Documento de Trabajo - 2017/06

Faster estimation of discrete time duration models with unobserved heterogeneity using hshaz2

David Troncoso Ponce (Universidad Pablo de Olavide)

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David Troncoso Ponce

Universidad Pablo de Olavide

RESUMEN

En este trabajo se presenta el nuevo comando de estimación hshaz2 y se describen sus principales características. Al igual que el anterior comando hshaz, desarrollado por el profesor Stephen Jenkins, el comando hshaz2 estima, mediante el método de máxima verosimilitud, un conjunto de modelos de duración en tiempo discreto con riesgos proporcionales que tiene en cuenta la presencia de heterogeneidad inobservable. La aportación principal del comando hshaz2 consiste en el desarrollo y programación de las expresiones algebraicas de las primeras y segundas derivadas parciales de la función de log-verosimilitud, que componen el vector gradiente y la matrix Hessiana, respectivamente.

De este modo, la diferencia entre hshaz y hashaz2 reside en el método empleado para alcanzar la convergencia de la función de log-verosimilitud: el comando hshaz usa el método d0, mediante el cual las derivadas del gradiente y el Hessiano son calculadas usando aproximaciones numéricas; mientras que hshaz2 usa el método d2, que provee las expresiones algebraicas de las derivadas de primer y segundo orden.

Las ventajas de emplear el nuevo comando hshaz2 son fundamentalmente dos: la mayor fiabilidad de los errores estándar de los parámetros estimados; y sobre todo, la reducción de los tiempos de computación requeridos para la estimación de este tipo de modelos, que posibilita la estimación de modelos de duración empleando grandes bases de microdatos longitudinales.

David Troncoso Ponce

Pablo de Olavide University

Seville, Spain

Email: dtropon@upo.es

Abstract. This article presents hshaz2, a new Stata command that uses d2 ml method to estimate discrete time duration models with unobserved heterogeneity. The main advantage of using hshaz2 is the gain in computation speed, that takes special relevance as the sample size increases. Estimation results show that, on a sample size of 568,042 observations, hshaz2 spends 0.42 and 1.13 minutes to achieve the convergence of a discrete time proportional hazard model with two and three points of support, respectively. Furthermore, hshaz2 allows for the estimation of multispell duration models, where individuals may be observed at risk of exiting more than once. Using, a sample with 1,547,507 observations, hshaz2 spends 1.17 and 2.17 minutes to achieve convergence of a multispell discrete time proportional hazard model with two and three points of support, respectively.

Keywords: Duration analysis, Unobserved heterogeneity, d2 ml method, hshaz, hshaz2, 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 and helpfile. I also thank financial support of research project SEJ-6882 from Junta de Andalucía. This article has been finished during a postdoctoral research stay at Fundación de Estudios de Economía Aplicada (FEDEA).

1 Introduction

Time required to estimate discrete time duration models that account for the presence of unobserved heterogeneity (hereafter, UH) uses to be an important concern for applied researchers when have to deal with large datasets.¹ Stata command hshaz, written by professor Stephen Jenkins, estimates proportional discrete time duration models taking into account the the presence of UH, following to [Heckman and Singer, 1984]. hshaz uses d0 ml method to achieve convergence of the log-likelihood function, which computes numerical approximation to first and second order derivatives, that composes gradient vector and Hessian matrix, respectively.

¹For the purpose of this article, I consider a large dataset as those with at least one million observations, as for example, longitudinal data that comes from Social Security administrative records.

This article presents hshaz2, a new Stata command that provides the algebraic expressions of both first and second order partial derivatives of the loglikelihood function estimated by hshaz command, and describes its main characteristics. Thus, hshaz2 also estimates, using maximum likelihood method, discrete time proportional hazard rate models with UH. However, hshaz and hshaz2 difer in the Stata ml method used to achieve convergence of the loglikelihood: hshaz uses d0 ml method, whereas hshaz2 uses d2 ml method.

The main advantage of programming the algebraic expressions of first and, above all, second order derivatives of the log-likelihood function are twofold: First, the reliability on the standard errors of parameters estimates. Second, and the most important issue, the gains obtained in computation speed to achieve the model convergence using d2 method [Gould et al., 2010].²

The rest of the article estructures as follows: Section 2 describes the database used to obtain estimation results; the econometric model and hshaz2 command syntax are explained in Sections 3 and 4, respectively; Section 5 presents estimation results, and Section 6 describes some details on the reparameterization of mass-points probability parameters. Finally, Section 7 concludes.

