Remarks on Structural Estimation The Search Framework

Transcription

1 Remarks on Structural Estimation The Search Framework Christopher Flinn NYU and Collegio Carlo Alberto November The Estimation of Search Models We develop a simple model of single agent search set in continuous time. We then discuss issues in identification and estimation of this type of model, that largely follow Flinn and Heckman (1982) and Flinn (2002). In the latter paper, you can find a discussion of estimation of a slightly more general version of this model that uses Italian data (from a special supplement of the ISTAT Labor Force Survey). 1.1 The Basic Job Search Model in Continuous Time The reservation wage in a continuous time stationary environment is derived as follows. For the moment, we consider the one-shot case in which an individual searchs for a job once and only once. To begin with,we set up the dynamic optimization problem as if it were a discrete time model, in which the length of a decision period was ε>0. [Typically discrete time models consider the decision period to be a month or a year.] V n denotes the value of continuing search given that an acceptable job hasn t been located yet. When such a job is located, the wage offer w is accepted, and, since nothing changes for the rest of eternity, the searcher keeps the same job. The present value of the job is then w/ρ, where ρ is the discount rate. The job offers are independent, identically distributed (i.i.d.) draws from a fixed distribution F, which we will assume to be continuous on a subinterval of(0, ). The offers arrive at random times, with the distribution of the times between offers given by g(t) =λ exp( λt), λ>0. Why make this assumption? Because for this distribution the hazard rate is given by h(t) = g(t) 1 G(t) = λ exp( λt) exp( λt) = λ 1

2 is a constant. This makes the process memory-less. In practical terms, it means that no matter how long the individual has been waiting for an offer, the likelihood that it will arrive in the next small interval of time is always the same. Given all of the other constancy in the environment, it means that she will not adjust her set of acceptable wages as she continues to search. The value of search over an arbitrarily small time period ε is given by Z V n =(1+ρε) 1 {bε + λε max( w ρ,v n)df (w)+(1 λε)v n + o(ε)}, where (1+ρε) 1 serves as an instantaneous discount factor, bε is the flow value of search accumulated over the period ε, λε is the approximate probability of receiving one offer over the period, and o(ε) represents all events that can occur over the period ε that involve more than one event, with lim ε 0 o(ε)/ε =0. The first term in the max operator is the value of accepting an offer of w, w/ρ, and the second argument is the value of rejecting it, which is the value of continuing search. The idea behind this representation is that all changes in the choice set and decisions are made at the end of the period ε. The decision rule is obtained by taking the limit of this expression as ε 0, so our assumption involves no loss of generality. Rewrite the expression as Z (1 + ρε)v n V n = bε + λε max( w ρ V n, 0)dF (w)+o(ε) ρεv n = bε + λε ρ Z ρv n (w ρv n )df (w)+o(ε). Dividing both sides by ε and taking limits as ε 0, we have ρv n = b + λ Z (w ρv n )df (w). ρ ρv n The reservation wage, w ρv n is defined by this implicit function. Due to the time homogeneity of all primitive parameters, the value w is constant. All behavior in this model is summarized by w. 1.2 Can We Take This to Data? The one shot search model has been taken to data a number of times, maybe most notably in Wolpin (Econometrica, 1987). Remark 1 We can take a model to data, no matter how inconsistent it is with empirical regularities and observation, as long as it is not probability 0 given the data used to estimate the model. 2

3 One thing that many empirical researchers forget, no matter how experienced or sophisticated they may be, is the following: Claim 2 All models are wrong. A model is a deliberate abstraction from reality. One model can be better than another, on one or several dimensions, but none are correct. They help us focus on the small set of phenomena in which we are interested, and/or have data regarding. When correctly developed and explained, it should be clear what set of phenomena are being excluded from consideration, and, at the end of the analysis, it is desirable to say how the omission of other relevant phenomena could have affected the results attained. An advantage of a structured approach to empirical analysis is that it should be immediately clear what factors have been considered exogenous and which endogenous, the functional form assumptions made, etc. So, the answer is a qualified yes. Wolpin used information on the search period for the first job for individuals coming out of school. The dependent variables of the analysis were the length of the search, and the wage obtained at the end of the search spell (if the end of search was observed). Since he used data only on the first spell of search after completing school and the first wage obtained, his model was consistent with the data he used. If he attempted to use information such as the length of subsequent unemployment spells (those experienced after taking the first job), wage changes on the first job, job-to-job mobility rates, etc., these events would not be consistent with his model, and hence could not be used with this information. 1.3 A Small Extension to the Model 1. We eventually want to take the continuous time search model to data. 2. All OECD countries are required to have a labor force survey. In all, respondents are asked to report their employment status. If not employed, they are asked if they are looking for work. If so, they are asked to report how long they have been looking for work. We will refer to the length of the on-going search spell as t u. 3. In many OECD countries (not Italia, unfortunately), individuals who are employed are asked to report their wage rate. We denote this piece of information by w. 4. Thus, in Italy, a random sample from the Labor Force Survey (ISTAT) contains information on { t u (i)} N. 5. A random sample from the Current Population Survey (CPS) in the U.S., conducted by the Bureau of Labor Statistics (BLS), contains the information { t u (i)} N u for the unemployed and {w(i)} N e, for the unemployed and employed sample members, respectively. 3

