Keywords: Gross worker flows, job finding rate, employment exit rate, matching function
Abstract:
JEL classifications: E24, E32, J63, J64
At the beginning of a recession, does unemployment go up because of fewer hirings, more job losses or both? What is the most effective policy to mitigate that increase, a firing tax, a hiring 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 fluctuations. In this paper, I study the relative importance of the two main driving forces of cyclical unemployment: vacancy posting, i.e. firms' recruiting efforts, and job separation.^{1}
An extensive literature has studied worker flows over the business cycle, and more recently Shimer (2007) focused on individual workers' transition rates and concluded that unemployment inflows contribute much less to unemployment fluctuations than unemployment outflows.^{2} This very influential conclusion led to a recent modeling trend that assumes that the job separation rate (JS) is acyclical.^{3} However, a flow rates decomposition exercise may underestimate the contribution of JS because the job finding probability does not only depend on firms' 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 finding rate (JF), one may attribute the high unemployment to a low JF, even though the true cause was an increase in job separation.
The first contribution of this paper is to address the endogeneity of JF by using a matching function to model the flow of new jobs.^{4} A matching function is extremely successful empirically 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 fluctuations in the job finding rate caused solely by changes in firms' job openings, I find that the contribution of the job separation rate to unemployment's variance is close to 40 percent instead of 25 percent using Shimer's (2007) methodology. Thus, not modeling the cyclicality of the job separation rate will lead researchers to understate the volatility of unemployment.
This second contribution of this paper is to extend the method pioneered by Shimer (2007), Elsby, Michaels and Solon (2009) and Fujita and Ramey (Forthcoming) and study the determinants of unemployment's higherorder moments. I find that JS plays an important role with respect to skewness and kurtosis. In particular, the steepness asymmetry of unemployment the fact that increases are steeper than decreases is due in large part to the job separation rate, which accounts for more than 60 percent of firstdifferenced unemployment skewness.^{5} Further, JS and vacancy posting contribute in roughly equal proportions to unemployment's mild kurtosis. However, this decomposition hides an important difference 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 fluctuations 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 find that job separation is responsible for almost all of the movements in unemployment during the first 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's method, its potential endogeneity bias and presents a way to address it; Section fits a matching function to the data and assesses the contribution of the job separation rate to unemployment's moments after controlling for the endogeneity of the job finding rate, Section 4 studies the behavior of the hazard rates at business cycles turning points; and Section 5 offers some concluding remarks.
In this section, I briefly review Shimer's (2007) methodology to identify the contributions of JF and JS to unemployment's variance and discuss the possible endogeneity of the job finding rate.
Denoting the unemployment rate at instant with and , Shimer (2007) postulates that during a "period " of one month  i.e.  all unemployed workers find a job according to a Poisson process with constant arrival rate and all employed workers lose their job according to a Poisson process with constant arrival rate . As a result, we have the firstorder differential equation:
The separation rate can then be estimated by solving (1) over and finding such that the solution equals for . Again, this estimation method relies on the assumptions that the job finding 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
Following Elsby, Michaels and Solon (2009) and Fujita and Ramey (Forthcoming), loglinearizing (3) gives
A potential bias in Shimer's approach was first emphasized by Fujita and Ramey (Forthcoming) who argue that Shimer's decomposition may understate the true contribution of the job separation rate. For example, if a high separation rate leads to a low job finding rate, one may attribute the low job finding rate to high unemployment, even though the separation rate was the true cause.^{6}
A simple way to think about this endogeneity problem is to consider a search and matching setup in continuous time. The job finding rate is defined as the ratio of new hires to the stock of unemployed, so that if denotes the number of new matches at instant , unemployed workers find a job according to a Poisson process with time varying arrival rate , where is the number of unemployed. An increase in the job separation rate will increase unemployment and mechanically lower the job finding rate, and a variance decomposition exercise that does not take into account the link between and will understate the contribution of JS.^{7}
Following the literature and assuming a CobbDouglas matching function with constant returns to scale, I can write such that with the number of job openings. Importantly, this specification is standard and is used 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 "exogenous" component of the job finding rate, i.e. the movements in JF that are due to fluctuations in vacancy posting, not to fluctuations in unemployment.
However, because is not constant over , one could worry that Shimer's (2007) method to recover and is not valid anymore because the differential equation satisfied by unemployment changes and takes the form
Fortunately, Shimer's approach still goes through if, within each month, movements in over are negligible compared to 's start of the period value. Indeed, if with , one can reasonably approximate the instantaneous job finding rate with the average one so that . Under this approximation, the differential equation reduces to (1) and one can recover as in Shimer (2007). In the Appendix, I show that this approximation is reasonable as it does not lead to any substantial bias in . Hence, from now on, I assume as in Shimer (2007) that at a monthly frequency, ,In this section, I study the contributions of vacancy posting and the job separation rate to unemployment fluctuations. I first argue that and are a natural set of variables to consider because they capture the decision variables of firms and workers and as such constitute the " primitive" variables that respond to shocks and determine unemployment. I then estimate a estimate a matching function to capture movements in the job finding rate, and I use the hazard rate decomposition approach to evaluate the contributions of vacancy posting and job separation to unemployment's variance, skewness and kurtosis.
