Estimating Labour Market Transitions and Continuations using Repeated Cross Sectional Data

Transcription

1 Esimaing Labour Marke Transiions and Coninuaions using Repeaed Cross Secional Daa Pierre Brochu Universiy of Oawa May 2009 Absrac This paper proposes a populaion cohor approach for esimaing labour marke coninuaions (or ransiions) using repeaed cross secional daa. I show ha he coninuaion probabiliy can be wrien as a raio of wo uncondiional means ha do no condiion on pas labour marke saus. As such, I can consruc a consisen se of sandard errors ha accoun for he full variabiliy of cross secional daa. Using Curren populaion Survey daa, I show ha exising mehods end o sysemaically underesimae he rue sandard errors which can lead he researcher o incorrecly conclude ha job sabiliy had decreased. JEL Classificaion: C41, J64 Key Words: repeaed cross secion daa, duraion analysis, employmen, job sabiliy Deparmen of Economics, Universiy of Oawa, 55 Laurier Avenue Eas, Desmarais Building, Room 10105, Oawa (Onario) K1N 6N5, Canada. Phone: (613) ex Fax: (613) pbrochu@uoawa.ca 1

2 1 Inroducion There is a long radiion of exploring labour marke ransiions in economics. Alhough he unemploymen-employmen ransiion has been he mos frequenly explored, oher ransiions or coninuaions have also been examined, such as he ransiion ou of he labour force (e.g. Jones and Riddell (1999)) and he coninuaion of a job (job sabiliy, e.g. Brochu (2008); Heisz (2005); Neumark, Polsky, and Hansen (1999)). While using panel daa o esimae hese labour marke ransiions is generally he preferred approach, here are circumsances where ha approach is problemaic. For example, limied hisorical coverage (Canadian panels) makes i impossible o differeniae beween cyclical and secular changes in job sabiliy. 1 Wih he absence of his differeniaion, one canno address he real quesion of ineres in he job sabiliy lieraure: how and why has job sabiliy changed? 2 In such insances, repeaed cross secional daa ses offer a valid alernaive. In his paper, I propose a populaion cohor approach for esimaing he coninuaion (or ransiion) probabiliy when using repeaed cross-secion daa. The proposed non-parameric approach is empirically racable, and is idenifying assumpions are relaively mild and easy o inerpre. 3 The approach akes advanage of he fac ha repeaed cross secional daa ses, like he Curren Populaion Survey (U.S.) and he Labour Force Survey (Canada), are represenaive of he counry s populaion. The exising empirical labour lieraure ha examines coninuaion probabiliies using nonparameric mehods have applied (or approximaed) panel ools when faced wih cross secional daa (e.g. Neumark, Polsky, and Hansen (1999)). Approximaing panel mehods makes he researcher focus exclusively on a specific group of individuals; for he job sabiliy lieraure which examines he employmen-coninuaion, is employed workers. Ye, heir cross secional samples of employed workers are drawn from a populaion ha changes over ime - some workers will lose heir jobs while ohers will find employmen. As such, researchers are faced wih he difficul ask of esimaing one ransiion probabiliy using samples drawn from wo differen populaions. The key insigh of his paper is ha by using a populaion cohor approach, one can wrie he labour marke coninuaion (or ransiion) as a raio of wo means; means ha do no condiion on pas labour marke saus. Based on his resul, I propose a cross secional esimaor for he coninuaion probabiliy, and anoher for is sandard errors. Using he proposed populaion cohor framework, I also re-examine mehods used in he empirical labour lieraure (e.g. job sabiliy lieraure). Through his lens, I clearly idenify he full 1 See Brochu (2006) for a deailed discussion of panel daa limiaions when exploring for changes in job sabiliy using Norh American daa. 2 This problem is no confined o Norh America daa. Researchers (e.g. Güell and Hu (2006)) ha have sudied he Spanish labour marke faced similar difficulies. Panel daa only became available afer significan labour reforms, i.e. he inroducion of fixed erm conracs. 3 See Moffi (1993) and Verbeek and Vella (2004) for a discussion of oher approaches o esimaing dynamic models using repeaed cross secions. 2

