Abstract
 Top of page
 Abstract
 1. INTRODUCTION
 2. THE MODEL
 3. ESTIMATION PROCEDURE
 4. EMPIRICAL RESULTS
 5. ROBUSTNESS
 6. THE IMPACT OF LEARNINGBYDOING
 7. CONCLUSION
 APPENDIX
 ACKNOWLEDGMENTS
 REFERENCES
 Supporting Information
We construct and estimate by maximum likelihood a job search model where wages are set by Nash bargaining and idiosyncratic productivity follows a geometric Brownian motion. The proposed framework enables us to endogenize job destruction and to estimate the rate of learningbydoing. Although the range of the observations is not independent of the parameters, we establish that the estimators satisfy asymptotic normality. The structural model is estimated using Current Population Survey data on accepted wages and employment durations. We show that it accurately captures the joint distribution of wages and job spells. We find that the rate of learningbydoing has an important positive effect on aggregate output and a small impact on employment. Copyright © 2009 John Wiley & Sons, Ltd.
1. INTRODUCTION
 Top of page
 Abstract
 1. INTRODUCTION
 2. THE MODEL
 3. ESTIMATION PROCEDURE
 4. EMPIRICAL RESULTS
 5. ROBUSTNESS
 6. THE IMPACT OF LEARNINGBYDOING
 7. CONCLUSION
 APPENDIX
 ACKNOWLEDGMENTS
 REFERENCES
 Supporting Information
Over the last few years, searchmatching models have become the subject of renewed scrutiny. This spurt of interest is largely motivated by the difficulty to reconcile both micro and macro features of the data with the theory. Central to the issue is the difference between market productivity and the value of leisure. Costain and Reiter (2008) or Hagedorn and Manovskii (2008) show that, by setting it close enough to zero, one can ensure that unemployment and posted vacancies fluctuate as much in the MortensenPissarides model as in the data. This solution, however, leads to a large discrepancy between predicted and observed wage inequality. Hornstein et al. (2007) illustrate this tension in a calibration exercise, finding that parameter values matching business cycle fluctuations greatly underestimate the degree of wage dispersion. Accordingly, recent research highlights the need to carefully analyze the micro predictions of the Mortensen–Pissarides framework.
This paper contributes to this project by estimating a searchmatching model where workers accumulate jobspecific skills through learningbydoing (hereafter LBD). This feature allows us to control for the effect of job tenure on the wage distribution; hence it lowers the amount of residual wage dispersion and consequently the cost of being unemployed. We find that LBD reduces the gap between the value of nonmarket activity derived from our structural estimation and the one needed to fit business cycle fluctuations, but only marginally so. Our analysis therefore concurs with the findings in Hornstein et al. (2007).
The rate of LBD cannot be estimated in a purely deterministic setup because the lower bound of the wage distribution increases with tenure when workers regularly progress along the learning curve. Given that some highly senior workers earn a wage close to their reservation wage, the estimated rate of LBD necessarily collapses to zero in a deterministic setup. This is why one needs to introduce some noise into the learning process. A possibility is to allow for measurement errors. The drawback of this approach is that it ignores the interaction between job destruction and the rate of LBD. We assume instead that jobs are affected by idiosyncratic productivity shocks. The addition of this realistic feature allows us to endogenize job separations so that our structural model takes into account the positive influence of LBD on job retention.1
Our analytical framework is therefore closely related to the canonical Mortensen and Pissarides (1994) model (hereafter MP94) with endogenous separations. We assume that productivity and thus human capital are purely matchspecific. We also assume that firms and workers cannot commit so that wages are set by Nash bargaining. We extend MP94 by letting initial productivities differ across jobs and by allowing workers to accumulate specific human capital. We also replace Poisson processes with Brownian motions in order to capture the high persistence of earnings shocks. Our model therefore builds on the framework described in Prat (2006). The contribution of this paper consists in structurally estimating the model on micro data. The analysis shows that the proposed extension of the MP94 model is able to match the joint distribution of wages and job spells, and identifies the parameter values required to achieve a good fit.