4 6. The labor market is in a steady state (SS) equilibrium only after sufficient time has elapsed since the market starts. In our case, we think of the market as consisting of a cohort. When a cohort of individuals enter the labor market, they all begin in the unemployment state. Eventually, they find jobs, lose jobs, get new jobs, etc., until the labor market is in the steady state. To get to the steady state, all individuals must lose jobs, so we have to allow for this possibility. The easiest way to do it is to allow for a constant risk (in continuous time) of losing ones job. Call this risk η, where η>0. Then, we can show that in the steady state, p(u) = Et u Et u + Et e. This is the probability that a randomly-sampled (at a point in time) individual would be unemployed, with p(e) =1 p(u). Given a constant rate of leaving the employment spell of η, it follows that the length of a compelete employment spell is exponentially distributed in the population, with density f e (t e )=η exp( ηt e ). Themeanlengthoftimeinemploymentis Z 0 t e f e (t e )dt e = η Computing the density of unemployment spells in the two state (i.e., U and E) stationary search model is only slightly more complicated. First, we must modify the decision rule unemployed searchers use in deciding whether to accept a job. From Flinn and Heckman (1982), for example, we find w = b + λ ρ + η Z w (w w )df (w). With η>0, searchers are less particular about the job offer since jobs last a finite amount of time. It is easy to show that w / η < The rate of leaving unemployment is computed as follows. The rate of receiving a job offer is λ, as we say before. The probability that it will be acceptable is the probability that the offer is at least as large as w, or p(w w )= F (w ), where F (x) 1 F (x) is called the survivor function. Since the time it takes to receive an offer and the value of the offer are independent, the rate of receiving acceptable offers is just the rate of receiving offers multiplied by the probability that 4

5 a randomly drawn offer would be acceptable. Thus the rate of leaving unemployment is given by h u = λ F (w ). Then completed unemployment spells are exponentially distributed, with f u (t u )=h u exp( h u t u ), and the mean length of time spent in unemployment is given by Et u = Z 0 t u f u (t u )dt u (λ F (w )) The wage offer distribution in the population is given by F (w), with the associated p.d.f. f(w). Since individuals only accept wages greater than w, the observed wage distribution is truncated from below at w. Then the accepted wage density is given by f A (w) = f(w) F (w ),w w. Clearly this density is proper in the sense of integrating to 1 and being nonnegative for all w w. 10. The last thing we have to consider before turning to identification of model parameters and estimation, is the relationship between the distribution of on-going spells of unemployment sampled at a point in time and the population distribution of unemployment spells. (a) Spells that are on-going at a point in time have not yet concluded, so the total length of the spell can be no less than t u, that is, t u t u. We say that these spells are right-censored. These spells are shorter, in a stochastic sense, than the completed spells. (b) Counterbalancing this is the fact that spells that are sampled at a random point in time are likely to be longer, once again, in a stochastic sense, than those drawn from the population distribution. This phenomenon is known as length-biased sampling. (c) We can see how this works by putting the two concepts together in directly deriving the density of right-censored, point-sampled unemployment spells. The likelihood of drawing an on-going spell of length t u is proportional to the probability that a completed spell is at least of length t u or Z f u ( t u ) = K f u (t u )dt u t u = K(1 F u ( t u )), 5