3 se of underlying assumpions of he radiional non-parameric esimaor and provide a consisen esimaor for is sandard errors boh of which fill gaps in he lieraure. I show wih Curren Populaion Survey daa ha he exising approaches end o underesimae he rue sandard errors because hey do no capure he full variabiliy of cross secional daa. I also show ha his maers a he inference sage; one can (incorrecly) conclude ha job sabiliy has changed when in fac i has no. Finally, a researcher may prefer he use of repeaed cross secions even when panel daa is available. Repeaed cross secions like he Curren Populaion Survey (CPS) and he Canadian Labour Force Survey (LFS) have much larger sample sizes ha allow for a more deailed analysis (e.g. Baker (1992)); one can focus on more narrowly defined groups wihou having o worry as much abou small samples. Depending on he quesion (and group) of ineres, panel ariion may become a significan problem. One can, for example, expec younger and more mobile workers o have higher ariion raes. For hese groups, a self-selecion bias may arise. The proposed esimaor, however, does no face his difficuly i is designed for cases where he same individual canno be followed over ime. The remaining secions of his paper are divided as follows: Secion 2 provides a discussion of sandard cross secional approaches; Secion 3 proposes an alernaive framework ha is suiable o cross secional analysis; Secion 4 re-inerpres he exising mehods using he proposed framework; Secion 5 provides some empirical examples; and finally, Secion 6 provides some final remarks. 2 Exising Approach Assume he researcher is ineresed in he reenion rae of an a-risk group, say, individuals wih ime-invarian characerisics c who have been employed for s periods a ime. 4 Following he exising cross secional lieraure (e.g. Neumark, Polsky, and Hansen (1999); Heisz (2005)), one can presen he reenion rae simply as he fracion of a-risk individuals in he populaion ha remains wih he same employer in he nex period R = N s+1,c +1 N (1) where N is he number of people in he populaion ha have ime-invarian characerisics c who have been unemployed for s periods a ime. 5 Alhough one canno follow individuals over ime wih repeaed cross secional daa, researchers (e.g. Baker (1992); Neumark, Polsky, and Hansen (1999)) ake advanage of he fac ha base weighs of represenaive cross secions, like he Labour Force Survey (Canada) and he Curren Populaion Survey (U.S.), sum up o heir respecive populaions. The cross secional esimaor 4 The framework can easily be generalized o explore any ransiion or coninuaion probabiliy. 5 I also follow he noaion of he lieraure by reaing N as a coun measure. 3

4 akes he form ˆQ = ñs+1,c +1 ñ (2) where ñ is he sum of he base weighs of all individuals wih characerisics c who have been employed s periods wih he same employer as of period. By using weighs as couns, he denominaor (numeraor) of Equaion (2) direcly esimaes he denominaor (numeraor) of Equaion (1). Direcly esimaing populaion couns is very inuiive, and i is reasonable o hink ha he accuracy of he esimaor will improve wih larger samples; ye, his canno be proven in any saisical sense. In addiion, one canno lay bare all underlying idenifying assumpions wihou such a proof. A clear undersanding of he consisency requiremens is criical considering ha an imporan selling poin of cross secional daa ses like he CPS and LFS have been heir large sample sizes. Mos imporanly, he lack of precision carries over o he inference sage. Given he funcional form of he esimaor, here is no sandard way o consruc sandard errors. In Secion 5, I use CPS daa o show ha he exising approaches end o sysemaically underesimae he rue sandard errors because hey do no capure he full variabiliy of cross secional daa. 3 Proposed Approach In his secion I propose an alernaive represenaion of he reenion rae (one ha differs from Equaion (1)). Based on proposed represenaion, I sugges an esimaor for he reenion rae (and for is sandard errors) ha is well-suied o cross secional daa. I sar by assuming a populaion cohor. 6 Having a populaion cohor simply means ha here is more han one period of informaion for each individual in he populaion. I does no impose any resricions on he ype of sample. One could have a panel daase where each sampled individual is followed over ime, or repeaed cross secions where each sample (cross secion) is drawn from he same populaion, bu a differen momens in ime. 7 Le X i be a vecor of ime invarian characerisics of individual i in period. Furher, le T EN i represen he lengh of enure, i.e. he number of periods he worker has been employed wih he same employer as of period. The reenion rae for a randomly chosen individual wih 6 Oher researchers (e.g. Deaon (1985), Moffi (1993)) who have esimaed dynamic models using repeaed cross secion have also relied on his assumpion. 7 Said differenly, I assume ha he cross secions are drawn from he same pool of individuals, wih each cross secion consising of is own random draw. 4

