We conclude this section of the paper by briefly discussing the index just presented. Firstly, it is not a comparison between the sequences that defines their distance. The index has a known beginning and end; each point is univocal and identifies one and only one combination of states in sequence. Two sequences which differ in the position of only one element will have different positions. From every point one can retrace the exact sequence that has produced it. A second characteristic of the index concerns its output. Each value of the index, in fact, can be conceived as a coordinate in the space of the multinomial sequence. This characteristic enables the researcher to adopt different methods to calculate the distance, but also to define forms of space other than Euclidean. Furthermore, the third characteristic is the natural way in which to handle multichannel sequences.

#### **3 The Data, Their Organization and the Coding of the Activities in a Multichannel Approach**

The goal of our analysis was to discover how 873 Italian dual-earner couples organized their daily activities during a typical work day from Monday to Friday.We used data from the Italian Time Use Survey 2008 (Istat 2011). We considered timeuse diaries of dual-earner couples' activities (His and Her) from 7.00 to 22.00.3 Each daily activity was observed every 10 min, and the data files for the sequence analysis consisted of 873 pairs of sequences, one for Him and one for Her, with

3Excluded from the sample were: (a) couples living with other couples (parents or others); (b) couples that filled in the questionnaire on different days, or during the weekend; (c) couples with incomplete information by one or both of the spouses; and, (d) couples in which his or her age was over 65.

a total of 90 points in time. Each couples of rows of this file corresponded to a cohabitation, while each variable corresponded to 10 min of observation and each cell of the row/column intersection stated the activities of Him or Her at time *t*.

In order to simplify the analysis, six different groups of activity were considered: Sleep; personal care—i.e. having a shower, eating, etc. (P.Care); paid work (Work); moving—any kind (Move); unpaid work—i.e. housework, child care, repair, etc. (H.Care); free time and other activities with or without others (F.Time).

Having defined the six daily macro-activities, the next step was to establish how to codify the daily activities of Him and Her in the couple. In this case, His activities and Her activities interact in time to give rise to the couple's daily activities. Taken individually, each of these two sequences takes the form of a series of mutually exclusive episodes. The problem is therefore how to codify two interacting sequences composed of a plurality of mutually exclusive events. To date, all the solutions proposed have been based on the generation of events combinations (Pollock 2007; Gauthier et al. 2010; Aisenbrey and Fasang 2017): that is, on the construction of a single sequence that combines the states of Him and Her.

This operation has several consequences. Firstly, as Abbott pointed out, using combinations of events requires one to pay ". . . the price of losing all information about the temporal 'shape' of events—their duration and their intensity in terms of producing occurrence—in short their time horizon" (Abbott 1990, p. 146). Secondly, there is the risk that distinct time-use patterns will be tied together, although the order of causality may be bi-directional.

There are various reasons to believe that daily activities of Him and Her cannot be reduced to a simple combination of states. Internally, moreover, each sequence consists of states regulated by their own mechanisms which operate differently in defining the timing and duration of each individual episode. For instance, consider the mechanisms that underlie the regulation of the states of free time and housework. In the former case, it is the working time that mainly regulates the time spent on these two activities; in the latter, we should expect a stronger interaction between gender roles.

It is therefore possible to hypothesize that the sequences of Him and Her and the states of which they are composed—have their own underlying generative mechanisms which establish the timing and duration of episodes. These generative mechanisms work independently of each other and interact in time: they stand in a coexistence relationship. Finally, the couple's daily activities are the result of a complex process of co-action between two sequences, that of Him and that of Her, regulated by different generative mechanisms resulting from the co-action between different states. Consequently, reducing everything to a combination of events means loss of a large part of information about the temporal 'shape' of events.

A couple's daily activities, or more correctly the couple sequences analyzed here, are therefore configured by the co-action of two multinomial sequences composed of mutually exclusive episodes. By extending the proposed application of the Lexicographical Index (see Sect. 2) to this case-study, 12 binary sequences can be defined, six for His states and six for Her states, each one of length *t* = 90, that is, the overall number of points of observation. The couple sequence is defined as a point in a 24-dimension space whose coordinates are the 24 lexicographic indexes defining the respective sequences of Him and Her. The distance between two couple sequences is given by the Euclidean distance between the two points of the two sequences in the 24-dimension space.

The coordinates defined for all 873 couples were analyzed using a *k*-means cluster algorithm.4 The joint exam of the scree plot (Makles 2012) of the kink in the curve generated from the within sum of squares (WSS), the *η*2 coefficient (0.43) and the proportional reduction of error, suggests that seven is the optimal number of groups drawn from a set of 20 cluster solutions with random starting points.

#### **4 From 7.00 to 22.00: A Typical Working Day of a Dual-Earner Couple in Italy**

It is not news that the everyday life of a dual-earner couple is complex. It involves a long and difficult schedule of: waking up, having a shower, breakfast, taking the car-bus-train, going to work, beginning work, lunch, resuming work, coming back home, then housework and family/child care for Her, relaxation for Him, dinner, and at the end of the day, before they go to sleep, some leisure activity. Overall (Fig. 1), this was also the typical daily routine followed by our 873 Italian dual-earner couples from 7.00 to 22.00. Looking at the most frequent activity combinations in the morning, at 7.00, 75.0% of couples were involved in personal care or going to work. From 8:00am to 6:00pm all the couples were at work.5 At 6:00pm, the couples started to be desynchronized: She was engaged in housework/children care; meanwhile He continued to work until 7.00pm. From 7.00pm to 7.30pm, He had some free-time activities, while She continued her housework activity. Finally, together, they had dinner and engaged in free-time activities.

Differences in the spouses' daily time-budgets for each activity (Table 1) also confirm a well-known finding on the unequal gender division of work-family activities (Gershuny and Robinson 1988; Raley et al. 2012; Craig et al. 2014). For instance, on average, She spends 2 and a half hours more than her partner on housework and childcare, while He spends 1 h and a half more than his partner on paid work.

However, the timing of this daily organization changes when we move from general into the seven clusters. In this case, there emerges a more composite picture of daily life, where "multiple equilibria" (Esping-Andersen et al. 2013) of time allocation during a typical workday and (de)synchronization strategies jointly explain the dual-earner couple's patterns of time use. The average time activities (Table 1) show a clear difference in the time spent on each activity by Him and Her within partners and among clusters.

4Our distance measure between sequences could as well be used for clustering with the propertybased and fuzzy methods addressed by Studer (2018) in this bundle.

5The absence of a break for lunch does not mean that spouses do not eat; only that, overall, there is not a common time interval for lunch due to the different work schedules.

**Fig. 1** Modal sequence graphs of the (de)synchronized patterns of His/Her activities (labels show Him activity/Her activity in that order)

Moreover, on shifting the focus to the schedules of each activity, the modal sequence graphs6 (Fig. 1), give us a clearer picture of how the strategies of dualearner time-use change over time in a typical workday. Both within the spouses and among the clusters the differences in time-use (Fig. 1) mainly occur in the second part of the workday. Until noon the couple's everyday lives are quite "synchronized". Him and Her show differences in the afternoon, when fewer women than men are at work and when the women shift their activities from paid to unpaid work (housework, child care, etc.). In other words, gender inequalities in the workfamily balance are generally set in the afternoon.

6For each cluster and for each point-in-time, the most frequent activities combination was identified. On this criterion, only 16 of all the 36 (six for each spouse) possible combinations were found to be frequently performed by the couples, suggesting a certain routine by couples in everyday life.

**Table 1** Mean time (in hours:minutes) spent on each activity, by cluster

Note: \* *p* < 0.1, \*\* *p* < 0.05, \*\*\* *p* < 0.01

Hence, the preliminary results seem to suggest that, on the one hand, gender specializations in different activities can assume different meanings when contextualized in the whole day and, on the other hand, that the partners' daily life seems to develop with socially shared, recognized, and identifiable patterns of combined time use. This insight raises two further questions. The first is how these patterns result from a complex process of adaptation to both work-social-family constraints and individual needs. The second question concerns how the daily times are combined by spouses, and how their performed combinations are random instead of being regulated by common generative mechanisms.

In order to investigate the complex process of adaptation of dual-earner couples' daily time organization, we ran a multinomial logistic regression model to verify if such patterns resulted from working-social-familial and individual constraints (Table 2). For this analysis, we used information about the couple's educational qualifications, social class, economic sector, and the presence of children.7

A joint reading of the modal multichannel sequence graphs (Fig. 1), the multinomial logistic regression parameters (Table 2) and the margins estimated probability (Table 3) quite clearly shows what are the (de)synchronization strategies adopted by couples and what may be the hidden generative mechanisms (Hallberg 2003). We highlight the importance of the presence of children, work sector and the educational level in explaining the cluster differences (Table 2).

Three different forms of time-use organization are highlighted by the graphs (Fig. 1). The first is characterized by a general synchronization of the spouses'

7Educational qualifications were classified as: (1) compulsory level—elementary school certificate (including no educational qualifications) and lower-secondary school certificate (including 2-to-3 year vocational certificates); (2) upper-secondary school diploma (including post-secondary diplomas); and, (3) university degree (including postgraduate qualifications). Social class was classified according to the EGP scale: (I+II) Service class; professionals, administrators, and managers; (IIIa) Routine non manual workers; (IVabc) Petty bourgeoisie; Farmers; (VI+VIIab) Skilled and non-skilled workers; Agricultural Labourers. The economic sector (agriculture and industry, private services and public services) of the couple was the combination of the main job sector of Him and Her. The couple's educational level (and social class) is defined as the highest educational level (social class position) between the spouses.

**Table 2** Multinomial logistic regression on the seven clusters by presence of children and sector, level of education and social class of couple. (Jackknife replication). Reference cluster (A)

Note: \* *p <* 0*.*1, \*\* *p <* 0*.*05, \*\*\* *p <* 0*.*01; Pseudo *R*2 = 0.05

different activities during the day. This maximally "egalitarian" (Esping-Andersen et al. 2013) gender participation in unpaid work seems able to preserve the free time of the spouses. Couples in clusters (A) and (B) are associated with the highest synchronization levels.

These dual-earner couples are characterized by a tertiary educational level for couples in cluster (A) and a secondary educational level for those in cluster (B) (Table 3). Thus, a low educational level seems to be an obstacle to an egalitarian strategy of synchronization.