2 Database: The Continuous Sample of Working Histories

I analyze a longitudinal sample composed of 44,077 unemployed workers, aged 16-37 year-old, in the Spanish labor market for the period 2000-2013, 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% non-stratified 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.³

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, etc); Second, job characteristics (type of labor contract, part-time coeficient, qualification level, etc); Third, information on the employer (firm size, activity sector, etc). 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.

²A detailed explanation, using an applied approach, on the estimation of duration models using hshaz command is available at the following link: http://www.iser.essex.ac.uk/teaching/stephenj/ec968/index.php

³[García-Pérez, 2008] and [Lapuerta, 2010] contain a deep exposition about features of CSWH as well as all necessary techniques to perform a duration analysis using working lives information.

For the unispell duration model estimation, that will be explained in Subsection 5.1, each unemployment episode has been expanded in monthly intervals. Thus, each unemployed worker is observed (and, therefore, contributes to the likelihood function) as many times as the number of months the unemployment episode lasts. After the database has been expanded, the sample size increases up to 568,042 observations. Moreover, the duration variables as well as all explanatory variables that vary with unemployment duration (such as, age, squared age, etc) have been generated in order to correctly measure timevarying covariates.

3 Econometric model

This Section briefly describes the main features of the econometric models that will be estimate in 5. The main goal of this kind of models is to analyze duration spent by a population in a specific state (in this example, unemployment state), as well as to analyze the set of factors, observable and above all unobservable, that afect time spent in that state.

Let’s consider an individual beginning an unemployment episode at time (time T is measured in month intervals). The unemployed worker is observed monthly during the unemployment episode until either he/she finds a new job, or the observation window ends. Unemployment duration is analyzed by estimating the hazard rate out of unemployment at each observed month.

The hazard rate out of unemployment estimated by hshaz2 (and hshaz) takes the following functional form:

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

As the expression above shows, the hazard rate at month depends on time (months) spent in the current unemployment state (i.e. duration dependence), captured by λ(t), as well as on a set of covariates summarized by x vector.⁴ Furthermore, the hazard rate also depends on an unobserved component given by η, that mesasures factors, such as job search efort, motivation, ability, etc, that are unobserved to the researcher and may afect the transition rate out of unemployment.

To estimate the unobserved heterogeneity distribution, following [Heckman and Singer, 1984], it is assumed the existence of diferent types of unemployed workers who differ between them in unobserved characteristics (such as, as mentioned above,

⁴x vector may contain both time-fixed and time-varying covariates.

motivation, ability, etc), that afect the transition rate out of unemployment. Therefore, the whole population is composed of a discrete mixture distribution of all the types of unemployed workers considered by the econometric model. The presence of each type of unemployed workers in the whole population is weighted by the probability of observing it, that is estimated jointly with the rest of the model parameters.

The contribution to the likelihood function of an individual i is given by the following expression:

\[L _ {i} = \sum_ {j = 1} ^ {P} \pi_ {j} \{\prod_ {t = 1} ^ {T _ {i}} \frac {h (T = t | \lambda (t) , x _ {i t} , \eta_ {j}) ^ {y _ {i t}}}{(1 - h (T = t | \lambda (t) , x _ {i t} , \eta_ {j})) ^ {(1 - y _ {i t})}} S (T = t | \lambda (t), x _ {i t}, \eta_ {j}) ^ {(1 - y _ {i t})} \}\tag{2}\]

Where and denote the hazard rate and the survival function⁵ observed at month respectively, conditional on the duration dependence , on the set of covariates , and on belonging to the type of unemployed workers with unobserved characteristics given by

The discrete probability distribution of unobserved heterogeneity is given by the estimation of the vector , with and . Each parameter estimates the probability of observing each type of (unemployed) workers (in) of the whole population.

Dependent variable 1 denotes a dummy variable that takes value 1 if worker i exits out of unemployment at month and takes value zero otherwise.⁷

Finally, the total likelihood function is given by:

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

hshaz command maximizes, using d0 ml method, the natural logarithm of L to estimate (the all) model parameters. The main contribution of hshaz2 command is that it provides the algrebraic expressions of both the first and second order (partial) derivatives of the natural logarithm of and therefore, achieves the convergence using d2 ml method.