5 characerisics c who has been employed for s periods a ime, R, is formally expressed as 8 R = P rob(t EN i+1 = s + 1 T EN i = s, X i = c) (3) Given is condiional srucure, Equaion (3) is a good saring poin for panel daa. One can condiion on an individual working in he firs period of a panel, and herefore esimae he sample analog of Equaion (3). Wih repeaed cross secions, however, his is no possible since one canno follow individuals over ime. I propose an alernaive represenaion ha is more conducive o cross secional analysis. One can rewrie Equaion (3) as 9 R = E( 1[T EN i+1 = s + 1, X i+1 = c] ) E( 1[T EN i = s, X i = c] ) (4) where 1[ ] is an indicaor funcion ha equals 1 if he condiions inside he bracke hold, and zero oherwise. The proof can found in Appendix A.1. One can esimae his reenion rae using wo repeaed cross secions by replacing he populaion means wih heir respecive sample analogs, i.e. ˆR = where n is he sample size in year. n+1 i=1 1[T EN i+1 = s + 1, X i+1 = c]/n +1 n i=1 1[T EN i = s, X i = c]/n (5) Equaion (4) is a key insigh of his paper. I is conducive o cross secional analysis because he numeraor does no condiion on period evens. This holds rue because an individual who has been wih he same employer for s + 1 periods as of ime + 1, had o have been wih he same employer in he previous period (and have one less period of enure). 10 Condiioning on only ime-invarian characerisics is a sufficien bu no a necessary condiion o be able o presen he reenion rae as a raio of wo means. One needs o be able o follow he a-risk group over ime. 11 More precisely, one needs o be able o infer - from a period + 1 cross secion - wheher an individual who remains wih he same employer as of period + 1, would have been par of he a risk group in period. As such, one can esimae a broad range of reenion raes. One can no only condiion on gender, race, educaion, bu also on age, indusry and occupaion. I elaborae on he laer hree caegories below. Assuming age invariance for a 1-year reenion rae, for example, would be unenable This is simply anoher way of wriing Equaion (1). 9 For ease of presenaion, I assume ha he [, +1] ime span is sufficienly shor as o rea age as ime-invarian. I laer show how o address ageing when and and + 1 are sufficienly far apar. 10 A similar argumen would hold rue if one were looking a oher labour marke coninuaion/ransiion probabiliies. 11 For R, he a-risk group consiss of individuals wih characerisics c who have been wih he same employer for s periods as of ime. 12 The Canadian job sabiliy lieraure has ypically focussed on 1-year raes (e.g. Heisz (2005); Brochu (2008)). For- 5

6 unaely, he above mehod can easily deal wih ageing. If one can idenify he age of a worker in year + 1, we know ha he was 1 year younger in year. As such, one can consisenly idenify he a-risk group over ime. Assume, for example, one is ineresed in he reenion rae of year old men ha have been wih he same employer 10 years as of The idenifying assumpion only requires ha we o be able o idenify - in he 1989 cross secion - male workers ha are 21 o 30 years old wih 11 years of job enure. Indusry saus is also no ime-invarian (from he researcher s poin of view). Indusry affiliaion is job relaed, and as such, one canno ypically idenify (in he daa) he indusry o which he unemployed worker previously belonged. Bu as previously menioned, one only needs o idenify wheher workers who coninued wih he same employer in period + 1 would have been in he a-risk group in he previous period. To do so, one mus assume ha job enure (i.e. he employer-employer relaionship) ends if he individual swiches indusry. I is a similar sory for occupaion. A change in occupaion classificaion mus signal he end of he employer-employee relaionship. This is a relaively mild assumpion as long as he occupaions are no oo narrowly defined. I should be emphasized ha he populaion cohor assumpion implies ha he samples (and heir underlying populaion) mus include boh workers and non-workers. Resricing he sample o only include employed workers (as is he case in he job sabiliy lieraure) would violae he populaion cohor assumpion. The pool of employed workers in he populaion does no remain consan over ime some individuals will lose heir jobs, while ohers will find employmen. The populaion cohor assumpion also requires ha one impose a differen age resricion for he period + 1 sample if he [, + 1] inerval is sufficienly large. The same ageing argumen ha affeced how one esimaed he reenion raes also applies o he sample. Assume one is ineresed in 1-year reenion raes of prime age (25-54) workers (and of is sub-groups). 13 The period sample will consis of all individuals 25 o 54 years of age. The period + 1 sample, however, should consis of 26 o 55 year olds. This will ensure ha he populaion cohor assumpion is me ha we are drawing from he same underlying populaion. A populaion cohor approach is possible wih boh he LFS and he CPS, because hey are boh represenaive of heir respecive populaion. The LFS and CPS are also carried ou a frequen inervals - on a monhly basis. As a resul, slippage (changes in populaion) due o immigraion, emigraion or deahs will be minimal if one carries ou monhly reenion raes. In Secion 4, I show how he proposed esimaor can be adjused when he cross secions are furher apar as is he case in he American lieraure. Given he simple funcional form of he proposed reenion rae esimaor, i.e. ˆR, one can easily generae consisen sandard errors. In Appendix A.3 I show how o do so by firs deriving For daa reasons, he American lieraure has focussed on 4-year (and 8-year) raes. The use of cross secions ha are 4-year (or 8-year) apar inroduced a furher complicaion: i becomes unreasonable o assume a populaion cohor. I will address his issue in he nex secion. 13 One could, for example, also be ineresed in he reenion rae of male workers 20 o 29 years of age. 6