6 where K is a factor of proportionality chosen so that f u is a proper density, or K 1 = Z 0 (1 F u (x))dx. Remark 3 If x is a positive random variable with density m and c.d.f. M, then Z 0 (1 M(x))dx = Ex. Thus K = Et u. Since t u is distributed according to a negative exponential distribution with mean h 1 u, Then K = h u. Then f u ( t u ) = h u (1 F u ( t u )) = h u exp( h u t u ). Proposition 4 If the population density of unemployment spells is negative exponential, then the distribution of right-censored, length-biased unemployment spells is the same as the population density of these spells, or f u (x) =f u (x), for all x Identification of Model Parameters The first step is to express the distribution of the data (that is, information on endogenous variables determined within the model) as a function of primitive (i.e., model-based) parameters. Given this distribution, we then take up the issue of identification. Let s start with the easiest case first The ISTAT data From the ISTAT data we only have access to { t u (i)} N. We are assuming that each individual s unemployment experience is independent from every others, and that all spells t u (i) aredrawnfromthesamedistribution. Thusthelikelihoodoftheentiresampleof on-going unemployment spells from a given monthly LFS is L( t u (1),... t u (N u )) = = YN u f u ( t u (i)) YN u f u ( t u (i)) = h N u u 6 YN u exp( h u t u (i)).

7 We typically work with the log likelihood function, which is XN u ln L = N u ln h u h u t u (i). This is the log (joint) density of the data. As we see, ln L is only a function of N u (exogenously determined), P N u t u (i), which is the sum of all of the point-sampled, rightcensored unemployment spells, and h u, which is a function of the primitive parameters of the model. Which parameters determine h u? It turns out that all of them do, since and h u = λ(1 F (w )), w = w (λ, b, η, ρ, F ). Thus h u = h u (λ, b, η, ρ, F ). Since the joint density only depends on P N u t u (i), we say that this sum is a sufficient statistic for the joint density of the point-sampled, right censored data under the model. All of the primitive parameters of the model enter the joint density only through the function h u, which is the only function of the parameters identified under the model from these data. A natural estimator for this function of parameters is the maximum likelihood estimator, which selects the parameter estimate ĥu that maximizes the log likelihood of the data. In this case, ln L(ĥu) = 0 h u N XN u u t u (i) = 0 ĥ u N u ĥu = P Nu t u (i). Nothing else can be said about model parameters given these data. However, what if we also include information on the employed in the sample? As we said before, these individuals do not report wages, but at least we know how many of them there are, N e. What we have considered so far is the conditional distribution of unemployment spells, conditional on being unemployed. The advantage of working with this conditional distribution is that it is valid whether or not we assume we are in the SS. If we are willing to assume that we are in the SS (because the workers whose behavior we are examining have been in the market for a sufficiently long period of time), then the number of unemployed becomes endogenous as well. 7

8 1. For the employed individuals, the likelihood that we found one of the employed at a random point in time is just which can be written or p(e) = p(e) = p(e) = Et e Et e + Et u, h 1 e h 1 e + h 1 u h 1 u η + h 1 u., 2. The joint likelihood of t u (i) and being unemployed for unemployed sample member i is just f u ( t u (i)) p(u). 3. The full likelihood for the ISTAT data is then L( t u (1),... t u (N u ),N e ) = YN u {f u ( t u (i)) p(u)} p(e) N e so the log likelihood is given by XN u ln L = N u ln p(u)+n u ln h u h u t u (i) +N e ln p(e) = N u ln η + N e ln h u N ln(h u + η) XN u +N u ln h u h u t u (i) = N u (ln η +lnh u )+N e ln h u XN u N ln(h u + η) h u t u (i) 4. In this log likelihood function, two parameters appear, h u and η. It is straightforward to show that they are identified. From the conditional log likelihood function, we know that we can identify and define a consistent estimator of h u. Given this estimator, it is straightforward to show that the unconditional log likelihood function allows recovery of the parameter η. 8

9 5. In terms of the maximum likelihood estimator of (h u,η), differentiating the unconditional log likelihood with respect to the two parameters, setting the derivatives equal to 0, and after some algebra, we get N u ĥ u = P Nu t u (i), same as before (with the conditional likelihood function), and ˆη = ˆP (U) 1 ˆP ĥ U, (U) where ˆP (U) =N U /N. 6. Only η and h U are identified, and can be consistently estimated, using the ISTAT Labor Force Survey data. Remark 5 Who cares if we cannot recover (i.e., identify) all of the primtive parameters? We can still fit the data with the single parameter h u under all of our model assumptions. The purpose of structural estimation in this context is threefold: 1. To give unambiguous interpretations to the parameters characterizing the model. In this case, we have succeded, but our knowledge does not get us far. We can estimate the alternating renewal process and chacterize the duration of successive employment and unemployment spells, but not much beyond this. 2. To conduct comparative statics exercises. We would normally want to conduct comparative statics exercises using primitive parameters only - but in this case, the only primitive parameter we have estimated is η. We could look at changes in η on the probability of being in unemployment for example, or the distribution of employment spells. Since there is no wage information, we cannot look at how a change in η affects the reservation wage, for example. 3. To conduct policy experiments. In this case, we typically need access to all of the primitive parameters. This is the case since our one decision variable, w, depends on all of the primitive parameters. Without consistent estimates of all of them, we cannot determine how a change in any one will change the endogenous outcome w The CPS Data This adds wage data to what we have from the ISTAT LFS. It turns out that this considerably increases the number of primitive parameters that can be identified and estimated. 9

