Vacancy Posting, Job Separation and Unemployment Fluctuations

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Vacancy Posting, Job Separation and Unemployment Fluctuations Regis Barnichon 2009-35 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Vacancy Posting, Job Separation and Unemployment Fluctuations (cid:3) Regis Barnichon Federal Reserve Board 20 July 2009 Abstract Thispaperstudiestherelativeimportanceofthetwomaindeterminantsofcyclicalunemployment (cid:135)uctuations: vacancy posting and job separation. Using a matching function to model the (cid:135)ow of new jobs, I draw on Shimer(cid:146)s (2007) unemployment (cid:135)ow rates decomposition and (cid:133)nd that job separation and vacancy posting respectively account for about 40and60percentofunemployment(cid:146)svariance. Whenconsideringhigher-ordermoments,I (cid:133)nd that job separation contributes to about 60 percent of unemployment steepness asymmetry, a stylized fact of the jobless rate. Finally, while vacancy posting is, on average, the most important contributor of unemployment (cid:135)uctuations, the opposite is true around businesscycleturningpoints,whenjobseparationisresponsibleformostofunemployment movements. JEL classi(cid:133)cations: E24, E32, J63, J64 Keywords: Gross Worker Flows, Job Finding Rate, Employment Exit Rate, Matching Function. (cid:3)I would like to thank Bruce Fallick, Shigeru Fujita, Mike Kiley, Chris Nekarda, Barbara Petrongolo, John M.Roberts,AysegulSahin,JaeW.Sim,AntonellaTutinoandseminarparticipantsforhelpfulsuggestionsand discussions. The views expressed here do not necessarily re(cid:135)ect those of the Federal Reserve Board or of the Federal Reserve System. Any errors are my own. E-mail: regis.barnichon@frb.gov 1

1 Introduction At the beginning of a recession, does unemployment go up because of fewer hirings, more job lossesorboth? Whatisthemoste⁄ectivepolicytomitigatethatincrease, a(cid:133)ringtax, ahiring subsidy or a combination of both? And why does unemployment increase faster than if goes down? The answers to these questions will depend for a large part on the determinants of unemployment (cid:135)uctuations. In this paper, I study the relative importance of the two main driving forces of cyclical unemployment: vacancy posting, i.e. (cid:133)rms(cid:146)recruiting e⁄orts, and job separation.1 An extensive literature has studied worker (cid:135)ows over the business cycle, and more recently Shimer (2007) focused on individual workers(cid:146)transition rates and concluded that unemploymentin(cid:135)owscontributemuchlesstounemployment(cid:135)uctuationsthanunemploymentout(cid:135)ows.2 This very in(cid:135)uential conclusion led to a recent modeling trend that assumes that the job separation rate (JS) is acyclical.3 However, a (cid:135)ow rates decomposition exercise may underestimate the contribution of JS because the job (cid:133)nding probability does not only depend on (cid:133)rms(cid:146)job openings but also on the number of unemployed workers. For example, if a higher separation rate leads to higher unemployment and to a lower job (cid:133)nding rate (JF), one may attribute the high unemployment to a low JF, even though the true cause was an increase in job separation. The (cid:133)rst contribution of this paper is to address the endogeneity of JF by using a matching functiontomodelthe(cid:135)owofnewjobs.4 Amatchingfunctionisextremelysuccessfulempirically and is used in almost all macroeconomic models that introduce equilibrium unemployment through search and matching frictions. By using a measure of vacancy posting to isolate (cid:135)uctuations in the job (cid:133)nding rate caused solely by changes in (cid:133)rms(cid:146)job openings, I (cid:133)nd that the contribution of the job separation rate to unemployment(cid:146)s variance is close to 40 percent instead of 25 percent using Shimer(cid:146)s (2007) methodology. Thus, not modeling the cyclicality 1In this paper, as in much of the literature on unemployment (cid:135)uctuations, I omit inactivity-unemployment (cid:135)ows, and focus only on employment-unemployment (cid:135)ows. See Shimer (2007) for evidence supporting this assumption. 2For work on gross worker (cid:135)ows, see Darby, Plant and Haltiwanger (1986), Blanchard and Diamond (1989, 1990),Bleakleyetal(1999),FallickandFleischman(2004)andFujitaandRamey(2006)amongothers. Shimer (2007), Elsby, Michaels and Solon (2009), Elsby, Hobijn, and Sahin (2008), Nekarda (2008) and Fujita and Ramey (Forthcoming) focus instead on transition rates between employment, unemployment and out of labor force. I abstract from important works on job (cid:135)ows (see Davis, Haltiwanger and Schuh, 1996) because some worker(cid:135)owsarenotmatchedbyacorrespondingjob(cid:135)ow. AsShimer(2007)pointsout,"(cid:133)rmscandestroyjobs by not hiring to replace workers who leave for other reasons" so that an increase in job destruction is in fact linked to a decrease in (cid:133)rm(cid:146)s hiring and in the job (cid:133)nding probability. 3See, among others, Blanchard and Gali (2008), Gertler and Trigari (2009) and Hall (2005). 4Fujita and Ramey (Forthcoming) also address the dynamic interactions between JS and JF by writing the contribution of each (cid:135)ow in a moving average form. The present paper follows instead a more structural approach to model the relationship between JS and JF. 2

of the job separation rate will lead researchers to understate the volatility of unemployment. ThissecondcontributionofthispaperistoextendthemethodpioneeredbyShimer(2007), Elsby, Michaels and Solon (2009) and Fujita and Ramey (Forthcoming) and study the determinants of unemployment(cid:146)s higher-order moments. I (cid:133)nd that JS plays an important role with respect to skewness and kurtosis. In particular, the steepness asymmetry of unemployment (cid:150)the fact that increases are steeper than decreases(cid:150)is due in large part to the job separation rate, which accounts for more than 60 percent of (cid:133)rst-di⁄erenced unemployment skewness.5 Further, JS and vacancy posting contribute in roughly equal proportions to unemployment(cid:146)s mild kurtosis. However, this decomposition hides an important di⁄erence between the two margins: vacancy posting presents a large negative excess kurtosis but JS presents a positive excess kurtosis. This result suggests that vacancy posting drives unemployment during normal times but that job separation is responsible for rare but violent (cid:135)uctuations in unemployment. To explore this idea further, I depart from an average decomposition and analyze the relative contributions of JS and vacancy posting at business cycles turning points. I (cid:133)nd that job separation is responsible for almost all of the movements in unemployment during the (cid:133)rst two quarters after unemployment reaches a low or a high, and that vacancy posting does not become the main contributor until a year later. Thus, ignoring the cyclicality of the job separation margin will lead researchers to downplay the asymmetric behavior of unemployment and understate the breadth and speed of adjustment of unemployment around turning points. The remainder of the paper is organized as follows: Section 2 reviews Shimer(cid:146)s method, its potentialendogeneitybiasandpresentsawaytoaddressit;Section3(cid:133)tsamatchingfunctionto the data and assesses the contribution of the job separation rate to unemployment(cid:146)s moments after controlling for the endogeneity of the job (cid:133)nding rate, Section 4 studies the behavior of the hazard rates at business cycles turning points; and Section 5 o⁄ers some concluding remarks. 2 The contributions of JF and JS In this section, I brie(cid:135)y review Shimer(cid:146)s (2007) methodology to identify the contributions of JF and JS to unemployment(cid:146)s variance and discuss the possible endogeneity of the job (cid:133)nding rate. 5A large literature has documented a non-trivial asymmetry in steepness for the cyclical component of unemployment; that increases in unemployment are steeper than decreases. See, among others, Neftci (1984), Delong and Summers (1986), Sichel (1993) and McKay and Reis (2008). 3

