Scheherazade Zerroukhat
email: szerroukhat@yahoo.fr

Redha Chellal
email: chellal.redha@gmail.com

Farid Bencherif
email: fbencherif@gmail.com

Amelioration of Glaisher's Congruence

Keywords:

come

Introduction

Let p be a prime odd number and for every integer a relatively prime to p , we define Fermat's quotient in basis a as:

q p (a) = a p-1 -1 p .
The study of Fermat's quotients and the congruances involving these quotients is related to the classical study of the Fermat's last theorem [START_REF] Ribenboim | 13 Lectures on Fermat's Last Theorem[END_REF]. The following confruence is due to Glaisher [START_REF] Glaisher | On the residues of the sums of products of the first p -1 numbers and their powers to modulus p 2 or p 3[END_REF]:

2 p-1 -1 p ≡ - 1 2 p-1 k=1 2 k k ( (mod p)).
Using many computations, B. Ronk [4] et [5] has discovered that this congruence was verified modulo p 2 for primes less or equals to 1000. He conjectured that this congruence was verified modulo p 2 for all primes p. In this paper, we will confirm this conjecture. We will also improve (ameliorate) the congruence of Glaisher by proving it

Theorem 1 For every prime number p ≥ 5, we have

2 p-1 -1 p ≡ - 1 2 p-1 k=1 2 k k (modp 2 ).

Proof of the theorem

We'll give a series of lemmas from which the proof of the result (theorem) will follow.

Lemma 2 For every prime number p ≥ 5

p-1 k=1 H k k2 k ≡ 0 (mod p).
Proof. This lemma was proved in [START_REF] Sun | Arithmetic theory of harmonic numbers[END_REF] Lemma 3 For every prime number p, p odd, we have

H p-k ≡ H k-1 (mod p).
Proof. This lemma was also proved in [START_REF] Sun | Arithmetic theory of harmonic numbers[END_REF].

Lemma 4 For every prime number p ≥ 5

p-2 k=1 2 k H k k + 1 ≡ 0 (mod p).
Proof.

p-2 k=1 2 k H k k + 1 = p-1 k=2 2 p-k H p-k p -k + 1 ≡ p-1 k=2 2 p-1-(k-1) H p-k p -(k -1) ≡ - p-1 k=2 H k-1 (k -1)2 k-1 ≡ - p-2 k=1 H k k2 k ≡ - p-1 k=1 H k k2 k + H p-1 (p -1)2 p-1 ≡ 0 (mod p). Lemma 5 x p -(x -1) p -1 p ≡ - p-2 k=0 x k+1 k + 1 + p p-2 k=1 H k x k+1 k + 1 (modp 2 ).
Proof. We have:

x p -(x -1) p -1 p = - 1 p p-1 k=1 p k x k (-1) p-k = p-1 k=1 (-1) k p -1 k -1 x k k = - p-2 k=0 (-1) k p -1 k x k+1 k + 1 .
Note that:

(-1) k p -1 k = (1 -p)(2 -p) . . . (k -p) 1.2. . . . k = (1 - p 1 )(1 - p 2 ) . . . (1 - p k ) ≡ 1 -p k j=1 1 j (modp 2 ).
Hence

x p -(x -1) p -1 p ≡ - p-2 k=0 (1 -pH k ) x k+1 k + 1 (modp 2 ). ≡ - p-2 k=0 x k+1 k + 1 + p p-2 k=1 H k x k+1 k + 1 (modp 2 ).
The proof followsimmediately. All we have to do is to replace x by 2. We'll have:


The Knowledge Account of Assertion and Moore's Paradox about Knowledge

Franck Lihoreau

To cite this version:

Franck Lihoreau. The Knowledge Account of Assertion and Moore's Paradox about Knowledge. 2007. halshs-00113478v3

Désolé, mais le brouillon de ce texte a été enlevé pour des raisons scientifiques. Merci de votre compréhension.


Patrice Gaubert

Metis-Matisse

NEURAL NETWORK AND SEGMENTED LABOUR MARKET 1

Introduction

In France, for administrative reasons, unemployed workers may actually be involved in occasional work while remaining identified as unemployed (and receiving the corresponding benefit). This is due to the fact that the unemployed are deemed to be seeking full-time jobs and non-fixed term contracts of employment. This situation may be analysed as evidence of a special type of secondary segment of the labour market in a context of massive unemployment. The authors consider the effects of this situation both on the duration of unemployment and its recurrence may be usefully investigated.

This paper is a contribution to the analysis of segmentation in the French labour market. It is based on a very different approach from the existing literature which is focused on career differentiation [START_REF] Favereau | Une approche longitudinale de la segmentation du marché du travail[END_REF][START_REF] Theodossiou | Wage determination for career and non-career workers in the UK : is there labour market segmentation ?[END_REF], returns to education [START_REF] Balsan | Analyse salariale des dispositifs d'aide à l'insertion des jeunes[END_REF], or, more generally, efficiency wage (Oi,1990 ;[START_REF] Albrecht | Dual labor markets, efficiency wages, and search[END_REF] or differentiation and discrimination [START_REF] Dickens | A test of dual market theory[END_REF][START_REF] Boston | Segmented labor markets : new evidence from a study of four race-gender groups[END_REF]. Basically labour market segmentation is described as comprising a primary and a secondary segment, differentiated mainly by job types : permanent in the primary segment, precarious in the secondary one. In contrast to the standard presentation, the high level of unemployment observed in France and the growth of precarious employment (part-time work, fixed-term contracts of employment, etc.), raises questions about the relationship between the secondary segment and unemployment. The rapid growth of these new job categories, often low skilled and low paid jobs, may be interpreted in two ways .Either they illustrate a change in the nature of the worker's transition from unemployment to employment (easing the search for a new job by a newly unemployed individual thanks to the closening of his links with the labour market), or, on the contrary, are they evidence of a rapid extension of the secondary segment of the labour market coupled with a symmetrical shrinkage of the primary segment, one of the characteristic features of recent economic growth trends. High unemployment in France raises questions as to what extent this model can be applied to the structure of unemployment. Is it possible to differentiate among categories of unemployed workers according to the duration and recurrence of unemployment ? How can these categories be defined, using information about educational and skill level, work experience and other aspects of previous jobs and the reasons for their termination ? These questions are analysed using a Kohonen network, to classify the unemployed. The data used are drawn from the register of unemployed kept by the Government Employment Service (ANPE). Two regions chosen were Nord and Rhone-Alpes. Some forty thousands individuals were randomly selected from the register in each of these two regions between July 1993 and August 1996. Our results show that unemployed workers who accept occasional work experience a significantly longer period of unemployment. For these workers the longer the time spent doing casual work during this period, the shorter the overall duration of the latter. The corresponding figure seems to be higher than that corresponding to the total monthly hours occupied in occasional work. The likelihood of recurrence of unemployment appears greater for such individuals than for other unemployed workers, while the length of their unemployment is close to that measured for people who do no occasional work.

Data

The complete register of the unemployed held by the ANPE contains information on all registered unemployed workers in the twenty-six administrative regions of France. The sample used for this study is made up of individuals who were looking for work between 1 July 1993 and 31 August 1996 and who were still registered as seeking a job or having found one by the latter date. As this data was originally collected for administrative purposes, it had to be adjust to be used here. One of the main problems we were faced with is that an individual may be registered more than once. This is what is meant by recurrence of unemployment. Each entry records the particular circumstances and category of job termination, how unemployment ended, duration of unemployment and the time spent in occasional work. This information is essential for determining the factors correlated to the duration of unemployment. Our investigation of individual workers revealed that some of them had been registered more than once and so this data was collated. This was possible when dealing with quantitative data (length of each period of unemployment, amount of months in occasional work, number of job offers received through the Government Employment Service). We also recorded the different periods of unemployment. But the qualitative characteristics of each search for work could not be taken into consideration. Only those concerning the latest search were available anyway. Individual characteristics are assumed to remain the same over this relatively short period of time. The proportion of individuals registered only once in two regions studied is recorded in the table below (the figures are similar for the other administrative regions) :

Variables created concerning occasional work

The data concerning casual work reveals some problems of measurement. The figures for total monthly hours are far from accurate, because only those concerning the first and last months of casual work are known and the exact dates of beginning and end are not available either. So a change in the total monthly hours worked from one month to the next one may mean a job change or that a job started for instance the twentieth of the previous month or ended the eighth of the following month.

Other information relating to other records is not known precisely either. For example, data is available only concerning the range within which the real measure is located (less than or greater than seventy-eight hours per month). The wage data for casual work is also sometimes missing.

For each period of unemployment involving occasional work we made a number of approximations. The missing information was replaced by computing the mean monthly hours for individuals belonging to the same administrative class. For each individual we then compared the average figure for the full duration of the occasional work with the maximum monthly hours declared during this period. As the difference is less than 10 % we decided to use the maximum observed in each period, because this figure actually exists and this avoids the problem of incomplete months. Similarly we used the maximum wage declared, which we assumed to be correctly associated with the monthly hours as defined.

Different variables were used in classifying and interpreting the results. First we used variables describing the individual (age, marital status, number of children, nationality) and professional characteristics (educational and skill level, seniority in the latest job). Second, we chose to examine different aspects of the worker's unemployment status, such as the reason for his job loss (lay off, voluntary quit, end of fixed term contract, first job search, return to labour market after a temporary exit), how his/her unemployment status ended (job, training program, exit from labour market, cancellation of registration for administrative reasons), duration and cumulated duration of unemployment measured in days and months (for individuals with more than one unemployment period), number of job offers received through the Employment Service. We also investigated the nature of the occasional work (total monthly hours, number of months in this work and cumulated duration, number of different periods of occasional work within a given period of unemployment, monthly wage).

