This proof consists of: …rstly, proving the following Lemma which, basically, requires that¢w0R(t) < 0for(i)and(ii)to hold, and, secondly, proving that¢w0R(t) < 0:
Lemma 5 If assumptions (A1)-(A7) are satis…ed and if, for everyt 2 [0; T );we have that¢w0R(t) < 0, then:
(i) wR(t) < wR0(t);
(ii) ¢wR(t) < 0:
Proof: Suppose that at somet 2 [0; T ) wR(t) ¸ w0R(t)holds. Then, because of the relationship between wR(t)and w0R(t) we will have that ¢wR(t) > 0:However, given thatwR(t)andw0R(t)are continuous functions and, by the Lemma’s assumptions,
¢wR0(t) < 0;it cannot be true thatwR(T ) = wR0(T );which must be veri…ed at time T given the assumptions of the model. Thus, the opposite must hold: wR(t) < w0R(t) and implied by this, that¢wR(t) < 0: Q:E:D:
Now we have to prove thatw0R(t)is a decreasing function oftunder all the assump-tions (K1)-(K4). The proofs under each of them are quite similar so we will show only the proof under (K1), i.e. forb(t) :
Given (7) we will have that:
wR0(t) ¡ wR0(t + 1) ¡ ®(t)
± + r
³G(wR0(t); t) ¡ G(w0R(t + 1); t)´= b(t) ¡ b(t + 1)
whereG(w0R(t); t) =Rw10 R(t)
¡w ¡ wR0(t)¢dF (w; t):
If b(t)is decreasing int, the right-hand side of this expression will be positive and since the function w0R(t) ¡ ®(t)½ G(wR0(t); t) is a increasing function of w0R(t); we will have that w0R(t) > wR0(t + 1); that is, wR0(t)is decreasing in the time the worker is unemployed.
Appendix B
To obtain monthly wages, the labor income and the unemployment bene…ts declared in the correspondent quarter have been compared. If there are no unemployment bene…ts, the monthly wage is the declared labor income divided by three. If there are unemploy-ment bene…ts, their amount is compared with the labor income: if the bene…ts are bigger
than 80% of the labor income (70% for periods posterior to 1992:2), then the monthly wage is the total amount declared as labor income. On the contrary, the monthly wage is the labor income divided by two. This rule is based on the characteristics of the un-employment bene…ts system in Spain, which lowered the replacement rate from 80% of the previous wage to 70% in the second quarter of 1992.
Calculation of monthly duration data is more di¢cult. The numbers of months of unemployment in the spell can be computed once it is established how many months of unemployment there are in the …rst quarter of unemployment and, if the worker exits unemployment, how many months he has been employed in the …rst quarter of employment. The general rule applied to compute the number of unemployment months in these two quarters is based on comparing the labor income of each quarter, if it is positive, with the unemployment bene…ts received that quarter or with the labor income of the following quarter. If there is no labor income in the …rst quarter the individual answers he is unemployed, it is considered that he is unemployed during all the quarter.
If the reported labor income is low enough a duration of two months is imputed in the correspondent quarter but if this income is su¢ciently large, it is considered that the worker has been only one month in unemployment in that quarter.
Appendix C
The likelihood function in equation (9) is based on the relationship between the hazard rate and the distribution function of a random variable. In a sample of unemployed workers, those with censored spells or completed spells but without an observed re-employment wage will have a likelihood contribution which is only a function of the unemployment hazard rateÁi(t)for thetperiods of unemployment:
Pr (Ti = t) = Ái(t)
t¡1Y
j=0
(1 ¡ Ái(j)) (C1)
Pr (Ti> t) = Yt j=0
(1 ¡ Ái(j)) (C2)
For those with completed spells and an observed re-employment wage the likelihood contribution is the following:
Pr(Ti= t; Woi) = Pr(Ti = t; Woi j Ti¸ t)
t¡1Y
j=0
(1 ¡ Ái(j)) (C3) where, in periodt;this contribution is the joint probability ofTi being equal totand of observing the wageWoi:
Here, an assumption about the wage o¤er distribution is needed. Like in other papers (Van den Berg, 1990 or Wolpin, 1987) it is assumed that wages have a lognormal distribution. In addition to this, as in Wolpin (1987) and justi…ed by the construction of the wage data, the re-employment wages are assumed to be measured with error. Thus, the observed re-employment wage has the following expression:
ln Woi = ln ¹Wi+ ui+ "i (C4) whereui is normal with zero mean and variance¾2u;and"i;the measurement error, fol-lows a normal distribution with zero mean and variance¾2":I assume that"iis distributed independently ofui .
