Documento de Trabajo - 2018/03

Estimation of competing risks duration models with unobserved heterogeneity using hsmlogit

David Troncoso Ponce (Universidad Pablo de Olavide)

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Las opiniones recogidas en este documento son las de sus autores y no coinciden necesariamente con las de FEDEA.

Estimation of competing risks duration models with unobserved heterogeneity using hsmlogit

David Troncoso Ponce

Universidad Pablo de Olavide

RESUMEN

Este artículo presenta hsmlogit, un nuevo comando de estimación que estima modelos de duración en tiempo discreto con riesgos en competencia y presencia de heterogeneidad inobservable. hsmlogit permite la estimación de uno, dos y hasta tres riesgos en competencia, así como la especificación (por parte del usuario) de un máximo de cinco puntos-masa para la identificación no paramétrica de la distribución de heterogeneidad inobservable (Heckman y Singer, 1984).

La contribución de este comando es mostrada en el presente trabajo mediante la estimación de un conjunto de modelos de duración, cuyo objetivo es evaluar el impacto que un determinado tipo de contrato laboral, introducido en España a principios de los años noventa, denominado Contrato para la Formación y el Aprendizaje, ha tenido sobre una muestra de trabajadores jóvenes en el mercado laboral español durante el periodo 2000-2014. Dicha muestra ha sido extraída de la Muestra Continua de Vidas Laborales (MCVL), y contiene los historiales laborales completos de un total de 48.246 entrantes al mercado laboral con edades comprendidas entre 16 y 24 años, y un bajo nivel educativo. Una vez expandido cada episodio laboral en observaciones de periodicidad mensual, la muestra final de estimación tiene 1.316.611 observaciones.

Los resultados obtenidos muestran la relevancia de modelizar más de un riesgo en competencia en la estimación del efecto potencial del contrato de formación y aprendizaje sobre las tasas de transición desde el empleo. Concretamente, la estimación de un modelo de duración con tres riesgos en competencia ha permitido descubrir el posible papel que juegan las subvenciones públicas a la conversión de los contratos de formación y aprendizaje en contratos indefinidos sobre las tasas directas (sin pasar por un episodio de desempleo intermedio) de salida hacia un empleo indefinido tras la finalización del contrato de formación.

Finalmente, hsmlogit proporciona las expresiones algebraicas tanto del vector gradiente como de la matriz Hessiana, lo cual reduce significativamente los tiempos de computación necesarios para alcanzar la convergencia del modelo, y además, mejora la precisión de los errores estándar de los coeficientes estimados. La posibilidad de estimar modelos de duración con riesgos en competencia y presencia de heterogeneidad inobservable, unido a la significativa reducción en los tiempos de computación, permiten al investigador explotar adecuadamente la complejidad y riqueza de bases longitudinales de microdatos de gran tamaño.

David Troncoso Ponce

Pablo de Olavide University

Seville, Spain

Email: dtropon@upo.es

Abstract.

This article presents hsmlogit, a new Stata command that estimates multispells discrete time competing risks duration models with unobserved heterogeneity. hsmlogit allows for the estimation of one, two and up to three competing risks, as well as a maximum of five points of support for the identification of unobserved heterogeneity distribution ([Heckman and Singer, 1984]). The main contribution of hsmlogit is that allows for exploiting the richness of large longitudinal micro datasets, by estimating competing risks duration models, instead of one-risk models (such as hshaz and hshaz2), as well as it takes into account the presence of unobserved heterogeneity afecting transition rates. In addition to this, and taking into account the larger size of longitudinal micro datasets used for the estimation of discrete time duration models, hsmlogit also provides the algebraic expressions of both first and second order derivatives that, respectively, define the gradient vector and Hessian matrix, which significantly reduce time required to achieve model convergence.

Keywords: Duration analysis, Unobserved heterogeneity, d2 ml method, hshaz, hshaz2, hsmlogit, Hessian matrix

Acknowledgement

I am greatful to professor Stephen Jenkins for his helpful comments and suggestions that have contributed to significantly improve this work, and for allowing me to use Stata code of his hshaz’s command syntax. I also thank financial support of research project SEJ-6882 from Junta de Andalucía.

