BAYESIAN EXPONENTIAL SURVIVAL MODEL IN THE ANALYSIS OF UNEMPLOYMENT DURATION DETERMINANTS
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1 A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 269, 2012 BAYESIAN EXPONENTIAL SURVIVAL MODEL IN THE ANALYSIS OF UNEMPLOYMENT DURATION DETERMINANTS Abstract. The prmary objectve of the work s to dentfy demographc and soco-economc factors nfluencng the unemployment duraton n the recent perod n Poland. Dfferent approaches to the problem have been appled. In ths paper we have used a survval parametrc model n Bayesan approach. The followng determnants have been concerned n the model: sex, martal status, educaton level, nformaton about contnung an educaton, regon of Poland, and age of respondent. The emprcal analyss s based on Household budgets n 2008 survey of Central Statstcal Offce and ndcates the man factors nfluencng unemployment duraton. Key words: unemployment; survval exponental model; Bayesan nference; MCMC method. I. INTRODUCTION The sgnfcance of unemployment results from ts economc, socal and poltcal aspects. To nvestgate the unemployment determnants, event hstory models (Drobn and Frtczak, 2001) and logt models (Daras and Jerzak, 2005; Coller, 2003) are usually appled. Another approach, based on standardsed unemployment rates can be found n (Socha and Sztanderska, 2000). The most frequently reported factors related to unemployment are: sex, age, educaton status and place of lvng. Besdes these, many other determnants are consdered such as: the technques of searchng for a job, the number of receved job offers, the perod of unemployment beneft, mnmal salary rates, etc. The prmary objectve of ths work s to dentfy the demographc and socoeconomc features, whch nfluence the unemployment. A Bayesan exponental survval model was appled to analyse the determnants affectng the duraton of unemployment perod. Ph.D., Insttute of Statstcs and Demography, Warsaw School of Economcs. [191]
2 192 II. THE SCOPE OF RESEARCH The emprcal analyss s based on Household budgets n 2008 survey of Central Statstcal Offce. Accordng to the am of ths research, we take nto consderaton unemployed persons, who were lookng for a job and were ready to take a job (Eurostat). As dfferent factors can nfluence unemployment dependng on ts duraton, a decson has been made to consder only persons who were unemployed maxmally for 24 months. In ths way we chose 2512 ndvduals. 214 of them already found a job and wated for startng work for these persons an event holds, whle the others are censored. In a model, a dependent varable s tme defned as the number of months of unemployment. The characterstcs of ndependent varables that potentally may have an mpact on the unemployment duraton has been dscussed below. The frst potental determnant s sex: 1 man (49.56%), 2 woman (50.44%). One can expect that hgher chances for fndng a job have men, then women, who more tme devote to ther famles. Martal status s one of the factors consdered by employers when hrng new employees. Hence, t s mportant to examne f unmarred people have more chance of fndng a job. Martal status was encoded as follows: 1 unmarred (49.32%), 2 marred (42.40%), 3 separated (1.04%), 4 a wdower, a wdow (2.07%), 5 dvorced (5.18%). We can suppose that educaton status s one of the most mportant determnants nfluencng the chance of fndng a job. Educaton level was encoded as follows: 1 hgher (10.19%), 2 post-secondary (2.95%), 3 secondary professonal (21.38%), 4 secondary general (12.18%), 5 basc vocatonal (34.63%), 6 prmary school (18.67%). A related factor potentally nfluencng unemployment perods s whether a respondent contnues educaton. The latter varable takes two values: 1 yes (8.16%), 2 no (91.84%). The regons of Poland dffer n the economc and technologcal development, hence we can suppose that the resdents of the west and central regon of Poland have more chance of fndng a job, then the resdents of the remanng regons. Regon of Poland was defned as follows: 1 central (provnce: ódzke, mazowecke) (18.95%), 2 southwest (provnce: dolnolske, opolske) (10.91%), 3 south (provnce: maopolske, lske) (14.81%), 4 northwest (provnce: welkopolske, zachodnopomorske, lubuske) (17.12%), 5 north (provnce: kujawsko-pomorske, warmsko-mazurske, pomorske) (17%), 6 east (provnce: lubelske, podkarpacke, wtokrzyske, podlaske) (21.22%). Next determnant whch has been taken nto consderaton s age (mn=17, max=66). It s mportant to examne f young persons have more chance of fndng a job.