2.1 The variance decomposition approach Denoting u t+(cid:28) the unemployment rate at instant t+(cid:28) R+ with t N and (cid:28) [0;1[, Shimer 2 2 2 (2007) postulates that during a "period t" of one month (cid:150)i.e. (cid:28) [0;1[ (cid:150)all unemployed 2 workers(cid:133)ndajobaccordingtoaPoissonprocesswithconstant arrivalratef(cid:22) andallemployed t workers lose their job according to a Poisson process with constant arrival rate s(cid:22). As a result, t we have the (cid:133)rst-order di⁄erential equation: du t+(cid:28) = s(cid:22) (1 u ) f(cid:22)u : (1) t t+(cid:28) t t+(cid:28) d(cid:28) (cid:0) (cid:0) By further assuming that the job (cid:133)nding rate is the same for all candidate workers, Shimer (2007) estimates the job (cid:133)nding rate separately by solving the (cid:133)rst-order di⁄erential equation du du<1 t+(cid:28) t+(cid:28) = f(cid:22) u u<1 d(cid:28) (cid:0) d(cid:28) t t+(cid:28) (cid:0) t+(cid:28) (cid:0) (cid:1) where u<1 denotes the stock of unemployed workers at date t+(cid:28) with duration less than one t+(cid:28) month. The estimated job (cid:133)nding rate over [t;t+1[ takes the form u u<1 f(cid:22) = ln(1 F(cid:22)) where F(cid:22) = 1 t+1 (cid:0) t+1. (2) t t t (cid:0) (cid:0) (cid:0) u t Note however that this result is only an approximation, as the job (cid:133)nding rate may not be constant over [t;t+1[. Equation (2) gives an estimate of the average job (cid:133)nding rate f(cid:22) over t [t;t+1[ and is valid under the assumption that movements in f (the job (cid:133)nding rate at t+(cid:28) time t+(cid:28)) are small over the month so that f f(cid:22), (cid:28) [0;1[. t+(cid:28) t ’ 8 2 The separation rate can then be estimated by solving (1) over [t;t+1] and (cid:133)nding s(cid:22) such t that the solution u equals u for (cid:28) = 1. Again, this estimation method relies on the t+(cid:28) t+1 assumptions that the job (cid:133)nding rate and the job separation rate are both constant over each time period and independent of unemployment. Shimer (2007) then argues that the measured magnitudes of the two hazard rates ensure that at a quarterly frequency, it is reasonable to use the following approximation s(cid:22) u t uss (3) t ’ s(cid:22) +f(cid:22) (cid:17) t t t Following Elsby, Michaels and Solon (2009) and Fujita and Ramey (Forthcoming), loglinearizing (3) gives dlnuss = (1 uss) dlns(cid:22) dlnf(cid:22) +(cid:15) (4) t t t t t (cid:0) (cid:0) (cid:2) (cid:3) 4

or duss = dusr +du jf +(cid:15) t t t t so that the deviations of unemployment can be decomposed into a component depending on the job separation rate, a component depending on the job (cid:133)nding rate and a residual term. Fujita and Ramey (Forthcoming) assess the separate contributions of the separation and job (cid:133)nding rates by noting that Var(duss) = Cov(duss;du jf )+Cov(duss;dusr)+Cov(duss;(cid:15) ): t t t t t t t so that (cid:12)jf = Cov(dus t s;duj t f) and (cid:12)sr = Cov(dus t s;dus t r) measure the contributions of the job Var(duss) Var(duss) t t separation rate and the job (cid:133)nding rate to unemployment(cid:146)s variance. 2.2 The endogeneity of the job (cid:133)nding rate A potential bias in Shimer(cid:146)s approach was (cid:133)rst emphasized by Fujita and Ramey (Forthcoming) who argue that Shimer(cid:146)s decomposition may understate the true contribution of the job separation rate. For example, if a high separation rate leads to a low job (cid:133)nding rate, one may attribute the low job (cid:133)nding rate to high unemployment, even though the separation rate was the true cause.6 Asimplewaytothinkaboutthisendogeneityproblemistoconsiderasearchandmatching set-upincontinuoustime. Thejob(cid:133)ndingrateisde(cid:133)nedastheratioofnewhirestothestockof unemployed,sothatifm denotesthenumberofnewmatchesatinstantt,unemployedworkers t (cid:133)nd a job according to a Poisson process with time varying arrival rate f = mt, where u is t ut t the number of unemployed. An increase in the job separation rate will increase unemployment and mechanically lower the job (cid:133)nding rate, and a variance decomposition exercise that does not take into account the link between s and f will understate the contribution of JS.7 t t 6Tobeprecise,FujitaandRamey(Forthcoming)arguethatthesteady-stateapproximationisresponsiblefor this bias as it suppresses the dynamic interaction between JS and JF. The present paper maintains the steadystateapproximationbutarguesthatthedynamicinteractionbetweenJSandJF(throughthematchingfunction) islikelytobere(cid:135)ectedinthequarterly (andafortioriyearly,seeFootnote15)steady-statedecompositionbecause unemployment converges to its steady-state value very rapidly (in about a month (Shimer, 2007)). 7NotethatthereisnosimilarmechanicallinkrunningfromJFtoJS.Atthe(cid:133)rmlevel,thejobseparationrate is de(cid:133)ned as the number of layo⁄s and quits divided by the (cid:133)rm(cid:146)s workforce (as mentioned in the introduction, I abstract from movements in and out of the labor force). Since the (cid:133)rm ultimately controls the size of its workforce,thereisnomechanicallinkbetweenJFandthe(cid:133)rmleveljobseparationrate. Asaresult,thereisno mechanical link from JF to aggregate JS. However, a relationship running from JF to JS could exist as a lower job (cid:133)nding rate may discourage quits and lower the job separation rate. However, Elsby et al (2009) show that in all but one recessions since 1969, the log job leaver in(cid:135)ow rate (i.e. quits) displays a delayed response and does not decline until 3 quarters after the beginning of the recession. Hence, the endogeneity of JS is unlikely tobeanissueforavariancedecompositionexerciseataquarterlyfrequency(asinShimer,2007). Thejobloser in(cid:135)ow rate on the other hand, increases right at the beginning of the recession and before the job loser out(cid:135)ow 5