In order to avoid level effects we use relative values to determine the impact of occasional work on the duration of unemployment: the total months of occasional work divided by the total months or total years of unemployment, total periods of occasional work divided by the total number of unemployment periods and by total years of unemployment, the number of job offers divided by total months of unemployment.

The figures obtained (expressed in means) are reported in the tables below. On the whole the Nord and Rhone regions are quite similar in terms of the variables presented, except, as far as the length of unemployment is concerned, which is slightly greater for the Nord region (twenty-one months versus eighteen). The distribution of this sample is reported in table 3, 4 and 5 by using frequencies computed with different variables to be developed later in our analysis. 4 2.6 9.6 7.1 13.3 62.4 25.3 12.3 62.4 11.5 14.9 11.3 Some regional differences should however be pointed out. These relate to educational levels (higher levels were found in the Rhone region) and to the relative shares of lay-offs, ends of contract and first jobs. Data concerning the unemployment allowance received by unemployed people was not available, though this is an important variable as far as our study is concerned.

Information on the kind of job found after unemployment was not available either. We were thus not able to carry out a detailed analysis of the duration of unemployment and of the overall situation of the unemployed embracing consecutive periods of work and unemployment. Hence our study will be limited to identifying the main categories of unemployed workers.

Classification and class characteristics

The Kohonen classification shown in table 6 [START_REF] Kohonen | Self-organization and Associative Memory[END_REF](Kohonen, , 1985 ; ;Cottrell, 1997, see in the Appendix I for some details about the method) is obtained using quantitative variables defined for each individual as age, job seniority, length of the latest period of unemployment, cumulated duration of unemployment since initial registration, number of job offers received per month of unemployment, number of periods of unemployment, the duration of occasional jobs per month of unemployment, number of periods spent doing occasional work per year of unemployment and per number of periods of unemployment, and finally, total monthly hours worked and total wages received for occasional work.

The variables defined earlier to describe occasional work are at the core of our analysis. The number of periods spent doing occasional work compared separately to the number of periods of unemployment and to the duration of this unemployment is used to differentiate between people presumably engaged in a work close to what they had doing before losing their job (occasional work occupying a large part of the period of unemployment and only one period of occasional work in each period of unemployment) and those people engaged in very occasional work for short periods of time and who change occupations within the same period of unemployment (presumably with no relation between these jobs and the one lost due to unemployment). Two maps are presented in the Appendix II, showing the code vector of each class (the line joining the different values is not intended to plot an evolution : the values are those obtained at the end of the algorithm for normalised variables ranked in alphabetical order according to name). These maps also include the ten super-classes resulting from a clustering classification applied to the Kohonen classification. The above table and those figuring in the appendix II show the main characteristics of the super-classes in terms of the means computed for the variables used in the classification. Our findings are significant in both regions studied. Two groups were found to be made up of older people. One group was characterised by lengthy seniority (Nord 3, Rhone 5) while seniority in the other was found to be relatively short (Nord 4, Rhone 4). This may be due to recent changes in the career of workers in the first group (unemployment caused by obsolete skills or because they were working in greatly depressed industrial sectors within the region) while in the second group workers experience long periods of unemployment soon after their entry into the labour market. Two groups comprising people of intermediate age (Nord 5 and 8, Rhone 6 and 9) were characterised by a single lengthy period of unemployment. It was observed that some people have already experienced a relatively long period of unemployment at the date of registration.

Other groups point to differences between the two regions, such as systematic recurrence of unemployment in all other groups in the Nord region, while this phenomenon is present only in some groups in the Rhone region. The reason for this lies in the high percentage of « ends of contract » in the Nord region among the causes of unemployment.

The main characteristics for the Nord region can be summarised as follows. One group is characterised by a high rate of recurrence ( 7) and there are two groups in which recurrence is a significant feature (2 and 9). One other group is both particularly young and numerous (1). Two other groups are characterised by a rather short period of unemployment (6 and 10). As for the Rhone region two groups show a high degree of recurrence (3 and 7), and two others show a particularly short length of unemployment (8 and 10). Another group is young and has experienced a single short period of unemployment (1), quite close to group 2 in age but different from it due to regular recurrence. The data furnished by the distribution of the population within these groups, using both initial qualitative variables and those constructed using ranges of quantitative variables, has proven invaluable in enabling a more accurate understanding of the different categories of unemployed workers. It should be pointed that in both regions there is one group (Nord 8 and Rhone 10) that has to be distinguished from the rest of the population studied. This group contains a high proportion of people corresponding to an administrative category of workers seeking work though already occupying a full-time job. This category is not deemed acceptable in so far as the French administration mixes these people with workers doing occasional work.

Another result, which lends support to our earlier interpretation obtained with quantitative variables, is that the two groups composing the first category (Nord 3 and Rhone 5) are principally made up of laid-off workers possessing high skills but low levels of education. Groups 2, 7 and 9 for the Nord region and group 3 for the Rhone region are characterised by a high percentage of « ends of contract », while groups 7 and 3 were also mentioned above to have employment profiles characterised by short duration. This would seem to provide evidence of a mechanism linking precarious work and unemployment. It can be seen that for some groups finding a job is an important reason for leaving unemployment (Nord 6, 9 and 10, Rhone 1, 2 and 8) and this is in turn linked to a shorter duration of unemployment (Nord 6, 7 and 10, Rhone 3 and 8). Doing occasional work also plays an important role for some of these groups (Nord 5, 9 and 10, Rhone 2, 7 and 9).

We will now turn to a closer examination of the different classes obtained.

Class interpretation

To obtain a more precise description of these classes and of what characterises the individuals belonging to them we carried out a multiple correspondence analysis using some of the qualitative variables already presented or which we created using some of the initial quantitative ones. We added the classification variable representing the super-class in which each individual is placed. We selected the following ten variables, in addition to the one furnishing the super-class number (the values plotted on the graphs figuring in the appendix II are in brackets)

-number of children : no child (0enf), one child or more (enf) -educational level : secondary school not completed (form0), secondary school completed (nbacc), post secondary school level (+bacc)

-skills : various classes of workers defined according to their skills (mano, ousp, op12, o3hq), unskilled clerks (emnq), skilled clerks (empq), technicians (tecd), foremen and executives (mait, cadr)

-reasons for unemployment : end of contract (fin), lay-off (lic), first job (prem), return to labour market (repr)

-duration in months of latest period of unemployment : less than 6 months (inf6), between 6 and 12 months (inf12), more than 12 months (sup12)

-type of exit from unemployment: job (empl), mostly job (pemp), training (stag), departure from labour market (retr), cancellation for administrative reasons (susp)

-recurrence of unemployment : single period of unemployment (rec1), two periods (rec2), three periods or more (rec3)

-average monthly hours of occasional work during the latest period of unemployment : no occasional work (0ar), between 0 and 39 hours (arm39), from 40 to 78 hours (ar78), from 79 to 117 hours (ar117), more than 117 hours (arp117)

-percentage of cumulated duration of unemployment doing occasional work : no work (0par), below 10 % (01par), between 10 and 30 % (13par), above 30 % (p3par)

-average number of periods of occasional work per year of unemployment : no work (0peran), less than one period (m1peran), one or two periods (12peran), more than two periods (p2peran). Some of these variables (duration in months of latest period of unemployment, recurrence of unemployment over the whole period, percentage of cumulated duration in occasional work, monthly hours spent in these jobs during the latest period of unemployment) are obtained by discretization of the quantitative variables used to produce the classification. Others were already qualitative ones. These eleven variables together with their fifty-five subcategories produce a complete cross tabulation table and a Burt table (extension of a contingency table when there are more than two variables). We applied a multiple correspondence analysis to this table producing successive planes with decreasing informational content representing simultaneously the fifty-five subcategories. Thanks to this data neighbouring categories can be interpreted, for the same or for different variables. The axes show the main characteristics of the links existing between variables, and provide a synthetic description of our ten super-classes. A supplementary variable was plotted, consisting of age divided into five subcategories : less than 25 years old (m25), from 25 to 35 (25a35), from 35 to 45 (35a45), from 45 to 55 (45a55) and over 55 years old (p55). The graphs corresponding to the first three axes for the two regions studied can be found in the appendix II. Only our findings for the Nord region will be commented on here as those obtained for the second region differ very slightly. The first axis relates essentially to the category of those doing occasional work. It reveals a sharp contrast between various measures of this work (arp117, arm117, arm78, p3par, 13par) and the absence of occasional work over the whole period (0par, 0ar, 0peran). The second axis is defined in terms of education, level of skill and the length of the latest period of unemployment. On the right of this graph there is clear evidence of a correlation between high skills and a high educational level on the one hand, and short duration of unemployment and the fact of an individuals first search for work on the other. On the left of the graph, very low skills (mano) and a poor educational level (form0) are correlated with a lengthy duration of unemployment (over 12 months). If we consider the supplementary variable, the second axis shows a major difference between the situation of young people (on the right) and older ones (on the left). It should be noted that involvement in occasional work during unemployment is clearly orthogonal to skills, education, age and length of unemployment. On the basis of the data available jobless people in this category seem very close to the others (except for the fact that the latter do occasional work). These findings should however be modulated concerning the duration of unemployment by taking into account the fact that different groups may be formed within the category of those involved in occasional work, as will be seen from our ensuing analysis. Though the third axis is not as well defined as the preceding ones, it does show the effect of a recurrence of unemployment which is not necessarily linked to the duration of the latest period. The fourth axis is defined by age whereas the fifth is defined by skill and not by educational level. The ten super-classes obtained with the Kohonen classification may be defined as follows:

-Group 1 consists of unemployed workers occupying occasional jobs, and who have a high level of education and skills. These main characteristics are also to be found in group 6.

-Group 2 mainly comprises poorly educated individuals, aged over 35, where average duration of unemployment exceeded 12 months during their latest jobless period. Some of them are involved in occasional work.

-Groups 3 and 4 are very close and are characterised by older poorly educated individuals, who do no occasional work. Group 3 is slightly different in so far as a significant number of individuals here are trying to return to the labour market.