What distinguishes the two clusters is the presence of children (Table 2), which increases the probability of being a member of cluster (B), while the couples in cluster (A) are more likely to be without a child. There are also differences of occupational sector and class between the couples in clusters (A) and (B). Couples in cluster (A) work in the private services sector, while those in cluster (B) are mainly employed in the public services sector. At the same time, couples in cluster (B) are mainly employees (IIIa or VI + VIIab), while in cluster (A) they are more likely to be self-employed (I + II or IVabc).

This particular combination of characteristics—and constraints—creates synchronized couples' patterns (Fig. 1). However, there are some substantive differences. In cluster (B), the spouses seem to have breakfast together before going

**Table 3** Predictive margins probability (in percentage) on the seven clusters by presence of children, sector, level of education and social class of couple. Marginal effects at reference profile∗

aReference profile: Children (No child), Sector (both Industry), Education (University), Social class (I + II)

to work and starting it synchronically. They both stop working quite early in the afternoon, favored probably by their kind of job and the economic sector in which they are employed. At 17:00 She is already at home, while He comes back at 17:40. Thereafter, both the spouses spend the rest of the day at home, doing housework and childcare before having dinner together and, finally, enjoying most of their free time synchronically. The only second part of the day in which they are not synchronized is the one immediately after dinner, when She postpones her free time for 20 min due to housework (Fig. 1).

In cluster (A) the absence of children and the type of work (self-employment in the private services sector) would seem to explain why the couples start their day in a manner differently from the others (Fig. 1). Both the spouses wake up together, and later than the couples in other clusters. They also have breakfast at the same time. Then He leaves the house while She quickly tides up before going to work. Job commitments fill equally most of their daily time. Moreover, their lunch and dinners are synchronized. Finally, the extent of their job commitments and the parallel absence of children seem to pull the spouses in cluster (A) directly to free and leisure time.8

8The absence of housework does not mean that spouses do not perform any housework. Simply, they are more likely to do it in a non-regular way, during brief and scattered moments of spare time.

Cluster (C) falls—although not completely—within the synchronized time-use patterns (Fig. 1). Couples in this cluster have some features in common with those of cluster (A). In particular, like dual-earner couples (A), those in cluster (C) are more likely to be childless. They also are mainly members of the upper class (I+II) (Table 3).

Cluster (C) has some characteristics in common with clusters (A) and (B) also in terms of daily time organization, even if its pattern ends with a longer tail of synchronized personal care: spouses may still be having dinner together at the end of the observation (22:00). However, what really makes cluster (C) unique is the time organization around lunch. While for cluster (B), there is no specific time for lunch, and for cluster (A) the time interval for lunch is well defined between two work 'segments', for cluster (C) the break from work is longer for Her. Moreover, around a certain synchronized lunch-time, there is a certain desynchronization due to His work commitments and Her housework tasks.  Here, the sequence of activity combinations around lunch is much more chaotic, fragmented and desynchronized compared with the clusters (A) and (B). However, except for this desynchronized part of the day, probably due to different work commitments, the rest of the day is mainly synchronized.

Alongside the synchronized patterns other desynchronized daily time-use patterns emerge. These strategies of desynchronization seem to be specialized into two forms, on the basis of the kind of tasks sequentially performed and combined by the two spouses during the day.

The first kind of strategy is called functionally desynchronized. Here, gender differences in activities-in-time appear to be an adaptation to structural desynchronization (Nock and Kingston 1984) of His and Her working schedules. The difference in work duration between men and women appears to produce a counterbalancing force by which—at the end of the work day—She 'compensates' the different spread of paid-work commitments of Him with unpaid work, in a quite calibrated way that preserves the free time of both the spouses. The gender division of work-family activities is "unstable" (Esping-Andersen et al. 2013), mostly due to "structural constraints" of the partners' working schedules.

Couples in clusters (D) and (E) are associated with the clearest functionally desynchronized patterns. For both clusters, Her working schedule is shifted forward in the afternoon (Fig. 1) and in most cases at least one spouse of these dual-earner couples is employed in the public sector. The most important differences between these couples is that (D) do not have a child while (E) do so (Table 2).

In cluster (D), He starts work much earlier than Her. On the other hand, She spends more time on personal care before going out to work. The probable absence of children may be helpful in this regard. At the end of the workday, these spouses come back home later and synchronically. Once at home, they desynchronize

Moreover, they may not necessarily do the housework every day, maybe postponing the chores to the weekend.

themselves again (Fig. 1) and while He takes a break to relax, She does some housework. It seems that there is some sort of compensation of daily time activities: He starts work much earlier than Her in the morning, and the gendered housework at the end of the day seems useful in establishing the balance, before dinner. Finally, they both eat and relax together (Fig. 1).

In the time-use pattern (E), She wakes up a little before Him, probably because of the young children's demands. They have breakfast together before going to work, and they start working synchronically. In the afternoon, She leaves the workplace much earlier than Him, perhaps in order to devote herself again to childcare and housework. After His return from work, they eat together, before spending synchronous free time. Again, the clear non-cooperation of Him in the household tasks may be due to the evident spread of work commitments during the whole day.

The second desynchronizations strategy is what we call traditional. Here, the couple's distribution of activities during the day does not seem to follow any compensatory mechanism. The overall desynchronization seems to be weakly linked to the "structure" of the spouses' work commitments (Nock and Kingston 1984). Conversely, it appears to be an outcome of a more "traditional" gender attitude to the work-family balance. Here, the result is a marked overload in paid/unpaid work for women (Mattingly and Blanchi 2003), with stronger evidence of the gendered leisure gap (Beblo and Robledo 2008).

Couples in clusters (F) and (G) are characterized by the presence of younger children, a low level of education, mainly compulsory level, and are mainly employed in the industrial sector. There are some differences in job features: couples in cluster (G) are mainly members of the white collar middle class (IIIa), while those in cluster (F) are mainly members of the petty bourgeoisie (IVabc). Moreover, cluster (G) shows a relatively high presence of couples where He works in the industrial sector and She in the public sector or He does so in the private sector and She in the public sector.

The time-use pattern of cluster (F) is apparently similar to that of cluster (E). In fact, She comes back home before Him and deals with domestic chores. However, compared with cluster (E) we note a greater extension of Her household commitments, from the early afternoon until the evening, when He has already finished his workhours. Thus, on one hand the desynchronization seems functional for the long time spent by Him at work; on the other hand, this couple's timeuse pattern does not show any cooperative or compensatory forms of time-use organization between the spouses (Fig. 1).

Last but not least, cluster (G) is certainly the maximum expression of the traditional desynchronization. The time-use pattern (G) describes a couple in which everything is on Her shoulders. The delay of the exit from home is followed by a long journey to work. Then, she continues to work until the late afternoon. Finally, when both the spouses return home, He takes a break and rests, while She continues to do housework and child care. The only synchronized moment in the final part of this couple's pattern is when they have dinner. Among all the time-use patterns, this is certainly the one with the highest level of gender inequality in regard to the daily work-family balance challenge (Fig. 1).

#### **5 Conclusions**

In the introduction of this paper, we pointed out the importance of adopting a multichannel sequence analysis approach to gain better understanding of the complexity of the work-family balance through holistic study of dual-earner couples' daily time use as an overall pattern. At the end of this paper, it is evident that the sequence analysis of time use diaries provides a rather clear and meaningful representation of the main patterns of the everyday organization of Italian dual-earner couples. The analysis shows the clear co-action of multiple generative mechanisms that give shape and relevance to each of seven patterns and define different forms of (de)synchronization in the everyday-life organization of both individuals and couples.

These patterns are attributable to three different strategies for organization of daily activities, and three types of equilibria (Esping-Andersen et al. 2013) within the family. In fact, these patterns describe three sets of work-family equilibrium strategies performed by dual-earner couples, with different expected levels of desirability. The first defines the synchronization strategies (clusters A, B and C). Hence, the housework division by gender is "egalitarian" because both partners participate in the housework and are able to share most of the free time available. The second defines the functional desynchronization strategies (clusters D and E). The division of housework by gender is "unstable" (Esping-Andersen et al. 2013) mostly because of "structural constraints" of the partners' work schedules (Nock and Kingston 1984). Nevertheless, the behavior of Him and Her reflects a collaborative complementarity which still tends to preserve the free time of both. The third pattern defines the traditional desynchronization strategies (clusters F and G). Partners are characterized by an unequal division of housework. They exhibit the classic features of a "traditional" equilibrium where the woman has heavy overexposure to home/child care tasks and limited free time availability (Mattingly and Blanchi 2003; Beblo and Robledo 2008).

The close relations with certain household features (the presence of children and the couple's level of education, social class and job sector) support the contention that such behaviors and patterns result, on the one hand, from the internal bargaining among each couple conditioned by the cultural-economic characteristics of the partners themselves, and, on the other, by external social constraints.9

The time-use patterns result from the complex co-action among individual, family and social factors whose combination defines the relevance and the shape of patterns. The time balance within His and Her activities, as well as its configuration across the day, is not random; rather, it changes according to multiple latent factors.

Dual-earner couples package their daily life mainly in accordance with their work and its schedules, and therefore mainly the type of job and the economic sector (Hamermesh 2002; Warren 2003; Lesnard 2008). Moreover, the analyses

9The solutions of these particular couples in daily scheduling affected the spouses' level of satisfaction as an outcome of daily life quality. For details see Bison and Scalcon (2016).

show that this time packaging changes in relation to the presence of children. We observed that the presence of children (especially young ones) introduces elements of desynchronization and specialization within the couples. The impact of young children, however, may differ according to both the couples' work schedules and their gendered attitudes to work-family activities. According to such a view, the last factor is the couples' level of education. We can assume it as a proxy for the predisposition towards egalitarian gender attitudes (Hakim 2003; Oláh et al. 2014). Not by chance, the most "egalitarian" strategies of synchronization are performed by high-educated couples, while the most "traditional" strategies of not functional desynchronization are performed by couples with a low level of education.

In conclusion, the presence of children, the level of education, and job characteristics are three dimensions that contribute to defining the patterns of couples' daily activities, already constrained by several social rhythms (i.e. school hours; lunch and dinner time; shop opening hours; etc.).

**Acknowledgements** The authors warmly thank the anonymous reviewers for their constructive comments.

#### **References**

West, C., & Zimmerman, D. H. (1987). Doing gender. *Gender & Society, 1*(2), 125–151.

Wu, L. L. (2000). Some comments on "sequence analysis and optimal matching methods in sociology: Review and prospect"˙I. *Sociological Methods & Research, 29*(1), 41–64.