3 Bayesan Exponental Survval Model n the Analyss 193 III. RESEARCH METHOD In ths paper we have used a Bayesan survval exponental model. The Bayesan methods combne subjectve pror knowledge wth the nformaton acqured from the data by usng Bayes theorem (Bolstad, 2007; Bernardo and Smth, 2004; Gelman et al., 2000). The proposed exponental model s one of the most mportant models n survval analyss (Blossfeld et al., 1989; Blossfeld and Rohwer, 1995). Ths survval parametrc model wll be presented n Bayesan approach. The Bayesan analyss of survval parametrc models has been dscussed n many works (Ibrahm et al., 2001). Suppose we have ndependent dentcally dstrbuted survval tmes y y,, y 1 n ; wth each y, 1,..., n havng an dentcal exponental dstrbuton wth parameter. The censorng ndcators we denote by v v,,v 1 n, where v 0 f y s rght censorng and v 1 f y s falure tme, 1,..., n. The densty functon for y s f y exp y, the survval functon Sy exp y. In regresson models we have one more addtonal element a matrx of ndependent varables X ( n p ). Let x denote th row of the matrx, then D n, y, X, v s the observed data. Let x, where x, ( p 1) s a vector of covarates,, ( p 1) s a vector of regresson coeffcents and s a known functon. For x exp x, we have the followng lkelhood functon: L v D f y Sy n 1 exp 1 v v expx exp y expx exp y expx n 1 n 1 vx exp n 1 y expx. 1 v Often for regresson coeffcents we choose unform mproper pror or a normal pror. In our model we take a p dmensonal normal pror N p 0, 0 for, where 0 denotes the pror mean vector, and 0 denotes the pror covarance matrx. Then the posteror dstrbuton for s gven by D L D p (1) p, (2) 0, 0 p 0, 0 denotes multvarate normal densty wth mean 0 matrx 0. and covarance
4 194 IV. MODEL ESTIMATION Estmaton and verfcaton of all the models has been performed usng SAS system. In order to obtan objectvely correct results, we have used a pror dstrbutons that have a mnmal mpact on a posteror dstrbuton. Moreover, we have nether results of statstcal modelng for the nvestgated tme perod nor the data for the entre country. Stll, only credble nformaton may mprove the qualty of model estmaton. Therefore, non-nformatve ndependent normal pror dstrbutons have been used for all regresson parameters to estmate all 6 the models: p ~ N0,10 I. The estmated models have been evaluated to assess the convergence of generated Markov chans. Inference n Bayesan analyss under unchecked convergence for some model parameters may result n wrong conclusons. Usng Geweke s test (Geweke, 1992) we have found that there s no ndcaton that the Markov chan has not converged for all the parameters of nvestgated models, at any sgnfcance level. The same result has been obtaned for Hedelberger-Welch test (Hedelberger and Welch, 1983), whch conssts of two parts.e. a statonarty test and a halfwdth test. The halfwdth test addtonally reports whether the sample sze s adequate to meet the requred accuracy for the mean estmate. Thus, t can be assumed that the obtaned posteror samples are approprate for statstcal nference. The results of model estmaton have been summarzed n table 1. Table 1. Posteror sample mean and nterval statstcs Parameter Mean Hghest Probablty Densty Interwal ( ) Exp(Mean) Exp(-Mean) Intercept Sex Educaton Educaton Educaton Educaton Educaton Regon Regon Regon Regon Regon Age
5 Bayesan Exponental Survval Model n the Analyss 195 Basng on the hghest probablty densty nterval (Bolstad, 2007), statstcally sgnfcant varables are sex, age and at least one level of other varables. V. SUMMARY AND CONCLUSIONS We obtaned that among varables chosen to model: sex, martal status, educaton level, nformaton about contnung an educaton, regon of Poland and the age at the moment of research, only two varables have been determned to be statstcally nsgnfcant: martal status and nformaton about contnung an educaton. In the case of frst determnant, prevous assumptons that unmarred people have more chance of fndng a job, were not confrmed. Informaton about contnung an educaton has turned out to be statstcally nsgnfcant, but we can state that by mprovng the educaton status, one can ncrease the chances for fndng a job n the future. We obtaned that the ndvduals, who had educaton hgher