7 he asympoic properies of ˆR. The sandard error esimaor is consisen because i accouns for he full variabiliy of cross secional daa. Finally, applying Equaion (5) o survey daa where he probabiliy of being seleced is no he same across observaions is sraighforward. One replaces he sample means wih weighed ones. 14 This reflecs he radiional use of weighs where he weighs are only used o reflec he varying probabiliy of selecion. 4 Links Beween Mehods In his secion I re-examine he exising reenion rae esimaor wihin a populaion cohor framework. I show ha his esimaor can be expressed as a funcion of sample means. As such, I can idenify he full se of underlying assumpions of his esimaor, and provide a consisen esimaor for is sandard errors. The American job sabiliy lieraure (e.g. Swinneron and Wial (1995); Neumark, Polsky, and Hansen (1999)) esimaed 4-year reenion raes. I would be unenable o assume ha he wo cross secions - drawn 4-years apar - come from he same underlying populaion; he American populaion will have changed due o deahs, emigraion and immigraion. 15 Forunaely, he populaion-cohor framework can be exended o deal wih such composiional changes. Assuming ha composiional changes break he enure spell, one can wrie he reenion rae as a funcion of wo populaion means R = adj E( 1[T EN i+1 = s + 1, X i+1 = c]) E( 1[T EN i = s, X i = c]) (6) where adj is he populaion growh (or adjusmen) facor. If, for example, he populaion size increased by 20%, he populaion growh facor would be 1.2. The argumens are similar o hose made in Secion 3 when I relaxed he ime-invarian characerisics assumpion. One needs o be able o idenify in he he American populaion who coninued wih he same employer in year +4, and would have been in he a-risk group in year. An inuiive proof is lef o Appendix A.3. A deah easily mees he idenifying assumpion; changes due o immigraion and emigraion, however, require furher aenion. One requires ha he migran change employer upon arrival in his new counry. This empirical sraegy would be appropriae if job ransfers (where workers says wih same employer) are no he driving force behind migraion paerns. The exising approach, i.e. Equaion (2), is in fac an esimaor of R as presened above. This become apparen if one rewries Equaion (2) as 14 Where he weighs are normalized o sum up o 1 in each sample period. 15 Birhs are no problemaic because we focus on he working age populaion. Assume one was ineresed in 4-year reenion raes for prime aged (25-54) workers. The year sample would consis of individuals aged 25 o 54, while he year + 4 sample would only look a a hose 29 o 58 years of age. 7

8 ( ˆQ = adj ˆ n+1 ) i=1 nw i+1 1[T EN i+1 = s + 1, X i+1 = c]/n +1 n i=1 nw i1[t EN i = s, X i = c]/n (7) where nw i is he normalized base weigh of individual i in year, 16 and bw i represening he base weigh). 17 ˆ adj = P n +1 i=1 bw i+1 P n i=1 bw i (wih Given ha he sum of he base weighs add up o he arge populaion in boh he CPS and Canadian LFS, adj ˆ is an esimae of he populaion growh. The second erm of Equaion (7) is simply he weighed version of ˆR (which was discussed a he end of Secion 3). By rewriing he exising esimaor as a funcion of he proposed one, I can make wo conribuions. Firs, I can clearly idenify he underlying assumpions of his esimaor - namely ha changes in populaion break he enure spell. Second, I can easily consruc consisen sandard errors. They will be similar o hose of ˆR, bu wih an adjusmen made for he populaion change. 18 In he nex secion, I use an empirical example o show ha exising approaches o esimaing sandard errors are downward biased, i.e. ha hey do no fully accoun for he variabiliy inroduced by no being able o follow individuals over ime. I also show ha his can lead he researcher o incorrecly conclude ha job sabiliy has changed. 5 Empirical Examples In his secion, I empirically compare exising and proposed approaches using wo large scale daa ses: he Canadian LFS and he U.S. CPS. This secion is divided ino wo pars. In he firs par, I focus on he reenion rae esimaors, and rely on repeaed cross secions from he LFS files o illusrae he differences. The second par focusses on he sandard errors. Using he CPS, I show ha he bias of exising approaches o esimaing sandard errors can lead he researcher o falsely rejec he null hypohesis of no change (or difference) in he reenion rae. 5.1 Example 1: LFS daa The LFS is a large monhly household survey of approximaely 54,000 households per monh, wih a focus on gahering informaion abou labour marke aciviies of Canadians. The LFS is a rich source of enure daa. As par of heir regular quesionnaire, respondens are asked when hey sared working for heir presen employer. Table 1 compares reenion rae esimaors using he maser LFS files. 19 I presen esimaes of he 1-year reenion rae for selec groups in he year 16 Where he base weighs are normalized o sum up o 1 in each sample period. 17 In Appendix A.4., I show ha Equaion (2) and (7) are numerically equivalen. 18 If one reas âdj as a consan, one can easily show ha se( ˆQ ) = âdj se( 19 These files were accessed on sie a he Briish Columbia Ineruniversiy Research Daa Cenre (BCIRDC). The BCIRDC is run and sponsored by he Universiy of Briish Columbia, Universiy of Vicoria and Simon Fraser Universiy, in collaboraion wih Saisics Canada. ˆR ) 8

