Studying employment pathways of graduates by a latent Markov model

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1 Studying employment pathways of graduates by a latent Markov model Fulvia Pennoni Abstract Motivated by an application to a longitudinal dataset deriving from administrative data which concern labour market and academic performances in Lombardy, we propose a multivariate latent Markov model with covariates for panel data. Our aim is to investigate how covariates influence labour market performance of the graduates which is measured through three type of response variables. The model is based on a Markov process to represent the latent characteristics of the subjects. Maximum likelihood estimation of the model parameters is based on the Expectation-Maximisation algorithm and it is performed by using a two-step approach first estimating a latent class model and then the latent Markov model. Key words: Expectation-Maximisation algorithm, human capital, labour market, latent variable model, panel data 1 Introduction In this paper we propose a model for the evaluation of the employment pathways in terms of wage, easiness in switching between types of position and employment skill of the graduates. In the present job market condition which is affected by the financial crisis, it is interesting to study university-to-work transition in terms of, human capital increase. As recently proposed by the OECD report [1] human capital is the knowledge, skill, competencies and attributes embodied in individuals that are relevant to economic activity. It is a complex, multifaced phenomenon which is not directly measurable only in terms of wage [2], [3]. Therefore the interest is on the evolution of a latent characteristic of an individual which is indirectly measured by certain response variables. Fulvia Pennoni Department of Statistics and Quantitive Methods, Via Bicocca degli Arcimboldi 8, 20126, Milano, fulvia.pennoni@unimib.it 1

2 2 Fulvia Pennoni A model that is suitable for the type of analysis above is the latent Markov model as proposed by [4]. The model represents the evolution of the latent characteristic of interest by an unobservable Markov chain which has a reduced number of states. The response variables are assumed to be conditionally independent given this latent process. In this paper the model is considered in its multivariate version including individual covariates in the latent process as proposed by [5]. Likelihood inference of the model is based on the EM algorithm [6]. 2 The dataset The dataset concerns 1624 individuals who graduated in 2007 from four universities of Milan. They have been followed along four quarters after the graduation date, covering one year. The choice of the specific 2007 cohort is motivated by the availability of the data coming from the following integrated databases: i) database of the observatory of the labour market in Lombardy, which has collected mandatory notices from public and private employers regarding changes in job status form 2000; ii) database of the Revenue office, which provides information about wages from 2004 to present of all subjects residing in Lombardy; iii) database of graduates from four universities of Milan, which provides information on the academic careers of graduates from The response variables are: i) employment status and type of employment contract indicating whether a subject is employed with a permanent or temporary contract, ii) job quality measured by the skill level of the job which is derived by a categorization of the job qualification made by the Italian National Institute of Statistics, iii) wages for each quarter. The available covariates concern the socio-demographic characteristics such as: i) age, ii) family income, iii) gender, iv) student employment, and academic characteristics such as v) type of degree, and vi) final grade. In Table 1 we report the descriptive statistics for the distribution of the available covariates, whereas in Table 2 we report the descriptive statistics for the response variables for each quarter of observation. 3 The proposed model With reference to a subject in the sample of n subjects observed at T time occasion, we introduce the symbol Y (t) to denote the vector of response of variables of interest at time occasion t, t = 1,...,T, which has elements Y (t) j, j = 1,...,r where each has c j levels. The symbol X (t) denotes the vector of all individual covariates available at the t-th time occasion. The proposed model assume the existence of a latent process U = (U (1),...,U (T ) ) which affects the distribution of the response variables. The main assumptions of the model are that the response variables in Y (t) are conditionally independent given the latent process U and that the latent process follows a first-order homogeneous Markov chain with state space {1,...,k}. We Y (t) j

3 Studying employment pathways of graduates by a latent Markov model 3 Covariate Category % Mean St.dev. age (in 2007) family income in Euro gender: male female employment before 2007 no yes type of degree technical architecture 8.93 business humanistic science final grade < cum laude Table 1 Descriptive statistics for the distributions of the covariates Type of contract Year st quarter 2nd quarter 3rd quarter 4th quarter none temporary permanent Skill none medium/low high Wage none less than 3750 e high than 3750 e Table 2 Frequency of every response variable for period of observation denote the conditional response probabilities by φ jy u = f (t) Y j U (t)(y u), j = 1,...,r,t = 1,...,T,u = 1,...,k,y = 0,...,c j 1. We admit that the covariates affect the distribution of the response variables given the latent process [5]. Therefore we have the following initial and transition probabilities of the latent process π u x = f U (t) X (t) (u x), u = 1,...,k π u ūx = f U (t) U (t 1),X (t) (u ū,x), t = 2,...,T, ū,u = 1,...,k,