1 Introduction

Empirical estudies on individual decisions have experienced an important increase in recent years due to the boost of large and rich longitudinal micro datasets put available to the research community. Specially for the field of empirical Labor Economics focused on the estimation of labor market transition rates, the recent availability of large longitudinal micro datasets allows for capturing the presence of unobserved heterogeneity (UH, hereafter) components that afect the estimated transition rates. However, an important number of these empirical estudies that incorporates the presence of UH mainly focused on one-risk duration models,¹ that analyze transition rates towards an only destination (for example, transitions from employment to unemployment, ignoring the existence of other destinations, such as inactivity, or finding another job). This article presents hsmlogit, a new Stata command that estimates multispells discrete time competing risks duration models with UH. hsmlogit allows for the estimation of one, two and up to three competing risks, as well as a maximum of five points of support for the identification of unobserved heterogeneity distribution ([Heckman and Singer, 1984]).

The main contribution of hsmlogit is that allows for exploiting the richness of large longitudinal micro datasets, by estimating competing risks duration models, instead of one-risk models (such as hshaz and hshaz2), as well as it takes into account the presence of unobserved heterogeneity afecting transition rates. In addition to this, and taking into account the larger size of longitudinal micro datasets used for the estimation of discrete time duration models, hsmlogit also provides the algebraic expressions of both first and second order derivatives that define the gradient vector and Hessian matrix, respectively, which significantly reduce time required to achieve model convergence [Gould et al., 2010].

The rest of the article estructures as follows: Section 2 describes the longitudinal database used to obtain estimation results; the econometric model and hsmlogit command syntax are explained, respectively, in Sections 3 and 4; Section 5 presents estimation results, and Section 6 shows the advantages of providing the algebraic expressions of both the gradient vector and Hessian matrix. Finally, Section 7 concludes.

2 Database: The Continuous Sample of Working Histories

I analyze a longitudinal sample of workers in the Spanish labor market that comes from the Continuous Sample of Working Histories database (CSWH, hereafter). The CSWH is a longitudinal database that provides the working histories records of more than one million people, who represent a 4% nonstratified random draw from a target population, composed of any person with a contribution relation with the Spanish Social Security Administration. It includes both wage workers and recipients of Social Security benefits, namely, unemployment benefits, disability, survivor pension and maternity leave.²

¹An exception is the work presented in [Troncoso-Ponce, 2016], where a two-states multispells discrete time competing risks duration model with UH is estimated to analyze the efect of apprenticeship contracts in the Spanish labor market.

²[García-Pérez, 2008], [Lapuerta, 2010], [Arranz and García-Serrano, 2011] and [Arranz, García-Serrano and Hernanz, 2013] contain a deep exposition about features of CSWH as well as all necessary techniques to perform a duration analysis using working lives information.

The CSWH contains detailed information on each employment and unemployment episodes experienced by workers through their entire working histories. The information provided by the CSWH can be grouped into several categories: First, personal characteristics of workers (gender, age, nationality, educational level, residence place, and other personal characteristics). Second, job characteristics (type of labor contract, part-time coeficient, qualification level, and other job characteristics). Third, information on the employer (firm size, activity sector, and other firm characteristics). Furthermore, an important feature of the CSWH is that provides the beginning and termination dates of all employment and unemployment episodes, which takes special interest for duration analysis.

The estimation sample is composed of 48,246 low-educated and low-qualified young workers in the Spanish labor market for the period 2000-2014. The average age is 22.5 years-old, and 75.5% of them are males. The average number of employment episodes per worker is 8.9, lasting, on average, 7.13 months. Indeed, more than 25% of all employment episodes last 2 months or less, and only 5% last at least 24 months, which highlights the high turnover rate experienced by these workers. Multispell estimation sample has 1,316,611 observations. A describe output is shown below, describing the estimation sample’s full varlist, as well as a summarize outuput to show the main descriptive statistics of the estimation sample’s varlist.

Description of the estimation sample’s varlist

Sorted by: codind spell j

Descriptive statistics of the estimation sample’s varlist

3 Econometric model

This Section briefly describes the main features of the econometric models that will be estimated in Section 5. The main goal of this kind of models is to analyze duration spent by a population in a specific state (in this example, employment state), as well as to analyze the set of factors, observable and specially unobservable, that afect time spent in that state (see [Lancaster, 1992], [Allison, 1982] and [Jenkins, 1995]).