Given the considerable influence of the MP94 model, it may seem surprising that it has not yet been structurally estimated. A likely explanation is that doing so raises several challenges. First of all, endogenous separations greatly complicate the derivation of the likelihood function because one has to deduce all the sample paths that breach the reservation productivity. We show how this problem can be solved through the introduction of geometric Brownian motions. We therefore assume that logwages follow a random walk with a deterministic trend. This specification accords well with the high persistence of earning shocks. Yet it substantially simplifies the canonical ARMA decomposition of the individual earnings process since it only retains its martingale component, disregarding transitory shocks.2 This simplification yields substantial analytical gains, allowing us to solve for equilibrium in closedform, and thus to estimate the model by fullinformation methods.3 Hence, although earnings processes of the ARMAtype are more accurate, we propose geometric Brownian motions as a useful firstorder approximation.
The other main technical difficulty is due to the nonstandard properties of the likelihood function: the reservation wage and consequently the range of the data is a function of the estimated parameters. This peculiarity of job search models is well known (see, for example, Flinn and Heckman, 1982). In order to circumvent this problem, they proposed to evaluate the likelihood function in two steps. Unfortunately, we cannot use their methodology because endogenous job destruction implies that workers and firms separate as soon as the match productivity breaches the reservation threshold. As a result, the lowest reported wage is no longer a superconsistent estimator because the density at the reservation wage is equal to zero. We are nonetheless able to establish the asymptotic property of the estimators. We adapt to our setup the proof by Greene (1980) that, in order to establish asymptotic normality, standard regularity conditions need not be satisfied when the likelihood function is equal to zero at the parameterdependent boundary. Hence endogenous separation actually simplifies the analysis since it enables us to estimate the likelihood function as if it were standard.
After having derived the equilibrium of the economy and characterized the likelihood function, we estimate the model using data from the January 2004 supplement of the Current Population Survey. The supplement contains data on accepted wages and employment durations for a supposedly random sample of the US labor force. Its representative dimension accords well with the macro orientation of our model and especially with our focus on the aggregate wage distribution.
We do not estimate the model on panel data for the three following reasons. First, given the relatively stylized structure of the model, we prefer to restrict our analysis to crosssectional patterns and leave a more thorough inspection of wage dynamics to extensions with general human capital and onthejob search. Second, macro panel data for the US economy are not readily available. The relatively small size of the PSID makes it difficult to accurately estimate the wage distribution among job entrants,4 while linking households over time using the CPS data leads to a significant proportion of mismatches with potential selfselection issues.5 Lastly, deriving the likelihood function for crosssectional data turns out to be a comprehensive task since we have to characterize the wage distribution conditional on job tenure. Thus our estimation procedure lays the ground for future empirical research on either panel or crosssectional data.
We restrict our attention to workers without tertiary education because the estimates do not capture the accumulation of general human capital, which is known to be much more significant for skilled workers.6 The estimation procedure returns estimates for the rate of LBD of around 2% per year. We assess the ability of the model to fit the joint distribution of wages and job spells and find that it reproduces the data surprisingly well given its parsimonious specification. Then we use the estimates to characterize the impact of the rate of LBD. We show that it shifts to the right the wage distribution and significantly increases its dispersion.
1.1. Related Literature
To the best of our knowledge, this paper is the first to structurally estimate the MP94 model. This gap in the literature is explained by the fact that jobs' outputs follow stochastic paths in MP94, while estimable search models are typically based on the premise that productivities remain constant through time. As a result, early structural models7 generated flat wage profiles. Only recently has the empirical literature begun to address the observed pattern of wage dynamics.
An influential approach proposed by PostelVinay and Robin (2002) consider that workers can bring potential employers into Bertrand competition. Employer competition generates upwardsloping wage profiles because it enables workers to gradually appropriate the output of their jobs. Two recent papers build on this wagesetting rule and combine it with human capital accumulation. Bagger et al. (2006) assume that wages are defined as piece rate contracts whose values are determined using the sequential auction model of PostelVinay and Robin (2002). Yamaguchi (2006) augments the sequential auction framework with bargaining as in Cahuc et al. (2006).
We focus instead on the Nash bargaining rule prevailing in the standard theory of unemployment so that wages follow changes in productivity. In that respect, our approach is more closely related to the paper by Nagypál (2007), which studies a model with both LBD and learning about match quality. Her analysis aims at disentangling the contributions of these two mechanisms. Her focus is quite different from ours: whereas Nagypál (2007) analyzes in great detail the hazard rate of job separation and the wage profile, we put greater emphasis on the aggregate wage distribution.