-Groups 5, 9 and 10 are composed of individuals who do occasional work, but who differ in terms of educational and skill level which is higher in group 9 than in group 5 and higher in group 10 than in group 9, though the duration of their unemployment is constantly decreasing. One major difference characterising group 10 is that these individuals appear to be often involved in occasional work (at least twice during each period of unemployment). It is also worth noting that the latter category experiences longer periods of unemployment.

-Group 7 is rather different from the rest of the population studied as these workers undergo at least three periods of unemployment and most of them do occasional work. This clearly demonstrates a mechanism we explained earlier, involving the link between, fixed-term contracts, on the one hand, and unemployment and occasional work on the other, the latter being presumably very similar to that done when working full-time;

-Group 8 is characterised by a very long duration of unemployment. These individuals are older and do no occasional work. These findings could be defined more precisely on the basis of further study of the different stages these groups went through over the three year period of observation. Overall, it can be observed that among those individuals working in occasional jobs there is a clear distinction between those who change jobs quite frequently and those for whom the occasional job becomes a permanent one.

Factors influencing the classification

We have been able to determine the most important variables used to construct the classification and the particular mix explaining each group obtained. This was done by applying a canonical discriminant analysis to the variable identifying the ten groups and the eleven quantitative variables used in the Kohonen classification. A number of significant canonical components for the Nord region (as well as for the Rhone region) were obtained. The first four (representing 79 % and 81 % of the sum of the eigen values) are easy to interpret and support our earlier findings. The first one consists of different variables relating to occasional work, and exerts a positive influence. This is the principal factor leading to the establishment of distinct classes. The second one is determined (for the Nord region) on the basis of age, length and recurrence of unemployment. The third one represents a brief period of unemployment without recurrence. The fourth one refers to the recurrence of unemployment and the amount of occasional work done over a given period of unemployment. The fifth canonical component reflects the number of job offers received per month of unemployment. On the basis of the means computed for each class using our canonical variables we are now able to define a broad classification of the main categories of the unemployed. The first category consists of young people experiencing a short period of joblessness and who do no occasional work (groups 1, 2, 6 and 7). The second is characterised by older people undergoing a lengthy period of unemployment and who do no occasional work (groups 3, 4 and 8). The third comprises individuals who often resort to occasional work, who stay unemployed and experience no recurrence of unemployment (group 5). The fourth one represents those workers who do a lot of occasional work, experience both relatively short periods of unemployment and regular recurrence of it (group 9 an 10). The last two groups would seem to be involved in a permanent precarious situation. They are either employed for a short period and then become jobless (and perhaps resort to occasional work in the same field) or are generally registered as unemployed but work significantly fewer monthly hours than in a regular job. These results are not contradictory with those obtained by [START_REF] Joutard | Les déterminants individuels de la durée du chômage: de l'intérêt de distinguer les emplois stables des emplois précaires[END_REF] on similar data, precisely on the question of the link between the total job seniority before the spell of unemployment and the greater opportunity to find a stable job. In fact, our results suggest that the practice of occasional jobs and the total duration of this practice is not perceived by future employers as a kind of accumulated human capital but as a negative signal of worker's potential. The consequence is that it exists a mechanism, very close in some way to the one enlighted by Joutard, between very occasional jobs during a spell of unemployment and a strong opportunity to find a precarious job. The recurrence of unemployment is a constitutive part of this mechanism and it illustrate the fact that the secondary segment is not a collection of heterogeneous activities produced by exits from the primary segment, but a coherent set of situations, possessing an internal logic and whose relations to the primary segment have to be explained. One of the results obtained here is that a longer duration of unemployment without practicing an occasional job seems to be the difficult way to return to the primary segment. With these precisions, it is noticeable that comparable results are obtained by Joutard with 1986-87 data and over the more recent period we studied, 1993-96.

Conclusion

Quantitative classification enables classes of unemployed workers to be clearly differentiated. These classes may be easily defined by the methods we used. Some of the variables used are essential to building a typology of jobless workers. This typology is built around the variables relating to occasional work coupled with varying periods of unemployment and some degree of recurrence as well as around the age factor variables. Granier (1998), using the whole population of the unemployed as if it was homogeneous, obtains biased results, mixing the different types of occasional jobs and, mainly, ignoring the segmentation of the labor market.

Our study provides only part of the evidence concerning the existence of segments in the labour market depending on entry-exit conditions. Much further investigation is required into the situation of the different classes of individuals who find themselves in and out of employment.

APPENDIX I

The Kohonen algorithm (Kohonen, 1984, 93, 95;[START_REF] Cottrell | Theoretical aspects of the Kohonen Algorithm[END_REF]) is a well-known unsupervised learning algorithm which produces a map composed by a fixed number of units arranged across a network, generally a two-dimensional grid, with n by n units, but the method can be used with any topological organization of the Kohonen network. A physical neighborhood relation between the units is defined and for each unit i, V r (i) represents the neighborhood with radius r centered at i. Each unit is characterized by a code vector C i of the same dimension as the input space (the data space). The learning algorithm takes the following form : * at time 0, C i (0) is randomly defined for each unit i; * at time t, we present a data vector x(t) randomly chosen among the rows of the data matrix and we determine the winning unit i*, which minimizes the Euclidean distance between x(t) and C i (t); * we then modify the C i in order to move the code vectors of the winning unit i* and its physical neighbors towards x(t), using the following relations :

)

) ( ) ( ( ) ( ) ( ) 1 ( t C t x t t C t C i i i for i in V r(t) (i*) (1) ) ( ) 1 ( t C t C i i for other i (2)
where (t) is a small positive adaptation parameter, r(t) is the radius of V r(t) and (t) and r(t) are progressively decreased during the learning.

In fact the Kohonen algorithm in a generalization of the Forgy algorithm, where the update of the code vectors involve also the neighbors of the winning unit and not only the winning unit.

After learning, each unit i is represented in the data space by its code vector C i .

Then each observation is classified by a nearest neighbor method : observation k belongs to class i if and only if the code vector C i is the closest to the data vector k among all the code vectors. The distance is the Euclidean distance in general, but it can be chosen in another way according to the context.

With respect to any other classification method, the main characteristic of the Kohonen classification is the conservation of the topology: after learning, « close » observations are associated to the same class or to « close » classes according to the definition of the neighborhood in the Kohonen network. This feature allows to consider the resulting classification as a good starting point to further developments and valuable representations as stated below.

As mentioned above, the first and raw result we get after learning, is a classification of the N observations into P = n×n classes. Eventually some classes can be empty. At the same time, we get the code vectors (associated to each unit). The first two figures in Appendix II shows standard representations of the code vectors inside their own unit.

The choice of the number P of units is arbitrary, and there does not exist any method to better choose the size of the network. We can guess that the « relevant » number of classes could often be smaller than P (which is commonly equal to 100). It is also difficult to give relevant interpretation of a too large number of classes. So we propose to reduce the number of classes by means of a hierarchical classification of the P code vectors using the Ward distance for example. As the code vectors are already organized across the grid, a standard method appears to be relevant.

In this way, we define two embedded classifications, and can distinguish the classes (Kohonen classes or « micro-classes ») and the « macro-classes » which group together some of the « micro-classes ». To make visible this two-levels classification, we affect to each « super-class » some colour or trame or grey level (here). Appendix II shows how the « micro-classes » are grouped together to constitute 10 « superclasses ».

The advantage of this double classification is the possibility to analyze the data set at a « super » level where general features emerge and at a « micro » level to determine the characteristics of more precise phenomena and especially the paths to go from one class to another one.

In the applications that we treated, the « super classes » create connected areas in the grid (one « super class » which contain some « micro class » contain also a neighbor of this class). This remark is very attractive because it confirms the topological properties of the Kohonen maps. Nevertheless, in some cases, it is possible to find a « super class » split into two pieces. In that case, one can guess that the data set is folded over and try to control it by studying the distortion.

APPENDIX II

Table 1

 1

	Region	Total nb	Total nb of	Individuals with	%
		of records	individuals	one registration	(3/2)

Table 2

 2

	Region	Obs	Duration	Cumulated duration Nb of periods	Nb of job Job/Unemp.
	NORD	38068	476.52		636.44	1.55	0.50	0.11
	RHONE	44165		410		547	1.48	0.52	0.09
	Region	Job/year	Offers/Unemp	Wages	Hours	Age	Seniority
				.
	NORD		0.54		0.12	767.08	37.17	31.32	4.32
	RHONE		0.50		0.11	1345.68	33.45	32.45	4.65

Table 3

 3

		Duration		Number of periods		Age bracket
		< 6 m.	6-12 m. > 12 m.	1 p.	2 p.	> 2 p.	< 35		35 à 55	> 55
	NORD	37.3	20.8	41.9	62.8	25.0	12.2	66.5		30.7	2.8
	RHONE	39.7	21.7	38.6	66.1	23.1	10.8	64.3		31.7	4.0
		Skill				Level of education
		Sup.	Clerks Indus S4	S2 Sec dip.	Second.	Prim.	None
	NORD	10.9	49.5	39.6 4.9 8.0		14.5	39.9		7.0	25.7
	RHONE	17.5	51.0	31.5 8.8 11.3		14.5	39.7		6.2	19.5
	Sup. : executive, technicians
	Indus : skilled and unskilled workers
	S4 : Master level and more
	S2 : Undergraduate level
	Sec. Dip. : Secondary level with high school diploma
					Table 4
		Reasons for job loss			Type of unemployment exit
		l. o. end	1st	return na	job	p. job quit training	cancel	f. job
			job
	NORD	15.1 54.5	16.9	10.3 3.2	25.8	13.2 10.5	6.8	13.4	30.3
	RHONE 23.9 46.7	11.2	12.7 5.5	27.8	18.8 9.6	6.4	14.1	23.3
	l.o. : lay-off
	end : end of fixed-term contract
	p.job : registration cancelled by administration, most of the time because job found
	f.job : individuals occupying a full-time job while seeking another one (not
	unemployed)