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## **Part VI Appraising Sequence Quality**

## **Measuring Sequence Quality**

**Anna Manzoni and Irma Mooi-Reci**

#### **1 Introduction: The Quality of Binary Sequences of Successes and Failures**

Examination of binary sequences, that is, sequences containing only two distinct characters, each of which represents a class of separate states or events of the process observed, is of key interest in social science research. Binary sequences, which can be seen as a series of positive versus negative characters – that is, of successes versus failures – are abundant in health and applied social science research. For instance, in the study of the course of a therapy, patients may show several kinds of unfavorable reactions, the failures, as well as various kinds of positive reactions, the successes. Similarly, pupils may make various kinds of mistakes, as a result, for example, of failures of the teaching process; alternatively, they may produce correct responses, the successes of the teaching process. Yet, another example refers to individuals' labor market careers, which can be represented as the succession of favorable and unfavorable labor market states; employment and vocational training, for example, can be considered as successes, while unemployment and inactivity as failures.

The presence of successes and the absence of failures defines the quality of a sequence. Over the course of the treatment, therapy quality is higher when the patient's unfavorable reactions – the failures – gradually disappear from the gamut of observed behaviors; the teaching quality is higher when inadequate responses disappear from the pupil's repertoire; and career quality improves if

A. Manzoni (-)

North Carolina State University, Raleigh, NC, USA e-mail<EMAIL_ADDRESS>
I. Mooi-Reci University of Melbourne, Parkville, Australia

© The Author(s) 2018 G. Ritschard, M. Studer (eds.), *Sequence Analysis and Related Approaches*, Life Course Research and Social Policies 10, https://doi.org/10.1007/978-3-319-95420-2\_15

261

unemployment is overcome by finding stable employment. However, available techniques to quantify sequence quality have been lacking with existing measures focusing predominantly on the comparison of ordered sequences (Abbott 1995) or the variability of sequences (Elzinga and Liefbroer 2007). Brzinsky-Fay (2007) and Gabadinho et al. (2011) already recognize the need of such a measure and make an attempt to quantify the quality of sequences. Our study builds on and expands this knowledge by proposing a new method which distinguishes between states of differing quality, that is, states with different characteristics. We operationalize the concept of positive and negative states, which define successful and less successful sequences, and take into account the variation in the frequency, duration, and recency of successes and failures over the course of a trajectory. We interpret binary sequences as series of Successes (*S*) and Failures (*F*) and encode them as strings consisting of the characters *S* and *F*. Of course, the quality or successfulness, of a therapy may also depend on various characteristics of the therapist, the therapy, and the patient; the quality of teaching may depend on the teacher, the teaching instructions, or the pupil; the quality of a career may depend on individual characteristics of the worker, or labor market opportunities, among other factors. Hence, in many situations, it would be interesting to model the quality of a binary sequence in terms of one or more independent variables.

In Sects. 2 and 3, we briefly review existing methods to compare and describe trajectories, outline elementary requirements of our proposed measure of sequence quality, and discuss its properties. Next, we show an application of our measure to model the quality of labor market careers. In doing so, we provide a direct test of unemployment "scarring" theories that expect the number, duration, and the most recent spell of unemployment to largely determine the path of subsequent employment career quality due to human capital depreciation and signaling processes (Arulampalam 2001; Mooi-Reci and Ganzeboom 2015). Specifically, we ask to what degree unemployment is negatively associated with the quality of one's future employment career, and at what rate, if at all, previously unemployed individuals eventually recover from it. Using data from the Household, Income, and Labour Dynamics in Australia (HILDA) Survey, we investigate the evolution of employment career quality after the occurrence of an initial spell of unemployment and as a function of an individual's attributes. We conclude with a discussion of potential extensions to our measure.

#### **2 Common Methods for Studying Sequence Trajectories**

Sequence analysis has advanced rapidly in the last few decades, and several developments have been proposed, among which extensions of Optimal Matching (Abbott 1995) based on alignment techniques (e.g., Lesnard 2008, 2010; Stark and Vedres 2006; Stovel et al. 1996; Stovel and Bolan 2004), as well as nonalignment techniques (e.g., Elzinga 2003, 2010); see Studer and Ritschard (2016) for a full review. Newly developed measures, such as for example the "turbulence" or "complexity" measures (Elzinga and Liefbroer 2007; Elzinga 2010; Gabadinho et al. 2011) have offered a quantification of the variability within rather than between sequences, but fall short in distinguishing between "good" or "bad" events over the course of a trajectory. This results in treating a series of positive events (e.g., upward job mobility) equally as a series of negative events (e.g., downward mobility), which may lead to serious substantive misinterpretations about the evolution of trajectories. Failing to attach a quality connotation to the states results in measures that ignore the variation in sequence quality.

An attempt to quantify the quality of sequences is proposed by Brzinsky-Fay (2007). The study uses data from the European Community Household Panel (ECHP) over the period 1994 to 2011 to examine sequences of school-to-work transitions of school leavers across ten European countries. It draws on explorative methods of optimal matching and cluster analysis to identify positive from negative variation of sequences where transition types with high volatility are considered negative and those with low volatility positive. While valuable, this volatility index accounts for the sequencing and variability of the transitions to infer indirectly about the nature of the transitions. However, the index suffers from the same 'old' shortcoming in which upward and negative job mobility are quantified equally because the index captures the volatility rather than the quality of the transitions. To our knowledge, one of the studies that comes closest to our measure of sequence quality is the "precarity" index proposed by Ritschard et al. (2018) in this bundle. The index draws on the complexity index as proposed by Gabadinho et al. (2011) that combines entropy with the number of transitions in the sequence and then uses a correction factor that reflects the penalizing versus rewarding quality of a transition. This correction factor is derived from the proportions of negative and positive transitions in a sequence. In doing so, the measure allows the user to determine the negative or positive transitions based on theory or data.

Different from the measure proposed by Ritschard et al. (2018) that is based on the quality of the transitions between states, our sequence quality measure is based on the quality of the states themselves and takes into account various dimensions of a particular state, such as the frequency, duration and its recency.

In the following section, we will introduce our new measure and its properties.

#### **3 Developing a Measure of Sequence Quality: Formal Properties**

Here, we introduce some notations and concepts. First, we define an alphabet, i.e., the set of states or characters that we deal with, as *A* = {*F,S*}, where *F* denotes a failure of some sort and *S* denotes a success of some kind; the specific interpretation will depend on the substantive meaning of the sequences. By concatenating the characters from the alphabet, we obtain sequences which we may denote as *x*, *y*, or *z*, for example. Let *x* denote such a sequence; then *xF* denotes the same sequence, elongated with a failure *F* and, similarly, *xS* denotes the same sequence, elongated with a success *S*. If *y* is another such sequence, then *xy* denotes the (right-)elongation of *x* with *y*. A run is a number of consecutive successes or failures. Thus, if *x* = *FFSSSF*, we say that *x* consists of three runs: *F*2, *S*3, and *F*1 = *F*. More generally, we say that e.g., *Sn* denotes a run of *n* successes where *n* is a nonnegative integer, the run-length. Nonnegative, since it is convenient to have an empty sequence *λ* = *S*0 = *F*0 that does not materially elongate any sequence: we have that *xλ* = *x* = *λx* for all sequences *x*.

These simple concepts and notations suffice to discuss some fundamental requirements that all quality measures of successfulness should adhere to. First, we require that such a measure, say *Γ (x)*, increases when the number of successes in a given sequence increases, independent of sequence length. Hence, we require that:

(1) For any *xy* = *λ*, 1 ≥ *Γ (xSy)* ≥ *Γ (xy)*, equality holding precisely when *x* = *Sn* and *y* = *Sm* for any nonnegative *n* and *m*.

Stating that there is an upper bound of 1 is a way of saying that *Γ (x)* is independent of the sequence length; however long the sequences, *Γ (x)* will not exceed the value of 1. Requirement (1) also states that wherever we put an extra success into a sequence, the result will be that *Γ (x)* increases, with the only exception that *Γ (Sn)* = 1 for all positive *n*. Substantively this effect can be seen as a "compensating effect" in which a positive event or state (e.g., employment) counteracts a previous negative state (e.g., unemployment or inactivity).

Our second requirement is the mirror image of the first as it pertains to the effect of failures on the value of *Γ (*·*)*:

(2) For any *xy*, 0 ≤ *Γ (xF y)* ≤ *Γ (xy)*, equality holding precisely when *x* = *Fn* and *y* = *F m* for any nonnegative *n* and *m*.

Requirement (2) implies that the lower bound of *Γ (*·*)* equals zero, which is attained for any sequence that consists exclusively of failures or that contains only the empty character *λ*. Furthermore, requirement (2) states that the quality of a sequence diminishes when we add more failures. Hence, requirements (1) and (2) jointly imply that in a given sequence, a quality measure: (a) has a fixed range of [0*,* 1]; (b) increases when the number of successes increases; and (c) decreases when the number of failures increases.

Now consider the sequences *x* = *SSF* and *y* = *FSS*. These two sequences only differ in the position of the failure *F*: *x* ends in a failure and *y* ends in a success. Whether *x* and *y* stood for sequences of responses of patients, pupils, or labor market states, in all cases we would consider *y* as the highest quality sequence because of the recency of the success. We believe this to be a general principle: the more recent the successes, the more positive the quality of the sequence. Therefore, we formulate a third requirement that precisely formalizes this principle:

(3) For any *xy*, *Γ (xF Sy) > Γ (xSF y)*.

While frequency refers to how often states occur in a sequence, we may also be interested in "durability", which is the consecutive frequency of states or, in other words, the length of spells in a sequence. Frequency refers to the occurrence of a state, independent of whether it is part of the same or of different spells. Durability refers to state frequency within a spell. Higher durability is manifested in higher frequency of a state. However, the reverse may not be the case, as high frequency of a state may be due not only to a single durable spell, but also to several spells of short duration, in which case we will see low durability despite high frequency. For example, take the sequence *x* = *S*2*F*2, which includes two consecutive states of success, that is a spell of success with a duration of two units, followed by a spell of failure with duration of two units. Then take a sequence *y* = *S*1*F*1*S*1*F*1 including alternating spells of successes and failures, each of the duration of one unit, for a total frequency of two states of failures and two states of success. Sequences *x* and *y* will have the same frequency of successes. However, the durability of success differs, reaching a value of two consecutive successes in sequence *x* and only one in sequence *y*. Note that differences in duration at equal frequency will affect the recency of success. Therefore, a measure fulfilling the above requirements (1–3) and accounting for both frequency and recency will capture duration differences as well. In sequence *x*, consecutive success durability is higher compared to sequence *y*, although success is more recent in sequence *y*. In the next section, we will discuss an implementation of *Γ* that satisfies the above requirements.