Let’s consider an individual beginning an employment episode at time (time T is measured in month intervals). The worker is observed monthly during the employment episode until either he/she exits to another modeled state (such as, unemployment, or finding a new job), or the observation window ends (right censored observations). Employment duration is analyzed by estimating the hazard rate out of employment at each observed month. Depending on the number of exits (i.e. risks) modeled by the command’s user, hsmlogit can estimate two diferent functional forms for the hazard rate.

Single-risk models use a Logit functional form to characterize the hazard rate, given by the following expression:

\[h (t | x, \eta) = \frac {e x p (\lambda (t) + x \beta + \eta)}{1 + e x p (\lambda (t) + x \beta + \eta)}\tag{1}\]

And competing risks models use a Multinomial Logit functional form to characterize the hazard rates:

\[h _ {d} (t | x _ {d}, \eta) = \frac {\exp (\lambda_ {d} (t) + x _ {d} \beta_ {d} + \eta)}{1 + \sum_ {d = 1} ^ {D} \exp (\lambda_ {d} (t) + x _ {d} \beta_ {d} + \eta)}\tag{2}\]

Assuming that , where and depending on the total number of risks modeled by the command’s user.

As the two expressions above show, the hazard rate at month depends on time (months) spent in the current unemployment state (i.e. duration dependence), captured by , as well as on a set of covariates summarized by x vector, that may contain both time-fixed and time-varying covariates. Furthermore, the hazard rate also depends on an unobserved component given by , that mesasures factors, such as job search efort, job networking, motivation, ability, etc, that are unobserved to the researcher and may afect the transition rate out of employment.

For the case of one-risk models, the contribution to the likelihood function of an individual i is given by the following expression:

\[L _ {i} = \sum_ {j = 1} ^ {P} \pi_ {j} \{\prod_ {t = 1} ^ {T _ {i}} \frac {h (T = t | \lambda (t) , x _ {i t} , \eta_ {j}) ^ {y _ {i t}}}{(1 - h (T = t | \lambda (t) , x _ {i t} , \eta_ {j})) ^ {(1 - y _ {i t})}} S (T = t | \lambda (t), x _ {i t}, \eta_ {j}) ^ {(1 - y _ {i t})} \}\tag{3}\]

Where dependent variable denotes a dummy variable that takes value 1 if worker i exits out from employment at month , and takes value zero otherwise.³ Expression given by denotes the hazard rate observed at month and denotes the survival rate observed at month , that estimates the cummulative probability of being employed (from the month until the month , and that is given by the following expression:

\[S (T = t | \lambda (t), x _ {i t}, \eta_ {j}) = \prod_ {s = 1} ^ {t} (1 - h (T = s | \lambda (s), x _ {i s}, \eta_ {j}))\tag{4}\]

As expressions 1 and 4 show, the hazard rate observed at month is conditional on the duration dependence and on the set of covariates And the survival rate at month is conditional on and on the set of covariates observed at months . Both the hazard and the survival rates also depend on belonging to the type of employed workers with unobserved characteristics given by 4

The total likelihood function of single-risk models is given by:

\[L = \prod_ {i = 1} ^ {N} \sum_ {j = 1} ^ {P} \pi_ {j} \{\prod_ {t = 1} ^ {T _ {i}} \frac {h (T = t | \lambda (t) , x _ {i t} , \eta_ {j}) ^ {y _ {i t}}}{(1 - h (T = t | \lambda (t) , x _ {i t} , \eta_ {j})) ^ {(1 - y _ {i t})}} S (T = t | \lambda (t), x _ {i t}, \eta_ {j}) ^ {(1 - y _ {i t})} \}\tag{5}\]

hsmlogit command maximizes, using d2 ml method, the natural logarithm of L to estimate the model parameters.

⁴It is assumed that unobserved characteristics do not vary with time and are not correlated to the rest of explanatory variables included in the specification of the hazard rate.

³Dependent variable refers to dead(deadvar) of hsmlogit command.

For the case of competing risks models, the contribution to the likelihood function of an individual i is given by the following expression:

\[L _ {i} = \sum_ {j = 1} ^ {P} \pi_ {j} \{\prod_ {t = 1} ^ {T _ {i}} \prod_ {d = 1} ^ {D} \{h _ {d} (T _ {d} = t | \lambda_ {d} (t), x _ {i t} ^ {d}, \eta_ {j}) ^ {y _ {i t} ^ {d}} \} S (T = t - 1 | \lambda (t), x _ {i t}, \eta_ {j}) ^ {(1 - \sum_ {d = 1} ^ {D} y _ {i t} ^ {d})} \}\tag{6}\]

Where denotes the hazard rate for the especific risk observed at month , conditional on the duration dependence , on the set of covariates , and on belonging to the type of employed workers with unobserved characteristics given by . Dependent variable ， for , denotes a dummy variable that takes value 1 if worker i exits out from employment towards the destination d at month , and takes value zero otherwise.

