is also the product of the chance of receiving a job offer and the probability of accepting, a fact that might be used as a starting point for a structural model. We shall, however, use the more common procedure of specifying the hazard directly. More specifically, we use a semiparametric approach to duration analysis that was Ž . proposed by Prentice and Gloeckler 1978 . The virtues of this method are that it is unnecessary to make parametric assumptions concerning the hazard’s time dependence, and account can be taken of the potential discrete nature of the data. Recently, this semiparametric approach has found several applications in the study of unemployment duration. 6 Assume that conditional on staying in the pool of unemployed persons until Ž . time t, individuals leave unemployment with rate r t,x , specified as r t ,x s u t exp x X b . 2.2 Ž . Ž . Ž . Ž . Ž . Ž . Eq. 2.2 is known as the Cox proportional hazards specification. Here, u t is the baseline hazard, which is some arbitrary nonnegative function of time, x is a 6 Ž . Ž . Ž . Examples are Meyer 1990 , Narendranathan and Stewart 1993b , Carling et al. 1996 . vector of covariates, and b is a coefficient vector. 7 Now define the discrete or grouped hazard, l , as the probability of a transition taking place in the interval t w . a s t y 1,t , conditional on survival until t y 1. Using the relation between the t hazard rate and the survivor function, it follows that t t X l s 1 y exp y r u,x d u s 1 y exp yexp x b u u d u . Ž . Ž . Ž . H H t ty1 ty1 2.3 Ž . w t Ž . x Ž . Defining g s log H u u d u , we can rewrite Eq. 2.3 as t ty1 X l s 1 y exp yexp x b q g 2.4 Ž . Ž . t t Time-varying covariates may easily be introduced if they are assumed to change only at the endpoints of time intervals: simply subscript x with t. The probability of surviving through any interval a after having survived the t Ž . preceding interval is 1 y l . Therefore the likelihood contribution of someone t who leaves unemployment in the t th interval is i t y1 i l 1 y l . Ž . Ł t t i ts1 We assume that censoring takes place in the beginning of intervals. 8 Then, defining d s 1 if individual i’s spell ends in a transition, 0 otherwise, i’s i likelihood contribution is Ž . d 1yd i i t y1 t y1 i i L s l 1 y l 1 y l . Ž . Ž . Ł Ł i t t t i ½ 5 ½ 5 ts1 ts1 Ž . Collecting terms, using Eq. 2.4 , taking logs, and summing over i, the log-likeli- hood of a sample of N observations is then t y1 N i X X log L s d log 1 y exp yexp x b q g y exp x b q g , Ž . Ž . Ž . Ý Ý i t t t t i ½ 5 is1 ts1 2.5 Ž . where we have subscripted x to indicate that some variables may be time-varying. The hazard model as outlined above has no error term, and is correct only if all differences in the individual durations are due to differences in the vector of 7 Ž . The exponential and Weibull models are also proportional hazard models with u t appropriately parameterised. 8 That is, someone who is censored at time t st y1 is assumed to have survived only through i interval t y1. i observed variables, x . However, there may also be unobserved sources of t heterogeneity. The standard way of trying to deal with this problem is to assume that an unobserved random variable ´ , which is time constant and independent of Ž . the observed covariates, enters the hazard multiplicatively. Eq. 2.2 is thus changed to r t ,x s u t exp x X b ´ , 2.6 Ž . Ž . Ž . Ž . With an additional assumption regarding the distribution of this unobserved variable, 9 such a model can be estimated. Usually a gamma distribution is chosen. Ž . Meyer 1990 implemented this approach in the semiparametric model. It may be shown 10 that when ´ is gamma distributed with unit mean and variance s 2 , the log-likelihood function becomes y2 ys t y1 n i X 2 log L s log 1 q s exp x b q g Ž . Ý Ý t t ½ is1 ts1 y2 ys t i X 2 yd 1 q s exp x b q g . 2.7 Ž . Ž . Ý i t t 5 ts1 Results from both specifications will be reported in this paper. 11 One final modification of the model is made: we ask whether the hazard increases as benefit exhaustion approaches. To include this aspect in the analysis, UB Ž we define I s 1 if benefits are received, 0 otherwise, and replace g by g q 1 t t . nonUB y I g . The interval length is four weeks, corresponding to the accuracy with t which the spells are measured in the unemployment register.