4 Command syntax

As has been explained in Section 1, the main contribution of hshaz2 command is the programming of both first and second order derivatives to achieve faster

\[\begin{array}{c} ^ {5} S (T = t | \lambda (t), x _ {i t}, \eta_ {j}) = ((1 - h (T = 1 | \lambda (1), x _ {i 1}, \eta_ {j}))) ((1 - h (T = 2 | \lambda (2), x _ {i 2}, \eta_ {j})))... ((1 \\ h (T = t - 1 | \lambda (t - 1), x _ {i t - 1}, \eta_ {j}))) ((1 - h (T = t | \lambda (t), x _ {i t}, \eta_ {j}))) \end{array}\]

⁶It 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.

⁷Dependent variable refers to dead(deadvar) of hshaz2 command.

estimations by using d2 ml method. The programming of gradient vector and Hessian matrix do not afect the estimation output reported by the former hshaz command, in the sense that the set of parameters to be estimated is the same, either by using d0 or d2 ml methods, respectively. Therefore, the command syntax of hshaz2 shares the same estructure that hshaz’s, and also makes use of the same terminology to refer to the estimation output. This section describes the main estructure of hshaz2 command syntax and explains the options allowed by hshaz2 for the estimation process.

The hshaz2 command syntax is:

As can be seen, the only element added by hshaz2 to the (hshaz’s) command syntax is an option(.), called spell(spellvar). The hshaz2 command allows for the estimation of multiple spells duration models, by which individuals may be observed at risk of exiting more than once. In such cases, it is necessary to correctly identify and to sort the diferent episodes experienced by each person in the estimation sample. This is the goal of the option spell(spellvar), where spellvar must be a numeric variable that identifies the sequential order of the spells experienced by each individual in the sample. This will be explained in detail in Subsection 5.2.

5 Estimation

This Section presents results of fitting discrete time duration models on the sample of unemployed workers described in Section 2. The aim of this Section is to show time saving involved by using hshaz2 command in comparison with hshaz, highlighting the importance of using d2 ml method to achieve convergence. In 5.1, I use both hshaz2 and hshaz commands to estimate two unispell duration models, with two and three mass-points, respectively. Once the results are obtained, the estimation speed of hshaz2 and hshaz commands are compared. Finally, in Subsection 5.2, I use hshaz2 command to estimate two multispell duration models, with two and three points of support for the identification of the unobserved heterogeneity, respectively.⁸

5.1 Fitting unispell duration models using hshaz and hshaz2

Pages 7 and 8 present the estimation output of fitting a duration model with two mass-points, running hshaz2 and hshaz commands, respectively. The rest of tables with detailed estimation results are shown at final Appendix. Comments on coeficients estimates will only focus mainly on the diferent duration dependence efect shown by estimation output with and without controlling for the presence of unobserved heterogeneity. The interpretacion of the rest of estimated coeficients are not commented, because of the main purpose of these regressions is to highlight that hshaz2 command replicates the results obtained by hshaz command, and therefore, it is not intended to address a regression analysis to properly estimate the efect of a set of covariates on the probability of exiting out of unemployment.

⁸I 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, and Stata 14 MP version.

The set of covariates is included in the specification of the hazard rate for control purposes, and summarizes: 1) personal characteristics of the unemployed workers, such as, gender, age and squared age,⁹ nationality,¹⁰ and educational level; 2) characteristics of the current unemployment spell, such as, a dummy variable to identify whether the unemployed worker receives unemployment benefits, as well as an interaction between this dummy variable and the natural logarithm of the duration of current unemployment spell; 3) the quarterly unemployment rate to capture bussiness cycle efects on the transition rate out of unemployment; 4) a set of dummy variables that identify the Spanish regions to capture regional efects. Additionally to the duration dependence specification (using a two order polinomial of the natural logarithm of the duration of current unemployment spell), three dummy variables are included to identify months 12, 18 and 24. These dummy variables are included to capture exit peaks, frequently observed in unemployment duration analysis, that may be due to unemployment benefits exhaustion efects.

As most empirical research on Labor Economics has found for many European labor markets, coeficient estimates show negative unemployment duration dependence, and reveal the importance of controlling for the presence of UH. Thus, the two-mass points duration model that does not control for the presence of UH ¹¹ underestimates (1.430829 and -0.5548032, for Log(t) and , respectively¹²) the efect of unemployment duration dependence. Controlling for UH, the efect of duration dependence increases up to 3.614478 and -0.9491475, for Log(t) and , respectively.