Unlike single-risk models, the survival function for competing risks takes into account the all possible risks faced by the individual at month , and therefore takes the following expression:

\[S (T = t - 1 | \lambda (t - 1), x _ {i t - 1}, \eta_ {j}) = \prod_ {s = 1} ^ {t - 1} (1 - \sum_ {d = 1} ^ {D} h _ {d} (T _ {d} = s | \lambda_ {d} (s), x _ {i s} ^ {d}, \eta_ {j}))\tag{7}\]

Similarly to single-risk models, the total likelihood function for competing risks is given by:

\[L = \prod_ {i = 1} ^ {N} \sum_ {j = 1} ^ {P} \pi_ {j} \{\prod_ {t = 1} ^ {T _ {i}} \prod_ {d = 1} ^ {D} \{h _ {d} (T _ {d} = t | \lambda_ {d} (t), x _ {i t} ^ {d}, \eta_ {j}) ^ {y _ {i t} ^ {d}} \} S (T = t - 1 | \lambda (t), x _ {i t}, \eta_ {j}) ^ {(1 - \sum_ {d = 1} ^ {D} y _ {i t} ^ {d})} \}\tag{8}\]

And likewise single-risk models, hsmlogit command maximizes, also using d2 ml method, the natural logarithm of L to estimate the model parameters for competing risks models.

3.1 The non-parametric identification of the UH distribution

Regarding the estimation of UH distribution, we assume the existence of unobserved factors afecting hazard rates, that if are ignored, may lead to spurious duration dependence, captured by λ(t) ([Van Den Berg, 2001]). A well known method to capture the efect of UH on the hazard rates is the proposed by [Heckman and Singer, 1984], by which the UH components are captured without imposing any parametric distribution function for the identification of UH distribution, but as a discrete mixture of several types of individuals with diferent values of UH components. Thus, it is assumed the presence of diferent types of workers who characterize themselves by having diferent levels of unobserved variables (such as, ability, cognitive and non cognitive skills, social and networking capabilities, etc.), captured by the set of parameters that are estimated as regression’s constant terms.⁵ For each Type of worker characterized by an associated probability of being observed in the data, given by , is also estimated jointly with the rest of the model parameters. Finally, the non-parametric discrete UH distribution is the result of the combination of these Types of workers, whose diferent values of UH are given by the vector and by their associated probabilities , are estimated jointly with the rest of the model parameters.

Furthermore, likewise hshaz2 command, when more than two mass-points are especified by the command’s user, hsmlogit also properly estimates masspoints probabilities using a Multinomial Logit function, rather than a Logit one, to compute the values of (see [Troncoso-Ponce, 2017]). For example, when the UH distribution is characterized by five points of support, the mass probabilty parameteres computed by hsmlogit take the following expression: , for , and . And for the computation of the standard errors of mass probability parameters, hsmlogit also provides to diparm() command the algebraric expressions of the first order derivatives of each , for each , with respect to each with

4 Command syntax

The hsmlogit’s command syntax follows the same design that hshaz and hshaz2’s. The only diference between hsmlogit’s command syntax and hshaz2’s is added by dead(deadvar) option. Unlike hshaz2, the dead(deadvar) option of hsmlogit command identifies whether the dependent variable typed by the command user in the dead(deadvar) option takes one, two or three risks. Therefore, hsmlogit, depending on the number of the values taken by the dependent variable, estimates, respectively, a single, a two, or a three competing risks duration model. The rest options of hsmlogit’s command syntax are the same that hshaz2’s (see [Troncoso-Ponce, 2017]).

The hsmlogit command syntax is:

hsmlogit varlist- weight -if exp -in range - , id(idvar) dead(deadvar) seq(seqvar) spell(spellvar) nmp(#) m2(#) p2(#) m3(#) p3(#) m4(#) p4(#) m5 eform nocons nolog nobeta0 maximize options

⁵As previously mentioned, hsmlogit allows for the estimation of a maximum of five points of support (ie. Types of workers) for the identification of the UH distribution.