#### **4 Using** *S***-Positions: Successes Weighed by Frequency and Recency**

For a binary sequence *x* over *A* = {*F,S*}, we write *x* = *x*1*,...,xn* when *x* has *n* characters and *xi* stands for the *i*-th position in *x*. Thus *xi* ∈ *A* for all *i* ∈ {1*,...,n*}. For example, with *x* = *SSFSF*, *x*1 = *x*2 = *x*4 = *S* and *x*3 = *x*5 = *F*. The length of *x* = *x*1*,...,xn* equals *n*; we denote this by writing |*x*| = *n*. For a sequence *x* with |*x*| = *n*, the *k*-th prefix of *x* is the sequence *xk* = *x*1*,...,xk* for 0 ≤ *k* ≤ *n* and *x*0 = *λ*.

We begin by discussing a simple example. Consider the sequence: *x* = *x*1*x*2*x*3*x*4*x*5 = *SSFSF*. First, we note that the first, second and fourth characters are successes. Then, we add the position-indices of these characters: 1 + 2 + 4 = 7. If these three successes had occurred later in the sequence, as for example in the sequence *FFSSS*, the sum of the position indices would have been bigger: 3 + 4 + 5 = 12. Therefore, we see that the sum of the position indices of the *S*observations quantifies the quality level: the more *S*-observations and/or the more recent these are, the bigger the sum will be. However, we cannot judge the size of this sum independently of the length of the sequence, since longer sequences will have larger position indices. To adequately quantify quality, we divide the sum of the observed *S*-positions by the sum of all positions and we denote this ratio as *Υ* to distinguish it from the general *Gamma*. Hence, we obtain:

$$\mathcal{T}(1+2+4)/(1+2+3+4+5) = 7/15 = 0.47.$$

Clearly, this ratio is always in the range [0*,* 1] as the numerator is non-negative and at most as big as the denominator. Formally and more generally, it is convenient to first define a position variable:

$$p\_l = \begin{cases} i & \text{if } x\_l = S \\ 0 & \text{otherwise} \end{cases} \tag{1}$$

and then define:

$$0 \le \mathcal{T}^w(\boldsymbol{\chi}^n) = \frac{\sum\_{l=1}^n P\_l^w}{\sum\_{l=1}^n i^w} \le 1 \tag{2}$$

for some nonnegative exponent *w*. Note that in the above example, we set *w* = 1. Clearly, *Υ* 1 quantifies sequence quality in the required way: the more failures and the later in the sequence, the smaller its numerator. As the numerator of Eq. 2 is nonnegative and cannot exceed the size of the denominator, *Υ w* will be tightly bound by 0 and 1; the upper bound will be attained when all states are *S*-states, while *Υ w* = 0 when the sequence shows no successes at all. Table 1 illustrates the behavior of *Υ w*. Specifically, the left part of the table illustrates the behavior of *Υ w* when failures get more recent for three different values of the parameter *w*; the right part illustrates the combined effect of recency and number of failures on *Υ w*.

Let us now turn to the meaning of the parameter *w* in Eq. 2. First, let us set *w* = 0. The denominator then reduces to *n i*=1 *i*0 = *n i*=1 1 = *n* and the numerator counts the number of *S*-states, regardless of their position in the sequence. Hence, we see that:

$$\mathcal{T}^0(\mathbf{x}^n) = \frac{f(\mathcal{S})}{n}$$

i.e., the measure calculates the fraction of *S*-states in the sequence *xn*.

**Table 1** Illustration of *Υ w* behavior with varying *w* and recency of failure

Next, we consider:

$$\mathcal{T}^1(\boldsymbol{x}^n) = \frac{\sum\_{i} p\_i}{\sum\_{i} i} = \frac{\sum\_{i} p\_i}{n(n+1)/2}$$

which is simply a formalization of the example above.

In Table 1, we illustrate how *Υ w(x)* reacts to changes in the recency of failures and to changes in the number of failures. To demonstrate the effect of the parameter *w*, Fig. 1 illustrates the dynamic behavior of *Υ w* for different values of *w*.

Specifically, we plot *Υ w* for various values of *w* applied to a fixed sequence *x* = *S*4*F*4*S*6*F*2*S*8*F*1*S*25 of length |*x*| = 50. To draw Fig. 1, we calculate, for each fixed value of *w*, *Υ w(xn)* for *n* = 1*,* 2*,...,* 50. Hence, the plots show how *Υ w* develops "over time", as the sequence develops from a single state to a sequence of 50 states. The sequence chosen has three runs of failures, each next run shorter than the previous one. The plots demonstrate that the bigger the parameter *w*, the more severe is the effect of failures, but recovery from the failures due to subsequent successes is also faster for bigger *w*. This holds independent of the order of success and failure runs and their length. Moreover, the plots suggest that, given enough time and enough subsequent successes, the sequences will "fully recover" from previous failures. Here, "full recovery" means that *Υ* will be arbitrarily close to 1. Formally, this can be seen from the fact that:

**Fig. 1** Plot of *Υ w(xn)* (vertical axis), for various values of *w* and *n* (horizontal axis) ranging from 1 to 50, and *x* = *S*4*F*4*S*6*F*2*S*8*F*1*S*25

$$\mathcal{T}^{w}(\mathbf{x}) = \frac{\sum\_{l} i^{w} - S\_{F}}{\sum\_{l} i^{w}} = 1 - \frac{S\_{F}}{\sum\_{l} i^{w}} \tag{3}$$

wherein *SF* denotes the sum of (the powers of) the positions on which failures occurred: given enough subsequent successes after the last failure, the fraction will converge to 1 since the quantity - *i iw* will increase while the quantity *SF* will remain constant. Full recovery from failures is a phenomenon that happens, for example, in labor market careers of former criminals: after enough clean time ("Success") after conviction ("Failure"), their chances of getting employed are equal to those never arrested (Blumstein and Nakamura 2009).

Calculating the quantity *Υ w* is straightforward; for the denominator of *Υ w*, closed expressions can be derived and *w* does not have to be an integer (Knuth 1993). However, in applications, such closed expressions are not required since the denominator is easily calculated simultaneously with the numerator while parsing the sequence. Thus, calculating *Υ w* is proportional to the length *n* of the sequences. The reader is aware that *Υ w* as defined in Eq. 2 is just one of the many possibilities to construct a quality measure on the basis of the position-numbers *pi*: in fact, for any function *f* that satisfies *f (a) > f (b)* precisely when *a>b*, the quantity

$$0 \le \|\Gamma(\boldsymbol{x}^n) = \frac{\sum\_{l=1}^n f(p\_l)}{\sum\_{l=1}^n f(p\_l)} \le 1$$

is a quality measure, i.e., satisfies the requirements (1) to (3) as formulated in the previous section. We believe that, by picking *f (pi)* = *pw i* , we have covered sufficient potentially useful possibilities.

Theoretical reasons may lead a user to define a *w* of 0 and get an index that does not correct for the recency of failures or successes. Alternatively, a *w* of 1 quantifies sequence quality in the required way: the more failures and the later in the sequence, the smaller its numerator. However, should theory assign deeper penalties to failures and faster recoveries to most recent successes, then a *w >* 1 would be preferable.

#### **5 An Application: The Quality of Labor Market Careers Among the Unemployed**

In this section, we apply *Υ w* to labor market careers in order to test whether unemployment has long-term negative effects on people's career outcomes, also termed in the literature as unemployment "scarring" (Arulampalam 2001; Mooi-Reci and Ganzeboom 2015). Two prominent theoretical mechanisms have been suggested to drive unemployment scarring. First, from a human capital perspective, loss and depreciation of firm and occupation specific skills over a spell of unemployment is thought to make workers less productive. This translates into job offers of a poor quality (e.g., non-standard or temporary type of jobs) that are more likely to be discontinued and result in recurrent unemployment spells in the future. This implies that the longer the elapsed time in unemployment the lower the chances of reemployment will be.

Signalling has also been suggested as an alternative mechanism driving scarring (Spence 1973). When employers have little information about one's qualifications or abilities they use a range of observable characteristics to infer an applicant's productive capabilities. Characteristics that are subject to individuals' own actions, such as previous unemployment spells, are likely to raise red flags about a worker's unobserved quality and the tasks that they are expected to perform. The views that employers hold of applicants with previous unemployment spells are most likely to translate into lower reemployment rates and wages for some age groups (Mooi-Reci and Muñoz-Comet 2016; Mooi-Reci and Wooden 2017; Pedulla 2016). However, not all "red flags" related to previous unemployment are of equal weight. When the reason for unemployment is believed to relate directly to a worker's poor performance with the previous employer, chances of reemployment will be much lower for reasons related to unobserved worker quality. Additionally, various studies provide considerable support that unemployment spells experienced far in the past – and thus no longer relevant to the current application – or during an economic downturn tend to carry less of a negative weight compared to spells experienced directly prior to the job application (Mooi-Reci and Ganzeboom 2015; Omori 1997). This means that more recent spells of unemployment will influence more negatively one's employment prospects.

These two potential scarring mechanisms raise the following two-fold question: if unemployment is negatively related to one's future career outcomes, do previously unemployed individuals eventually recover from it, and if so, at what rate? This question will guide our analyses in the next sections.

#### *5.1 Data*

We use the Household, Income, and Labour Dynamics in Australia (HILDA) Survey, one of the richest and longest running household longitudinal survey in Australia (Summerfield et al. 2015). The HILDA survey commenced in 2001 and data is collected every year among household members 15 years and over. In 2001, the initial sample counted 13,969 persons across 7,682 households, with the main sample remaining about the same in the subsequent data collection waves. The wave-by-wave response rates are exceptionally high in the HILDA survey, with retention rates varying between 86.9% in wave 2 to 96.4% in 2013. The retention rate of the first sample was 67.4% in 2013 (i.e. this is the number of people responding in both the initial wave and the 2013 wave), which is noticeably high. Every year respondents are asked to identify their labor market status at the time of interview, that we recoded into: employment (E), unemployment (U), nonparticipation/inactivity (N) and retirement (R) (see Summerfield et al. 2015 for a complete list of the questionnaires). Since the focus of this study is on career sequences, we have restricted the sample to those men and women who are not in full-time education and for whom information on labor force status is available at any interview date over the period 2001–2013. This selection leaves us with a sample of 22,081 person-year observations. Of these 22,081 observations, by far the largest concentration (i.e. 44.42%) is among the group for which we have complete labor force status information for the entire observation period between 2001 and 2013. All the analyses are performed using Stata 14.1

*Measures*: Our dependent variable, *binary sequence quality*, is a time-varying measure defined as discussed in Sects. 3 and 4 and captures the quality of career sequences starting from a first unemployment experience up until *t*. To investigate the evolution of employment careers after the occurrence of an initial spell of unemployment, we define a variable counting the years elapsed since the first unemployment experience (i.e. our reference category). As we observe respondents up to 12 years following a first unemployment experience, we construct 12 dummies capturing each of the years in which a respondent was observed in any of the labor market statuses. These time-specific dummies allow us to trace the rate of recovery since the initial episode of unemployment.