Regarding the estimation of the unobserved heterogeneity distribution, 71.6% (0.7163904) of the sample are Type I unemployed workers, and the rest 28.4% (0.2836096) belongs to Type II group. For the correct interpretation of UH coeficients, it is necessary to take into account that, as η₁ is set to zero, then η₂ estimates the diferential unobserved efect (of being Type II unemployed workers) on the probability of exiting out of unemployment (state), with respect to the (estimated coeficient of the) regression constant term, -5.159466. Therefore, the estimated unobserved efect of being Type I and Type II unemployed workers are given by -5.159466 and -6.0860925 (=-5.159466-0.9266265), respectively.

⁹Age 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.

¹⁰Nationality efect is captured using a dummy variable, that takes value one if the unemployed worker is not Spanish, to identify whether the unemployed worker is not Spanish.

¹¹It refers to Discrete time PH model without frailty firstly estimated by hshaz and hshaz2 commands, shown in pages 14 and 18 at Appendix.

¹²These coeficients are available at final Appendix, in pages 14 and 18.

. display "Started at $S_TIME"

. hshaz2 `varsaleU´ , id(codind) seq(j) d(exit) nmp(2) difficult (output omitted )

Started at 19:19:53

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. display "Started at $S_TIME"

. hshaz `varsaleU´ , id(codind) seq(j) d(exit) nmp(2) difficult (output omitted )

Started at 19:20:35

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Table 1: Estimation of unispell duration models (Sample size: 568,042 observations)

Table 1 reports time spent by running both hshaz and hshaz2 commands to achieve the convergence of the fitted duration models mentioned above, with two and three mass-points, respectively.¹³ Results of Table 1 highlight two relevant diferences between hshaz2 and hshaz commands: First, hshaz2 provides a significant reduction in time required to achieve the convengence: to fit a two mass-points model, hshaz spends thirty seven minutes and fifty five seconds minutes, while hshaz2 spends only fifty one seconds, which means thirty seven minutes less. And, second, unlike hshaz2, time required by hshaz to achieve the convergence strongly depends on the number of points of support especified by command’s user: hshaz needs one hour and fourteen minutes to fit a three mass-points model, while hshaz2 spends only one minute and thirty one seconds.

In order to show the convergence process followed by running both hshaz and hshaz2 commands, Figures 1 and 2 plot the log-likelihood values taken at each iteration by hshaz and hshaz2 commands during the convergence process of the estimation of two mass-points and three mass-points unispell models, respectively. As can be seen in Figures 1 and 2, the values taken by the loglikelihood functions of hshaz and hshaz2 at first iterations slightly difer, but when they aproximate to the maximum, the two log-likelihood functions converge to the same value: -122,078.61 for two mass-points models, and -121,861.5 for three mass-points models.

¹³For this specific sample of youth unemployed workers, the identification of unobserved heterogeneity is not possible when more than three mass-points are specified, which leads to not achieve convergence, neither running hshaz, nor hshaz2 commands. This is the reason why only results from two and three mass-points duration models are shown in Table 1.

Figure 1: Iteration process of unispell duration models estimation (two masspoints)

Figure 2: Iteration process of unispell duration models estimation (three masspoints)

Table 2: Estimation of multispell duration models using hshaz2 (Sample size: 1,547,507 observations)

5.2 Fitting multispell duration models using hshaz2

As was previously mentioned, hshaz2 allows for the estimation of multispell duration models, by which individuals may be at risk of exiting more than once. And this feature implies, in our example, that each individual may experience several unemployment episodes. Time saving advantages given by hshaz2 takes special interest in multispell duration models because of the increasing of the number of observations of the estimation sample, and therefore, the increase in required estimation time that it implies. To highlight the importance of this, a multispell duration model is fitted using another version of the sample, in which multiple unemployment spells are observed for each individual. This sample has 1,547,507 observations. The number of indivuals is 44,077, and the total number of unemployment spells is 146,851, that means an average number of spells per individual of roughly 3.33. The econometric especification includes the same covariates specified by the rest of the models estimated in the previous Section.

Table 2 reports time spent by hshaz2 to achieve convergence for the two estimated models, with two and three points of support, respectively: the first one includes two points of support, and the second one specifies three points of support. Estimation output at final Appendix, in reports detailed estimation output.