5 Estimation results

This Section shows results from the estimation of three duration models, each of them depends on the number of exits modeled. The first model, presented below in the first estimation output (Single risk model with UH using hsmlogit), simply estimates the transition rate out of employment without diferentiating the destination state. The second one, shown below in the second estimation output (Two competing risks model with UH using hsmlogit), estimates a two risks duration model, by which the two modeled risks are: i) exiting to unemployment; and ii) a job-to-job transition to another employment. Finally, the third model, shown below in the third estimation output (Three competing risks model with UH using hsmlogit), allows for distinguishing the type of labor contract of the new employment found in the job-to-job transition. Specifically, the model difers between finding a fixed-term contract, and an open-ended one. Therefore, these three competing risks are: i) exiting to unemployment; ii) finding a fixed-term contract; and iii) finding an open-ended contract.

As mentioned in Section 3, the functional form of the hazard rate estimated in the first model is given by a Logit function, whereas the hazard rates of the second and third models are given by Multinomial Logit functions with two a three competing risks, respectively. The mentioned three tables with the estimation output show estimation results of fitting multispells duration models with two mass-points of unobserved heterogeneity.⁶

For the three estimated models, the set of covariates included in the specification of the hazard rates controls for the efect of: i) personal characteristics of the employed workers, such as, gender, age (age16tv) and squared age (age16tv2),⁷ nationality,⁸ and educational level⁹; ii) business cycle efects, by including the quarterly unemployment rate (unrate) and the product of the unemployment rate with the natural logarithm of the current employment spell (unratexlnjemp), and its squared (unratexlnjemp2); iv) a set of dummy variables that identify the Spanish regions (andal-rioja) to capture regional efects. Additionally to the duration dependence specification (using a three order polynomial of the natural logarithm of the duration of current employment spell), three dummy variables are included to identify months 6, 12, 18 and 24. These dummy variables are included to capture exit peaks related to the duration of temporary contracts in the Spanish labor market. Finally, to capture the efect of holding an apprenticeship contract on the employment exit rate, the dummy variable (cf) takes value one whether the worker is holding an apprenticeship contract, and takes value zero whether the worker has another type of temporary contract diferent from the apprenticeship one.¹⁰ The regression coeficients not shown in the estimation output tables are omitted due to space reasons, and are available to the interested reader upon request.

⁶All estimation results, with and without UH, shown in this article are available to the interested reader upon request.

⁷Age covariates measure the diference between the current age (time-varying age) with respect to the legal working age in the Spanish labor market, 16 years old.

⁸Nationality efect is captured using a dummy variable, called inmigra, that takes value one if the employed worker is not Spanish, and zero otherwise.

⁹The efect of educational level is captured by including two dummy variables: educcompul1 and educcompul2. Dummy variable educcompul1 (educcompul2) takes value one whether the worker has a primary (secondary) compulsory education degree, and takes zero otherwise.

¹⁰Hence, the regressions’ constant term contains male native employed workers holding a temporary contract in the Spanish regions Madrid and Catalonia, with less than primary compulsory education.

Single risk model with UH using hsmlogit

. hsmlogit_v4 `varsaleE´ , id(codind) spell(spell) seq(j) d(exit1) nmp(2) difficult Discrete time competing risks hazard model without frailty

Note: m1 = 0

Two competing risks model with UH using hsmlogit

. hsmlogit_v4 `varsaleE´ , id(codind) spell(spell) seq(j) d(exit2) nmp(2) difficult Discrete time competing risks hazard model without frailty

(exit2==no exit is the base outcome)

Discrete time competing risks hazard model, with discrete mixture Log likelihood = -840603.57 Number of obs = 1,316,611

Note: m1 = 0

Three competing risks with UH using hsmlogit

. hsmlogit_v4 `varsaleE´ , id(codind) spell(spell) seq(j) d(exit3) nmp(2) difficult Discrete time competing risks hazard model without frailty

(exit3==no exit is the base outcome)

Discrete time competing risks hazard model, with discrete mixture Log likelihood = -860570.7 Number of obs = 1,316,611

The estimation exercise shown in this Section is addressed only to highlight the importance of allowing for modelling more than one single risk in a duration model that also takes into account the presence of UH. For that reason, analogously to [Troncoso-Ponce, 2017], the main purpose of these regressions is not intended to address a rigorous regression analysis to properly estimate the efect of a set of covariates on the probability of exiting out of employment. Therefore, in this Section, comments on detailed estimation results will be focused mainly on the impact of holding an apprenticeship contract (captured by the covariate cf in the three estimation outputs presented above) when we allow for modelling more than one single risk.