Following the literature and assuming a Cobb-Douglas matching function with constant re- 1 (cid:27) turnstoscale,Icanwritem = m u(cid:27)v1 (cid:27) suchthatf = m vt (cid:0) withv thenumberofjob t 0 t t(cid:0) t 0 ut t openings. Importantly, this speci(cid:133)cation is standard and is u(cid:16)sed(cid:17)in almost all macroeconomic models that introduce equilibrium unemployment through search and matching frictions (see e.g. Pissarides, 2001). Using a measure of vacancy posting, I can then isolate the (cid:147)exogenous(cid:148) component of the job (cid:133)nding rate, i.e. the movements in JF that are due to (cid:135)uctuations in vacancy posting, not to (cid:135)uctuations in unemployment. 1 (cid:27) However, because f = m vt+(cid:28) (cid:0) is not constant over [t;t+1[, one could worry that t+(cid:28) 0 ut+(cid:28) Shimer(cid:146)s (2007) method to recov(cid:16)er f(cid:22) t(cid:17) and s(cid:22) t is not valid anymore because the di⁄erential f g equation satis(cid:133)ed by unemployment changes and takes the form (cid:8) (cid:9) du t+(cid:28) = s(cid:22) (1 u ) f u : t t+(cid:28) t+(cid:28) t+(cid:28) d(cid:28) (cid:0) (cid:0) Fortunately, Shimer(cid:146)s approach still goes through if, within each month, movements in f t+(cid:28) over[t;t+1[arenegligiblecomparedtof (cid:146)sstartoftheperiodvalue. Indeed,iff = f +" t t+(cid:28) t t+(cid:28) with " << f , one can reasonably approximate the instantaneous job (cid:133)nding rate with the t+(cid:28) t average one so that f f(cid:22). Under this approximation, the di⁄erential equation reduces to t+(cid:28) t ’ (1)andonecanrecovers(cid:22) asinShimer(2007). IntheAppendix,Ishowthatthisapproximation t is reasonable as it does not lead to any substantial bias in s(cid:22) . Hence, from now on, I assume t f g as in Shimer (2007) that at a monthly frequency, f f(cid:22), (cid:28) [0;1[: t+(cid:28) t ’ 8 2 3 The contributions of vacancy posting and job separation In this section, I study the contributions of vacancy posting and the job separation rate to unemployment (cid:135)uctuations. I (cid:133)rst argue that v and JS are a natural set of variables to t t f g f g considerbecausetheycapturethedecisionvariablesof(cid:133)rmsandworkersandassuchconstitute the(cid:147)primitive(cid:148)variablesthatrespondtoshocksanddetermineunemployment. Ithenestimate a estimate a matching function to capture movements in the job (cid:133)nding rate, and I use the hazard rate decomposition approach to evaluate the contributions of vacancy posting and job separation to unemployment(cid:146)s variance, skewness and kurtosis. 3.1 Focusing on vacancy posting and JS While the literature has traditionally studied the properties of job (cid:135)ows and worker (cid:135)ows, it also natural to consider the behavior of v ;JS because these variables are the control t t f g variables that economic agents ((cid:133)rms and workers) adjust in response to shocks, and that rate, consistent with the causal relationship put forward in this paper. 6

policy can directly in(cid:135)uence (through e.g. a hiring subsidy or a (cid:133)ring tax). Starting with the (cid:133)rm(cid:146)s problem, a (cid:133)rm can adjust its number of workers through two channels: hirings and (cid:133)rings. For example, a (cid:133)rm faced with a positive TFP shock can choose to increase hirings, decrease (cid:133)rings or use a combination of both. Put di⁄erently, hirings and (cid:133)rings are the two control variables of the (cid:133)rm (with respect to employment). However, by focusing on gross worker (cid:135)ows, one cannot rely on the hazard rate decomposition approach to quantitatively estimate the contribution of each margin of adjustment. Fortunately, for a (cid:133)rm, choosing the number of new hires and (cid:133)res is isomorphic to choosing the number of job openings (assuming that they ultimately all get (cid:133)lled) and choosing the percentage of the workforce to be shed, i.e. the job separation rate due to layo⁄s. Turning to the worker(cid:146)s problem, an employed worker can decide whether to quit and as a result can in(cid:135)uence the job separation rate due to quits. As a result, the total job separation rate (de(cid:133)ned as the number of layo⁄s and quits over the number of employed workers) captures both (cid:133)rms and workers decisions.8 In the rest of the paper, I will only report the contributions of the aggregate job separation rate and vacancy posting, but in the Appendix, I present a variance decomposition exercise that treats separately the three main decision variables of economic agents: vacancy posting, layo⁄s and quits. 3.2 Modeling JF with a Cobb-Douglas matching function To model the job (cid:133)nding rate, I estimate a Cobb-Douglas matching function that can capture movements in the monthly job (cid:133)nding rate. Under the assumption that f f(cid:22) over each t+(cid:28) t ’ month [t;t+1[, I can use Shimer(cid:146)s estimate of the job (cid:133)nding rate f(cid:22) = ln(1 F(cid:22)), and I t t (cid:0) (cid:0) estimate the following equation v lnf(cid:22) = (1 (cid:27))ln t +c+" (5) t t (cid:0) u t after detrending all variables with an HP-(cid:133)lter.9 Seasonally adjusted unemployment u is constructed by the BLS from the Current Poput lation Survey (CPS). More di¢ cult is the choice of a measure for vacancy posting v . There t are two standard measures of job openings; the Help-Wanted advertising Index (HWI) and the 8As mentioned in the introduction, I abstract from movements in and out of the labor force. 9Davis, Faberman and Haltiwanger (DFH, 2009) study the behavior of vacancies and hirings in JOLTS and (cid:133)nd that one in six hires occur outside of the matching function framework, i.e. without a prior vacancy. Regression (5) could then be subject to an omitted variable bias. Denoting z the fraction of hires outside t the matching function framework, total hires equals m =(1 z ) so that I can write lnf(cid:22) = ln(1 z )+ t t t t (cid:0) (cid:0) (cid:0) (1 (cid:27))ln vt +c+" : Assuming the worse case scenario in which corr(ln(1 z );ln vt) = 1 and (roughly) (cid:0) ut t (cid:0) t ut estimating the standard-deviation of z t from DFH, Figure 10 to be a(cid:12)t most 0:04, I get a m(cid:12)aximal bias for (cid:27) of (cid:12) (cid:12) 1:var(ln(1 z t ))=0:012, suggesting that the omitted variable bias i(cid:12)s small. (cid:12) (cid:0) 7