We also control for socio-demographic characteristics, including variables for age and age squared, and account for human capital with a variable capturing the highest attained educational level at the time of interview. Education is specified as a categorical variable with Year 11 and below (i.e., early childhood education and primary school) as the reference category and six additional categories for Year 12 (i.e., lower secondary school), Cert III or IV (upper secondary school), Advanced diploma (i.e., post-secondary non-tertiary education), Bachelor or honors, Graduate diploma, Postgraduate education. We also include a variable for Gross Domestic Product per capita as proxy of economic growth in a specific year (GDP). Finally, to guard against the possibility that observed career outcomes are driven by career fluctuations that existed prior to the first unemployment (e.g., due to periods of inactivity), we include a variable specifying the career quality before the first unemployment experience in our models. Table 2 shows descriptive statistics for our person-year sample.

#### *5.2 Method*

A key statistical challenge for our analysis is non-random selection into the initial employment state that is correlated with unobservable traits. To solve this issue, we estimate a model with correlated random effects, also known as the 'hybrid' model (Allison 2009). The hybrid model allows time-varying covariates to be decomposed into individual specific means and deviations from these individual-specific means.

1The sqsuccess package in Stata implements our quality measure as described in this paper. We thank Ulrich Kohler for developing it.

**Table 2** Summary descriptive statistics

The advantage of the hybrid model is that it corrects for unobserved heterogeneity across the time-varying variables and allows for inclusion of time-constant variables which otherwise would have been dropped from a fixed effect specification (Allison 2009). The model takes the form:

$$\Upsilon\_{li}^{w} = \alpha + \beta \Upsilon\_{\text{BEFORE}\,I}^{w} + D(d\_{li} - \bar{d}\_{l}) + \chi \bar{d}\_{l} + \delta(\mathbf{x}\_{li} - \bar{\mathbf{x}}\_{l}) + \eta \bar{\mathbf{x}}\_{l} + \mu\_{l} + \varepsilon\_{li} \tag{4}$$

where *Υ w it* is the measure for sequence quality of worker *i* in year *t* (which covers the period 2001 to 2013); *Υ w* BEFORE *i* controls for the career quality of individual *i* before the time of the first unemployment occurrence and *β* captures its effect. *D* is a vector of coefficients associated with the deviations of the specific time dummies from the overall time cluster mean denoted in the bracket *(dit* − *d*¯ *i)*, which gives us the within-effect estimates. In addition to these time deviations, the model adds a vector of coefficients (*γ* ) associated with the time cluster mean (*d*¯ *i*) to control for dependency of the repeated observations. Next, within-effect estimates for individual characteristics that are supposed to influence the overall sequence quality are captured through *δ*, while *(xit* − ¯*xi)* refers to the difference between timevarying variables expected to be associated with one's sequence quality (including age, education, and GDP) and their individual specific cluster means. Next, a vector of coefficients (*η*) associated with the individual specific cluster means *x*¯*i* is added to the model. As described by Allison (2009) and later by Schunck (2013) this addition is necessary to estimate the evolution of the variations around the mean and to control for correlations between level 1 (i.e., respondents) and level 2 errors (i.e., occasions). Note that interpretation of cluster means is not of interest because we aim at predicting career quality based on the variations of individual characteristics around the mean, which in essence, mimics the logic behind fixed effect models. Finally, *μi* refers to the individual specific error (i.e., level 2 error) with *εit* referring to the level 1 error. To capture sex-specific unemployment effects, we estimate separate models for men and women.

#### *5.3 Findings*

Table 3 shows, separately for men and women, the estimated coefficients from three hybrid models that are based on three weight specifications of our career quality measure, that is: *w* = 1, *w* = 0*.*5 and *w* = 2. In each model, the year coefficients show how the rate of career quality of previously unemployed workers evolves in each year following unemployment. To ease the interpretation, Fig. 2 plots the key results from such models. As we clarified above, the weight parameter determines the extent to which the quality measure is affected by a failure, with stronger penalties for failures, but faster recovery for successes the bigger the parameter *w*.

Although coefficient estimates from both individual-specific means and their deviations are shown in Table 3, only coefficients from deviation specific means are used for interpretation, because coefficients pertaining to individual specific means have no substantive interpretation (see above and for a review, see Allison 2009).

The large positive coefficient estimates for the year dummies in all three models suggest that, for both men and women, there is a trend of recovery. Specifically, using a *w* of 1 among the male sub-sample in Model 1, coefficient estimates indicate an improvement in the career quality with 0.36 points in the first year following the initial unemployment spell. This improvement in the career quality continues in the second and third year with 0.49 and 0.55 points, respectively, before flattening nine years after the first unemployment spell. We find a similar progressively improving trend among women. Interestingly, in the longer run women's patterns of recovery exceed men's. However, from the confidence intervals shown in Fig. 2 (and from tests of interactions, not shown but available upon request) we can establish that gender differences are not statistically significant. Other model specifications with different weights follow expected trends of recovery: using a weight of 0.5 (Model 2) coefficient estimates are lower, indicating less penalizing failures but also slower recovery, while weights of 2 show faster recovery in the years following first unemployment. As mentioned earlier, the choice of the *w* estimator is determined by theory and varies depending on the purpose of research. All other parameters included in the model act in the expected direction. Age, which can be interpreted as a proxy for experience, improves the career quality, which is also positively

**Table 3** Coefficients from hybrid models predicting career quality (*Yit*) over time since first unemployment, by sex

(continued)

**Table 3** (continued)

∗∗ = *p <* 0*.*01; ∗ = *p <* 0*.*05

associated with higher levels of education and economic growths; that is, the higher the education level and economic prosperity the higher the probability for previously unemployed men and women to find re-employment. Finally, a high level of career quality before the first observed unemployment is related to higher career quality later in careers.

#### **6 Conclusion and Discussion**

In this study, we proposed a novel measure of sequence quality, which differs from existing methods of sequence analysis in that it is the first to quantify sequence quality by accounting for the variation in the frequency, duration and recency of failures and successes over the course of a trajectory. The possibility to differently weigh such measure also allows researchers the flexibility to adjust it based on their theoretical considerations. While quality may encapsulate more than positive or negative states, our measure captures a major dimension of quality.

**Fig. 2** Post unemployment career quality since first unemployment. Coefficient estimates from hybrid models with different weights, by sex

Drawing on theories about unemployment scarring, we defined states of unemployment and inactivity as failures and those of employment as successes to predict whether prior unemployment lead into descending spirals into inactivity and joblessness or whether patterns of full career recovery exist. We used data from the HILDA survey in Australia over the period 2001–2013 to illustrate the usefulness of our measure and investigate whether unemployment has adverse effects on future careers. Our results reveal no full career recovery among previously unemployed men and women, even over an extended period of 12 years. These results can be explained in the light of human capital depreciation and job mismatching, but signaling-related factors may be equally important in driving labor market disparities.

Some limitations with regard to the application of our measure should be mentioned. First, we applied our measure to the labor force status reported at the time of interview. Though labor force information at the time of interview is less likely influenced by inaccuracy and recall errors than retrospective types of data, it misses important information about labor force changes that occurred in-between the interview dates. Therefore, the measure of failure may be underestimated if workers experienced unemployment or inactivity spells in-between interviews. Further, the data used here are illustrative of binary sequences in which we distinguish between four possible crude labor force states. With more detailed data on people's careers it is possible to study more fine-grained labor force outcomes such as job-to-job changes in the same hierarchical position or moves to better or worse positions. Finally, an interesting issue for future research is to move beyond binary sequences to include more categories and variation in the sequences. Not all outcomes are binary. People can have neutral outcomes or outcomes that are more continuous. Therefore, quantifying the quality of categorical or continuous types of outcomes remains an important avenue for future research.

**Acknowledgements** Professor Cees Elzinga has highly contributed to writing the methods part of the manuscript and to developing the algorithms that we use in this. Ulrich Kohler developed the ado file to implement our success measure in Stata. We would also like to thank Professor Tim Liao for reading and commenting on an earlier version of this manuscript, as well as the participants to the LaCOSA II conference and two anonymous reviewers for valuable comments.

This work was supported (partially) by the Australian Government through the Australian Research Council's Discovery Projects funding scheme (project DP160101063). It uses unit record data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey. The HILDA Survey Project was initiated and is funded by the Australian Government Department of Social Services (DSS) and is managed by the Melbourne Institute of Applied Economic and Social Research (Melbourne Institute). The findings and views reported in this paper, however, are those of the authors and should not be attributed to either DSS or the Melbourne Institute.

#### **References**

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## **An Index of Precarity for Measuring Early Employment Insecurity**

#### **Gilbert Ritschard, Margherita Bussi, and Jacqueline O'Reilly**

#### **1 Introduction**

Young people have found it increasingly difficult to access work or have cycled between precarious employment, inactivity or unemployment since the stark increases in youth unemployment following on from the financial crisis of 2008 (O'Reilly et al. 2015). Therefore policy approaches have sought to address the consequences of social exclusion and "scarring effects" caused by precarious trajectories (Bell and Blanchflower 2011; Gregg and Tominey 2005; Tumino 2015). In the light of this concern, we aim at developing an index of precarity that allows us to assess the quality of early employment trajectories. At the same time, this index allows us to investigate to what extent the quality of early employment trajectories is linked with future youth employment outcomes.