Table 6 -

 6 Kohonen classification (super-classes)

	SUPER-	Size	Age	Seniority	Duration	Cumulated	Number of
	CLASS				last period	duration	periods
				NORD
	1		24.23	1.30	183.49	241.24	1.26
	2		29.75	2.19	485.12	930.82	1.77
	3		47.01	22.05	440.69	539.29	1.28
	4		45.00	3.59	320.31	376.18	1.17
	5		33.25	4.55	1072.32	1159.83	1.16
	6		28.33	4.20	168.05	236.03	1.41
	7		28.20	2.62	164.90	639.94	3.57
	8		37.31	4.56	1617.14	1673.05	1.03
	9		27.57	2.76	354.32	525.66	1.77
	10		28.03	3.39	158.16	206.79	1.32
	Total		31.32	4.32	476.52	636.44	1.55
				RHONE
	1		25.28	1.50	261	264	1.02
	2		25.75	1.62	212	325	1.51
	3		28.89	2.38	191	623	2.62
	4		42.91	4.82	220	304	1.20
	5		48.57	19.34	446	545	1.27
	6		36.48	2.30	1380	1415	1.05
	7		31.20	3.24	462	725	1.80
	8		28.76	4.81	140	182	1.29
	9		31.94	2.74	1069	1118	1.09
	10		30.58	4.72	257	307	1.30
	Total		32.45	4.65	410	547	1.48

1 This study was conducted with financial assistance from the Government Employment Service (ANPE). The authors accept full responsability for the ideas expressed herewithin. I, and two referees for valuable suggestions and Leslie


Patrice Gaubert
email: gaubert@univ-paris1.fr

Marie Cottrell
email: cottrell@univ-paris1.fr

A DYNAMIC ANALYSIS OF SEGMENTED LABOR MARKET

Keywords: Wage inequality and mobility, Internal labor markets and labor relations segmented labor market, unemployment, trajectories, Kohonen algorithm, Markov chain. JEL-Code: J31, J41, J49

Using the Panel Study of Income Dynamics data on the period 1982-1992, this paper investigates some mechanisms of the labor market in the United States. This market is analyzed as a stable structure constituted of segments which present contrasted characteristics under the usual distinction between primary and secondary sectors. Using a neural network algorithm applied on quantitative variables measured at the level of heads of household, a broad classification in four classes of situations is constructed. It shows a clear hierarchy going from situations of very precarious work or no work at all, to situations of stable jobs with higher wages than the average. A Markov chain, constructed with the trajectories between the different situations of these workers, shows a very stable structure of this segmented labor market.

OBJECT OF THIS STUDY

The key idea of approaches of segmented labor markets1 is that jobs have to be distinguished from a qualitative point of view, even if different distinctions not overlapping are used like internal -external markets or primarysecondary segments. The first one addresses the rules governing the relations between the workers and their occupation (hiring, career, exit from employment). The second one is based on a qualitative comparison of existing jobs (level of earnings, stability, career).

These two distinctions use numerous common factors but they introduce two different theoretical explanations of the labor market as well as their empirical validations.

Various recent examples may be found of this diversity: In a theoretical way may be cited the contributions proposed by [START_REF] Bulow | A Theory of Dual Labor Markets with Application to Industrial Policy, Discrimination and Keynesian Unemployment[END_REF] in terms of efficiency wages or by [START_REF] Albrecht | Dual Labor Markets, Efficiency Wages and Search[END_REF] in a more institutional approach. On the empirical ground one may cited the observation of a significant bias between the levels of earnings and the characteristic of stability of jobs situated in large firms, as in [START_REF] Oi | Employment Relations in Dual Labor Markets ("It's Nice Work If You Can Get It[END_REF]. Besides the existence of efficiency wages, a suggested explanation, is the presence of higher fixed costs induced by a specialized on the job training, or the practice of higher wages induced by a more constrained work associated to the discipline of a team production.

The simplest way to take into account a segmentation in an empirical manner is to use a subjective determination as in [START_REF] Theodossiou | Wage Determination for Career and Non-career Workers in the UK: Is There Labour Market Segmentation?[END_REF] using the results of a survey: are belonging to the primary segment all the workers answering that they have a career in their job. This subjective information is controlled using a more objective variable, having received a specific on the job training more than one year. These elements, career, special training, are only partial characteristics of a segmentation. They do not constitute sufficient conditions of its existence and the method of identification seems to be a weak one.

The main objective here is to produce a robust representation of the segmentation of the labor market using both qualitative and quantitative variables to express its different dimensions. These variables have to be significant of the induced quality of a job, in terms of earnings (level and time variation), of duration (hours per week and weeks per year), and of seniority (in the current job compared to the number of years of presence on the labor market since the beginning of the working life). The combination of these quantitative variables may produce a qualitative evaluation of different jobs. This have to be completed by some qualitative characteristics like the type of occupation and the industry. Of course some personal information like the level of education has to be taken in account.

The final objective is not to obtain a representation of the segmentation, but this representation is a step to identify the trajectories followed by workers between these segments over a long period.

After a presentation of the data and the method used to identify a broad classification (section 2), the initial results are characterized in order to justify the final classification (section 3), then the main trajectories are presented and used to produce a measure of a stationary state representing the 'virtual' structure of the segmented labor market (section 4). Finally the main factors influencing the transitions between states characterizing the main trajectories are identified (section 5).

THE DATA

The data are obtained from the PSID, using the heads of households present in their family during the period 1982-1992. This panel contains much information about their job and also about their personal characteristics, At this time no more filtering has been used to select the individuals2 , even considering the over-representation of low-income black people. Among other results the classification produced is supposed to group this particular population.

Among the initial 4 000 heads of households obtained, only 2 507 are kept due to missing values concerning the major variables used. At the same time, in order to produce information on the evolution of some important characteristics of their job, the two first years are used only to construct rates of growth or variables in difference. So the trajectories are identified on the period 1984-1992. In order to produce the classification 15 variables have been selected, 12 are observed and 3 constructed with observed variables. According to the methodology used by the authors of the panel data, personal variables are measured for the year of the survey, while data concerning the participation to the labor market are evaluated for the preceding year: the dimension of the family in 1992 is measured at the time of the survey and the number of work hours evaluated for this year is actually the number of hours worked in 1991. So the effective period of observation of the labor market is 1983-1991.

One may assume that there is an important distinction, concerning the quality of a job, to be considered along with the durability expressed by seniority in a specific place, that is the duration over each year expressed by the number of weeks worked and also the number of hours per week. This is a decisive characteristic to separate 'normal' jobs from various types of contingent work whose influence is growing at the end of the studied period3 . Two personal variables used are:

-the age of head (AGEH) -the family size (SIZFAM)

Several variables describe the major characteristics of the main job:

-the seniority in the current job (SENH) -the number of years of work since the age of 18 (ANCH) -the annual number of work hours (HMJH) -the number of weeks worked in a year (WMJH) -the number of hours per week (HWMJH) -the hourly wage obtained (RSALH), expressed in real terms in order to eliminate the effect of inflation when evaluating the existence of a career 4-the annual rate of growth of the hourly wage which is computed using the two last years (GRSALH) and two other variables expressing a change in the duration of the main job, variation of the number of work hours per week (VHWMJH) and variation of the number of weeks worked in a year (VWMJH) since 1982 -two variables expressing a situation of non-work during the year, the number of weeks out of the labor force (WOUTH) and the number of weeks unemployed (WUNEH).

A NEURAL NETWORK CLASSIFICATION

In order to obtain a valid classification for the whole period, the data used to construct the neural network is constituted of the observations made for the 15 variables for the three years, 1984 1988 1992, as if they were different individuals observed at the same time. So the classification is conducted on 7521 observations (2 507 x 3) and 15 variables. Each variable being measured in its specific units, the very different sizes of the numbers for these variables could disturb the classification, so all the variables are standardized before the construction of a Kohonen grid. The dimensions of this grid are 8 x 8.

It has been establish, [START_REF] Kohonen | Self-organizing Maps[END_REF][START_REF] Cottrell | Theoretical aspects of the SOM algorithm[END_REF], Kaski & Oja (1999), that a Kohonen classification is an extension of k-means method with an imposed structure of neighborhood. One of the improvements resulting is that observations which are close or similar are placed in the same class or in neighbor classes. The neighborhood structure selected here is a two-dimensional grid. It produces 64 classes, each one being represented by its code vector as in the following figure 1. It can be seen that close code vector have similar profiles. With this classification giving 64 classes with the whole population observed over three years, it is possible to identify the complete trajectory followed by each individual from 1984 to 1992. This is obtained using the data available for each of the six other years, which are not used for the classification, to compute, for each individual the class of the closest code vector in terms of Euclidean distance 7 .

For each individual the observed trajectory is represented by a sequence of 9 integers in the interval 1-64 and it is represented, according to the suggestion made by Serrano-Cinca in Deboeck & Kohonen (1998). (eventually no more than the number of individuals in the sample). So it is very difficult to describe and interpret a great number of different trajectories, as it is not easy to interpret even the 64 classes obtained. This is why it is useful to reduce this number to something closer to the theoretical distinction of the labor market.

THE FINAL CLASSIFICATION (7 SUPER-CLASSES)

Different methods are used in order to reduce the number of classes [START_REF] Cottrell | The Kohonen Algorithm : A Powerful Tool for Analysing and Representing Multidimensional Quantitative and Qualitative Data[END_REF], for instance a hierarchical classification applied on the 64 code vectors. The result obtained is an unordered set of classes.