than prmary, have more chance of fndng a job. The persons havng a secondary professonal educaton have 98.7% more chance of fndng a job comparng to those who have attended prmary schools only, the persons havng a hgher educaton have ths chance more than twce as hgh as the members of the former group. Our research confrms the prevous assumpton and s consstent wth the results of other studes (Daras and Jerzak, 2005). The results for sex varable also confrm our prevous speculatons, we obtaned that men have 42.6% more chance of fndng jobs than women. The results of other research (Daras and Jerzak 2005; Socha and Sztanderska, 2000) ndcate worse stuaton of women n the labor market, even f women are better educated and are more actvely searchng for a job. Moreover employers are more lkely to hre men, than women, due to the role women play n ther famles.e. they more frequently take care of chldren. One of mportant factors nfluencng the unemployment duraton s age; we obtaned that the chances of fndng a job decrease by about 1.5% as the age of a respondent ncreases by one year. Accordng to other researchers (Daras and Jerzak, 2005) the unemployment rate s also dependant of the age, the lowest chances for fndng a job have persons aged over 44. The results for regon of Poland varable are that only one level of ths varable s statstcally sgnfcant beng central regon.e. provnces ódzke mazowecke. We obtaned that the mean unemployment duraton for the resdents of ths regon s shorter by 37.4% then for the resdents of the east regon. The model appled n ths artcle enables the dentfcaton of demographc and soco-economc factors nfluencng the unemployment duraton. The advantage of survval models s the fact they nclude all the hstory of an ndvdual. But these models demand data, whch are frequently not provded by commonly made surveys.
6 196 REFERENCES Bernardo J.M., Smth A.F.M. (2004), Bayesan Theory, John Wley & Sons, New York. Blossfeld H.P., Hamerle A., Mayer K. (1989), Event hstory analyss, Statstcal theory and applcaton n the socal scences, Hllsdale, NJ: L. Erlbaum. Blossfeld H.P., Rohwer G. (1995), Technques of event hstory modelng, New approaches to causal analyss, Hllsdale, NJ: L. Erlbaum. Bolstad W.M. (2007), Introducton to Bayesan statstcs, John Wley & Sons, New York. Coller W. (2003), The mpact of demographc and ndvdual heterogenety on unemployment Duraton: A regonal study, Studes n Economcs, Daras T., Jerzak M. (2005), Wpyw cech spoeczno-demografcznych osób bezrobotnych na molwo znalezena pracy, Materay Studa, Zeszyt nr 189. Drobn S., E. Frtczak (2001), Employment patterns of marred women n Poland, Careers of couples n contemporary socety, New York. Gelman A., Carln J.B., Stern H.S., Rubn D.B. (2000), Bayesan data analyss, Chapman & Hall/CRC, London. Geweke J. (1992), Evaluatng the accuracy of samplng-based approaches to calculatng posteror moments. In Bernardo J., Berger J., Dawv A., Smth A., Bayesan Statstcs, 4, Hedelberger P., Welch, P. (1983), Smulaton run length control n the presence of an ntal transent, Operaton Research, 31, Ibrahm J.G., Chen M-H, Snha D. (2001), Bayesan survval analyss, Sprnger-Verlag, New York. Socha M., Sztanderska U. (2000), Strukturalne podstawy bezroboca w Polsce, PWN, Warszawa. BAYESOWSKI WYKADNICZY MODEL PRZEYCIA W ANALIZIE DETERMINANT DUGOCI CZASU POZOSTAWANIA BEZ PRRACY Celem nnejszego opracowana jest dentyfkacja czynnków demografcznych oraz spoeczno-ekonomcznych wpywajcych na dugo czasu pozostawana bez pracy. Zbór danych wykorzystany w badanu pochodz z bada Gównego Urzdu Statystycznego Budety Gospodarstw Domowych Do analzy determnant dugoc czasu pozostawana bezrobotnym wykorzystano bayesowsk wykadnczy model przeyca. W estymacj modelu wykorzystano metody Monte Carlo oparte na acuchach Markowa, a w szczególnoc próbnk Gbbsa. W wynku przeprowadzonej analzy otrzymano, e wród wybranych do modelowana zmennych objanajcych: pe, stan cywlny, pozom wyksztacena, nformacja o tym, czy respondent nadal s doksztaca, regon Polsk, który zameszkuje respondent oraz wek w momence badana, tylko dwe okazay s statystyczne nestotne: stan cywlny oraz nformacja o tym, czy respondent nadal s doksztaca.
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