9 Two imporan conclusions can be drawn from Table 1. One, he wo mehods generae very similar 1-year reenion rae esimaes. As Equaion (7) indicaes, he wo esimaes will only differ by a scaling facor, i.e. he populaion growh facor. Table 1 shows he 2000 growh facor only slighly exceeded 1. As a robusness check, I esimaed he populaion growh facor for each year over he period. The growh facor averaged over his period, and was close o 1 in all years despie he fac ha he Canadian populaion has increased over ime due o immigraion. This is because one is only looking a wheher an individual remains wih he same employer in he nex year, and no, say, 10 years from now. The second conclusion is ha he sandard errors are relaively small. The loss in efficiency due o he fac ha one canno follow individuals over ime in a cross secional approach is compensaed by he large samples of he LFS. As a resul, he he probabiliy esimaes are very precise. Boh ses of sandard errors were esimaed using proposed mehods. 5.2 Example 2: CPS daa Wihin a reenion rae approach, esing for differences in job sabiliy across ime or groups is sraighforward only a single resricion needs o be esed. For ease of exposiion, I focus on ime differences; he argumens are similar when esing across groups. The null hypohesis is H 0 : R j R 1 = 0, where R j R 1 is he difference in reenion rae over a j 1 period. I use a -es approach. The -saisic, n, is 21 n = ˆR j ˆR 1 (8) ˆVR j R /n 1 where ˆV R j R is he esimaor of Avar( ˆR 1 j ˆR 1 ). The sandard errors esimaor mus be able o accoun for he fac ha one does no follow individuals over ime - as is he case wih he proposed approach. The lieraure has used wo approaches. The firs approach (e.g. Swinneron and Wial (1995); Diebold, Neumark, and Polsky (1997)) apply a sandard error esimaor designed for panel daa o he cross secional case, i.e. i implicily assumes ha one can follow individuals over ime. The second approach which was firs proposed by Neumark, Polsky, and Hansen (1999), and subsequenly applied o Canadian daa by Heisz (2005), reas he denominaor of ˆQ in Equaion (2) as a consan. 22 As such, hey fail o 20 Following Brochu (2008), all coninuaion probabiliies condiion on being 20 o 54 years of age. This imposes he following sample resricions: he 2000 sample only included hose 20 o 54 years of age, while he 2001 sample was resriced o individuals 21 o 55 years of age. See Brochu (2006) for more deails. 21 To simplify he presenaion I assume ha he cross secions are all of size n. The asympoic properies of he reenion rae differenial are lef o Appendix A More precisely, hey presen heir reenion rae esimaor as a raio of wo esimaed couns (see Equaion (2) and rea he numeraor as a random variable, bu he denominaor as a consan when hey consruc heir sandard error esimaor. 9

10 accoun for he full variabiliy of he cross secional approach. The sandard errors esimaor mus also be able o accoun for he possible correlaion beween ˆR j and ˆR 1. More precisely, ˆR 2 and ˆR 1 may be correlaed since boh he denominaor of ˆR 2 and he numeraor of ˆR 1 are funcions of he same (year 2) observaions. 23 By allowing boh he numeraor and denominaor in Equaion (3) o have sampling disribuions, he proposed approach can easily generae he necessary covariance erm. This is no he case for exising mehods. The Neumark-Polsky-Hansen (NPH) mehod, for example, rules ou he possibiliy of any correlaion beween ˆR 2 and ˆR 1. I use CPS daa o illusrae how he choice of sandard errors esimaor can maer a he inference sage. As wih he LFS, he American CPS is a large monhly household which asks respondens abou heir labour marke aciviies. Bu conrary o he LFS, a enure quesion is no par of he regular CPS quesionnaire i is only included in selec supplemens. I herefore rely on 4-year reenion raes as was previously done in he American job sabiliy lieraure. Finally, I rely on he Q esimaor (insead of R ). Assuming no change in he underlying populaion would be oo resricive an assumpion. Tables 2 and 3 examine changes in 4-year reenion raes from 1996 o 2000 for males and females, respecively. 24 Sandard errors are calculaed for he NPH mehod, he DNP mehod, 25 he proposed mehod, and he proposed mehod wih no covariance erm. In all cases, weighs were used o make a clearer comparison of he various mehods. Robusness checks for differen years and sub-populaions indicae ha weighs do no significanly affec he resuls. Tables 2 and 3 indicae ha accouning for he covariance erm can increase or decrease he sandard errors. 26 Even when he covariance erm is negaive, a consisen paern emerges wih he proposed mehod generaing sandard errors consisenly larger han eiher DNP or NPH mehods. This paern was found o be robus for oher ime periods and oher sub-populaions. The proposed mehod generaes sandard errors ha are up o 172.9% larger han he DNP esimaes and up o 55.6% larger han he NPH esimaes. From Tables 2 and 3 one can observe a sysemaic differenial in he sandard errors; he gap is larger for longer enured groups - groups wih higher job sabiliy. In general, he exen o which he NPH mehod underesimaes he correc sandard errors will be correlaed wih he size of he employmen coninuaion probabiliy. 27 As a resul, he DNP and NPH approaches o esimaing sandard errors may lead he researcher o falsely rejec he null hypohesis of no change in job sabiliy. Calculaing -saisics for males 23 A similar difficuly occurs when esing across groups, say, groups A and B. The numeraors (and denominaors) of Rj A and Rj B may also be correlaed. 24 All U.S. coninuaion probabiliies condiion on being a leas 16 years of age, and no being self-employed. 25 The DNP mehod, refers o he applicaion of he longiudinal sandard errors esimaor o cross secional daa. Diebold, Neumark, and Polsky (1997) may no have been he firs o use he mehod wih cross secional daa, bu hey were one of he firs o provide a deailed explanaion of he approach. 26 From Equaions (33) and (35) one can see ha he covariance erm will be posiive if and only if he covariance beween D i2 and N i2 is also posiive. 27 Focussing on Equaion (14) illusraes his poin. Condiioning on a sample disribuion for D i1, a larger E(N i+1 ) is associaed wih a larger firs erm; a variance erm no accouned for by he NPH mehod. 10