The additional features included in these three papers greatly complicate the analysis. This is why they all rely on simulation techniques to estimate their structural models. To the contrary, the framework proposed in this paper can be solved analytically and estimated by maximum likelihood. This reflects our focus on the relatively stylized but nevertheless very influential MP94 model.
Lastly, this paper is naturally connected to the large body of empirical research using Mincer equations to evaluate the rate of LBD.8 On the one hand, our model is too stylized to contribute to the debate about the relative importance of job tenure versus experience since it does not include general human capital. On the other hand, our structural framework allows us to quantify the aggregate impact of LBD. We find that it has a significantly positive effect on aggregate output but a small effect on employment.
1.2. Structure of the Paper
The rest of the paper is organized as follows. Section 2 discusses the model setup and characterizes the equilibrium. The econometric procedure and the asymptotic properties of the estimates are detailed in Section 3. Section 4 describes the data and discusses the estimation results. Section 5 assesses the robustness of the estimates. In section 6, we introduce an aggregate matching function to close the model and evaluate the impact of LBD on the equilibrium. Section 7 concludes and the Appendix contains the proofs of the propositions.
3. ESTIMATION PROCEDURE
 Top of page
 Abstract
 1. INTRODUCTION
 2. THE MODEL
 3. ESTIMATION PROCEDURE
 4. EMPIRICAL RESULTS
 5. ROBUSTNESS
 6. THE IMPACT OF LEARNINGBYDOING
 7. CONCLUSION
 APPENDIX
 ACKNOWLEDGMENTS
 REFERENCES
 Supporting Information
We now discuss how to estimate the model's parameters. Our structural framework leads to similar identification problems to the deterministic search model. We therefore adopt the identification strategy laid out by Flinn and Heckman (1982). First, the truncation of the wage offer distribution implies that the model is fundamentally underidentified. We have to impose a parametric restriction on the distribution of job offers F(·) such that it belongs to the class of distribution functions which can be recovered from truncated observations. Section 3.3 discusses this problem in further detail.
The searching costs s and discount rate r are not individually identified because they both enter the likelihood function only through their impact on R. For this reason, we treat the reservation output R as if it were a parameter and estimate it directly. This may seem inconsistent with the fact that R is not an exogenous primitive of the model but rather an endogenous variable. Thus one may wonder whether we are actually discarding the information contained in the optimal stopping rule (5). However, when all parameter estimates have been obtained, equation (5) remains central to the estimation procedure as it allows us to ‘back out’ the locus relating r and s.15 Thus one should think of equation (5) as a generalization of the reservation rule in deterministic settings, the only difference being the additional term α/(α − 1) which captures the value of waiting. Given that the point estimates of the identified parameters pin down the value of α, we can follow, without loss of generality, the same procedure as Flinn and Heckman (1982).
Lastly, the bargaining power β has to be fixed prior to the estimation. Although β is theoretically identified due to the highly nonlinear likelihood function, trials show that its estimates often do not converge to an interior solution. In the absence of information about firms' profits, it is not surprising that our dataset does not allow us to recover both sizes and allocations of job surpluses. This wellknown difficulty is now gradually overcome by research based on matched employer–employees data (see Cahuc et al., 2006). Given the onesided nature of the CPS data, we stick to the usual practice of assuming symmetric bargaining and then perform some robustness tests with respect to β.
3.1. The Likelihood Function
Following these preliminary steps, the likelihood of the sample can be expressed as a function of the remaining set of parameters. We slightly restrict the generality of the problem by assuming that the distribution of job offers F(·) can be completely parametrized in terms of a finitedimensional vector Ω so that the set of estimated parameters Θ = {ζ, δ, σ, Ω, R, λ}.