The joint distribution ofWo andT j T ¸ tis given by the following equations:
Pr(T = t; Wo j T ¸ t) = fWo(Wo j T = t) Pr(T = t j T ¸ t) = fWo(Wo j t) £ Á(t) (C5) Note that the distribution of Wo conditional on t is the truncated distribution of the observed wages, with the reservation wage att + 1being the truncation point, thus, fWo(Wo j t) = fWo(Woj W ¸ WR(t + 1)). For an expression for this density, see Wolpin (1987).
Finally, we can express the likelihood function in logarithms and in the usual way of expressing likelihood functions for discrete data (see Jenkins, 1995). This is exactly the second equation in (9).
References
[1] Baker, M. and A. Melino (2000), “Duration Dependence and Nonparametric Heterogeneity: A Monte Carlo Study”, Journal of Econometrics, 96, 357-393.
[2] Berkovitch E. (1990), “A Stigma Theory of Unemployment Duration” in: Y.
Weiss and G. Fishelson, eds., Advances in the Theory and Measurement of Unemployment (Macmillan, London), 20-56.
[3] Bover, O., M. Arellano and S. Bentolila (1997), “Unemployment Duration, Bene…t Duration and the Business Cycle”, CEMFI Working Paper 9717.
[4] Devine T. J. and N. M. Kiefer (1991), Empirical Labor Economics: The Search Approach, Oxford University Press, New York.
[5] Elbers, C. and G. Ridder (1982), “True and Spurious Duration Dependence:
The Identi…ability of the Proportional Hazard Model”, Review of Economic Studies, 49, 403-409.
[6] Eckstein Z. and K. Wolpin (1990), “Estimating a Market Equilibrium Search Model from Panel Data on Individuals”, Econometrica, 58-4, 783-808.
[7] Flinn, C. and J. Heckman (1982), “New Methods for Analyzing Structural Models of Labor Force Dynamics”, Journal of Econometrics, 18, 115-168.
[8] García-Pérez, J. I. (1997), “Las Tasas de Salida del Empleo y el Desempleo en España (1978-1993)”, Investigaciones Económicas, XXI(1), 29-53.
[9] García-Pérez, J. I. (1998), “Non-stationary Job Search with Firing: a Struc-tural Estimation”, CEMFI Working Paper 9802.
[10] Heckman, J. and B. Singer (1984), “A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data”, Econometrica, 52, 271-320.
[11] Jenkins, S. (1995), “Easy Estimation Methods for Discrete-Time Duration Models”, Oxford Bulletin of Economics and Statistics, 120-138.
[12] Lancaster, T. and A. Chesher (1983), “An Econometric Analysis of Reser-vation Wages”, Econometrica, 51, 1661-1676.
[13] Lippman, S. and J. McCall (1976), “The Economics of Job Search: A Sur-vey”, Economic Inquire, 14, 155-189.
[14] Meyer, B. (1990), “Unemployment Insurance and Unemployment Spells”, Econometrica, 58, 757-782.
[15] Miller, R. (1984), “Job Matching and Occupational Choice”, Journal of Po-litical Economy, 92, 1086-1120.
[16] Narendranathan W. and S. Nickell (1985), “Modelling the Process of Job Search”, Journal of Econometrics, 28, 29-49.
[17] Narendranathan W. and M. Stewart (1993), “How Does the Bene…t E¤ect Vary as Unemployment Spells Lengthen?”, Journal of Applied Econometrics, 8, 361-381.
[18] Van den Berg, G. J. (1990), “Nonstationarity in Job Search Theory”, Review of Economic Studies, 57, 255-277.