11 wih 12+ years of enure illusraes his poin. Using eiher he DNP or NPH mehods, one srongly rejecs he null hypohesis a he 5% significance level. In fac, my mehod suggess ha he null hypohesis should no be rejeced, no even a he 10% level. 6 Conclusion In his paper, I propose a populaion cohor approach for esimaing labour marke ransiions. Based on his framework, I propose a non-parameric esimaor for he coninuaion probabiliy and anoher for is sandard errors. Using he populaion cohor framework, I also re-examine exising cross secional mehods. I idenify he underlying assumpions of he non-parameric approach used in he job sabiliy lieraure, and propose a consisen esimaor for is sandard errors. Finally, I show using CPS daa ha he choice of mehod (for consrucing sandard errors) can make a difference a he inference sage. In paricular, ha using exising mehods can lead o he researcher o conclude ha job sabiliy has changed when in fac i has no. 11

12 A Appendix A.1 Proposiion 1 The reenion rae of a worker wih ime-invarian characerisics c who has been employed for s periods a ime can be expressed as R = E( 1[T EN i+1=s+1,x i+1 =c] ) E( 1[T EN i =s,x i =c] ). proof: A.2 R = P rob(t EN i+1 = s + 1 T EN i = s, X i = c) (9) = P rob(t EN i+1 = s + 1, T EN i = s, X i = c) P rob(t EN i = s, X i = c) (10) and since T EN i+1 = s + 1 implies T EN i = s, one can rewrie R as = P rob(t EN i+1 = s + 1, X i = c) P rob(t EN i = s, X i = c) (11) = P rob(t EN i+1 = s + 1, X i+1 = c) P rob(t EN i = s, X i = c) (12) = E( 1[T EN i+1 = s + 1, X i+1 = c] ) E( 1[T EN i = s, X i = c] ) (13) For ease of exposiion, define N i+1 = 1[T EN i+1 = s + 1, X i+1 = c] and D i = 1[T EN i = s + 1, X i = c]. Proposiion 2 Assuming iid samples for each year ha are drawn from a populaion cohor, in- = 1, hen n ( ˆR R ) d N(0, V ) where V dependence across years, and lim n,n +1 is wih n n +1 V = φ 2 1V (D i ) + φ 2 2V (N i+1 ) (14) φ 1 = E(N i+1) [E(D i )] 2, φ 2 = 1 E(D i ) (15) proof: a) (consisency) Apply he Lindberg-Levy Cenral Limi Theorem and use he resul of Proposiion 1 b) (Asympoic normaliy) n +1 i=1 n N i+1 /n +1 p E(Ni+1 ) (16) p D i /n E(Di ) (17) i=1 12