The likelihood of the sample is computed as follows. Let Y denote the set of observations, so that Y≡{y^{1}, y^{2}, …, y^{n}}, where n is the total number of workers in the sample. The individual observations are defined using four variables: e^{i}, w^{i}, T^{i}, τ^{i}. Let e^{i} denote an indicator function which takes value 0 when worker i is currently unemployed, 1 when he is employed but fails to report his tenure, and 2 when worker i is employed and reports both his wage and job spell. The variables w^{i} and T^{i} are the current hourly wage and job tenure. In the case where worker i fails to report the length of his job spell, i.e. e^{i} = 1, T^{i} is obviously ignored. If worker i is currently searching for a job, i.e. e^{i} = 0, y^{i} is set equal to the unemployment duration τ^{i}. The likelihood function is therefore made of three distinct components. The individual contribution of a job searcher is equal to the density associated with an ongoing unemployment spell of length τ conditional on unemployment times the probability of observing an unemployed worker:
where F̄(P)≡1 − F(P). The likelihood of observing an employed worker paid wage w is given by υ(x(w)). The expression can be further decomposed reinserting (11) into (9) to obtain
Note that output is defined as a function of the observed wage. Its implicit value follows from combining (5) with (6). Similarly, the joint likelihood of observing a worker paid wage w with a job tenure equal to T is given by
Putting together these three components, the loglikelihood for the observed sample reads
 (12)
where n_{S} is the number and S is the set of indices of job searchers in the sample, W is the set of indices of employees who only report their current wages and H is the set of indices of employees who report both wages and job spells. Although the parameters ζ and Ω do not appear in the analytical expression of the likelihood function, they are implicitly identified: ζ determines the values of µ, γ, φ(·) and ψ(·), while the parametric vector Ω obviously influences F(·). Note that, as discussed before, the reservation output is treated as a primitive parameter of the model.
3.2. Asymptotic Properties
The likelihood function is continuously differentiable and its parameters belong to a compact support. Yet it does not satisfy all the standard requirements for a wellbehaved likelihood function since the support of the distribution is a function of the parameters. Furthermore, the density functions f(e, w(R)) and f(e, w(R), T) are both equal to zero at the reservation wage. Hence the lowest reported wage is not a superconsistent estimator of the reservation wage, so that we have to rely on a different estimation method from the twostep approach proposed by Flinn and Heckman (1982).
Our problem bears similarities to the estimation of optimal production frontiers. Optimal frontiers models also imply that the range of observations changes with the parameters being estimated. Moreover, they share with our model the additional implication that agents are never exactly on the optimal frontier.16 As firms cannot perfectly counteract random perturbations, they remain within the neighborhood of the optimal combination of inputs without ever achieving it perfectly. Given that the estimation of optimal frontiers is one of the most popular areas of applied econometrics, great attention has been devoted to the econometric solutions for this kind of problem. In an influential paper, Greene (1980) showed that when the density at the parameterdependent boundary is equal to zero standard regularity conditions need not be satisfied in order to produce standard asymptotic distribution results. We adapt Greene's proof to our setup and establish that the estimators satisfy asymptotic normality under standard requirements.
Proposition 5Suppose that: (i) the parameter space Γ is compact and contains an open neighborhood of the true value Θ_{0}of the population parameter; (ii) the distribution of job offersF(P) is continuously differentiable. Then the maximum likelihood estimator
converges in probability to Θ_{0}so that, where H is the Hessian of the likelihood function and J is the information matrix.
The proof of Proposition 5 relies on the fact that f(1, w(R)) and f(2, w(R), T) are both equal to zero. This property justifies interchanging the order of integration and differentiation so that the asymptotic property of the estimators can be characterized by linear approximation. Our problem is slightly less standard than the one considered by Greene (1980) because the derivatives of the density functions with respect to Θ are not equal to zero when evaluated at the reservation output. Thus interchanging the order of integration and differentiation is justified solely for the first derivative. This is why the hessian matrix H is not equal to − J, so that the asymptotic covariance matrix cannot be simplified and set equal to J^{−1}. In any case, as explained in Newey and McFadden (1994), the information matrix equality is not essential to asymptotic normality. The only complication is technical, as reflected by the intricate form of the asymptotic variance.
3.3. Lognormal Distribution of Job Offers
We have characterized the estimation procedure for general distributions of job offers. The econometric implementation of the model requires to narrow the analysis to a particular family of distributions. We hereafter assume that F(·) is lognormal. Lognormal distributions are commonly considered because they satisfy the ‘recoverability condition’ defined by Flinn and Heckman (1982), meaning that their location and scale parameters can be recovered from truncated observations. The class of functions which satisfy the ‘recoverability condition’ also encompasses, among others, gamma distributions.17 Lognormality is eventually justified by its good fit of the data. In our case this assumption has a more crucial role. Given the intricate expression of the likelihood function, there is little hope to derive it in closed form. Solely when initial productivities are lognormally distributed in the population does the likelihood function admit an analytical expression so that approximation errors due to numerical integrations can be avoided.18 Given its length, we do not include the expression of L(Θ) in the body of the paper.