Job Openings and Labor Turnover Survey (JOLTS). The Help-Wanted Index is constructed by the Conference Board and measures the number of help-wanted advertisements in 51 major newspapers. This index is only a proxy for vacancy posting but has the advantage of dating back to 1951, thus providing a long time series. However, this (cid:147)print(cid:148)HWI index has become increasingly unrepresentative as advertising over the internet has become more prevalent. In fact, the Conference Board stopped publishing its print HWI in May 2008 and publishes insteadsince2005ameasureofon-linehelpwantedadvertising. Tobuildanindexthatcombines information on (cid:147)print(cid:148)and (cid:147)online(cid:148)advertising, I follow Fallick(cid:146)s (2008) approach and estimate the share of print help-wanted advertising as the ratio of a trend in the HWI to the value of that trend in 1994, which roughly corresponds to the introduction of the World Wide Web. After 2005, when both the online and print HWI are available, I calculate the index by weighting the growth rates of the two indexes by the estimated print share.10 JOLTS is produced by the BLS and contains monthly data on job openings from 16,000 establishments since December 2000. Since JOLTS provides a more direct, and arguably better, measure of vacancy posting than HWI, I construct a composite job openings index using print-online help wanted advertisements until December 2000 and using JOLTS data thereafter.11 Figure 1 presents the di⁄erent measures of vacancy posting. I (cid:133)rst estimate (5) with monthly data and using the composite HWI-JOLTS index from 1951:M01 until 2009:M02. All data were previously detrended with an HP (cid:133)lter. Table 1 presents the result. The elasticity (cid:27) is precisely estimated at 0:59, and apart from JF(cid:146)s highfrequency movements (probably due to measurement errors), a matching function does a very good job at capturing movements in the job (cid:133)nding rate. Indeed, after taking quarterly averages, Figure 2 shows that a matching function tracks the empirical job (cid:133)nding rate very closely. Since JOLTS and HWI are two di⁄erent dataset, I verify the robustness of the results using only one data source at a time. Further, to make sure that the results are not biased by the strong low-frequency movements in HWI before 1977 that are unrelated to the labor market, I estimate (5) with the print-online help-wanted index over 1977:M01-2009:M02 only. We can see that the estimated (cid:27) is unchanged at 0:59. Finally, I use JOLTS data only over 2000:M12-2009:M02 and (cid:133)nd a slightly lower (cid:27) at 0:57. Encouragingly, these estimates lie in the middle of the plausible range reported by Petrongolo and Pissarides (2001). 10Another problem with the HWI is that it is subject to low-frequency (cid:135)uctuations that are related only tangentiallytothelabormarket;notably,thedeclineinprintadvertisinginthe1990sandthe1960snewspaper consolidation that may have increased advertising in surviving newspapers. Fortunately, detrending all series with a low frequency trend (since Iam only focusing on business cycle (cid:135)uctuations) should remove the e⁄ect of such secular shifts. 11Since JOLTS reports the number of job openings at month(cid:146)s end, I use vJOLTS as the time t measure for t 1 the number of vacancies. This allows me to be consistent with vHWI, which(cid:0)measures the total number of t help-wanted advertisements from the 14th of the previous month to the 13th of the current month. 8

A legitimate concern with this regression exercise is that equation (5) may be subject to an endogeneitybias. Theuseofamonthlyfrequencyandthefactthatu denotesthebeginningof t period unemployment rate should minimize the problem, but it is still important to verify that there is no signi(cid:133)cant bias. To do so, I estimate (5) using lagged values of vt as instruments.12 ut Encouragingly, Table 1 shows that the endogeneity bias is likely to be small as the coe¢ cient is little changed at 0:58.13 The robustness of the results over di⁄erent measures of vacancies and over di⁄erent sample periods is promising and suggests that a matching function provides a good approximation of the job (cid:133)nding rate and can be reasonably used to control for the endogeneity of JF. For the rest of the paper, I will use the composite HWI-JOLTS measure of vacancy posting with a matching function elasticity (cid:27) = 0:59 but the results do not rely on this speci(cid:133)c choice. 3.3 Variance decomposition Writing the steady-state approximation for unemployment (3) at a quarterly frequency (as in Shimer, 2007) and modeling the job (cid:133)nding rate with a matching function, I get s s s uss t t t : (6) t (cid:17) s +f ’ 1 (cid:27) ’ 1 (cid:27) t t s +m vt (cid:0) s +m vt (cid:0) t 0 ut t 0 us t s (cid:16) (cid:17) (cid:16) (cid:17) where all variables now denote quarterly averages of their monthly counterparts.14 Thisapproximationreliesontheimplicitassumptionthatmovementsins haveane⁄ecton t steady-state unemployment (which is the case by de(cid:133)nition) as well as on the job (cid:133)nding rate within the time period, so that the quarterly average of the monthly job (cid:133)nding rate re(cid:135)ects the in(cid:135)uence of the job separation rate. Fortunately, in the US, unemployment converges to its steady-state value in about a month (Shimer, 2007), so that the dynamic interactions between JS and JF (through the matching function) are likely to be re(cid:135)ected in the quarterly (and a fortiori yearly) steady-state decomposition.15 Moreover, I can track the validity of my approach by looking at the contribution of the residual. Indeed, after log-linearizing (6) and 12Such instruments are valid if the residual is not serially correlated. The Durbin-Watson statistics for regression (1) in Table 1 is 1.83. To verify that serial correlation is de(cid:133)nitely not an issue, I performed a GMM regression over 1951-1990 for which the Durbin-Watson statistics is 2.02. Results are unchanged. 13An issue that I brushed aside is the timing of the measurements of unemployment, vacancy and the job (cid:133)ndingrate. IntheAppendix,Ipresentamorerigorouswaytoaddressthesemeasurementissues,butestimates of (cid:27) are unchanged by these timing considerations. 14It is important to note that (6) is only an approximation and does not de(cid:133)ne steady-state unemployment. Steady-state unemployment is still determined from Shimer(cid:146)s (2007) job (cid:133)nding rate measure. I only use a matching function to approximate JF and isolate movements due to changes in vacancy posting. 15As a robustness check, I conduct a variance decomposition exercise at a yearly frequency and (cid:133)nd that the results are unchanged. 9

using the fact that lnf = lnm +ln(cid:18)1 (cid:27) +" , I can rewrite (4) as t 0 t(cid:0) t dlnuss = (1 uss)[dlns (1 (cid:27))(dlnv dlnuss)]+(cid:17) (7) t t t t t t (cid:0) (cid:0) (cid:0) (cid:0) with (cid:17) the sum of successive approximation errors due to the (cid:133)rst(cid:150)order log-linearization, the t use of a matching function to model JF, and the fact that I enter steady-state unemployment inside the matching function. Rearranging (7), I have 1 uss (1 (cid:27))(1 uss) dlnuss = (cid:0) t dlns (cid:0) (cid:0) t dlnv +(cid:17) (8) t 1 (1 (cid:27))(1 uss) t (cid:0) 1 (1 (cid:27))(1 uss) t t t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) or dlnu = du~sr +du~ jf +(cid:17) (9) t t t t with du~s t r = 1 (cid:0) (1 (cid:0) 1 (cid:0) (cid:27)) u ( s t 1 s (cid:0) us t s) dlns t and du~ j t f = (cid:0)1 (cid:0) (1 ( (cid:0) 1 (cid:0) (cid:27)) (cid:27) ( ) 1 ( (cid:0) 1 (cid:0) us t u s s t ) s) dlnv t : The latter can be interpreted as movements in unemployment solely due to changes in vacancy posting, so that dlnf~ = (1 (cid:27))dlnv captures (cid:147)exogenous(cid:148)changes in the job (cid:133)nding rate, or equivalently t t (cid:0) movements in the job (cid:133)nding rate holding unemployment constant. Henceforth, I will refer to jf du~ as movements in unemployment due to (cid:135)uctuations in job openings. t I now proceed with the variance decomposition exercise by using the fact that Var(duss) = Cov(duss;du~ jf )+Cov(duss;du~sr)+Cov(duss;(cid:17) ) t t t t t t t sothat(cid:12)sr = Cov(dus t s;du~j t f) and(cid:12)sr = Cov(dus t s;du~s t r) measurethecontributionsofjobseparation Var(duss) Var(duss) t t and the (cid:147)exogenous(cid:148)(i.e. independent of unemployment) component of the job (cid:133)nding rate to unemployment(cid:146)s variance. A back-of-the-envelope calculation can readily give an idea of the revised contribution of the job separation rate when I take into account the endogeneity of JF. With (cid:27) 0:6 and ’ u 0:05, the endogeneity of JF biases the contribution of JS downwards by 60 percent (from ’ 1 1:6). Insteadofacontributionofabout25percentasreportedinShimer(2007), 1 (1 (cid:27))(1 u) ’ J (cid:0) S w (cid:0) ould (cid:0) in fact contribute to about 40 percent, a far from negligible amount.16 Using the log-deviation from trend du = ln us t s where uss and sss denote the trend t fl us t s flt flt component of us t s and s t , I can rewrite (8) as (cid:16) (cid:17) uss 1 u(cid:22)ss s (1 (cid:27))(1 u(cid:22)ss) v ln t = (cid:0) t ln t (cid:0) (cid:0) t ln t +(cid:17) : (10) uss 1 (1 (cid:27))(1 u(cid:22)ss) s (cid:0) 1 (1 (cid:27))(1 u(cid:22)ss) v t (cid:18)flt (cid:19) (cid:0) (cid:0) (cid:0) t (cid:18)flt(cid:19) (cid:0) (cid:0) (cid:0) t (cid:18)flt(cid:19) 16As a robustness check, if I span the plausible matching function elasticities estimated in the literature 0:5 0:7 (Petrongolo and Pissarides, 2001), the contribution of JS is 10 to 20 percentage points larger after (cid:0) taking into account the endogeneity of JF. 10