[19] Vishwanath T. (1989), “Job Search, Stigma E¤ect, and Escape Rate from Unemployment” Journal of Labor Economics, 7-4, 487-502.
[20] Wolpin, K. I. (1987), “Estimating a Structural Search Model: The Transition from School to Work”, Econometrica, 55(4), 852-874.
Table 1
Distribution of unemployment duration and other variables in the sample
Completed Spells Censored Spells Number Percentage Mean Accepted Number Percentage
Wage Months
0-1 88 10.13 95 13.61
1-2 109 12.54 87 12.46
2-3 157 18.07 110 15.76
3-4 162 18.64 58 8.31
4-5 105 12.08 37 5.30
5-6 59 6.79 57 8.17
6-7 39 4.49 18 2.58
7-8 50 5.75 9 1.29
8-9 27 3.11 31 4.44
9-10 21 2.42 24 3.44
10-11 21 2.42 13 1.86
11-12 12 1.38 32 4.58
12-13 11 1.27 17 2.44
13-14 5 0.58 16 2.29
14-15 3 0.35 94 13.47
Age1830 136 15.65 125,788 84 12.03
Age3045 418 48.10 131,546 269 38.54
Skill 63 7.25 153,888 67 9.60
With bene…ts 517 59.49 126,442 460 65.90
TOTAL 869 127,294 698
Note: Mean accepted wages are in 1996 Spanish pesetas (exchange rate:
130.6 pesetas/dollar).
29
Table 2
Identi…cation with the estimation procedure: some simulations results
Without unobserved heterogeneity
Coef. a0 a1 a2 a3 e0 e1 v0 b0 b1 b2 b3 r0 ° p ´1
True -1 -0.14 0.5 0.4 11.5 0.1 -3.4 12.5 -0.1 0.5 -0.2 1 - -
-Estim. -1.06 -0.137 0.55 0.42 11.48 0.10 -3.39 12.40 -0.12 0.69 -0.19 0.98 St. Er. 0.16 0.02 0.14 0.14 0.024 0.03 0.23 0.215 0.04 0.23 0.15 0.32 With unobserved heterogeneity
True -0.8 -0.14 0.5 - 11.5 0.1 -3.4 12.4 -0.1 0.5 - 1 -0.5 0.5 -0.2
Estim. -0.70 -0.143 0.58 11.46 0.09 -3.20 12.26 -0.13 0.80 1.39 -0.28 0.57 -0.40
St. Er. 0.25 0.02 0.26 0.03 0.03 0.25 0.32 0.05 0.35 0.46 1.40 0.26 0.36
Notes: The parameters take the following form:
®(t) = 1 ¡ exp(¡ exp(a0+ a1£ t + a2£ var2 + a3£ var1));
E(w) = ee0+e2£var2; b(t) = eb0+b1£t+b2£var3+b3£var1; V ar(w) = ev0; ½2 = V ar(w)+V ar(")V ar(w) = 1+eer0r0:
The distribution of the unobserved heterogeneity term is a discrete one with two mass points. The probability of´ = ´1isp:In the estimation with unobserved heterogeneity, the e¤ect ofvar1 is dropped and taken as the one of unobserved heterogeneity. The di¤erential e¤ect of this on b(t)is measured by the parameter°:
30
Table 3
Functional forms of the estimated parameters
Job o¤ers arrival rate:
®(t; ´) = 1 ¡ exp(¡ exp (¯1+ ¯2£ dur + ¯3£ skill + ¯4£ bene…ts + ´)) Distribution of wages:
Wo= ¹W eue" with u » N¡0; ¾2u¢
" » N¡0; ¾2"¢
W = exp (¯¹ 5+ ¯6£ skill + ¯7£ age1830 + ¯8£ age3045 )
¾2u = exp (¯9)
¾2" = exp (¯10)
Value of time for the unemployed worker:
b(t) = exp (¯11+ ¯12£ dur + ¯13£ bene…ts + ¯14£ ´)
31
Table 4
Main results of the structural estimation
without Unob. Het. with Unob. Het.