Here the Kohonen algorithm has been used again, but the one dimensional version, producing an ordered sequence of broad classes which may be easily interpreted as going from the worst situation (unemployment and precarious jobs with low pay) to the better one (only one full time job with a great stability) via different intermediate situations (two or more extra jobs, partial time work, real wages under the average).

After several tries, the more significant structure seems to be a set of seven super-classes. The means of the 15 variables used to construct the Kohonen classification are given in table 1, for the whole sample and for each of the super-classes. For each variable the greatest value is in bold and the lowest in italics. The different super-classes have very different sizes (See Figure 4), but the most important fact is that each one is clearly distinguished from the others using their quantitative characteristics. The limits of these super-classes have been represented on the Kohonen map (Figure 5), using different tones of gray. The Kohonen algorithm allows the visualization of the neighborhood between individuals and between classes.

Figure 5 : The 7 super-classes

The main characteristics presented by the super-classes show the growing quality of the situation on the labor market, when moving from the lower left corner towards the upper right one. It is possible to summarize the superclasses as follows:

-class 1 (lower left) is made of individuals younger than average population, with very short seniority in the current job, working a number of hours per week and a number of weeks per year neatly smaller than the average, with no extra job, low paid, being out of the labor force much more often than the average and whose variations of hourly earnings and of hours of work over the period are mostly negative.

-class 2 (center of the map) contains individuals younger than the average, with a full-time job and one or more extra job, with earnings obtained in the main job severely lower than the average and no real changes in their situation over the period studied. This seems correspond to a situation where the main occupation, even a full-time one, produce an insufficient income and leads to the obligation to exert an extra job.

-class 3 (on top of class 1) is a very small one, approximately 50 persons each year, and presumably not really representative of a part of the whole population. Nevertheless this class is well defined with very old people having a kind of reduced activity which may be linked to a possible preparation to retirement.

-class 4 (below class 2) is made of young people with short seniority in their job, wages slightly below the average, an important augmentation in the number of hours worked but not in the wage obtained.

-class 5 (on top of class 2) is close from class 2 with respect to the fact that people have two or more occupations at the same time but very different when earnings are considered: it seems to represent in this class the possibility for skilled workers to obtain supplemental benefits from their special ability. This may be confirmed using information on the type of occupation and of industry.

-class 6 (upper left) appears to be made of older people with stable employment, having only one job and earnings that are close to the average. This class may be interpreted as a part of the primary segment, some lower tier or half, with respect to the presence at the same time of a great stability and a modest compensation.

-class 7 (upper right) is constituted of people of middle age, with large families, a good stability in their employment, working a longer duration than the average and for an hourly wage clearly above the average, with a steady growth of their compensation. It looks like the upper part of the primary segment. This sequence of classes seems to be well ordered with respect to a growing quality of the situation occupied on the labor market. This may be seen with the graphical representation (figure 6) of the code vectors obtained for the super-classes with a one-dimensional algorithm. A first representation of the trajectories over the period of observation may be traced using these 7 classes, each individual being located each year in one of the class. A trajectory is represented by a sequence of 9 integers, as 234774344 for instance.

A transition is a sequence of two situations expressing that the individual has moved to another class. Considering the above example this individual is staying two consecutive years in the same situation and this two times, states 7 and 4, and is changing each year for the other years. The diversity of situations may be summarized by presenting the frequencies of situations identifying the proportions of the different states occupied by an individual who is present most of the time in one specific class (5 years or more). They are in fact the probabilities to be in one other class when the dominant situation is a given class (table 2).

Table 3 : Probabilities when no class has a dominant position

The main result is that the individuals stay most of the time in the same class, except for this group representing less than 15% of the whole sample. That means that the structure that appears constituted by segments with very different properties with respect to stability, existence of a career, seems to present the quality of a permanent state over a long period. This will be even clearer with the construction of a Markov chain.

Note that except for the very small group 3, the less stable classes, relatively, are those corresponding to lower situations, and precisely the two classes having extra job(s).

CLUSTERING INTO 4 MAIN CLASSES

At this stage, we could study the frequencies of the different trajectories, the frequencies of the category changes, the transition probabilities, etc; But the number of possibilities is still too large with 7 super-classes. So, to study more precisely the trajectories, a last grouping is realized, dividing the initial Kohonen map in four main classes :

-super-classes 1 and 3 are grouped to produce the main class A: it is made of the more precarious conditions, recurring unemployment and low pay;

-super-classes 2, 4 and 5 represent intermediate conditions: important duration of work and moderate wages. They constitute the main class B; -super-classes 6 and 7 are still separated to produce the main classes C and D, respectively the lower part and the upper part of the primary segment.

A complimentary information is obtained with a principal component analysis concerning the role of the different variables used to construct the classification. The 5 first components 2/3 of the explanation. The projection of the variables on the first 3 axes are clearly significant (Appendix graphs 1-3). The first axis is defined by the variables of activity: the number of work hours, the number of weeks, opposed to the number of weeks of unemployment and out of the labor force. The second one oppose age, seniority to the family size (younger family are larger). The third one is only defined with the extra job variables. The level and the growth of wage and the variables in variation appears only as fourth and fifth axes. That means that the separation of the different situations is mainly explained by other factors than the differentiation of wages.

Even with this new grouping the main classes are well defined using the 15 quantitative variables (table 4), on the whole observations used to construct the initial classification (3 years pooled). The major characteristics observed above with the more detailed partition are still visible: work duration, seniority, level and growth of real wages, the practice of extra jobs. . For each variable the greatest value is in bold and the lowest in italics. -the proportion of white people is very weak in class A and significantly above the average in class D -a similar result is obtained with the highest level of education concerning these two classes -a special mention have to be made concerning the distribution of class C among the levels of education: the importance of the secondary level (high school without the final grade) and a post-graduate level slightly greater than the average are to be considered together with the great stability of this sub-population in a class constituted by older people with an important seniority in the same firm and moderate wages. This could be interpreted as a working population with a traditional qualification or on-thejob training implying a great stability but not a real career in their job.

-not surprisingly, managers and professional workers are merely present in class D, while class A is constituted mainly with unemployed and unqualified workers (Others are principally laborers) -class C is made of subordinate occupations. The classification used to represent the sectors of activity seems to be too aggregate to be conclusive 8 .

THE TRAJECTORIES BETWEEN THE 4 MAIN CLASSES

Over the 2 507 individuals only 1 028 different trajectories are found, to be compared to the 4 9 possible trajectories, it is clear that a trajectory cannot be conceived as a random process between the four classes. It means that there is a clear difference between the hierarchy of the diverse situations observed and the dynamics of transitions: there is no reason for these transitions to follow the hierarchy, in a positive direction or the inverse. Class C is not a step towards the best state, class D.

The total number of the transitions showed in table 9 is 4665, that is a little less than 25% over the total number of possible transitions (8 x 2507), the major part is then constituted by the unchanged situation from one period to the next one.

These empirical probabilities may be used to build a transition matrix in order to model the changes of classes by a Markov chain. See in We can conclude that the situation observed, presented in table 10 for only 3 years of the period, is quite close from the limit situation computed with the Markov chain. This confirms the great stability of the main structure, even if a visible proportion of changes are observed mostly at the beginning and in the middle of the chain. The same conclusion is obtained when using the transition matrix produced with the classification in 7 classes.

We will continue the identification of the main factors influencing the transitions and determining the duration observed before these transitions.

CONCLUSION

A first of results establishes clearly the pertinence of a segmentation of the labor market in terms of intensity and stability of the occupation. A first half of the whole studied population exerts a very stable job, 60 % of this segment having a set of valuable conditions associated to this job (level and growth of the real wage).

Considering the trajectories followed by the individuals among the main possible situations some results are obtained: -a great stability is observed during the whole period studied, this means that most of the time each individual stays in the same situation from one year to the next one -the inferior part of what may be named the primary segment (class C) is the most stable situation -most of the observed transitions are between the two lower classes but a significant number of transitions is observed between these classes and the upper part of the primary segment. A study of the factors influencing these different types of transitions would be valuable to understand the process of the trajectories observed. Frequencies of some qualitative variables (year 1984)

Figure 1 :

 1 Figure 1 : Profiles of the 64 code-vectors

Figure 2 :Figure 3 :

 23 Figure 2 : Trajectory of an individual staying in good job situation during the whole period

Figure 4 :

 4 Figure 4 : Sizes of the 7 super-classes.

Figure 6 :

 6 Figure 6 : Code-vectors of the 7 super-classes

Table 1 : General mean and mean by super-class

 1

		Whole	Class	Class	Class	Class	Class	Class	Class
		Population	1	2	3	4	5	6	7
	AGEH	40.12	36.4	35.32	59.41 33.18 40.58	52.69	39.46
	ANCH	15.43	10.56	11.26	30.32 8.65	16.18	28.20	14.86
	GRSALH	0.06	-0.18	0.02	0.07	0.06	0.03	0.02	0.19
	HEXJH	60.70	12.98	562.12 56.01 0.25	215.01 7.39	4.74
	HMJH	1974	663	1994	901	2040	2136	2008	2348
	HWMJH	42.18	24.69	41.88	22.95 42.09 44.34	42.09	48.72
	NBXJH	0.18	0.05	1.24	0.28	0	1.03	0.06	0.03
	RSALH	13.35	6.47	10.60	10.95 11.30 14.77	13.88	17.70
	SENH	91.14	19.51	64.02	41.05 58.28 118.81 173.39 93.04
	SIZFAM	3.17	2.92	2.93	2.04	2.67	3.88	2.57	4.08
	VHWMJH	0.59	-6.43	0.06	-17	-0.13	0.23	-0.52	5.23
	VWMJH	0.65	-15.66 2.77	-3.83	3.89	0.17	1.05	2.92
	WMJH	44.61	15.29	47.51	40.81 48.48 48.23	47.60	48.10
	WOUTH	0.69	5.76	0.09	1.37	0.13	0.05	0.06	0.11
	WUNEH	2.09	16.08	0.80	3.29	0.40	0.13	0.41	0.53
	Size	7521	772	588	79	1932	416	1495	2240