The (cid:133)rst column of Table 2 compares the values of the betas over 1951-2008 with and without controlling for the endogeneity of the job (cid:133)nding probability. Controlling for unemployment (cid:135)uctuationsisimportantasthecontributionofthejobseparationrateincreasesfrom24percent to 39 percent.17 The successive approximations naturally increase the error component in the log-decomposition, and the contribution of the residual amounts to about 5 percent. To evaluate the bias introduced by a matching function, the middle row of Table 2 presents a variance decomposition exercise between JF and JS but using the matching function to model JF. We can see that the use of a matching function increases the contribution of the residual to 4 percent and correspondingly biases downwards the estimate of JF(cid:146)s contribution. As a result, thecontributionofvacancypostingislikelytobeunderestimatedandisprobablycloser to 60 than 55 percent. Overall, the residual contribution remains small. This con(cid:133)rms that the matching function does a good job at approximating the job (cid:133)nding rate, and that my approach provides a reasonable framework to evaluate the respective contributions of vacancy posting and layo⁄s/quits. Usinga(cid:133)rst-di⁄erencedlog-decompositionasinFujitaandRamey(Forthcoming)andusing du = (cid:1)lnuss = ln us t s , I have t t uss t 1 (cid:0) 1 uss (1 (cid:27)) 1 uss (cid:1)lnus t s = 1 (1 (cid:0) (cid:27))( t 1 (cid:0) 1 uss ) (cid:1)lns t (cid:0) 1 ( (cid:0) 1 (cid:27))( (cid:0) 1 t u (cid:0)s 1 s ) (cid:1)lnv t +(cid:17) t : (cid:0) (cid:0) (cid:0) t 1 (cid:0) (cid:0) (cid:0) (cid:0) t(cid:1)1 (cid:0) (cid:0) The second column of Table 2 presents the result. This time, the contribution of JS increases from 40 percent to 63 percent, while the contribution of JF drops to only 35 percent. The contribution of the residual remains small at around 2 percent. To sum up, controlling for the endogeneity of the job (cid:133)nding rate raises the contribution of JS to unemployment(cid:146)s variance by 60 percent; with a 40=60 split between vacancy posting and job separation for a decomposition in level and a 60=40 split for a decomposition in (cid:133)rst-di⁄erences. As a result, modeling the job separation probability as acyclical will lead researchers to understate the volatility of unemployment.18 17IntheAppendix,IextendthisapproachbyusingCPSdatafromtheBLSonthereasonsforunemployment (layo⁄s,quitsorlaborforceentrants)over1968-2004asusedinElsby&al(2008). I(cid:133)ndthatlayo⁄scontribute to 45 percent of unemployment (cid:135)uctuations but quits, being procyclical, lower the contribution of JS by 10 percentage points, a point originally made qualitatively by Elsby et al. (2008). 18Indeed, Shimer(2005)shows in a very in(cid:135)uentialpaperthatthe Mortensen-Pissarides (1994)modelwith a constant job separation rate lacks an ampli(cid:133)cation mechanism because it generates less than 10 percent of the observed business cycle (cid:135)uctuations in unemployment given labor productivity shocks of plausible magnitude. 11

3.4 Higher-order moments Whiletheliteraturehasfocusedonunemployment(cid:146)svariancetoevaluatetheimportanceofthe job separation rate, higher-order moments could paint a di⁄erent picture. Notably, a stylized fact about unemployment is its asymmetric behavior, and a large literature has documented a non-trivial asymmetry in steepness for the cyclical component of unemployment, i.e. that increases are steeper than decreases.19 To evaluate the respective contributions of job separation and vacancy posting, I extend Fujita and Ramey(cid:146)s (Forthcoming) approach to higher-order moments and notably to the concept of skewness. E(X (cid:22))n Let us denote the mean of X as (cid:22) = E(X) and its nth moment (cid:11) n (cid:17) (E(X (cid:0) (cid:22))2)n=2 for n N: (cid:0) 2 Further, let us assume that X (cid:22) can be written as a sum of terms so that X (cid:22) = (cid:0) n (cid:0) (X (cid:22) ): By noting that (X (cid:22))n = X (cid:22) = (X (cid:22))n 1(X (cid:22) ), I have i i i i (cid:0) i i (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) i (cid:18) i (cid:19) i EP(X (cid:22))n = E(X (cid:22))n 1(X (cid:22) ) so tPhat I can write P (cid:0) (cid:0) i(cid:0) i (cid:0) i i P E(X (cid:22))n E(X (cid:22))n 1(X (cid:22) ) (cid:11) = (cid:0) = (cid:0) (cid:0) i (cid:0) i : (11) n (E(X (cid:22))2)n=2 (E(X (cid:22))2)n=2 (cid:0) X i (cid:0) Dividing (11) by (cid:11) , I get n E(X (cid:22))n 1(X (cid:22) ) 1 = (cid:0) (cid:0) i (cid:0) i E(X (cid:22))n i (cid:0) X and I interpret (cid:13) i = E(X (cid:0) E (cid:22) ( ) X n (cid:0) 1 (cid:22) ( ) X n i (cid:0) (cid:22) i ) as a measure of the contribution of X i to X(cid:146)s nth moment. Indeed E(X (cid:22))n 1(cid:0) (X (cid:22) ) captures the fraction of E(X (cid:22))n that is due to (cid:0) i i (cid:0) (cid:0) (cid:0) movements in X . i I can now estimate the contributions of vacancy posting and job separation to the steepness asymmetry of unemployment.20 To do so, I consider the skewness of (cid:133)rst-di⁄erence log-unemployment. Table 3 shows that over 1955-2008, (cid:133)rst-di⁄erenced log-unemployment has a skewness coe¢ cient of 1:2, signi(cid:133)cant at the 5% level.21 Vacancy posting and JS also present a signi(cid:133)cant asymmetry in steepness with coe¢ cients of 0:79 and 0:42. Using the (cid:0) 19See, among others, Neftci (1984), Delong and Summers (1984), Sichel (1993) and McKay and Reis (2008) for evidence of asymmetry at quarterly frequencies. 20I(cid:133)rstdetrendunemployment,vacancyandthehazardratesbeforestudyingtheskewnessof(cid:133)rst-di⁄erenced variables as trends may bias the skewness coe¢ cient. 21Over1951-1954,unemploymentexperienced verylargequarterlymovementsthatdramaticallyincreasethe skewnesscoe¢ cient(by0.4)and con(cid:133)denceinterval. Sincetheskewnessestimateisotherwisestableover1955- 2008, I omit the 1951-1954 time period for clarity of exposition. Nonetheless, my results remain valid over 1951-2008. 12