While the literature has traditionally studied the properties of job flows and worker flows, it also natural to consider the behavior of because these variables are the control variables that economic agents (firms and workers) adjust in response to shocks, and that policy can directly influence (through e.g. a hiring subsidy or a firing tax). Starting with the firm's problem, a firm can adjust its number of workers through two channels: hirings and firings. For example, a firm faced with a positive TFP shock can choose to increase hirings, decrease firings or use a combination of both. Put differently, hirings and firings are the two control variables of the firm (with respect to employment). However, by focusing on gross worker flows, one cannot rely on the hazard rate decomposition approach to quantitatively estimate the contribution of each margin of adjustment. Fortunately, for a firm, choosing the number of new hires and fires is isomorphic to choosing the number of job openings (assuming that they ultimately all get filled) and choosing the percentage of the workforce to be shed, i.e. the job separation rate due to layoffs. Turning to the worker's problem, an employed worker can decide whether to quit and as a result can influence the job separation rate due to quits. As a result, the total job separation rate (defined as the number of layoffs and quits over the number of employed workers) captures both firms 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, layoffs and quits.
To model the job finding rate, I estimate a CobbDouglas matching function that can capture movements in the monthly job finding rate. Under the assumption that over each month , I can use Shimer's estimate of the job finding rate , and I estimate the following equation
Seasonally adjusted unemployment is constructed by the BLS from the Current Population Survey (CPS). More difficult is the choice of a measure for vacancy posting . There are two standard measures of job openings; the HelpWanted advertising Index (HWI) and the Job Openings and Labor Turnover Survey (JOLTS). The HelpWanted Index is constructed by the Conference Board and measures the number of helpwanted 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 "print" 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 instead since 2005 a measure of online help wanted advertising. To build an index that combines information on " print" and "online" advertising, I follow Fallick's (2008) approach and estimate the share of print helpwanted 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 printonline help wanted advertisements until December 2000 and using JOLTS data thereafter.^{11} Figure 1 presents the different measures of vacancy posting.
I first estimate (5) with monthly data and using the composite HWIJOLTS index from 1951:M01 until 2009:M02. All data were previously detrended with an HP filter. Table 1 presents the result. The elasticity is precisely estimated at 0.59, and apart from JF's highfrequency movements (probably due to measurement errors), a matching function does a very good job at capturing movements in the job finding rate. Indeed, after taking quarterly averages, Figure 2 shows that a matching function tracks the empirical job finding rate very closely. Since JOLTS and HWI are two different 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 lowfrequency movements in HWI before 1977 that are unrelated to the labor market, I estimate (5) with the printonline helpwanted index over 1977:M012009:M02 only. We can see that the estimated is unchanged at 0.59. Finally, I use JOLTS data only over 2000:M122009:M02 and find a slightly lower at 0.57. Encouragingly, these estimates lie in the middle of the plausible range reported by Petrongolo and Pissarides (2001).
A legitimate concern with this regression exercise is that equation (5) may be subject to an endogeneity bias. The use of a monthly frequency and the fact that denotes the beginning of period unemployment rate should minimize the problem, but it is still important to verify that there is no significant bias. To do so, I estimate (5) using lagged values of as instruments.^{12} Encouragingly, Table 1 shows that the endogeneity bias is likely to be small as the coefficient is little changed at 0.58.^{13}
The robustness of the results over different measures of vacancies and over different sample periods is promising and suggests that a matching function provides a good approximation of the job finding rate and can be reasonably used to control for the endogeneity of JF. For the rest of the paper, I will use the composite HWIJOLTS measure of vacancy posting with a matching function elasticity but the results do not rely on this specific choice.