The single risk duration model estimates a statistically significant negative efect (−0.9136) of holding an apprenticeship contract on the probability of exiting out from the employment state, which may suggest that apprenticeship contracts last longer (ie. seem to be more stable) than regular fixed-term contracts. And when we allow for modelling two competing risks (exiting to unemployment, or a job-to-job transition to another job), the efect of apprenticeship contracts remain negative and statistically significant on both the two risks modeled: exiting to exit to unemployment (−0.8244), and a direct transition to another job (−1.0252).

However, interestingly, the estimated efect of apprenticeship contracts turns positive when we allow for modelling the job-to-job transition separately in two diferent, and mutually exclusive, destinations: i) a direct transition to a fixedterm contract; and i) a direct transition to an open-ended contract. As the third estimation output shows, apprenticeship contracts increase the probability of experiencing a job-to-job transition towards an open-ended contract (0.7109). The main reason of observing this possitive efect is the role played by public financial incentives addressed to the conversion of apprenticeship contracts into open-ended ones. Apprenticeship contracts in Spain benefit from public subsidies for the conversion into open-ended contracts. These subsidies mainly consist of a significant reduction in Social Security contributions paid by the employer during a maximun period of three years, from the starting date of conversion of the apprenticeship contract into an open-ended one. The main goal of these financial incentives is to favour employment stability, and to foster the accumulation of employment experience of apprentices by allowing them to put in practice the work-specific skills acquired during the apprenticeship period. Thus, the possitive coeficient found (0.7109) may be capturing the efect of these public financial incentives provided by Spanish policy makers addressed to the conversion of apprenticeship contracts into open-ended ones.

An axhaustive analysis of the apprenticeship contracts in the Spanish labor market is presented in [Troncoso-Ponce, 2016] and [Jansen and Troncoso-Ponce, 2017]. The first one estimates a multispell and multistate competing risks duration

Table 1: Interpretation of UH coeficients (three competing risks model)

model with UH especific to both each state and to each destination state, as well as a selection equation that estimates the transition rates to the entry into the labor market holding three diferent types of labor contract: an apprenticeship contract, a fixed-term contract, and an open-ended contract. The second, and more recent, work also estimates a multispell and multistate competing risks duration model with UH, but the selection equation consists of an initial conditions equation, rather than a transition rate equation, that controls for the efect of a set of observable covariates on the probability of having an apprenticeship contract just in the first employment spell of the individual’s working life. Moreover, the empirical strategy followed in this work allows us to disentangle two types of efect: an instant efect, and a subsequent efect of apprenticeship contracts on the employment and unemployment transition rates.

5.1 Some insights on the interpretation of UH coeficients

Regarding the estimation and interpretation of UH coeficients, as we assume that is set to zero,¹¹ the estimated regression’s constant terms (−1.537957, −0.9171136 and −5.338236, for the exit to unemployment, to a fixed-term, and to an open-ended contract, respectively) capture the UH component specific to Type I workers, whereas η captures the unobserved diferential efect of Type II workers with respect to Type I workers. Therefore, the estimated value of UH component specific to Type II workers are the result of the sum of the regression’s constant terms and the estimated coeficient value of η₂.

Table 1 shows the estimated coeficients of the UH components of Type I and Type II workers from the estimation results of the three competing risks model. The estimation of the non-parametric UH distribution, characterized by the presence of two types of workers (two points of support), captures Type I and Type II workers who represent, respectively, 84.88% and 15.12% of the estimation sample. As Table 1 shows, Type II workers have unobserved characteristics that positively correlate to the employment hazard rates, which implies that Type II workers face employment transition rates (towards all the three modeled risks) higher than Type I workers’.

¹¹It explains the footnote shown at the estimation output tables with the message ”Note: m1 = 0”, where m1 denotes UH component given by η₁. See also [Troncoso-Ponce, 2017] and hshaz command’s oficial Stata helpfile.