A vast body of literature acknowledges the increased precarity experienced by young people in the labour market; however, there is a lack of quantitative tools able to grasp the complexity of trajectories and, at the same time, assess their effects in the long run. Filling this gap implies developing a tool that evaluates quantitatively the desirability of employment trajectories and adapts to the longitudinal feature of life-course analysis. We suggest using a modified version of the original "index of complexity" elaborated by Gabadinho et al. (2010), by applying weights to differentiate between the value of labour market transitions.

G. Ritschard (-)

J. O'Reilly University of Sussex, Brighton, UK

NCCR LIVES and Geneva School of Social Sciences, University of Geneva, Geneva, Switzerland e-mail<EMAIL_ADDRESS>
M. Bussi University of Louvain, Louvain-la-Neuve, Belgium

The added value of the index of precarity is to assess the degree of insecurity of employment trajectories experienced by individuals. Furthermore, it can be used to predict, together with other covariates, to what extent the precarity embedded in complex employment trajectories has an impact on the type of labour market positions in the future. Another contribution of our index is its transferability: it can be used across ages, social groups and countries and for any transition across relevant states whose succession leads to a cumulative (dis-)advantage.

This contribution shortly presents the social challenges faced by young people in their first labour market experiences. Then it presents how the index is constructed and justifies this choice. We finally calculate the index using a dataset that comes from a study by McVicar and Anyadike-Danes (2002) on transition from school to work using the Status Zero Survey 2000. The dataset includes a time series sequence of 72 monthly labour market activities for each of 712 young people in a cohort survey. These young people, living in Northern Ireland were followed up from July 1993 to June 1999. We find that young people with precarious trajectories in the first three years after leaving school have dramatically higher risk of experiencing negative labour market positions two years later.

#### **2 Rising Precarity Among Young People**

Changes in the skills structure and the education provision, new forms of employment and the recent economic crisis have contributed to making youth first transitions in the labour market more unstable, complex and individualised (Mills and Blossfeld 2005; Kalleberg 2009; Smithson and Lewis 2000; Gardiner 2016). These changes, crystallised in the multifaceted phenomenon called globalisation, affect young people's first transitions directly and indirectly. Directly these have increased competition among workers and imposing a fast-changing technological environment. Indirectly the effects have affected institutions that shape young people's first employment transitions such as education and welfare systems, employment relations and family structures.

These changes have turned out to be more detrimental for those groups in the labour market with low skills and educational attainment and more likely to be discriminated, such as young people, women, ethnic minorities and migrants. Moreover young people's low negotiating power is worsened by the lack of seniority or work experience (Mills and Blossfeld 2005). According to Standing (1999) these groups have been increasingly experiencing a process of churning between various forms of employment precarity and inactivity.

Protracted experiences of precarious employment—i.e. "employment that is uncertain, unpredictable, and risky from the point of view of the worker" (Kalleberg 2009) and often of poor quality—and joblessness have become more common (Gebel 2010; Ortiz 2010; Worth 2005; Scherer 2001). These contribute to add further precarity when individuals are expected to make critical decisions that shape their life-course (Mills and Blossfeld 2005).

Volatility of employment trajectories can result in stigmatisation, negative signalling to employers. In the long run, this is likely to entail a "scarring" effect. This means that past negative and precarious experiences in the labour market can have long-term negative consequences in terms of repeated periods of unemployment, lower wages and lower levels of human capital attainment over the life cycle (Ayllón 2013; Manzoni and Mooi-Reci 2011; Cable et al. 2008; Schmelzer 2011; Weich and Lewis 1998), but also in terms of well-being (Daly and Delaney 2013).

Rising levels of precarious forms of employment as typified in the UK by zero hours contracts where working hours are not guaranteed (Bussi and O'Reilly 2016) or mini-jobs in Germany (Palier and Thelen 2010) are a reflection of this increasingly fragmented and fragile labour market. In these circumstances Schmid and Schömann (2004) have argued that it is more useful to think of (early-) employment insecurity rather than of (early-) job insecurity. Employment insecurity is better at capturing how precarity affects labour market integration and how individual negotiated choices are embedded in multiple changes and transitions over time. Employment security can involve changing employer and job but maintaining an employment relationship and stable income (Chung and Van Oorschot 2011). Schmid (2015) argues that the welfare state needs to actively contribute to secure employment over the life course rather than a focus on job security. Employment security requires an investment in skills to reinforce employability and access to work (Muffels and Luijkx 2008).

On the basis of this body of literature, we propose to develop an index of precarity to capture a range of labour market precarity comprehensively and test the hypothesis that young persons will be unemployed, inactive or in temporary employment in latter periods if they have a trajectory dominated by non-employment or precarious employment.

#### **3 Conceptualising Precarity**

Measuring precarity and its impact is not straightforward. Barbier (2005) explains that the experience of employment precarity does not translate in the same way across institutional and cultural contexts. Most of the existing literature has focused on different aspects of precarious experiences in the labour market: among others the quality of jobs (Leschke and Watt 2008); the capacity of precarious work contracts acting as bridges towards more stable jobs (Booth et al. 2002; Scherer 2004; de Graaf-Zijl et al. 2011; Cockx and Picchio 2012); and the impact of precarious work on health and well-being. Furthermore, studies on the impact of spells of inactivity and unemployment have mostly looked at single spells, or cumulative spells of unemployment/inactivity. They have rarely assessed the disadvantage derived by a succession of negative spells and downward changes in the labour market, i.e. of a protracted precarity.

Our index aims to provide a quantitative tool accounting for the cumulative process of precarity and its impact. Different summaries of a sequence more or less related to employment precarity can be found in the literature. The diversity of the states visited, i.e. here of the labour market positions experienced, is often considered as an indicator of the uncertainty—in the sense of unpredictability—of the trajectory. It can be measured, for example, by the entropy (lack of predictability) of the distribution of the states within the sequence or by the inverse of the variance of the time spent in the successive distinct states. However, such measures do not account for the sequencing, i.e. order of the states. For example, the sequences *FFUU* and *FUFU* have same entropy but the order of states is different. At least two composite measures that combine a diversity indicator with something sensitive to this sequencing have been proposed in the literature. The turbulence index of Elzinga and Liefbroer (2007) combines the inverse of the variance of the durations in the distinct successive states with the number of subsequences that can be extracted from the sequence of distinct successive states. The complexity index of Gabadinho et al. (2010) combines entropy with the number of transitions in the sequence.

These indexes are intended to measure the unpredictability or instability of sequences, but this is done without taking the nature of the states into account. For example, letting *E*, *W*, and *U* stand for education, work, and unemployment, the sequences *EEWWW* and *EEUUU* would get the same entropy, turbulence and complexity values, while the second is evidently a more precarious trajectory.

Although sequence instability as measured by the turbulence or complexity index certainly contributes to understanding the precarity of a sequence, a precarity indicator has to account for the nature of the states that constitute the trajectory. A first simple solution is to distinguish between positive (e.g. employed or in education) and negative (e.g. unemployed) states like in the volatility indicator of Brzinsky-Fay (2007). This indicator is defined as the ratio between the number of positive and negative positions in the sequence. Based on this same distinction between positive (success) and negative (failure) states, Manzoni and Mooi-Reci (2018) propose a refined solution where precarity increases with the recency of failures. However, these two solutions do not explicitly account for the instability of the sequence.

A general approach to get a precarity index accounting for both the instability of the sequence and the nature of the states is to apply a correction factor based on the nature of the states to any measure of sequence instability. For example, we would get such an indicator by multiplying either the turbulence or the complexity index by Brinzky-Fay's volatility indicator.

Instead of just distinguishing between positive and negative states, we could assign degrees of precarity to the different states. A temporary job would, for instance, get a higher degree of precarity than a full time job and a lower degree compared to unemployment. Moreover, the precarity of a trajectory depends on the evolution within the sequence and we should, therefore, not only account for the nature of the states but also for the type of the state transitions—changes of states in the sequence.

Here, we consider that the precarity of a sequence

1. increases with the sequence instability due to the lack of predictability of the different states experienced;

In addition, we consider that the transitions may have different advantageous or disadvantageous degrees that should be taken into account when computing the proportion of negative and positive transitions. For instance, we could consider that a transition from full time employment to unemployment is more damaging than a transition from a full time to a part time job. We would also expect that a transition from full time to unemployment generates a higher precarity weight than a transition from a temporary job to unemployment when the former is less likely to occur than the latter.

#### **4 The Precarity Index**

We use the complexity index (Gabadinho et al. 2010, 2011) as a measure of the sequence instability and propose to qualify—penalize/reward—it by means of a correction factor derived from the proportions of negative and positive transitions in the sequence.

#### *4.1 Defining the Index*

The index of complexity of a sequence is a composite index defined as the geometric mean between the normalized entropy of the sequence and the normalized number of transitions in the sequence. The entropy is normalized by dividing it by the logarithm of the size of the alphabet, i.e. the logarithm of the number of all possible states that the person can experience, which is the maximal possible entropy given the alphabet. The number of transitions is normalized by − 1, i.e. the length of the sequence minus one. Formally, the complexity *c(s)* of a sequence *s* reads

$$c(\mathbf{s}) = \sqrt{\frac{h(\mathbf{s})}{\log(n\_a)} \frac{n t\_s}{(\ell\_s - 1)}}$$

where *h(s)* is the entropy, *na* the size of the alphabet, *nts* the number of transitions, and *s* the length of the sequence.

Now, assuming the states—the labour market positions—can be ordered from the best to the worst state (see Sect. 4.4 for how to relax this strict order requirement), we say that a state transition *A* → *B* in a sequence is negative when there is a deterioration, i.e. when the difference rank*(A)* − rank*(B)* between the ranks of the origin and destination states is negative. Likewise, the transition is said positive in case of improvement, i.e. when rank*(A)* − rank*(B) >* 0. Letting *q*−*(s)* be the (weighted) proportion of negative transitions in the sequence *s* and *q*+*(s)* the (weighted) proportion of positive transitions, we define the correction factor in terms of the difference between the two:

$$q(\mathbf{s}) = q^-(\mathbf{s}) - q^+(\mathbf{s}).$$

Since *q*−*(s)* and *q*+*(s)* are proportions, we have −1 ≤ *q(s)* ≤ 1. The correction factors will be 1 + *q(s)* and the proposed qualified complexity index reads

prec*(s)* = *λ a(s*1*)* + *(*1 − *λ) c(s)α(*1 + *q(s))β* (1)

where *c(s)* is the complexity index of the sequence and *a(s*1*)* ∈ [0*,* 1] the starting cost, i.e. the degree of precarity associated to the starting state *s*1 in the sequence. The correction factor *(*1 + *q(s))* is non negative. It is greater than 1 when *q(s)* is positive, i.e. when there are more negative than positive transitions. Thus the greater *q(s)* the stronger the penalization of the original complexity index. The parameter *λ* serves to control the trade-off between the starting cost and the corrected complexity while the exponents *α* and *β* control the respective importance of the complexity and the correction. The choice of the values of these parameters is addressed in Sect. 4.2.