Writing the steadystate approximation for unemployment (3) at a quarterly frequency (as in Shimer, 2007) and modeling the job finding rate with a matching function, I get
This approximation relies on the implicit assumption that movements in have an effect on steadystate unemployment (which is the case by definition) as well as on the job finding rate within the time period, so that the quarterly average of the monthly job finding rate reflects the influence of the job separation rate. Fortunately, in the US, unemployment converges to its steadystate value in about a month (Shimer, 2007), so that the dynamic interactions between JS and JF (through the matching function) are likely to be reflected in the quarterly (and a fortiori yearly) steadystate decomposition.^{15} Moreover, I can track the validity of my approach by looking at the contribution of the residual. Indeed, after loglinearizing (6) and using the fact that , I can rewrite (4) as
Rearranging (7), I have
or
I now proceed with the variance decomposition exercise by using the fact that
so that and measure the contributions of job separation and the "exogenous" (i.e. independent of unemployment) component of the job finding rate to unemployment's variance.A backoftheenvelope 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 and , the endogeneity of JF biases the contribution of JS downwards by 60 percent (from ). Instead of a contribution of about 25 percent as reported in Shimer (2007), JS would in fact contribute to about 40 percent, a far from negligible amount.^{16}
Using the logdeviation from trend where and denote the trend component of and , I can rewrite (8) as
Using a firstdifferenced logdecomposition as in Fujita and Ramey (Forthcoming) and using , I have
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 finding rate raises the contribution of JS to unemployment's variance by 60 percent; with a split between vacancy posting and job separation for a decomposition in level and a split for a decomposition in firstdifferences. As a result, modeling the job separation probability as acyclical will lead researchers to understate the volatility of unemployment.^{18}
While the literature has focused on unemployment's variance to evaluate the importance of the job separation rate, higherorder moments could paint a different picture. Notably, a stylized fact about unemployment is its asymmetric behavior, and a large literature has documented a nontrivial 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's (Forthcoming) approach to higherorder moments and notably to the concept of skewness.
Let us denote the mean of as and its th moment for
Further, let us assume that can be written as a sum of terms so that By noting that , I have so that I can write
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 firstdifference logunemployment. Table 3 shows that over 19552008, firstdifferenced logunemployment has a skewness coefficient of 1.2, significant at the 5% level.^{21} Vacancy posting and JS also present a significant asymmetry in steepness with coefficients of 0.79 and 0.42. Using the logdecomposition (9) and using , I have
so that I can interpret and as the contributions of the job separation and vacancy posting margins to the skewness of firstdifferenced unemployment. Table 4 shows that while the job separation rate contributes to less than half of unemployment's variance, this is hardly the case with unemployment asymmetry since the job separation's contribution stands at more than 62 percent. Thus, a model that would not consider fluctuations 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 first 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 .Table 3 presents another new fact pertaining to the fourth moment of unemployment and its hazard rates. While unemployment has a mild (but significant) negative excess kurtosis ( 0.34), vacancy posting and job separation have kurtosis of opposite signs. Vacancies present a large negative excess kurtosis ( 0.94) but JS presents a positive excess kurtosis ( 0.54). Recall 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 finding is not surprising if we think of job separation as capturing (among other things) bursts of layoffs. 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'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's fourth moment, with a slight advantage for vacancy posting. Given the lower contribution of JS to unemployment's variance, the mild negative kurtosis of unemployment despite the large negative kurtosis of vacancy posting is consistent with an interpretation of job separation influencing 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 . While only indicative, this fourthmoment decomposition suggests that vacancy posting drives unemployment during normal times but that job separation is responsible for rare but violent fluctuations in unemployment.
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 fluctuations 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 fluctuations.
After detrending unemployment using an HPfilter with , 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 steadystate unemployment rate with identified turning points.^{22}
The first rows of Figure 5 and 6 plot the average dynamics of the logdeviation from trend of steadystate unemployment, the job separation rate, and the job finding rate in a window of 3 and 6 quarters before and after the highs and lows of unemployment.^{23} As first shown by Elsby et al. (2009) with NBER recessions dates, an interest of this approach is that the logdecomposition (4) allows us to directly observe the relative contributions of JS and JF to unemployment fluctuations. The second rows of Figure 5 and 6 display the same average dynamics but using the log of vacancy times instead of JF.^{24} From (10), we can directly observe the relative contributions of job separation and vacancy posting as corresponds to movements in unemployment caused by changes in vacancy posting.
A first observation is that, while the previous section showed that vacancy posting was, on average, the most important contributor of unemployment fluctuations, this is hardly the 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's (2009) findings for NBER recessions: once unemployment reaches a low, JS is responsible for most of the initial increase in unemployment, 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 "exogenous" component of JF, job separation accounts for more than 50 percent of unemployment movements for as much as 6 quarters after a high or a low, and for almost all of the initial response. Interestingly, this result is consistent with the decomposition of unemployment's fourth moment in the previous section, which suggests that extreme values of unemployment are due to the job separation rate. Looking at the contribution 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'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 first quarters after a turning point. Second, after unemployment highs, vacancies lag job separation by a quarter. This is in line with Fujita and Ramey (Forthcoming), who find that the job separation rate lags the job finding rate.
An implication of these last findings is that ignoring the job separation margin when modeling unemployment will lead researchers to underestimate the breadth and speed of adjustments in unemployment around turning points.