In conclusion, the estimation of not only a single or a two competing risks, but a three competing risks duration model has allowed for capturing a positive and statistically significant efect of apprenticeship contracts on the probability of transiting directly (via job-to-job) to an open-ended contract, that otherwise would have remained hidden to the empirical researcher if only one risk, or even two, would have been estimated. Furthermore, given the relevance of UH, and its non parametric identification, in discrete time duration models with multispell observations (see, for example, [Gaure, Roed and Zhang, 2007] and [Abbring and Van den Berg, 2004]), the new Stata command hsmlogit takes especial relevance, as allows for the estimation of discrete time competing risks duration models with UH.

6 The advantages of using ml d2 method

As mentioned in Section 1, hsmlogit provides the algebraic expressions of both the gradient vector and Hessian matrix, allowing for using d2 ml method to achieve the model convergence. An important advantage of programming the Hessian matrix is that allows applied researchers to deal with large longitudinal microdata sets (see for example [Troncoso-Ponce, 2017]). To show the savyings in estimation time, this Section presents time required to estimate multispell both single and competing risks duration models (with 2, 3 and 4 mass-points) using d0, d1 and d2 ml methods,¹². Comments in this Section will be focused only on the comparison between d1 and d2 ml methods. The comparison between d0 and d2 ml does reinforce the same conclusions obtained below.

Table 2 reports time spent¹³ by each of the three ml methods in achieving the models’ convergence.¹⁴ Results from Table 2 highlight two relevant diferences between d1 and d2 ml methods: Firstly, d2 method significantly reduces time required to achieve the all models convergence. Diferences in time required seem to be less evident in the estimation of single risk models: for instance, for fitting the two mass-points model, d2 (d1) method needs 46 seconds (6.28 minutes). However it becomes more important as both the number of risks and the number of mass-points increase: for fitting the three competing risks model with four mass-points, d2 method only requires 8.02 minutes, whereas d1 method needs 1.58 hours. On its part, d0 method not even achieve the model convergence:

¹²The all estimations, whose time required are shown in Table 2, include a set of twenty eight covariates that, as results shown in Section 5, control for duration dependence, personal characteristics, type of labor contract, regional efects and economic cycle. The detailed estimation results are available upon request to the interested reader.

¹³I work with Stata 14.0 MP - Parallel edition 64 bits. The machine employed to obtain estimation results incorporates an Intel(R) Core(TM) i7-6700HQ CPU at 2.60 GHz, and 12 Gb RAM memory. The operating system is Windows 10 Home.

¹⁴In this sample composed of youth Spanish employees, the estimation of five points of support for the identification of the non-parametric unobserved heterogeneity distributuion is not possible, neither fitting single risk models, nor competing riks models.

Table 2: Time required for the estimation of multispell competing risks duration models (Sample size: 1,316,611 observations)

after eleventh iteration, it gets into a backed up loop. Secondly, unlike d2 method, time required by d1 method to achieve the model convergence strongly dependes both on the number of exits modeled, and on the number of points of support for the identification of the UH. Table 2 shows that, using d2 (d1) method, the diference in time spent between the less time-demanding model (the single risk model with two mass-points) and the most time-demanding model (the three competing risks with four mass-points) reaches 7.16 minutes (1.52 hours).

7 Concluding remarks

This article presents hsmlogit, a new Stata command that estimates multispells discrete time competing risks duration models with unobserved heterogeneity. hsmlogit allows for the estimation of one, two and up to three competing risks, as well as a maximum of five points of support for the identification of the nonparametric unobserved heterogeneity distribution [Heckman and Singer, 1984]. The relevance of modelling more than one risk has been highlighted by estimating the efect of apprenticeship contracts on a sample composed of low educated young workers in the Spanish labor market for the period 2000-2014. Thus, the estimation of a three competing risks duration model has been the only way to find out the potential efect of public financial incentives for the conversion of apprenticeship contracts into open-ended ones on the direct (via job-to-job) transition rates towards an open-ended contract. Moreover, since hsmlogit allows for the estimation of non-parametric UH distribution ([Heckman and Singer, 1984]), our results capture the presence of two types of workers with diferent values of unobserved characteristics that afect the estimated hazard rates.

Finally, hsmlogit provides the algebraic expressions of both the gradient vector and the Hessian matrix, which significantly reduces time required to achieve the model convergence, and also improves the standard errors’ accuracy of the estimated coeficients. The possibility of estimating competing risks duration models with the presence of UH, along with time savyings provided by the use of d2 ml method may allow the applied researchers to easily and properly exploit the richness and complexity of large longitudinal microdata sets.

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