6 Reparameterization of mass-points probabilities

Functional form followed by hshaz command to compute mass-points probability parameters is a Logit, regardless of the number of points of support specified by the command’s user, whereas hshaz2 computes the mass-points probabilities using a Multinomial Logit. For the estimation of two mass-points models, as only one parameter must be estimated, both hshaz and hshaz2 compute the values of mass probability parameters using the same functional form, with and , providing the same coeficient estimates of and as well as for their standard errors. However, when more than two points of support are especified, the functional forms used by hshaz and hshaz2 to compute the values of mass probability parameters are diferent. For example, for three mass-points models, hshaz computes and whereas hshaz2 computes and

This explians the diferences found in mass-probability parameters estimates, given by and , depending on whether have been estimated using hshaz or hshaz2. Thus, as can be seen in the estimation output of Appendix (in pages 18 to 21), estimated values of and provided by hshaz (hshaz2) are -0.915 (-0.629) and -1.533 (-1.105), respectively. And standard errors of and p provided by hshaz (hshaz2) are 0.016 (0.025) and 0.081 (0.087), respectively. Both hshaz and hshaz2 provide and coeficients statistically significant at 1% level.¹⁴ However, estimated values of mass probability parameters, given by and , do not show significant diferences.

7 Conclusion

hshaz2 command provides the programming of gradient vector and Hessian matrix of the log-likelihood function of hshaz command, written by professor Stephen Jenkins. The programming of the algebraic expressions of both the first and second derivatives allows to use d2 ml method evaluator to achieve faster estimations of discrete time proportional duration models with unobserved heterogeneity. Furthermore, in contrast to hshaz, hshaz2 allows for the estimation of multispell duration models, by which individuals may be observed at risk of exiting more than onece. The gains achieved in saving time provided hshaz2 are stricky: Estimation results show that, on a sample size of 568,042 observations, hshaz2 (hshaz) spends 0.42 (22.2) and 1.13 (43.08) minutes to achieve the convergence of a duration model with two and three points of support, respectively. Finally, The gains in estimation time involved by hshaz2 command takes special relevance as the sample size increases: Using a sample with 1,547,507 observations, the estimation of a multispell duration model with two (three) points of support requires 1.17 (2.17) minutes.

¹⁴To estimate the standard errors of mass probability parameters, hshaz2 provides to diparm command the algebraric expressions of first order derivatives of each , for each , with respect to each with

References

[Gould et al., 2010]

1. Gould, W., Pitblado, J. and Poi, B. Maximum Likelihood Estimation with Stata, Fourth Edition, Stata Press, 2010.
2. [Heckman and Singer, 1984] Heckman, J. J. and Singer, B._{, A} Method for Minimizing the Impact of the Distributional Assumptions in Econometric Models for Duration Data, Econometrica, Vol. 52, pp. 271-320.
3. [Arranz and García-Serrano, 2011] Arranz, J.M. and García-Serrano, C. ⁽²⁰¹¹⁾ Are the MCVL tax data useful? Ideas for mining, Hacienda Pública Española, Vol. 199(4), 151-186.
4. [Lapuerta, 2010] Lapuerta, I. (2010) Claves para el tra- bajo con la Muestra Continua de Vidas Laborales, DemoSoc working paper (2010-37), Universitat Pompeu Fabra

[García-Pérez, 2008] García-Perez, J.I., 2008 ´ La Muestra Continua de Vidas Laborales: Una guía de uso para el anlisis de transiciones, Revista de Economía Aplicada, N. E-1, Vol. XVI, pp. 5-28.

1. [Jenkins, 2005] Jenkins, S., ²⁰⁰⁵ Survival Analysis, manuscript.

Appendix

Estimation output of fitting a two mass-points model using hshaz2 and hshaz

Note: m1 = 0 . display "Finished at $S_TIME" Finished at 19:20:35

Note: m1 = 0 . display "Finished at $S_TIME" Finished at 19:42:55

. display "Started at $S_TIME" Started at 19:42:55

. hshaz2 `varsaleU´ , id(codind) seq(j) d(exit) nmp(3) difficult Discrete time PH model without frailty

Note: m1 = 0 . display "Finished at $S_TIME" Finished at 19:44:08

Note: m1 = 0 . display "Finished at $S_TIME" Finished at 20:27:16

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