4 4 Fulvia Pennoni where x denotes a realization of X (t), u denotes a realization of U (t), and ū denotes a realization of U (t 1). An interesting way to allow the initial and transition probabilities of the latent Markov chain to depend on the individual covariates is by adopting the following parameterization and log π u x π 1 x = β 0u + x β 1u, u = 2,...,k, (1) log π u ūx πū ūx = γ 0ūu + x (γ 1u γ 1ū ) u = 1,...,k, u ū. (2) This formulation is of interest when we want to understand how covariates affect the latent characteristic which is indirectly measured by the response variables. For classifying the sample of subjects on the basis of categorical response variables we rely on a latent class model [7]. According to this model we also select the number of latent classes by using the Bayesian Information Criterion (BIC, [8]). Then, the estimation is performed by maximizing the log-likelihood [ ] l(θ) = log i fỹ X (ỹ i x i ), where x i and ỹ i are vectors of observed data i = 1,...,n, and θ is the vector of all model parameters. Function l(θ) is efficiently computed by using a recursion which is known in the hidden Markov literature. Likelihood maximization is performed by an EM algorithm ([6], [9]) based on the complete data log-likelihood, that is the loglikelihood that we could compute if we knew the latent state of each subject at every occasion. Differently from a standard EM algorithm under this maximization we do not update the conditional response probabilities which are held fixed. In such a way, the algorithm is faster to converge as the number of iterations needed are much less. Once parameter estimates have been computed, standard errors are associated at this estimates. They are computed on the basis of nonparametric bootstrap [10] which consists of repeatedly drawing samples from the observed sample and computing the maximum likelihood estimates for every bootstrap sample. Then the standard error corresponding to the parameter estimate is found through the bootstrap distribution of the estimators. 4 Main results In applying the basic latent class model to the dataset, we chose the number of latent states k = 3. The maximum log-likelihood of the model is equal to ˆl = with 20 parameters. The corresponding value of BIC is The estimates of the conditional response probabilities according to this model are reported in Table 3.

5 Studying employment pathways of graduates by a latent Markov model 5 ˆφ jy u Type of contract u = 1 u = 2 u = 3 none temporary permanent Skill none medium/low high Wage none less than 3750 e high than 3750 e Table 3 Estimated conditional probabilities of labour condition under the selected model We observe different types of labour conditions given the latent classes. In particular, the first class, which is the largest with about 53% of subjects, corresponds to unemployed subjects which may have income from other sources. For the second class, including about the 27% of subjects, we have those subjects with temporary contract and with some qualified work but with low wage. For the third class, including 20% of subjects, we have subjects with stable high quality jobs and high income. According to the selected number of classes the analysis is focused on studying the dependence between the latent classes and the observable covariates by fitting the multivariate latent Markov model with covariates. The results of this fitting in terms of the parameters affecting the initial and the transition probabilities are reported in Table 4. Effect ˆβ12 ˆβ13 ˆγ 12 ˆγ 13 female student employee age grade grade grade grade grade grade family income/ architecture business humanistic science Table 4 Estimates of the regression parameters affecting the latent process ( minus the sample average, 95% bootstrap interval does not contain 0)

6 6 Fulvia Pennoni In order to properly interpret the results in this table we have to consider that the adopted model is based on the parameterizations (1) and (2). On the basis of the estimates of the regression coefficients for the multinomial logit of the latent classes, at the beginning of the time occasions, to be female has a positive effect on being in the second latent class with respect to a male. It means that females find more easily a low quality job compared to males. Having work experience during university has a strong positive effect on finding a first job and also a high quality employment. Students from high income families opt to continue their education or simply avoid search effort to find a job compared to those from low income families. Students with a technical degree have much more chance of getting a job position with respect to the other degrees. Even for the most qualified jobs people with a degree in architecture and humanistic are disadvantaged compared to those with a technical degree. Considering the subsequent periods of observation females are more likely to accept a low quality employment compared to their male counterparts, as well as student employees compared to students. We notice also that, less young graduates tend to have more difficulty to find a job. Moreover, technical degree helps to reach a high quality employment compared to the other degrees followed by business and science degrees. Acknowledgements We are grateful to Prof. M. Mezzanzanica and to Dr. M. Fontana, of the Interuniversity Research Centre on Public Services, University of Milano-Bicocca, for providing the dataset. We also acknowledge Finite mixture and latent variable models for causal inference and analysis of socio-economic data (FIRB - Futuro in ricerca) funded by the Italian Government (RBFR12SHVV). References 1. OECD (1998). Human Capital Investment. An International Comparison. Paris: Centre for International Research and Innovation. 2. Folloni, G. and Vittadini, G. (2010). Human capital measurement: a survey. Journal of Economics Surveys, 24, Wößmann, L. (2003). Specifying human capital. Journal of Economic Surveys, 17, Wiggins, L.M. (1973). Panel Analysis: Latent probability models for attitude and behavious processes. Elsevier. 5. Bartolucci, F. and Farcomeni, A. and Pennoni, F. (2013). Latent Markov Models for Longitudinal Data. Chapman & Hall/CRC, Boca Raton. 6. Baum, L.E. and Petrie, T. and Soules, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Annals of Mathematical Statistics, 41, Lazarsfeld, P.F. and Henry, N.W. (1968). Latent Structure Analysis. Houghton Mifflin, Boston. 8. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics 6, Dempster, A.P. and Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, Efron, B. and Tibshirani J. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC, New York.