Table 2 : Probabilities to be one year in a class being most of the time in a given class

 2

	Dominant	Size	Proba	Proba	Proba	Proba	Proba	Proba	Proba
	Position			of 1	of 2	of 3	of 4	of 5	of 6	of 7
	1	157	0.75	0.03	0.01	0.11	0.00	0.03	0.08
	2	115	0.04	0.70	0.00	0.13	0.07	0.00	0.05
	3	10	0.16	0.01	0.64	0.01	0.00	0.16	0.02
	4	599	0.07	0.06	0.00	0.77	0.02	0.00	0.08
	5	65	0.01	0.11	0.00	0.01	0.70	0.05	0.12
	6	498	0.03	0.01	0.02	0.01	0.02	0.86	0.06
	7	732	0.03	0.02	0.00	0.07	0.03	0.03	0.82
	A group of people are not in a dominant position, that means that over 9
	years no one of the 7 classes appears at least 5 times: 337 individuals (over
	2507). For this group the probabilities are given in table 4.
	No dominant	Size Proba	Proba	Proba	Proba	Proba	Proba	Proba
	position			of 1	of 2	of 3	of 4	of 5	of 6	of 7
			331 0.14	0.16	0.03	0.22	0.13	0.08	0.23

Table 4 : Means of the whole sample and by main class

 4 Using the Kohonen algorithm supply an order on the 4 situations A, B, C et D, from the more precarious and degraded to the more stable and with the best wages. Some clear information may be obtained from 4 available qualitative variables, in terms of frequencies (see Table5for the data related to year 1992, and Appendix 3 for years 1988 and 1984).

	in 1992	Whole	Class A	Class B	Class C	Class D
		sample
	RACE
	1 Whites	69.1 %	51.6	69.2	69.5	74.5
	2 Blacks	29.7	48.0	29.8	28.9	23.8
	EDUCATION
	1 Primary	0.9	1.4	0.2	2.6	0.4
	2 Secondary	18.9	32.7	13.1	27.8	15.1
	3 Sec. achieved	40.2	42.7	44.4	37.0	36.9
	4 Post-sec.	28.6	18.5	32.7	20.7	32.6
	5 BA & more	11.3	4.6	9.5	12.0	15.0
	OCCUPATION
	0 No	2.0	17.4	0	0	0.1
	1-2 Managers, professionals	36.2	15.7	36.7	32.9	44.9
	4 Clerks	12.0	14.2	13.2	14.0	8.8
	5 Craftsmen	17.1	14.9	17.0	18.5	17.1
	6 Operatives	15.1	14.9	15.1	15.6	14.9
	7 Others	12.6	20.6	12.2	15.7	8.5
	ACTIVITY SECTOR
	0 No	2.4	18.9	0.3	0.4	0.4
	1 Agricultural, Mining, Construction	13.1	14.2	12.9	10.0	14.8
	2 Manufacturing	22.8	17.4	20.6	25.4	25.3
	3 Transportation, Communication	9.1	4.6	8.1	7.7	12.6
	4 Wholesale and retail Trade	14.9	17.4	14.9	16.5	13.1
	5 Finance, Insurance	4.3	2.5	4.0	5.1	4.6
	6 Business and Personal Services	9.1	12.8	9.4	7.9	8.4
	7 Professional Services	16.6	9.6	20.6	20.5	12.3
	8 Public administration	7.7	2.5	9.1	6.5	8.6

Table 5 : Frequencies of some qualitative variables (year 1992)

 5 Some simple results are visible in this table:

Table 6 : Probabilities to be one year in a class being most of the time in a given class

 6 Besides, 971 persons stay in the same class during 8 or 9 years over the 9 years observed. More precisely, only 34 persons (1.4%) stay in the main class A during the 9 years, 423 (17%) stay 8 or 9 years in class B, 218 (9%) stay in class C and 306 (12%) are present 8 or 9 years in class D. The rest of the sample, 1 536 persons (61%) are less stable but the changes observed are not numerous, of course less numerous than what have been measured on the classification in 7 classes. This is well establish with the probabilities computed with the same idea of a dominant position (table 6): 5 years or more in a class over the 9 years.

	Dominant	Size Proba	Proba	Proba	Proba
	Position		class A	class B	class C	class D
	A	179	0.75	0.13	0.06	0.07
	B	951	0.07	0.82	0.01	0.10
	C	498	0.05	0.04	0.86	0.06
	D	732	0.04	0.11	0.03	0.82
	A very small group appears to be in no dominant position over the whole
	period (table 7)
	No dominant	Size	Proba	Proba	Proba	Proba
	position		class A	class B	class C	class D
		147	0.34	0.33	0.13	0.29

Table 7 : Probabilities when no class has a dominant position 8

 7 But using a more detailed classification is limited by the size of the sample used.A very stable situation is visible and specially for class C. It may be seen on table7that the significant changes are only between A and B, B and D and D and B. For the little group with noticeable changes it is clear that class C is mostly isolated from the rest of the labor market.This study may be continued with the measure of the frequencies computed over the transitions from a class to another one (table8)

		AB AC AD BA BC BD CA CB CD DA DB DC
	Size 554 177 242 492 159 1036 175 150 262 241 871 306
	%	0.12 0.04 0.05 0.11 0.03 0.22 0.04 0.03 0.06 0.05 0.19 0.07

Table 8 : Frequencies of transitions

 8 Again it is clear that the transitions occurred mainly between classes A, B et D. The number of improvements of the working situation (AB, BD) is close to the number of deterioration (BA, DB) and the moves from and to class C are very few.

Table 9

 9

	the

Table 9 : The Markov transition matrix This

 9 model relies on some important hypotheses concerning the factors influencing the transitions, precisely that the only factors which are taken into account for the constitution of the main classes are stable over the considered period, Then it is possible to compute the stationary distribution 9 and to compare it to the distribution for the observed years. See Table10.

		class A	class B	class C	class D
	1984	.138	.400	.181	.281
	1988	.110	.381	.199	.309
	1992	.112	.356	.203	.329
	stationary	.106	.363	.209	.322

Table 10 : Distributions of three years and stationary distribution

 10

Table : Quantitative variables used in the Kohonen Classification Repartition of the qualitative variables for 1984 in the 4 main classes

 :

	APPENDIX 1
	AGEH in 1984 ANCH GRSALH RACE 1 Whites	Whole age of the head of the household Class A Class B sample number of years worked since the age of 18 Class C annual growth of hourly wage 67.2 49.1 69.6 70.9	Class D 70.2
	HEXJH 2 Blacks	annual work hours in extra jobs 29.9 48.6	27.6	26.0	26.4
	HMJH EDUCATION HWMJH 1 Primary NBXJH 2 Secondary 3 Sec. achieved	annual work hours in main job 1.4 1.7 annual number of weeks of work in main job 0.3 3.3 19.4 31.8 13.7 25.8 number of extra jobs 39.0 43.9 41.6 35.8	1.4 17.2 34.9
	RSALH 4 Post-sec. SENH 5 BA & more SIZFAM OCCUPATION 0 No	34.7 real hourly wage 5.5 seniority in main jobs (in month) 20.5 2.0 family size 4.1 28.6	40.1 4.2 0.1	28.0 7.1 0.2	38.4 8.1 0.3
	VHWMJH 1-2 Managers, professionals variation of work hours per week in main job, since 1982 32.6 8.1 34.1 35.3 40.8
	VWMJH 4 Clerks WMJH 5 Craftsmen 6 Operatives	12.9 variation of weeks of work in main job, since 1982 12.1 14.2 13.9 17.0 12.7 18.2 16.8 number of weeks of work in main job 15.7 17.6 13.8 16.1	10.8 17.5 17.0
	WOUTH 7 Others	number of weeks out of labor force 12.4 19.1 13.0	12.4	8.1
	WUNEH ACTIVITY SECTOR 0 No	number of weeks of unemployment 4.7 29.2 0.9	0.9	0.6
	1 Agricultural, Mining, Construction	12.7	11.6	12.9	9.7	14.8
	2 Manufacturing		24.1	19.1	21.8	28.0	27.1
	3 Transportation, Communication	9.7	5.8	9.7	8.8	12.1
	4 Wholesale and retail Trade	15.4	11.6	17.1	15.5	14.9
	5 Finance, Insurance		4.1	1.7	4.3	5.1	4.5
	6 Business and Personal Services	7.6	10.4	7.4	7.1	7.0
	7 Professional Services	14.1	7.5	16.8	17.7	11.2
	8 Public administration	7.6	3.2	9.1	7.3	7.8

see Doeringer & Piore (1971).

except for those whose accuracy indices concerning major variables like earnings or work hours are worse than 2 in the PSID codification.

it would be of a greater importance for a study conducted on the 90's. This work is possible using the Early Releases of PSID data for the period 94-97, at this time it is only in progress and it is too early to indicate some results.

using the implicit deflator of GDP.

in the following tables and in the Kohonen maps the variables are in an alphabetical order: a table ordered of the variables is found in the appendix.

At this time the gender characteristic is not used due to the fact that the head of the household is a woman only when no man is present in the family; a further study is to be conducted using the data, available in the PSID, concerning the wife involved in the labor market.

when some values are missing this distance is computed with the existing values.

For an irreducible Markov chain (i.e. starting from any state it is possible to go to any state), the stationary distribution is defined as the invariant distribution, it is also the limit distribution when the time goes to infinity.