Proposition 6Under the assumption that the initial productivities are drawn from a lognormal distribution, so that
 (13)
the likelihood functionsL(Θ) has a closedform solution. The resulting expression is reported in the Appendix.
6. THE IMPACT OF LEARNINGBYDOING
 Top of page
 Abstract
 1. INTRODUCTION
 2. THE MODEL
 3. ESTIMATION PROCEDURE
 4. EMPIRICAL RESULTS
 5. ROBUSTNESS
 6. THE IMPACT OF LEARNINGBYDOING
 7. CONCLUSION
 APPENDIX
 ACKNOWLEDGMENTS
 REFERENCES
 Supporting Information
In this section we introduce an aggregate matching function to close the model and evaluate the impact of LBD on labor market outcomes. We assume that the matching process is similar to the one described in Pissarides (2000). Since the matching function has become the workhorse for the study of equilibrium unemployment, the exposition can be brief. Firms post vacancies that are randomly matched and incur a flow cost equal to c. The number of job matches per unit of time is a function of the number of vacancies and job seekers. When the aggregate matching function is homogeneous of degree one, the rate at which a vacancy meets a worker depends only on the unemployment rate u and on the ratio v of vacant jobs divided by the size of the labor force. The transition rate for vacancies is given by a function q(θ) where the labor market tightness parameter θ denotes the vacancy–unemployment ratio. Similarly, jobs seekers meet firms at the rate θq(θ).
As opposed to the labor force whose size is fixed and normalized to one, new firms enter the market until arbitrage opportunities are exhausted. Therefore the FreeEntry condition is given by
 (14)
Similarly we can replace in (2) the exogenous contact rate λ by θq(θ) to obtain
Reinserting (14) into the previous equation allows us to solve for the asset value of being unemployed as a function of labor market tightness;
The job's surplus follows from replacing the previous equation in (3). Accordingly the Optimal Separation rule is such that
 (15)
The equilibrium values of the two endogenous variables θ and R are determined by the equilibrium conditions (14) and (15).32 Since the aggregate matching function defines a onetoone mapping between λ and θ, a parametric assumption allows us to retrieve the values of the labor market tightness and searching costs using the estimates reported in the previous section. As is common in the literature, we assume that the matching function is Cobb–Douglas. We further restrict our attention to the case where the allocation is efficient and consequently use the ‘Hosios condition’ to set the elasticity of the matching function equal to β, so that q(θ) = θ^{−1/2}.
Table V contains the implied costs of search and equilibrium labor market tightness for the deterministic and stochastic models as well as for the restricted estimates. The high values of equilibrium tightness are unreasonable if interpreted as the ratio of vacancies to job seekers. Thus one should interpret θ as measuring the ratio of recruitment effort to search effort. The relatively high searching costs are required to offset the important surpluses of the matches in the right tail of the distribution of job offers.
Table V. Point estimates of remaining variables  Deterministic  Stochastic  Entrants  Tenure < 1 year 


Tightness 
θ  3.86  3.80  3.86  3.95 
 (0.320)  (0.286)  (0.321)  (0.302) 
Flow costs of search 
s  98.2  92.7  80.2  96.0 
 (4.36)  (4.45)  (7.29)  (6.37) 
c  26.8  25.4  21.8  25.2 
 (1.26)  (1.25)  (2.23)  (1.85) 
By keeping the values of the environmental parameters constant and varying the rate of LBD, we can simulate its impact on labor market outcomes. The results are reported in Figure 4. The upperleft panel shows the effect on aggregate wage distribution. Not surprisingly, a higher rate of LBD increases the mass in the right tail. This does not necessarily lead to higher inequality, however, because the left tail of the wage distribution is truncated by the increase in the reservation wage. The ratio of standard deviation to average wage reported in the upperright panel shows that the latter effect dominates when the rate of LBD is close to zero.
Unemployment rate as a function of ζ is reported in the lowerleft panel. As expected, the function is decreasing but its elasticity is close to zero. To understand why, it is useful to recall that the opportunity cost of employment rU is equal to −s + cθβ/(1 − β). This implies that, for the estimated value of the recruitment costs c, the impact of θ on the worker's outside option is amplified by more than an order of magnitude. Small adjustments of the vacancy–unemployment ratio have drastic effects on wages, which explains why employment remains remarkably stable.