log-decomposition (9) and using du = (cid:1)lnuss = ln us t s , I have t t uss t 1 (cid:0) 1 uss (1 (cid:27)) 1 uss (cid:1)lnus t s = 1 (1 (cid:0) (cid:27))( t 1 (cid:0) 1 uss ) (cid:1)lns t (cid:0) 1 ( (cid:0) 1 (cid:27))( (cid:0) 1 t u (cid:0)s 1 s ) (cid:1)lnv t +(cid:17) t : (cid:0) (cid:0) (cid:0) t 1 (cid:0) (cid:0) (cid:0) (cid:0) t(cid:1)1 (cid:0) (cid:0) E(duss)2du~sr E(duss)2du~jf so that I can interpret t t and t t as the contributions of the job separation E(duss)3 E(duss)3 t t andvacancypostingmarginstotheskewnessof(cid:133)rst-di⁄erencedunemployment. Table4shows that while the job separation rate contributes to less than half of unemployment(cid:146)s variance, this is hardly the case with unemployment asymmetry since the job separation(cid:146)s contribution stands at more than 62 percent. Thus, a model that would not consider (cid:135)uctuations in the job separation rate would seriously downplay the asymmetric behavior of unemployment. Reassuringly, the contribution of the residual remains low and stands at around 5 percent. A comparison of the (cid:133)rst two rows of Table 4 indicates that a matching function biases upwards the contribution of JF as the latter increases from 60 to 63 percent. As a result, the contribution of vacancy posting is likely to be overestimated, and a rough split between job separation and vacancy posting is 60=40. Table 3 presents another new fact pertaining to the fourth moment of unemployment and its hazard rates. While unemployment has a mild (but signi(cid:133)cant) negative excess kurtosis ( 0:34), vacancy posting and job separation have kurtosis of opposite signs. Vacancies present (cid:0) alargenegativeexcesskurtosis( 0:94)butJSpresentsapositiveexcesskurtosis(0:54). Recall (cid:0) that a high kurtosis distribution such as that of JS has a sharper peak and longer, fatter tails, i.e. extreme values are drawn more often than with a normal distribution. This (cid:133)nding is not surprising if we think of job separation as capturing (among other things) bursts of layo⁄s. On the other hand, a low kurtosis distribution such as that of vacancies has a more rounded peak and shorter thinner tails, i.e. fewer extreme values. To visualize the distribution of steadystate unemployment, vacancy posting and the job separation rate, Figure 3 plots the kernel density estimates of these variables using a Gaussian kernel with optimal bandwidth. The dashed lines represent the corresponding (i.e. mean and variance) normal distributions. While unemployment(cid:146)s distribution is very close to being normal, this is hardly the case for vacancy posting and job separation. Vacancy posting has almost a bimodal distribution with rapidly decreasing tails but the job separation rate has a small mass of points around the mean and very fat tails. Looking at the contributions of each hazard rate, Table 4 shows that vacancy posting and job separation contribute in roughly equal proportion to unemployment(cid:146)s fourth moment, with a slight advantage for vacancy posting. Given the lower contribution of JS to unemployment(cid:146)s variance, the mild negative kurtosis of unemployment despite the large negative kurtosis of 13

vacancy posting is consistent with an interpretation of job separation in(cid:135)uencing unemployment through rare but violent episodes of job separation. The contribution of the residual amounts to less than 4 percent, and the second row of Table 4 indicates that the use of a matching function biases the contribution of JF downwards. As a result, the split between job separation and vacancy posting is roughly 45=55. While only indicative, this fourth-moment decomposition suggests that vacancy posting drives unemployment during normal times but that job separation is responsible for rare but violent (cid:135)uctuations in unemployment. 4 The contributions of vacancy posting and job separation at business cycle turning points The evidence from the kurtosis decomposition exercise suggests that vacancy posting drives unemployment during normal times but that job separation is responsible for rare but violent (cid:135)uctuations in unemployment. To explore this idea further, I depart from an average decomposition to analyze the relative contributions of the job separation rate and vacancies around the turning points of unemployment (cid:135)uctuations. After detrending unemployment using an HP-(cid:133)lter with (cid:21) = 105, I follow McKay and Reis (2008) and identify highs and lows in unemployment using the algorithm of Bry and Boschan (1971). Figure 4 plots the steady-state unemployment rate with identi(cid:133)ed turning points.22 The (cid:133)rst rows of Figure 5 and 6 plot the average dynamics of the log-deviation from trend of steady-state unemployment, the job separation rate, and the job (cid:133)nding rate in a window of 3 and 6 quarters before and after the highs and lows of unemployment.23 As (cid:133)rst shown by Elsby et al. (2009) with NBER recessions dates, an interest of this approach is that the log-decomposition (4) allows us to directly observe the relative contributions of JS and JF to unemployment (cid:135)uctuations. The second rows of Figure 5 and 6 display the same average dynamics but using the log of vacancy times (1 (cid:27)) instead of JF.24 From (10), we can directly (cid:0) observe the relative contributions of job separation and vacancy posting as (1 (cid:27))dln(v) (cid:0) corresponds to movements in unemployment caused by changes in vacancy posting. A (cid:133)rst observation is that, while the previous section showed that vacancy posting was, on average, the most important contributor of unemployment (cid:135)uctuations, this is hardly the 22See McKay and Reis (2008) for a presentation of possible methods to identify the peaks and troughs of a series. All the results are robust to using the alternative methods reported in McKay and Reis (2008). 23More speci(cid:133)cally, for each quarter t around an unemployment turning point T, I plot ln us T s +t ln us T s ,[(1 u(cid:22)ss )dlns (1 u(cid:22)ss)dlns ],and [(1 u(cid:22)ss )dlnf (1 u(cid:22)ss)dlnf ]. u(cid:22)s T s +t (cid:0) u(cid:22)s T s (cid:0) T+t T+t (cid:0) (cid:0) T T (cid:0) (cid:0) T+t T+t (cid:0) (cid:0) T t Ahcc(cid:16)ording(cid:17)to (4)(cid:16), the(cid:17)(cid:133)irst term is the sum of the last two so that, for each quarter t around a turning point T, Figure 5 and 6 show the contributions of JF and JS to deviations of unemployment from its low or high. 24Comparing carefully the two rows of Figure 4 (or Figure 5), the two unemployment rates are not exactly equal. This small di⁄erence comes from the approximation error when modeling JF with a matching function. 14