APPENDIX 2 (Principal Component Analysis)


Marie Cottrell
email: cottrell@univ-paris1.fr

Patrice Gaubert
email: gaubert@univ-paris1.fr

CLASSIFICATION OF RECURRING UNEMPLOYED WORKERS AND UNEMPLOYMENT EXITS

Keywords: Unemployment, Labor Market, Kohonen Maps

This study focuses on recurring unemployment, that is people with two or more spells of unemployment during the period of observation (July 1993 -August 1996). First, a classification is obtained which is then used to examine the specific role of occasional jobs during a spell of unemployment and, in this context, the influence of the received unemployment benefits on the duration of this spell. This paper is a continuation of previous analyses of unemployment in France, based on long-term data from the unemployed register held by ANPE (National Employment Bureau). The present analysis conducted using additional information about unemployment benefits received by the unemployed from UNEDIC (Unemployment Benefits Office).

1.

INTRODUCTION

A first study of unemployment in France (Gaubert & Cottrell, 1999) has produced a broad classification of the unemployed. This classification is built using quantitative variables characterizing the unemployed (duration of unemployment, number of months of unemployment spent doing occasional work, job seniority, number of job offers received per month of unemployment).

Practicing an occasional work (AR) while being searching a durable job has been found to be the major discriminating factor in the studied population, while producing three broad classes (a fourth one is constituted by those looking for their first job, obviously different from the rest of the population):

-those looking for a durable job and not having had any occasional work during their search; -those who, while searching a durable job, are having a part time activity which constitute a significant proportion of the duration of the spell of unemployment ; their situation seems to be closer to that of people having a precarious part-time occupation than to the unemployed according to the usual definition;

-those having had very few occasional jobs.

Simultaneously three types of exits from unemployment, using the usual definition, have to be distinguished: durable job, part time job with a limited duration and withdrawal from the labor market.

One of the characteristics of those having a precarious part time job is that they are looking for a durable job after the completion of a fixed-term contract. For them the duration of a spell of unemployment is significantly shorter than on the average. An interpretation may be that those employed under short-term employment contracts are having successive spells of full-time and parttime employment, with the same type of activity, possibly in the same firm. This could illustrate, on the individual's side, some effects of the firm's search for more flexibility.

This result would be an interesting one interpreted in terms of labor market segmentation, showing the major difficulty to find again a job in the primary segment after having unemployed.

Starting with these previous results, one purpose of this paper is to study the possible trajectories of the unemployed, emphasizing the successive observed transitions.

The data concerns unemployed people with two or more spells of unemployment, including information about the allowance received (periods and corresponding amounts) and the former wage in the last job occupied before the spell of unemployment.

One possible direction is to study the duration of the transition, the types of successive transitions followed, the role of the way unemployment benefit is designed in France (with a progressive reduction of the amount over time) on the specific path followed, in order to evaluate the main factors which determine recurring unemployment.

This new work is then a study of the segmentation of the French labor market in a way not frequently found in recent literature

For instance, some analyses may be indicated : in terms of career differencing [START_REF] Favereau | Une approche longitudinale de la segmentation du marché du travail[END_REF][START_REF] Theodossiou | Wage determination for career and non-career workers in the UK : is there labour market segmentation ?[END_REF], returns on human capital [START_REF] Balsan | Analyse salariale des dispositifs d'aide à l'insertion des jeunes[END_REF], or studies oriented towards firm organization, efficiency wage [START_REF] Oi | Employment relations in dual labor markets[END_REF][START_REF] Albrecht | Dual labor markets, efficiency wages, and search[END_REF], or discriminating practices [START_REF] Dickens | A test of dual market theory[END_REF][START_REF] Boston | Segmented labor markets : new evidence from a study of four race-gender groups[END_REF].

DATA AND VARIABLES.

The initial data is the complete register of the unemployed held by the ANPE; information on unemployment benefits and compensations in latest job is added from the data collected by UNEDIC. The studied period goes from July 1993 to August 1996. The population is constituted of all the unemployed who were looking for a job at the beginning of this period2 , or who became unemployed later (but before the end of August 1996); at the end of the period they are either unemployed or their status has changed in some way.

For the present study we use a 1% sample of the unemployed registered in the administrative region of Ile-de-France (Paris and suburbs) having two or more spells of unemployment (590 000 individuals on a population of more than 2 millions 167 000), those named recurring unemployed. The resulting data is constituted by 19 246 individuals and 44 405 spells of unemployment.

This sample has been matched with the unemployment benefit register to retrieve the corresponding information.

Three types of information are available:

-individual characteristics: age, marital status, number of children, nationality, level of education, seniority in the latest job, professional skill level; -characteristics of the unemployment: reason of job loss, how the unemployment status has ended, number of spells and duration of each one, number of job offers received, characteristics of the casual job if they has been some (number of hours per month and monthly wage) -characteristics of the unemployment allowance: average amount on a daily basis, compensation received in the latest job.

An occasional job (AR) is defined as any activity occupied by an individual who receives a compensation while he is registered as looking for a durable job. In France, this does not put an end to being registered as unemployed if the activity is very occasional or the number of hours per month remains small. Since 1995, the ANPE distinguishes among the registered unemployed those working for more than 78 hours per month. They constitute the so-called 6th category, excluded from some official statistics of unemployment, even though the person remain registered as looking for a job. In terms of unemployment benefits, if the characteristics of the latest job comply with the rules of unemployment benefits, they may perceive a compensation while its amount is not greater than 70 % of the latest job compensation. This occasional activity will be included to compute the next unemployment benefit.

The rules of computation of the unemployment allowance, on a daily basis, are quite complex. The rate is decreasing with thresholds defined differently according to the age and the professional history of the individual. For instance a young unemployed is submitted to a reduction rate of his benefits of 17 % each period of 122 days. In order to obtain a better homogeneity with the latest job compensation and the one obtained in an occasional activity, we have calculated an average allowance has been computed over the spell duration. This simplification removes the influence of the diminishing rate. We intend to investigate more thoroughly that point in a another study will.

The first step is a simplification of the raw data by two means. First, using analysis of variances and simple regressions, some variables have been removed as not significant (marital status, number of children, nationality, for instance). Then various qualitative variables have been coded with a number of modalities severely reduced. For instance the reasons of job loss are grouped in 4 classes instead of 11 in the administrative table, and 4 classes of exits from unemployment instead of 63.

The following tables summarize the variables used, with their name, definition and the codification for qualitative variables. withdrawal from the labor market (illness, retirement, military service); 4.

Quantitative Variables

administrative cancellation (discouraged worker)3 .

Similarly, the causes of registration at the ANPE have been coded in 4 modalities:

1. lay off 2.
end of fixed-term contracts 3.

voluntary quit 4.

first job search.

TRANSITIONS BETWEEN TWO STATUS.

The two next tables show the figures obtained, when the types of exits from unemployment are crossed with the causes of the former registration and, conversely, the causes of further registration crossed with the types of exits from current unemployment. Each cell shows the figure obtained by dividing the frequency of a particular transition by the total number of transitions (first number) and the one obtained by dividing this frequency by the row sum (second number) which is the total number of a type of registration. For instance the transition coded (1,1), from lay-off to a new job, represents 20.73 % of all the transitions and 59.08 % of those coming from lay-offs (which is itself 35.09 % of the different transitions The main weakness of the results presented in table 3 is that there is nothing about the kind of job found for the four cells of the first column. This is due to the fact that the information is not available in ANPE register, so it is impossible to separate durable jobs from fixed-term contracts (CDD).

The table presenting the further transitions gives a more precise information. People we refer to here are those who become unemployed again after an exit from the first spell of unemployment. Four types of transition are very significant; they are in bold characters and are coded (1,1), (1,2), (3,1), (3,2). Respectively they constitute the following transitions:

RMOTIFI

-job ==> lay-off -job ==> end of fixed-term contracts of employment (end CDD) -exit from the labor market ==> lay off -exit from the labor market ==> end of fixed-term contracts (end CDD) It is important to notice that the major transition to unemployment results from fixed-term contracts coming to an end. This means that a path constituted of consecutive periods of employment and unemployment is going to be the reference situation. The figures would be even greater if computed with data more recent than 1996.

With these transition tables, two new quantitative variables have been defined. RSS11 measures the proportion of transitions from job to lay-off among the total number of transitions 'off then back into unemployment'. RSS12 is obtained by dividing the number of transitions 'find a job then end of fixed-term contract' by the total number of transitions exit-registration. These variables are not used to perform the classification but they add some information to help the interpretation of the classes obtained.

4.

CLASSIFICATION OBTAINED USING QUANTITATIVE VARIABLES.

A Kohonen classification have been constructed using the quantitative variables described in section 2. A Kohonen classification [START_REF] Kohonen | Self-organization and Associative Memory[END_REF][START_REF] Kohonen | Self-organization and Associative Memory[END_REF], 1995[START_REF] Cottrell | Theoretical aspects of the Kohonen Algorithm[END_REF]) is a neural technique [START_REF] Ripley | Pattern Recognition and Neural Networks[END_REF] similar to the moving centers method [START_REF] Lebart | Statistique exploratoire multidimensionnelle[END_REF] but using a notion of neighborhood between the classes. This implies a very useful property: neighbors are associated to the same class (as with any classification method) or in close classes. This produces a quantization of the data space which preserves the topology. This property allows very illustrative graphical representations [START_REF] Kaski | Data Exploration Using Self-Organizing Maps[END_REF][START_REF] Cottrell | The Kohonen algorithm : a powerful tool for analysing and representing multidimensional quantitative et qualitative data[END_REF][START_REF] Cottrell | Nouvelles techniques neuronales en analyse des données. Application à la classification, à la recherchede typologie et à la prévision[END_REF],Cottrell et al., 1999) where the closeness between individuals is evaluated in a global way. The map obtained may be interpreted as a non linear projection of the data. The study of centroids (or code-vectors) of each class conducts to a clear interpretation of the classification. The successive steps are to begin with a detailed classification (100 classes in this study), then these classes are grouped using a clustering classification, to produce a small number of broad classes constituted with neighbor classes. The description of the broad classes defines a typology of the individuals.