The rigidity of the unemployment rate and the flexibility of wages are even more pronounced in the deterministic model. This is because the model without LBD does not control for the effect of job tenure on wage dispersion. As a result, the opportunity cost of employment is lower in the deterministic model. Introducing LBD reduces the gap between the value of nonmarket activity derived from structural estimations and the one required to fit business cycle fluctuations of the unemployment rate. However, the adjustment remains rather modest, so that the conclusion reached by Hornstein et al. (2007) continues to hold in our setup.
For the same reasons the estimates imply that labor market policies, such as employment subsidies or unemployment benefits, affect mostly wages and leave employment nearly unchanged. The predictions of the deterministic model are similar with an even stronger employment rigidity.33 Accordingly, the model suggests that policies aimed at reducing the rate of unemployment should focus on lowering both recruitment and search costs.
The lowerright panel contains a plot of the aggregate output as a function of the LBD rate. We normalize aggregate output to one when ζ = 0, for ease of interpretation. The model predicts that an increase of the LBD rate from 0 to 4% raises aggregate output by around 45%. These gains arise due to three reinforcing effects: (i) the direct impact of LBD obviously leads to a higher average output for a given job spell; (ii) the increase of the reservation wage implies that, ceteris paribus, ongoing job relationships have a higher average productivity; (iii) the higher rate of employment mechanically raises aggregate output. Decomposing the relative importance of these effects shows that the first one accounts for nearly 98% of the total gains. The increase in the reservation productivity explains 1.8% of the total gains, while the slight increase in employment accounts for the remaining 0.2%.
Before concluding, we would like to discuss two ways in which the simulation exercise could be improved. First, the results are partly driven by the low estimates for the variance parameter σ, since a higher volatility would magnify the impact of the reservation wage on job destruction and thus unemployment. One might suspect that estimating the model on panel data would lead to higher values for σ. Hence the model's implication might differ when estimated on a data source with richer information about wage dynamics. Second, our structural model does not take into account prevalent labor market regulations such as firing costs and employment subsidies. A recent paper by Silva (2008) illustrates how our framework could be used to evaluate the impact of labor market institutions. She extends the setup by introducing a minimum wage and severance payments, before estimating the resulting model using data on employment histories from Chile. She finds that severance payments are more efficient than minimum wages in reducing distortions due to low bargaining power for workers.
7. CONCLUSION
 Top of page
 Abstract
 1. INTRODUCTION
 2. THE MODEL
 3. ESTIMATION PROCEDURE
 4. EMPIRICAL RESULTS
 5. ROBUSTNESS
 6. THE IMPACT OF LEARNINGBYDOING
 7. CONCLUSION
 APPENDIX
 ACKNOWLEDGMENTS
 REFERENCES
 Supporting Information
It has been shown in this paper how the production side of the Mortensen–Pissarides model with endogenous job separation can be estimated by maximum likelihood using crosssectional data. The analysis establishes that the parsimoniously specified model convincingly fits the joint distribution of wages and job spells once LBD is taken into account. A concrete contribution of the analysis is to identify the rate of LBD in an equilibrium setup, whereas the estimates available in the literature are typically based on ‘reducedform’ estimations. Taking into account the accumulation of jobspecific skills raises the opportunity cost of employment. Yet the increase is too small to noticeably augment the elasticity of the unemployment rate. For example, we find that the rate of LBD has a significantly positive effect on aggregate output but a small impact on employment.
As we have deliberately tipped the balance in favor of tractability over realism, the model lends itself to several theoretical extensions. We conclude by briefly discussing some of these. The most obvious refinement would be to introduce general human capital. Although not so demanding at the conceptual level, this extension will come at the cost of closedform solutions. More promising is the introduction of onthejob search since it would connect the model with the burgeoning econometric literature based on employers' competition. Until recently, uncertainty and onthejob search have been considered in isolation. But, as attested by a series of recent papers (Bagger et al., 2006; Yamaguchi, 2006), the importance of combining both dimensions is now widely recognized. Such a research project raises serious technical challenges. For the moment, these structural models treat job separations as exogenous and available estimates are based on indirect inference methods. This paper suggests that stochastic calculus helps to alleviate some of the difficulties. Finally, we also hope that the derivations of the asymptotic properties of the estimators will be of some interest to researchers working in areas other than labor economics since our result can be applied to a wide class of models with endogenous exit.