Parameter Coef. t-ratio Coef. t-ratio
®i(t; ´)
Constant -0.754 -6.014 -0.442 -2.784 Duration -0.139 -8.513 -0.165 -6.742
Skill 0.152 0.839 0.237 1.554
Bene…ts -0.539 -4.340 -0.296 -1.934 W¹i
Constant 11.659 458.986 11.652 480.729
Skill 0.035 1.226 0.058 1.617
Age18-29 -0.014 -0.789 -0.001 -0.052 Age30-45 0.032 1.747 0.039 2.020
¾2u
Constant -5.996 -5.907 -6.302 -6.806
¾2"
Constant -2.178 -32.471 -2.179 -33.112 bi(t)
Constant 12.064 85.451 12.083 83.947 Duration -0.028 -2.582 -0.027 -3.085
Bene…ts 0.246 2.093 0.267 2.228
Unobs. Het. -0.723 -1.690
Unobs. Het.
´1 -0.304 -1.525
p 0.347 5.361
Log-likelihood -7,992.82 -7,987.09
No. of observ. 8,520 8,520
32
Table 5
Predicted values for the main elements of the model t ®(¢) Á(¢) F (w¹ R(¢)) wR(¢) b(¢)
Mean values for all variables:
0 41:08 3:13 7:62 135; 902 219; 667 4 23:99 18:12 75:53 117; 809 196; 943 14 5:13 5:13 100:00 58; 691 149; 892 For the group with ´1 :
0 32:00 0:00 0:00 151; 919 270; 079 4 18:04 1:66 9:20 129; 070 242; 140 14 3:79 3:79 100:00 84; 351 184; 291 For the group with ´1 :
0 45:90 4:79 10:44 127; 390 192; 878 4 27:16 26:86 98:89 111; 825 172; 925 14 5:87 5:87 100:00 45; 054 131; 612 With access to Unempl. Bene…ts:
0 38:94 1:27 3:26 140; 489 234; 107 4 22:57 16:11 71:38 121; 249 209; 889 14 4:79 4:79 100:00 65; 929 159; 745 Without access to Unempl. Bene…ts:
0 48:36 11:89 24:59 125; 874 179; 213 4 29:06 28:48 98:00 111; 019 160; 674 14 6:38 6:38 100:00 38; 772 122; 288 Skilled workers:
0 48:11 3:25 6:75 141; 506 219; 667 4 28:88 21:50 74:45 124; 879 196; 943 14 6:34 6:34 100:00 62; 535 149; 892 Unskilled workers:
0 40:49 3:11 7:68 135; 443 219; 667 4 23:60 17:80 75:42 117; 237 196; 943 14 5:04 5:04 100:00 58; 504 149; 892
Notes: ¹F (wR(¢)) = 1 ¡ F (wR(¢)) : The …rst three columns are percentages and the other two are expressed in 1996 pesetas. The predictions are carried out using the model with unobserved heterogeneity.
33
Figure 1:E¤ect of the separation rate on reservation wages at period 0
-1,00 -0,80 -0,60 -0,40 -0,20 0,00
B0=100
0,00 0,20 0,40 0,60 0,80 1,00
B0=100
0,00% 20,00% 40,00% 60,00% 80,00% 100,00%
B0=100 B0=150 B0=200
Note: The baseline parameters in each graph are b(t) = b0 ¤ exp(b1 ¤ t)
= 100 ¤ exp(¡0:03t); ®(t) = 0:3 ¤ exp(¡0:05t); E(w) = 150; CV (w) = 40%; ± = 2%:
34
Figure 2:Kaplan-Meier estimates of the hazards
0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2
0 2 4 6 8 10 12 14
Months
Empirical Hazards
Figure 3: Histogram of the reemployment wages
35
Figure 4: Estimated acceptance probabilities
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 2 4 6 8 10 12 14 16
Month
with Benefits without Benefits
Figure 5: Estimated hazard rates: the e¤ect of unemployment bene…ts
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35
0 2 4 6 8 10 12 14
Month
with Benefits without Benefits
36
Figure 6: Estimated hazard rates: the e¤ect of unobserved heterogeneity
0,00 0,05 0,10 0,15 0,20 0,25 0,30
0 2 4 6 8 10 12 14
Month
with eta1 with eta2
37