In order to obtain the Kohonen classification, 10 quantitative variables are used (those listed in table 1 with the exception of the salary associated to the last job (SRREVAL) which is (by definition) not available for young people looking for a first job). From 100 classes at the beginning, 5 broad classes are eventually constituted for an easier description.

Two code-vectors are presented in Figure 1 and Figure 2 (selected from two very distant Kohonen classes). The line joining the values of the 10 variables represents the profile of this class. All the variables used are standardized over the whole observations in order to eliminate a level effect due to different magnitudes of the variables. Figure 4 shows the inter-classes distances [START_REF] Cottrell | A Kohonen map representation to avoid misleading interpretations[END_REF].

Figure 4 : Inter-classes Distances and broad-classes

In order to characterize the five broad classes, the means and standard deviations are examined (see table 6).

Classe 1

Classe A more complete description of the classification may be obtained by using eight qualitative variables. Some of them are obtained by discretization of quantitative variables used to produce the classification (AGEC, CTINDMOY, DURC, HAR, PPARC). Others are exogenous (DIPL3, RMOTIFA, RMOTIFI).

5.

TYPOLOGY OF CLASSES.

Figure 5 shows a representation of 8 qualitative variables for each of the broad classes and for the whole population. Class 1 is made up of young people, having a very short seniority, a duration of unemployment lower than the whole population mean, an average amount of unemployment benefits close to 0. Most of them are still looking for their first job.

Class 2 comprises people highly involved in occasional work. The duration of their unemployment and the benefits obtained are slightly above the average. The main cause of this unemployment is the end of fixed-term contracts.

Class 3 is characterized by the very long duration of unemployment. People are older than the average with benefits from unemployment slightly above the average. Most of them are not exerting any occasional work, except a sub-group which may be seen on figure 6.

Class 4 is constituted with young people, with no occasional work, and a short seniority in past employment. They are different in some points from those defined as class 1: the cumulated duration is longer, the greater recurrence of unemployment is greater, as the relative importance of ends of fixed-term contracts. Their situation seems a good illustration of a of a typical trajectory with successive periods of fixed-term contracts and unemployment.

Class 5 is made up of older people, with a lengthy experience and a corresponding higher averaged amount of benefits, rarely exerting an occasional work. The duration of their unemployment is slightly above the average but clearly lower than the one of class 3.

Thus it appears that there is no direct link between unemployment duration and the involvement in an occasional work or the amount of benefits perceived while unemployed. These results are clearly inconsistent with some usual conclusions applied to the unemployed as a whole. One of thoseis the presentation of various types of precarious jobs as a step leading to more durable jobs. Another one is the traditional positive link between the amount of benefits and the duration of unemployment. It appears that these conclusions are not verified for several subgroups of the unemployed.

More precisely it seems necessary to distinguish two types of occasional jobs. One with both an important amount of hours per month and a large proportion of the spell of unemployment occupied by this activity. This is a pattern very similar with that of precarious jobs, combining fixed-term employment contract and part time work. This type is represented here by class 2; the duration of unemployment is somewhat shorter than observed for people having some experience of the labor market4 . One other type is constituted with people grouped in the classes 4 or 5, rarely exerting this kind of activity and with an amount of monthly hours frankly lower than the one of the first type. Their duration of unemployment is slightly longer than the population average. On the subject of the effect of unemployment benefits on unemployment duration it is possible that the reduction resulting from the use of calculated average amounts perceived results in an important attenuation of this effect. This may be of some importance for the first months of a spell of unemployment, when the diminishing rate used to define the amount of benefit implies for the unemployed a real reduction of the perceived amount.

Another general observation is that the obtained classes are not differentiated when considering the transition matrix 'exit from unemployment then back to unemployment': the main transition is clearly the end of a fixed-term contract found after a spell of unemployment. This analysis may be improved with a study of the distribution of the population observed using qualitative variables : the distribution of each one within the Kohonen classes and its transformation when describing the whole grid. This transformation may induce specific groupings of classes, possibly with an easier interpretation. An example is produced by Figure 6 representing the different types of exercise of an occasional job. Another one is Figure 7 representing the different levels of daily benefits.

ANALYSIS OF MULTIPLE CORRESPONDENCES.

A complete description of the characteristics of the unemployed and their trajectories is conducted using the qualitative variables presented earlier plus a new one defined as the broad class in which each individual is placed. A classical factor analysis is carried out [START_REF] Lebart | Statistique exploratoire multidimensionnelle[END_REF], the multiple correspondence analysis. It used 8 qualitative variables, representing 32 modalities.

The following figure is the projection of the set of modalities obtained using the two first dimensions. It shows some results already found in the previous steps. The first axis is clearly defined with the amount of hours corresponding to an occasional job.

The second one shows a contrast between young people with a high educational level and older people with an educational level below from the average. This axis is very close to the one defined by the duration of unemployment, causes of unemployment and types of exits from unemployment. Young people are associated with shorter duration of unemployment, search for a first job and exit from unemployment with a job. Causes of unemployment are placed along this axis with the following order, first job, quit, end of fixed-term contract, lay-off. The types of exit from unemployment are similarly projected along the same axis showing the exit by a job sharply separated from the three other ones, almost stacked at the opposite side of the axis.

Involvement in an occasional work and the whole set of other factors are orthogonal, as it has been observed earlier.

The projection on the axes 1 and 3 leads to the same conclusions. See Figure 9.

CONCLUSION.

A typology of recurring unemployed may be defined. It is based on age, level of education, experience, involvement in an occasional work, amount of benefits from unemployment. But it seems that there is no link between the practice of an occasional work and the duration of unemployment. About the question of unemployment recurrence, the main fact is that the exit from unemployment is typically obtained with a fixed-term job, which in turn determines the next unemployment spell. The observed cumulated duration of unemployment is 23 months for the whole population studied and respectively, for the broad classes, 13, 24, 39, 22 and 23 months.

A further step is to analyze the impact of unemployment benefits and the diminishing rate applied in a more precise way. This means a study of the successive periods constituting a spell of unemployment and the possible impact of the decreasing amount of allowance on the possible exit from unemployment.

This study would be largely improved with a thorough analysis of the type of occupation attached to the occasional job and its possible correspondence to the one of the regular job. This is an important feature to understand the phenomenon of recurrence. The problem is that, at this time, this information is not available in the data furnished by the various institutions involved.

From a technical point of view, it could be very useful to apply another neural technique called KACM [START_REF] Ibbou | Classification, analyse des correspondances et techniques neuronales[END_REF], Cottrell, et al., 1999), in order to obtain the modalities of qualitative variables on a unique grid, instead of the successive planes furnished by the traditional factor analysis.

Figure 1 :Figure 2 :A

 12 Figure 1 : Code vector in class 2

Figure

  Figure 5 : Qualitative Variables Distribution

Figure 6 :Figure 7 :

 67 Figure 6 : Distribution of Monthly Hours of Occasional Work (AR) in the broad classes

Figure 8 :

 8 Figure 8 : The two first axes.

Figure 9 :

 9 Figure 9 : The axes 1 and 3

Table 2

 2 The different types of exits from unemployment have been grouped in 4 categories:

	AGE	Age of the individual
	CMDUR	Cumulated duration of unemployment since initial registration in months
	CPPAR	Proportion of the duration of the occasional work within the period of unemployment
	DUR	Length of the latest period of unemployment in days
	EXPER	Job seniority in years
	INDUR	Cumulated duration of unemployment allowance in days
	MGAIN	Monthly wage received in an occasional work
	MXMHEUR	Amount of hours per month worked in an occasional job
	NCHOM	Number of periods of unemployment
	SRREVAL	Daily wage perceived in the latest job
	TINDMOY	Average amount of daily benefits computed on the cumulated duration of
		unemployment
		Table 1

Table 3 : Registration towards exit

 3 ).

	RMOTIFA	1	2	3	4	Total
	RMOTIFI
	1	20.73 %	1.21 %	4.81 %	8.33 %	35.09 %
		59.08 %	3.45 %	13.71 %	23.75 %
	2	29.33 %	1.57 %	5.95 %	9.95 %	46.81 %
		62.67 %	3.35 %	12.72 %	21.26 %
	3	4.94 %	0.18 %	0.86 %	1.05 %	7.03 %
		70.29 %	2.50 %	12.31 %	14.90 %
	4	8.36 %	0.41 %	0.98 %	1.32 %	11.07
		75.51 %	3.72 %	8.87 %	11.90 %
	Total	63.37 %	3.37 %	12.61 %	20.65 %	100 %

Table 4 : Transition matrix: latest exit to new registration

 4

		1	2	3	4	Total
	RMOTIFA
	1	27.08 %	41.20 %	4.58 %	5.37 %	78.23 %
		34.62 %	52.66 %	5.86 %	6.86 %
	2	0.86 %	0.68 %	0.06 %	0.18 %	1.79 %
		48.22 %	38.22 %	3.56 %	10.00 %
	3	7.83 %	4.86 %	0.55 %	0.72 %	13.96 %
		56.11 %	34.79 %	3.93 %	5.18 %
	4	3.05 %	2.21 %	0.27 %	0.48 %	6.01
		50.76 %	36.75 %	4.49 %	8.00 %
	Total	38.83 %	48.95 %	5.47 %	6.75 %	100 %

Table 6 : Quantitatives Variables according to the 5 classes

 6

	2	Classe 3	Classe 4	Classe 5	Population

We thanks ANPE and UNEDIC for a financial support for this study and the permission to use the data.

This explains that a duration of unemployment greater than

months may be found.

Among the types defined by the ANPE corresponding to the last main category, class 90, involving a major difficulty. This class corresponds to people not complying the formal rules to remain registered; according to the ANPE services, most of these people have found a job by their own means and neglect to send an advice to the ANPE. So this class has finally been included in the first category.

Thus excluding those looking for their first job characterized with a significant shorter duration of unemployment.