13 V D n+1 Le ˆN +1 = n 1 +1 i=1 N i+1, N +1 = E(N i+1 ) and V N+1 = V (N i+1 ), and define ˆD, D and in a similar fashion. n ( ˆR R ) = ( ) ˆN+1 n N +1 (18) ˆD D = ( n ˆN +1 N +1 )D ( ˆD D )N +1 (19) D ˆD n n n+1 +1 ( ˆN +1 N +1 )D n ( ˆD D )N +1 = + o p (1) (20) D 2 = φ 1 n ( ˆD D ) + φ 2 n+1 ( ˆN +1 N +1 ) + o p (1) (21) d N ( 0, φ 2 1V D + φ 2 ) 2V N+1 (22) Replacing he populaion momens in Equaion (22) wih he corresponding sample analogs generaes a consisen esimaor for he asympoic variance. Taking he square roo of he esimaed variance will generae he sandard errors. A.3 Proposiion 3 Assume ha he composiion of a counry s populaion changes from period o period + 1. Furher assume ha hese composiional changes break (or inerrup) he spell of ineres. The reenion rae can be expressed as R = adj E( 1[T EN i+1 = s + 1, X i+1 = c] ) E( 1[T EN i = s, X i = c] ) (23) where adj is he populaion growh facor. proof: To ease he presenaion, I assume ha he change in populaion is due o he arrival of one new immigran in year + 1. Similar argumens would hold rue for oher populaion changes. Wihou loss of generaliy, assume a populaion of size N in year, and N + 1 in year + 1. Order he year + 1 populaion so ha he new immigran is las. By Proposiion 1, he reenion is N R i=1 = 1[T EN i+1 = s + 1, X i+1 = c]/n N i=1 1[T EN i = s, X i = c]/n (24) By assuming ha he change in populaion resuls in breaks in he spell of ineres, one can conclude ha 1[T EN N+1,+1 = s + 1, X N+1,+1 = c] = 0. As a resul, R can be rewrien as ( ) N + 1 N+1 i=1 = 1[T EN i+1 = s + 1, X i+1 = c]/n + 1 N N i=1 1[T EN i = s, X i = c]/n adj E(1[T EN i+1 = s + 1, X i+1 = c]) E(1[T EN i = s, X i = c]) (25) (26) 13

14 A.4 Proposiion 4 Given repeaed cross secions where he base weighs sum up o he arge populaion, hen ( Q = adj ˆ n+1 ) i=1 nw i+1 1[T EN i+1 = s + 1, X i+1 = c]/n +1 n i=1 nw (27) i1[t EN i = s, X i = c]/n proof: The exising cross secional esimaor is presened as a raio of wo populaion couns. I akes form Q = ñs+1,c +1 ñ where ñ is he sum of he base weighs of all individuals wih characerisics c who have been employed speriods as of period. For ease of exposiion, define N i+1 = 1[T EN i+1 = s + 1, X i+1 = c] and D i = 1[T EN i = s + 1, X i = c]. One can rewrie his esimaor as A.5 Q = = = n+1 i=1 bw i+1n i+1 n (28) i=1 bw id i (29) n+1 i=1 bw i+1 n i=1 bw i n+1 i=1 bw i+1 n i=1 bw i n+1 adj = ˆ n+1 i=1 bw i+1 Pn +1 i=1 bw i+1/n +1 N i+1 /n +1 n (30) Pn bw i i=1 i=1 bw D i /n i/n n+1 i=1 nw i+1n i+1 /n +1 n i=1 nw (31) id i /n i=1 nw i+1n i+1 /n +1 n i=1 nw id i /n (32) Proposiion 5 Assuming iid samples for each year, samples of equal size, independence across years, and no change in populaion, hen n(( ˆR j ˆR 1 ) (R j R 1 ) d N(0, V ) where V depends on j, an ineger greaer han or equal o 2. Case 1: j = 2 Case 2: j 3 V = φ 2 1V (D i1 ) + φ 2 2V (N i2 ) + φ 2 3V (D i2 ) + φ 2 4V (N i3 ) + 2φ 2 φ 3 µcov(d i2, N i2 ) (33) V = φ 2 1V (D i1 ) + φ 2 2V (N i2 ) + φ 2 3V (D ij ) + φ 2 4V (N ij+1 ) (34) wih φ 1 = E(N i2) [E(D i1 )] 2, φ 2 = 1 E(D i1 ), φ 3 = E(N ij+1) [E(D ij )] 2, φ 4 = 1 E(D ij ) (35) and µ is he probabiliy ha a random chosen person in he populaion from which D i is drawn, is also par of he populaion from which N i is drawn. 14

15 proof: For ease of noaion le ˆN j = n 1 nj j i=1 N ij, N j = E(N ij ) and V Nj = V (N ij ), and define ˆD j, D j and V Dj in a similar fashion. Finally, le C 2 = Cov(D i2, N i2 ) Case 1: j = 2 n(( ˆR2 ˆR 1 ) (R 2 R 1 ) = (( ) ( )) ˆN3 n N 3 ˆN2 N 2 ˆD 2 D 2 ˆD 1 D 1 (36) = n ( ˆN 3 N 3 )D 2 ( ˆD 2 D 2 )N 3 D 2 ˆD2 n ( ˆN 2 N 2 )D 1 ( ˆD 1 D 1 )N 2 D 1 ˆD1 (37) = n ( ˆN 3 N 3 )D 2 ( ˆD 2 D 2 )N 3 D2 2 n ( ˆN 2 N 2 )D 1 ( ˆD 1 D 1 )N 2 D1 2 + o p (1) (38) = φ 1 n( ˆD1 D 1 ) φ 2 n( ˆN2 N 2 ) φ 3 n( ˆD2 D 2 ) + φ 4 n( ˆN3 N 3 ) + o p (1) (39) d N ( 0, φ 2 1V D1 + φ 2 2V N2 + φ 2 3V D2 + φ 2 ) 4V N3 + 2φ 2 φ 3 µc 2 (40) Case 2: j 3. The proof is similar o Case 1, wih one excepion. Since he four componens of he es saisics, i.e. ˆNj+1, ˆDj, ˆN2 and ˆD 1 are funcions of differen yearly samples when j 3, he covariance erm is zero. Replacing he populaion momens wih corresponding sample analogs generaes a consisen esimaor for each asympoic variance. 15