case at business cycle turning points. Around highs and lows, JS is the prime determinant of movements in unemployment. Without controlling for the endogeneity of JF, the results shown in Figure 6 are in line with Elsby et al(cid:146)s (2009) (cid:133)ndings for NBER recessions: once unemploymentreachesalow,JSisresponsibleformostoftheinitialincreaseinunemployment, but after two quarters JF becomes the dominant contributor of the increase in unemployment. The same conclusion holds for unemployment highs. However, the second row of Figure 6 shows that when I consider only the (cid:147)exogenous(cid:148)component of JF, job separation accounts formorethan50percentofunemploymentmovementsforasmuchas6quartersafterahighor a low, and for almost all of the initial response. Interestingly, this result is consistent with the decomposition of unemployment(cid:146)s fourth moment in the previous section, which suggests that extremevaluesofunemploymentareduetothejobseparationrate. Lookingatthecontribution of the residual, the approximation is relatively good three quarters before and after a turning point but deteriorates slightly thereafter. However, assigning all of the residual(cid:146)s contribution to vacancy posting (a worst case scenario for JS) does not change the main conclusion; JS still accounts for more than 50 percent of unemployment movements a year after a high or low. Two other observations are worth noting. First, the asymmetric nature of unemployment is clearly apparent in Figure 5 and 6 as unemployment increases faster than it decreases. This asymmetry can be linked to the asymmetric response of JS. Vacancy posting reacts slowly, and the slope of vacancy posting is much weaker than that of job separation in the (cid:133)rst quarters after a turning point. Second, after unemployment highs, vacancies lag job separation by a quarter. ThisisinlinewithFujitaandRamey(Forthcoming), who(cid:133)ndthatthejobseparation rate lags the job (cid:133)nding rate. Animplicationoftheselast(cid:133)ndingsisthatignoringthejobseparationmarginwhenmodelingunemploymentwillleadresearcherstounderestimatethebreadthandspeedofadjustments in unemployment around turning points. 5 Conclusion In this paper, I study the relative importance of the two main determinants of unemployment (cid:135)uctuations: vacancy posting and job separation. By isolating (cid:135)uctuations in the job (cid:133)nding rateduesolelytochangesinvacancyposting,Itakea(cid:133)rststeptoaddresstheendogeneityofthe job (cid:133)nding rate, and I (cid:133)nd that the contribution of the job separation rate to unemployment(cid:146)s variance is close to 40 percent instead of 25 percent using Shimer(cid:146)s (2007) methodology. I also extend Shimer (2007), Elsby et al (2009) and Fujita and Ramey (Forthcoming) variance decomposition exercise to higher-order moments, and I (cid:133)nd that job separation contributes to about 60 percent of unemployment steepness asymmetry, a stylized fact of the jobless rate. 15

Finally,whilevacancypostingis,onaverage,themoreimportantcontributorofunemployment (cid:135)uctuations, the opposite is true around business cycle turning points, when job separation is responsible for most of unemployment movements. These results imply that modeling the job separation margin as acyclical will lead researchers to (i) understate the volatility of unemployment, (ii) seriously downplay the asymmetric behavior of unemployment, and (iii) underestimate the breadth and speed of adjustments in unemployment around business cycle turning points. 16

Appendix A1 The timing of u , v , and f t t t An important issue when using measures for unemployment, vacancy posting and job (cid:133)nding probability concerns the precise de(cid:133)nition of each variable. In particular, while some variables are beginning or end of month values, others are monthly averages. In the CPS, the BLS surveys the number of unemployed during the reference week, de(cid:133)ned as the week including the 12th day of the month. The Help-Wanted Index vHWI measures t the total number of advertisements (print or online) from the 14th (t)of the month to the 13th of next month (t+1). JOLTS, on the other hand, indicates the number of job openings vJOLTS on the last day of month t. Finally, Shimer(cid:146)s (2007) de(cid:133)nition of F(cid:22) implies that F(cid:22) t t t measures the average job (cid:133)nding probability between two unemployment measurement dates, i.e. between the week including the 12th of next month and the week including the 12th of the current month. To be as consistent as possible with these measurement dates, the average job (cid:133)nding probability should depend on the average unemployment rate and the average number of postedvacancybetweentworeferenceweeks. Sinceu measurestheunemploymentrateduring t the (cid:133)rst reference week, the correct measure of unemployment inside the matching function should be 1 (u +u ). Since vHWI already corresponds to an average over a period and 2 t t+1 t vJOLTS measures the number of job openings at a date roughly in between two reference t weeks, vJOLTS corresponds to vHWI as those two measures would be equal if the number of t t job openings remained constant in between two reference weeks. As a result, a more consistent regression would be v lnf(cid:22) = (1 (cid:27))ln t +c+" (12) t (cid:0) 1 (u +u ) t 2 t t+1 after detrending all variables with an HP-(cid:133)lter. Of course, such a regression is clearly subject to an endogeneity bias as u is a function of f(cid:22) Therefore, to estimate (12), I use GMM as t+1 t in column (4) of Table 1. Encouragingly, the regression results are virtually identical to the ones obtained using (5).25 A2 Identifying s with an endogenous job (cid:133)nding rate t f g In this appendix, I describe a more rigorous way to recover the job separation rate without the need to assume that f f(cid:22) over [0;1[. While this approach is quite sensitive to the t+(cid:28) t ’ 25The results are available upon request. 17

parameterization of the matching function and the value of (cid:27), it allows me to verify that assuming f f(cid:22) has almost no consequences on the estimation of s : Instead of assuming t+(cid:28) t t ’ f g that f remains constant over [t;t + 1[, I make the weaker assumption that only v is t+(cid:28) t+(cid:28) constant over [t;t + 1[ and equals v . This assumption is consistent with the de(cid:133)nition of t vHWI; the total number of vacancies over [t;t+1[ (see Appendix A1). The law of motion for t unemployment (1) now takes the form du t+(cid:28) = s(cid:22) (1 u ) f u t t+(cid:28) t+(cid:28) t+(cid:28) d(cid:28) (cid:0) (cid:0) = s(cid:22) (1 u ) m v1 (cid:27)u(cid:27) t t+(cid:28) 0 t(cid:0) t+(cid:28) (cid:0) (cid:0) Similarly to Shimer (2007), I then solve this di⁄erential equation for di⁄erent values of s(cid:22) t until the solution at time t+1 equals u . In Figure 7, I compare the estimates of s obtained t+1 t withandwithoutassumingconstanthazardrates. Aswecansee, bothestimatesareextremely similar suggesting that the approximation f f(cid:22) over [0;1[ is reasonable as it does not lead t+(cid:28) t ’ to any substantial bias in s(cid:22) . t f g A3 The contributions of layo⁄s and quits In this section, I study the separate contributions of layo⁄s and quits to unemployment(cid:146)s variance by using CPS data from the BLS on the reasons for unemployment (layo⁄s, quits or labor force entrants) over 1968-2004 as in Elsby et al. (2009). Denoting u(cid:21), u q and ue the t t t unemployment rates by reason, I have u = u(cid:21)+u q +ue and dlnu = ! dlnu(cid:21)+! dlnu q + t t t t t (cid:21) t q t ! dlnue, with u(cid:21) = s(cid:21) t et, u q = sq t et and ue = se t it where e is the employment rate and i the e t t f t (cid:21) t f t q t f t e t t labor force participation rate. Looking at Elsby et al. (2009) decomposition, we can see that businesscycle(cid:135)uctuationsine andi aresmallcomparedtocyclical(cid:135)uctuationsinthehazard t t rates, and that (cid:135)uctuations in se are small compared to movement in the other in(cid:135)ows rates t (see Elsby et al. (2009), Figures 9 & 11). As a result, I can write the following approximation dlnuss = ! dlns(cid:21) ! dlnf(cid:21)+! dlns q ! dlnf q t (cid:21) t (cid:21) t q t q t (cid:0) (cid:0) +! dlnse ! dlnfe+(! +! )dlne +! dlni e t e t e q t e t (cid:0) ! dlns(cid:21)+! dlns q ! dlnf(cid:21) ! dlnf q ! dlnfe (cid:21) t q t (cid:21) t q t e t ’ (cid:0) (cid:0) (cid:0) ! dlns(cid:21)+! dlns q dlnf (cid:21) t q t t ’ (cid:0) And using a matching function to model the job (cid:133)nding rate, I can write ! (cid:1)lns(cid:21)+! (cid:1)lns q (1 (cid:27))(cid:1)ln(v ) (cid:1)lnuss (cid:21) t q t (cid:0) (cid:0) t t ’ 1 (1 (cid:27))(1 uss ) (cid:0) (cid:0) (cid:0) t 1 (cid:0) 18