10 The unconditional (or full-information ) likelihood in this case is L = Y i S U {f u ( t u (i)) p(u)} Y f(w(i) θ) p(e), i S e F (w ) where S j is the set of sample member indices for state j, j = e, u. The density f(w θ) F (w θ) is the density of accepted wage offers. For now, we have assumed that the distribution F belongs to a parametric family, characterized by the parameter vector θ. Note that the accepted wage density is only well-defined when w w. Also note that we will now be in a position to decompose the hazard rate out of unemployment h u = λ F (w θ). With this expression, we have L = Y λ F (w )exp( λ F (w η ) t u (i)) η + λ F (w i S ) U Y f(w(i)) λ F (w ) i S e F (w ) η + λ F (w ), with log likelihood given by ln L = N ln λ + N U ln F (w ) λ F (w ) X i S u t u (i) +N U ln η + X i S e ln f(w(i)) N ln(η + λ F (w ) 1.5 Identification in this Canonical Model We can say much about identification issues simply by examining the log likelihood function. Note that 1. The primitive parameters that explicitly appear in the function are λ, F, η. 2. The functions of data that appear are N U, P i S u t u (i), and P i S e ln f(w(i)). 3. The other primitive parameters b and ρ, only enter through the scalar critical value w, which is a function of all structural parameters. 10

11 4. We are, very importantly, assuming no measurement error. That is, the durations and wages are perfectly measured. Flinn and Heckman (1982) first noted that the search model, with these data, is fundamentally underidentified. Given appropriate parametric assumptions on the wage offer distribution F, they noted that it was possible to uniquely identify the primitive parameters λ, η, F. It was also possible to identify the decision rule w, in the following manner. 1. Accepted wages are considered to be i.i.d. draws from the acceptable wage offer density f A (w) = f(w) F (w ), and the support of this distribution is [w, ). Thelowerpointofsupportisan unknown parameter. 2. The first order statistic of a sample of N draws is simply min(w 1,w 2,...,w N ). The density of the first order statistic is f(w (1) )=Nf(w (1) ) F (w (1) ) N If we set it is straightforward to show that ŵ = w (1), plim (ŵ )=w. N 4. Their idea was to first estimate w using the smallest observed wage in the sample (we canmodifythingsherebyfirst trimming the wage data to throw out very small and very large values of w). It so happens that this extreme value estimator converges to the value w at rate N rather than the more customary rate of N. 5. They then condition on the estimated value of the decision rule - by plugging the estimate ŵ into the log likelihood function. Then define the m.l. estimators by solving first order conditions for the parameters λ, η, and θ. Denote the (unique) m.l. estimates by ˆλ, ˆη, and ˆθ. 6. Given consistent estimates of λ, η, and θ, and the decision rule w, insert these values into the functional equation determining w, ŵ = b + ˆλ Z (w ŵ )df (w ˆθ). ρ +ˆη ŵ 11

12 We can consistently estimate this functional equation, which is a smooth function of all of the estimated parameters, but it is clear that there are 2 parameters to be determined from 1 equation. This shows that the model is fundamentally underidentified. 7. What is identified is a locus of points (b, ρ) consistent with the equation. If we fix either variable, we can uniquely identify the other. One last point on this. Can we identify the population wage offer distribution nonparametrically? Yes, in two cases: 1. We assume that all wage offers are acceptable. Then the accepted wage distribution is the same as the population offer distribution by assumption. We can form a nonparametric m.l. estimator of F simply from the empirical distribution function of observed wages. (a) This assumption is not great in all settings, particular when there is bargaining over match values, and match values are drawn from a productivity distribution with support R +. (b) This assumption poses some problems with conducting comparative statics exercises. For example, if η increases, the critical value w should decrease, but w is taken to be the lower support point of a primitive parameter - the wage offer distribution. This creates some logical consistency problems 2. If we were to observe rejected job offers, we could use the empirical distribution of all rejected and accepted offers to nonparametrically estimate F. (a) Problem with this is that it is likely that some rejected offers are larger than accepted offers. This has zero probability under the model. I will conclude this talk with some computational examples of estimating a simple structural search model using the GAUSS language. 12