In this paper, I study the relative importance of the two main determinants of unemployment fluctuations: vacancy posting and job separation. By isolating fluctuations in the job finding rate due solely to changes in vacancy posting, I take a first step to address the endogeneity of the job finding rate, and I find that the contribution of the job separation rate to unemployment's variance is close to 40 percent instead of 25 percent using Shimer's (2007) methodology. I also extend Shimer (2007), Elsby et al (2009) and Fujita and Ramey (Forthcoming) variance decomposition exercise to higherorder moments, and I find 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 more important contributor of unemployment fluctuations, 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.
An important issue when using measures for unemployment, vacancy posting and job finding probability concerns the precise definition 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, defined as the week including the 12th day of the month. The HelpWanted Index measures the total number of advertisements (print or online) from the 14th ()of the month to the 13th of next month (). JOLTS, on the other hand, indicates the number of job openings on the last day of month . Finally, Shimer's (2007) definition of implies that measures the average job finding 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 finding probability should depend on the average unemployment rate and the average number of posted vacancy between two reference weeks. Since measures the unemployment rate during the first reference week, the correct measure of unemployment inside the matching function should be . Since already corresponds to an average over a period and measures the number of job openings at a date roughly in between two reference weeks, corresponds to as those two measures would be equal if the number of job openings remained constant in between two reference weeks.
As a result, a more consistent regression would be
In this appendix, I describe a more rigorous way to recover the job separation rate without the need to assume that over . While this approach is quite sensitive to the parameterization of the matching function and the value of , it allows me to verify that assuming has almost no consequences on the estimation of Instead of assuming that remains constant over , I make the weaker assumption that only is constant over and equals . This assumption is consistent with the definition of ; the total number of vacancies over (see Appendix A1). The law of motion for unemployment (1) now takes the form
Similarly to Shimer (2007), I then solve this differential equation for different values of until the solution at time equals . In Figure 7, I compare the estimates of obtained with and without assuming constant hazard rates. As we can see, both estimates are extremely similar suggesting that the approximation over is reasonable as it does not lead to any substantial bias in .
In this section, I study the separate contributions of layoffs and quits to unemployment's variance by using CPS data from the BLS on the reasons for unemployment (layoffs, quits or labor force entrants) over 19682004 as in Elsby et al. (2009). Denoting , and the unemployment rates by reason, I have and , with , and where is the employment rate and the labor force participation rate. Looking at Elsby et al. (2009) decomposition, we can see that business cycle fluctuations in and are small compared to cyclical fluctuations in the hazard rates, and that fluctuations in are small compared to movement in the other inflows rates (see Elsby et al. (2009), Figures 9 & 11). As a result, I can write the following approximation
Table 1: Estimating the matching function from Shimer's Job Finding rate
Dependent variable: Sample  1951:M12009:M02  1977:M12009:M02  2000:M122009:M02  1951:M12009:M02 

Regression  (1)
Composite index: HWI  JOLTS 
(2)
HelpWanted Index 
(3)
JOLTS 
(4)
Composite index: HWI  JOLTS 
Estimation  OLS  OLS  OLS  GMM 
0.59***
(0.01) 
0.59***
(0.01) 
0.57***
(0.02) 
0.58***
(0.02) 

R^{2}  0.81  0.81  0.73   
Table 2b: Contribution of JF and JS to unemployment variance, Yearly, 19512008
Variance:  Variance:  Variance:  Variance:  Variance:  Variance:  

Matching fct : No
Control Endog: No 
22.2%  77.9%  0.1%  28.4%  69.4%  2.2% 
Matching fct : Yes
Control Endog: No 
22.9%  77.2%  0.1%  28.4%  72.2%  0.2% 
Matching fct : Yes
Control Endog: Yes 
37.0%  64.1%  1.1%  45.8%  56.8%  2.6% 
Table 3: Higherorder moments of unemployment and hazard rates, 19552008
JS  

Skewness  1.21**
(0.53) 
0.79**
(0.24) 
0.42**
(0.09) 
Kurtosis  2.66
(1.16) 
2.06**
(0.40) 
3.54**
(1.40) 
Table 4: Contribution of JF and JS to higherorder moments of unemployment, 19552008
Skewness:  Skewness:  Skewness:  Kurtosis:  Kurtosis:  Kurtosis:  

Matching fct : No
Control Endog: No 
38.8%  60.1%  1.1%  27.4%  73.1%  0.4% 
Matching fct : Yes
Control Endog: No 
38.8%  63.5%  2.3%  27.4%  69.7%  3.0% 
Matching fct : Yes
Control Endog: Yes 
62.5%  42.7%  5.2%  44.0%  52.5%  3.5% 