16 References Baker, M. (1992): Unemploymen Duraion: Composiional Effecs and Cyclical Variabiliy, American Economic Review, 82(1), Brochu, P. R. (2006): An Exploraion in Job Sabiliy, PhD Thesis, Universiy of Briish Columbia. (2008): Rising Job Sabiliy in he 1990s: he Impac of Composiional Change, Unpublished Manuscrip, Universiy of Oawa. Deaon, A. (1985): Panel Daa from Time Series of Cross-Secions, Journal of Economerics, 30, Diebold, F. X., D. Neumark, and D. Polsky (1997): Job Sabiliy in he Unied Saes, Journal of Labor Economics, 15(2), Güell, M., and L. Hu (2006): Esimaing he Probabiliy of Leaving Unemploymen Using Uncompleed Spells from Repeaed Cross-secion Daa, Journal of Economerics, 133(1), Heisz, A. (2005): The Evoluion of Job Sabiliy in Canada: Trends and Comparisons wih U.S. Resuls, Canadian Journal of Economics, 38(1), Jones, S. R. G., and W. C. Riddell (1999): The Measuremen of Unemploymen: An Empirical Approach, Economerica, 67(1), , Noes and Commens. Moffi, R. (1993): Idenificaion and Esimaion of Dynamic Models wih a Time Series of Repeaed Cross-secion, Journal of Economerics, 59, Neumark, D., D. Polsky, and D. Hansen (1999): Has Job Sabiliy Declined Ye? Evidence for he 1990s, Journal of Labor Economics, 17(4), S29 S64. New Swinneron, K. A., and H. Wial (1995): Is Job Sabiliy Declining in he U.S. Economy?, Indusrial and Labor Relaions Review, 48(2), Verbeek, M., and F. Vella (2004): Esimaing Dynamic Models from Repeaed Cross Secions, Journal of Economerics, 127,

17 Table 1: 1-year Employmen-Coninuaion Probabiliies: Canada, 2000 Group Specificaion Proposed Mehod Exising Mehod Populaion Growh Facor ( R 1 ) ( Q 1 ) ( adj ˆ 1 ) Overall (0.0028) (0.0029) Male (0.0050) (0.0051) Female (0.0052) (0.0053) Tenure less han 1 year (0.0070) (0.0071) 17

18 Table 2: U.S. 4-year Male Employmen-Coninuaion Probabiliies - Time Differenials Tenure Group Difference Sandard Errors Specificaion Mehod (0.0076)** DNP (0.0106) ** NPH (0.0116) ** proposed (0.0118) * proposed (no covariance erm) (0.0092) DNP (0.0133) NPH (0.0148) proposed (0.0151) proposed (no covariance erm) (0.0111)** DNP (0.0197)** NPH (0.0250)** proposed (0.0244)** proposed (no covariance erm) (0.0083)** DNP (0.0149)** NPH (0.0227) proposed (0.0194)* proposed (no covariance erm) oal (0.0045)* DNP (0.0060) NPH (0.0083) proposed (0.0073) proposed (no covariance erm) ** The esimaed difference is significan a he 5% level * The esimaed difference is significan a he 10% level 18

19 Table 3: U.S. 4-year Female Employmen-Coninuaion Probabiliies - Time Differenials Tenure Group Difference Sandard Errors Specificaion Mehod (0.0074)** DNP (0.0097)** NPH (0.0111)** proposed (0.0113)** proposed (no covariance erm) (0.0092) DNP (0.0125) NPH (0.0148) proposed (0.0150) proposed (no covariance erm) (0.0106)** DNP (0.0206)** NPH (0.0270)* proposed (0.0272)* proposed (no covariance erm) (0.0098)** DNP (0.0172)** NPH (0.0268)** proposed (0.0231)** proposed (no covariance erm) oal (0.0047) DNP (0.0060) NPH (0.0085) proposed (0.0075) proposed (no covariance erm) ** The esimaed difference is significan a he 5% level * The esimaed difference is significan a he 10% level 19