and ln us t s ! (cid:21) lnln fl s s t t +! q (cid:1)lnln fl s s t t (cid:0) (1 (cid:0) (cid:27))(cid:1)ln fl v v t t : uss ’ (cid:16) (cid:17) 1 (1 (cid:27)(cid:16))(1(cid:17) uss) (cid:16) (cid:17) flt (cid:0) (cid:0) (cid:0)flt Using this extended methodology, I (cid:133)nd that layo⁄s contribute to 45 percent of unemployment (cid:135)uctuations but quits, being procyclical, lower the contribution of JS by 10 percentage points, a point originally made qualitatively by Elsby et al. (2009). The contribution of vacancy posting is 63 percent, close to that reported in Table 4 despite the shorter time period. 19

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Table 1: Estimating the matching function from Shimers’ Job Finding rate Dependent f f f f variable: Sample 1951:M1 2009:M02 1977:M1 2009:M02 2000:M12 2009:M02 1951:M1 2009:M02 (1) (2) (3) (4) Regression Composite index: Help Wanted Index JOLTS Composite index: HWI JOLTS HWI JOLTS Estimation OLS OLS OLS GMM σ 0.59*** 0.59*** 0.57*** 0.58*** (0.01) (0.01) (0.02) (0.02) R2 0.81 0.81 0.73 Notes:In all regressions, all variables were previously detrendedwith an HP filter (λ=107). Standard errors are reported in parentheses. For equation (4), 3 lags used for instruments. Table 2:Contribution of JF and JS to unemployment variance, 1951 2008 Variance Variance βJS βJF βη βd(JS) βd(JF) βη Matching fct°: No 24.4% 75.9% 0.3% 39.6% 59.6% 0.8% Control Endog: No Matching fct°: Yes 24.2% 71.8% 4.0% 39.6% 59.2% 1.2% Control Endog: No Matching fct°: Yes 39.3% 55.4% 5.3% 63.4% 34.8% 1.9% Control Endog: Yes Notes:“Matching fct°” indicates whether I use Shimers’ (2007) estimate forjf or if I instead modeljf usinga matching function (with a matching elasticityσ=0.59). “Control Endog” i ndicates whetherf captures all movements in JF or only those due to changes in vacancies. Table 3: Higher order moments of unemployment and hazard rates, 1955 2008 uss v JS 1.21** 0.79** 0.42** Skewness (0.53) (0.24) (0.09) 2.66** 2.06** 3.54** Kurtosis (1.16) (0.40) (1.40) Notes: All variables are expressed in log.For skewness,variables are detrended with an HP filter (‚=105). ). Newey West standard errors are reported in parentheses and ** indicates significance at the 5% level.The Skewness is measured with variables in first difference while the Kurtosis is measured with variables in levels. The job finding rate is modelled with σ=0.59. 22

Table 4: Contribution of JF and JS tohigher order moments of unemployment,1955 2008 Skewness Kurtosis γd(JS) γd(JF) γη γJS γJF γη Matching fct°: No 38.8% 60.1% 1.1% 27.4% 73.1% 0.4% Control Endog: No Matching fct°: Yes 38.8% 63.5% 2.3% 27.4% 69.7% 3.0% Control Endog: No Matching fct°: Yes 62.5% 42.7% 5.2% 44.0% 52.5% 3.5% Control Endog: Yes Notes:“Matching fct°” indicates wh ether I use Shimers’ (2007) estimate for jf or if I instead model jf using a matching function (with a matching elasticity σ=0.59). “Control Endog” indicates whether f captures all movements in JF or only those due to changes in vacancies. The Skewness is measured with variables in first difference while the Kurtosis is measured with variables in levels. 120 100 80 60 40 Print HWI Composite HWI 20 HWI JOLTS 0 1 6 1 6 1 6 1 6 1 6 1 6 5 5 6 6 7 7 8 8 9 9 0 0 9 9 9 9 9 9 9 9 9 9 0 0 1 1 1 1 1 1 1 1 1 1 2 2 Figure 1: Di⁄erent indexes of vacancy posting, 1951M01-2009M02. 23

0.4 0.3 0.2 0.1 0 0.1 0.2 m q 1 s 0.3 0 t JF t 0.4 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 2011 Figure 2: Empirical and model Job Finding rate. 24

7 uss v 6 JS 5 4 3 2 1 0 1 0.4 0.2 0 0.2 0.4 0.6 Figure 3: Kernel density estimates (Gaussian kernel) for steady-state unemployment, vacancy posting and the job separation rate. Dotted-lines represent the corresponding normal distributions. All variables are logged and detrended with an HP-(cid:133)lter((cid:21) = 105). 0.12 0.10 0.08 0.06 0.04 0.02 0 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 2011 Figure 4: Steady-state unemployment with identi(cid:133)ed highs and lows, 1951-2008. 25

0.4 ln(Uss) ln(JF) 0.3 ln(JS) 0.2 0.1 0 3 2 1 0 1 2 3 4 5 6 0.4 ln(Uss) 0.3 (1 s )ln(V) ln(JS) 0.2 h 0.1 0 0.1 3 2 1 0 1 2 3 4 5 6 Figure 5: Average business cycle dynamics for steady-state unemployment, the job separation rate, the job (cid:133)nding rate, vacancies, and the residual near unemployment lows. 0.1 0 0.1 0.2 ln(Uss) ln(JF) 0.3 ln(JS) 0.4 3 2 1 0 1 2 3 4 5 6 0.1 0 0.1 0.2 ln(Uss) (1 s )ln(V) 0.3 ln(JS) h 0.4 3 2 1 0 1 2 3 4 5 6 Figure 6: Average business cycle dynamics for steady-state unemployment, the job separation rate, the job (cid:133)nding rate, vacancies, and the residual near unemployment highs. 26

0.055 endogenous JF constant JF 0.05 0.045 0.04 0.035 0.03 0.025 0.02 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 2011 Figure 7: Estimates of the job separation rate with and without assuming f = f over t+(cid:28) t [t;t+1[: 27