Different variants of *q(s)* may result depending on whether we take into account the transition costs, and, if so, on how these costs are determined.

Let *w(st, st*+1*)* be the cost of a transition from state *st* to state *st*+1 over two successive time positions *t* and *t* + 1. To get *q(s)*, we first compute the total cost *nw(s)* of the successive transitions in the sequence *s*, the total cost *nw*−*(s)* of the negative transitions, and the total cost *nw*+*(s)* of the positive transitions:

$$\begin{aligned} nw(s) &= \sum\_{l=1}^{\ell-1} w(s\_l, s\_{l+1}) \\ nw^-(s) &= \sum\_{l=1}^{\ell-1} I^-(s\_l, s\_{l+1}) \, w(s\_l, s\_{l+1}) \\ nw^+(s) &= \sum\_{l=1}^{\ell-1} I^+(s\_l, s\_{l+1}) \, w(s\_l, s\_{l+1}) \end{aligned}$$

where *I* −*(st, st*+1*)* is a deterioration indicator taking value 1 for state deterioration and 0 otherwise, and *I* +*(st, st*+1*)* a similar function for state improvement. The proportions of cost-weighted negative and positive transitions are then:

$$q^{-}(\mathbf{s}) = \frac{n w^{-}(\mathbf{s})}{n w(\mathbf{s})} \qquad q^{+}(\mathbf{s}) = \frac{n w^{+}(\mathbf{s})}{n w(\mathbf{s})}.$$

#### *4.2 Tuning the Index*

The formula (1) has three tuning parameters *λ*, *α*, and *β* that allow the formula to encompass a whole family of indexes. We should chose these parameters such that 0 ≤ *λ* ≤ 1, *α* ≥ 0 and *β* ≥ 0.

The parameter *λ* determines the trade-off between the precarity of the starting state and the corrected complexity. With *λ* = 0, the precarity level of the starting state in the sequence would simply be ignored while with *λ* = 1 the precarity of the whole sequence would just be that of its starting state. It seems reasonable to give less importance to the starting state than to the corrected complexity and we suggest, therefore, a value *λ* = 0*.*2. With this value the precarity degree of the starting state receives a 20% weight while the corrected complexity counts for 80% of the index.

Parameters *α* and *β* are exponential weights that allow for some control on the respective importance of the complexity and the correction factor in the corrected complexity term. For instance, we would get an index that does not account at all for the complexity by setting *α* = 0. Setting in addition *λ* = 0, the index would reduce to the mere correction factor. Likewise, *β* = 0 suppresses the correction. We get unweighted effects with *α* = *β* = 1. With *α >* 1 we increase the importance of the complexity. Likewise, a value *β >* 1 strengthens the correction, which may prove useful in case we feel the correction is insufficient. We got good results in our experiments with *α* = 1 and *β* = 1*.*2.

Alongside the values of the tuning parameters, the analyst has also to make a choice regarding the precarity degree *a(s*1*)* of the starting state—the offset in formula (1)—and the weights *w(st, st*+1*)* of the transitions that impact the correction factor 1 − *q(s)*. Different strategies can be envisaged for these choices including defining them on theoretical grounds, on the hypothesized rank order of the states, or deriving them from the data.

Regarding the precarity degree of the states to be used as *a(s*1*)* value, a solution using the hypothesized state order is to assign the *na* equally spaced values between 0 and 1 as precarity degree to the sorted states. The *i*th state would in that case get a precarity degree of *(i* − 1*)/(na* − 1*)*. As a data-driven approach, we could, for example, set the precarity degree of each state as the probability to visit at least *k* bad states during the next *m* periods.

Likewise, referring to the rank order of the states, we could set the costs *w(st, st*+1*)* of the transitions as the difference between the ranks of the origin and destination states. Data driven approaches for the transition costs could be for instance

1. Give higher cost weights to rare transitions by defining the weight of each transition as a decreasing function (e.g. 1 − *p*) of its estimated transition probability *p(st*+1|*st)*. Here, we could either compute this probability on the original sequences or ignore the durations of the successive states in the sequences and compute the probabilities of transition on the sequences of distinct successive states (DSS).

2. Define the cost of each negative transition as the estimated probability to be in a bad state say *k* periods after the transition, and for each positive transition as the estimated probability to be in a good state *k* periods after the transition.

For the illustrative example in Sect. 4.3 and the application in Sect. 5, we retain the values *λ* = 0*.*2, *α* = 1, and *β* = 1*.*2, set the precarity degree *a(s*1*)* as the *na* equally spaced values between 0 and 1, and we use the complement to 1 of the transition probabilities in the DSS sequences as transition costs.

#### *4.3 Behavior of the Precarity Index*

To illustrate the behaviour of the index, we consider the set of fictitious sequences shown in Table 1 where the states are *F*, Full time permanent employment, *P*, Part time, *T* , Temporary employment, *U*, unemployment, and *I* , inactivity. We assume that there is a decreasing hierarchical order of the first four states, namely *F,P,T,U*. We also assume that *I* is neither better nor worse than the other states and can therefore not be classified.

So far we have assumed the states are strictly ordered. Therefore, we consider for now only the first 15 sequences where the non-comparable state *I* does not appear. In Table 1, the first two sequences are the most complex with three transitions each. The first one is slightly more complex with four different states as opposed to three states in the second sequence. Nevertheless, the second looks more precarious with two downward transitions compared to only one in the first sequence. Sequences 14 and 15 are composed of only one state. They have a zero complexity degree. However, sequence 14 in which the stable state is unemployment

**Fig. 1** Precarity and its components under the strict state order assumption

(*U*) looks more precarious than sequence 15 composed of one favourable state (*F*, full-time employment). We expect the index of precarity to reflect these differences in quality and complexity.

The values of the precarity index (computed with *λ* = 0*.*2 and *β* = 1*.*2), as well as of the complexity index are shown in the right panel of Fig. 1. In the middle panel the figure shows for each sequence *s* the starting cost *a(s*1*)*, i.e. the precarity degree of the first state in the sequence, and the correction factor *(*1 + *q(s))β*.

The index behaves as expected regarding the first two sequences: The second sequence gets a slightly higher precarity value than the first despite its lower complexity. Sequence 14 with zero complexity—i.e. no transitions—gets a nonzero precarity value because the cost of its starting state—*U*, being unemployed—is high in terms of precarity. This is not the case for sequence 15 the other sequence with zero complexity, which starts off with employment, i.e. zero cost in terms of precarity. We observe that the sequences with highest precarity are not the more complex ones but those with only negative transitions. The index of precarity is highest for sequences 3 and 4. It is slightly lower for sequences 8, 9, 10 and 13 that also have only negative transitions, but lower complexity due to only one transition. In addition, as expected sequences with only positive transitions (7, 11, 12) get low values of precarity.

#### *4.4 Relaxing the Strict State Ordering Requirement*

A strict order of the states is hardly compatible with the complexity of reality, especially when some states could not be clearly ranked with respect to the others (e.g. should education be considered as worse or better than employment?). We consider two situations that depart from the strict state order assumption: equivalent classes of states and non-comparable states.

A *class of equivalent states* is a subset of states that are considered equivalent, i.e. states with no ordering between them. In our example, *F* and *P* could be considered as equivalent if we assume that the choice between a full-time and a part-time job is a pure employee's choice.

A *non-comparable state* is a state that cannot be ordered, i.e. a state that is neither better nor worse than any other state. This is the case of the inactivity state *I* in our example.

We need solutions to handle these cases at two levels: for the computation of the proportions of negative and positive transitions, and for determining the state costs in the case where we want to derive them from the state order as we do in our example.

For equivalent classes, we do not penalize or reward any transition between states of a same equivalent class. In other words, all transitions between elements of an equivalent class get a zero weight in the weighted proportion of negative and positive transitions. As for transitions from or to any non-comparable state, we also chose to neither penalize nor reward them by giving them zero weight. In addition, however, subsequences such as *PIIIU*, where non-comparable states—the *I* s in the example—occur in-between two regularly ranked states will be counted as a transition from its first element to the last, e.g. *PIIIU* is counted as a transition *P* → *U*, i.e. as a negative transition.

Regarding starting costs based on the state order, we assign to each state (labour market position) in an equivalence class the mean cost of the states in the class, and the overall mean starting cost (i.e. 0.5) to each non-comparable state. A consequence is that in case the highest ranked state belongs to an equivalence class, its cost would be the non-zero mean value of the class and there would be no zero starting cost.

To illustrate we have computed the precarity index for the full set of sequences shown in Table 1, i.e. including those with the non-comparable state *I* and assuming in addition that full-time, *F*, and part-time working, *P*, form an equivalence class. We used again a trade-off value *λ* = *.*2 and an exponent weight *β* = 1*.*2. Figure 2 shows the obtained precarity values and their components the complexity index *c(s)*, the weighted correction factor *(*1 + *q)β*, and the starting cost *a(s)*.

Looking at the first 15 sequences, we see a few differences with what we found in Fig. 1 using the strict order assumption. We first observe that there is now no sequence with a zero precarity value. Sequence 15 that had a zero value under the strict order assumption gets now a small positive value. This is because of the equivalence class between the two best ranked states in the state order. The starting state *F* gets here the non-zero mean value between *F* and *P* as precarity degree.

**Fig. 2** Precarity and its components in presence of the non-comparable state *I* and the equivalence class {*F,P*}

Considering *F* and *P* as equivalent also has consequences on the ranking of the sequences. Sequence 1 gets here a higher precarity value than sequence 2. Because of the equivalence class, the two sequences have the same number of positive and negative transitions, which is reflected by correction factors close to 1. The main difference between the two sequences is the worse starting state in sequence 1.

Among the sequences with the non-comparable state *I* , sequence 16 appears to be the most precarious. It is made of a single downward transition and has maximal complexity for a sequence with 3 out of 5 states. Sequences 17 and 18 have only an upward transition and get therefore low precarity values. Finally, sequences 19, 20, and 21 have only zero weighted transitions—hence a neutral correction factor of 1—and get mid precarity values.

#### **5 Application to the School to Work Transition**

We now show how the index can be used on a real world dataset using the Status Zero Survey data of McVicar and Anyadike-Danes (2002) on the school to work transition of young Northern Irish.1 This cohort survey was used to establish a link between individual, family and school characteristics and types of trajectories. The aim was to identify those young people who are more likely to experience unsuccessful trajectories in the adult labour market. The survey provides monthly information on the labour market activities of 712 young people for 72 months (6 years) after they left compulsory schooling. Despite the fact that these data refer to the period between July 1993 and June 1999, they represent a good testing ground for our index of precarity as they focus on early employment trajectories of young people who just left education. We complete the study from McVicar and Anyadike-Danes (2002) by assessing the quality of trajectories of young people and testing whether this contributes, beyond static individual school and family characteristics, to predicting future labour market positions. This is particularly relevant in the current economic and labour market situation, where young people have been hit hard by the crisis and are often overrepresented in temporary and precarious employment.

Due to its collection structure, all individuals are aged 16 at the start of the trajectories. Here we shall ignore the first two holiday months and retain the sequences from September 1993 to June 1999, i.e. sequences of length 70. The data distinguishes between six labour market activities: school (SC), training (TR), further education (FE), higher education (HE), employment (EM), and joblessness (JL).

We use the dataset to study how the degree of precarity of the trajectory during the first 36 months—from September 1993 to August 1996—impacts the situation of the young person two years later, i.e. the 6th year after the end of compulsory school. Hence, we aim to measure the scarring effect of early precarious trajectories in mid-term labour market outcomes. More specifically, we examine the chances to be at least one month in one of the states JL, TR, or SC during this 6th year, i.e. between September 1998 and June 1999.

In order to study the precarity of the trajectories during the first 36 months, we consider the three equivalence classes:

$$C\_1 = \{\text{FE, HE, EM}\}, \ C\_2 = \{\text{SC, TR}\}, \ C\_3 = \{\text{JL}\}.$$

In addition, we assume the decreasing order *C*1 *> C*2 *> C*3 of the equivalent classes. Thus, for example, changing from employment (EM) to training (TR) will be considered a downward transition, a change from school (SC) to further education (FE) as an upward transition, and a change from further education (FE) to employment (EM) as neutral.

1The data ship with the R package TraMineR (Gabadinho et al. 2011).

**Table 3** Ten sequences with highest precarity value

We compute the index using *λ* = *.*2, *β* = 1*.*2, and transition weights based on the transition probabilities in the DSS. The weights are shown in Table 2. Weights of upward transitions are displayed with a negative sign to recall their reducing effect on the correction factor. Table 3 lists the ten sequences with the highest values for the precarity index. All these most precarious sequences have at least as many upward than downward transitions except sequence 377 that starts with the worst state JL. Sequence number 377 appears to be precarious because it starts with the worst state JL, i.e. a high starting cost. We can also observe that among the ten most precarious trajectories the upward transitions are typically transitions with lower weights than the downward transitions. E.g., the transition JL→EM that occurs frequently in the precarious sequences has a 0.44 weight, while SC→JL present in half of the ten sequences has almost twice that weight.

In order to predict the impact of the quality of the trajectory during the initial 36 months on the future situation we build two variables from the 10 last months of the observed sequences, i.e. months 61 to 70—September 1998 to June 1999—of our sequences. The first, *bad.dur*, is the total time spent in one of the states JL, TR, or SC during the last 10 months (6th year), and the second is a binary variable, *is.bad*, taking value 1 when *bad.dur>* 0, i.e. when the individual has spells of negative states during the 6th year. There are 19*.*4% of the followed individuals who spent at least a month in a negative state, and Fig. 3 shows the distribution of the total number of months for those 121 cases. Clearly most of those who had bad spells spent all ten months in undesirable states. Figure 4 shows that the distribution of

**Table 4** Linear regression for time in bad states during last 10 months and logistic regression for 'More than 0 months in bad states during the last 10 months'

the precarity degree during the first 36 months greatly differs between those in bad situation two years later and the others.

The impact of the precarity degree can be measured through a logistic regression using *is.bad* as the dependent variable and the precarity as predictor. Table 4 shows the effect of the precarity degree when controlled for two other covariates, namely whether students gained good qualification at the end of compulsory school (*gcse5eq*) and whether they are males (*male*). The results evidently demonstrate the strength of the precarity degree of the early trajectory as predictor of the future situation. An increase of the precarity index by 0.1 unit for example multiplies the odd of experiencing a negative situation a few years later by about 60. From the linear regression, an increase of the precarity degree during the first 3 years of 0.1 unit leads on average to an increase of the time in negative states during the 6th year by almost one month.

#### **6 Conclusion**

In this study, we set out to develop an index to quantify the degree of precarity of individual sequences and to predict future insecurity in professional careers. Starting from the assumption that the complexity of the sequence contributes to the precarity of the trajectory, we have defined the index as a corrected complexity index. There is a multiplicative correction based on the difference between the proportions of downward and upward transitions in the sequence and an offset correction to account for the degree of precarity at the start of the sequence. Despite its relative simplicity, the index proved to be able to effectively grasp precarity. Using the data from McVicar and Anyadike-Danes (2002) on the school to work transition of school leavers in Northern Ireland, we also demonstrated the usefulness of the index for studying how precarity during the first years after compulsory schooling impacts future outcomes. There is certainly room for further improvements, for instance, by accounting for the time elapsed between transitions and/or the timing of the transitions. The concept of recency used by Manzoni and Mooi-Reci (2018) for their own quality index is also an interesting dimension to consider.

The precarity index is very flexible and can be tuned by choosing the transition weights, the degree of precarity of the starting costs, the trade-off parameter *λ*, and the exponent weights *α* and *β* that determine the respective importance of the complexity and the correction factor. These weights and parameters offer the analyst the possibility to adapt the index to specific contexts. However, making choices is not indispensable. The index provides most often sensible results with default parameter values and automatic methods for setting transition weights and starting precarity degrees. The only necessary information that the user has to specify is the rank order of the states with possible equivalence classes and non comparable states.

Although the indicator was specifically developed for measuring the precarity of sequences of labour market activities in order to predict future insecurity in professional careers, the index could as well be used for sequences of other domains of the life course such as family or health trajectories. The only requirement is the existence of some (partial) order between the states of the alphabet adopted, i.e. at least some states should be preferable to some others. Moreover, it could also be of interest to use the index as the dependent variable to study how precarity depends on personal characteristics such as sex, social origin or previous educational trajectory.

The index has been implemented as a beta version in TraMineRextras and should be made available in a next release of TraMineR (Gabadinho et al. 2011).

**Acknowledgements** Margherita Bussi and Jacqueline O'Reilly acknowledge financial support from the Horizons2020-funded NEGOTIATE project, grant agreement No 649395. Gilbert Ritschard acknowledges the support of the Swiss National Centre of Competence in Research LIVES - Overcoming vulnerability: Life course perspectives, which is financed by the Swiss National Science Foundation (grant number: 51NF40-160590). The authors warmly thank the anonymous reviewers for their constructive comments.

#### **References**

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## **Correction to: Unpacking Configurational Dynamics: Sequence Analysis and Qualitative Comparative Analysis as a Mixed-Method Design**

**Camilla Borgna and Emanuela Struffolino**

**Correction to: Chapter 10 in: G. Ritschard, M. Studer (eds.),** *Sequence Analysis and Related Approaches***, Life Course Research and Social Policies 10, https://doi.org/10.1007/978-3-319-95420-2\_10**

In the original version of this book, the second author Emanuela Struffolino was missed to be added as the corresponding author and the affiliation of the author Camilla Borgna was incorrect. The second author Emanuela Struffolino has now also been included as the corresponding author and the affiliation of Camilla Borgna is corrected as Collegio Carlo Alberto, Turin, Italy.

© The Author(s) 2018 E1

The updated online version of this chapter can be found at https://doi.org/10.1007/978-3-319-95420-2\_10

G. Ritschard, M. Studer (eds.), *Sequence Analysis and Related Approaches*, Life Course Research and Social Policies 10, https://doi.org/10.1007/978-3-319-95420-2\_17

## **Index**

#### **B**

Binary sequences, 261

#### **C**

Career recovery, 262 Career trajectory, 52 Childhood co-residence structures, 84 Cluster analysis, 71, 84, 131, 155, 185, 203, 223, 241 Compressing sequence data, 185 Configurations, 168 Continuous data, 203 Cox regression, 70

#### **D**

Departure from the parental home, 84 Disability, 70 Discrete-time event history analysis, 56 Dissimilarity measure between sequences, 149, 241 Divisive clustering, 225 Dual-earner couples, 241

#### **E**

Employment insecurity, 279, 281 Event history analysis, 16, 56, 64, 84

#### **F**

Failure, 261 Fuzzy clustering, 223

#### **G**

Gender, 122 Gender inequality, 50 Glass ceiling, 50 Glass escalator, 50 Growth mixture model, 203

#### **H**

Hidden Markov model, 185, 203 HMTD model, 203

#### **I**

Index of complexity, 279 Index of precarity, 279 Internet addiction test, 203

#### **K**

Kinship, 122

#### **L**

Latent Markov model, 186 Lexicographic index, 241 Life course analysis, 35 Life history calendar, 36 Life trajectories, 69 Longitudinal data, 1, 204

#### **M**

Migration, 122 Mixed-methods, 167 Monothetic, 224

© The Author(s) 2018 G. Ritschard, M. Studer (eds.), *Sequence Analysis and Related Approaches*, Life Course Research and Social Policies 10, DOI10.1007/978-3-319-95420-2 **N** Network analysis, 103, 121

#### **O**

Occupational mobility, 49 Optimal matching, 1

#### **P**

Phase, 149 Polythetic, 223 Prediction probability, 36 Probabilistic model, 36 Professional competitions, 149

#### **Q**

QCA, 168 Quality measure, 261, 279

#### **S**

Scarring, 262, 279 Sequence analysis, 1, 19, 69, 84, 104, 121, 167, 223 Sequence history analysis, 83 Sequence networks, 103, 121 Sequence quality, 261, 274 Sequence structure, 103 Social network analysis, 16, 121 Social sequence analysis, 185 Success, 261

#### **T**

Temporal structures, 149 Time use, 113, 241 Trajectories, 168, 279

**U**

Unemployment scarring, 261, 279

#### **V**

Vertical sex segregation, 50 Visualizing sequence data, 149, 186