The Wooldridge method is based on a very simple and novel strategy as a solu-tion for the initial values problem in nonlinear dynamic random-e¤ects panel data models. This character of the Wooldridge method has attracted many researchers.
However, nothing is known about its performance in comparison with the other al-ternative Heckman’s reduced-form approximation. In this paper, using the dynamic
random-e¤ects probit and tobit (type I) models, the …nite sample performance of the Wooldridge method is investigated in comparison with the ideal case in which the initial values are known constants, the worst case which emerges with the ex-ogenous initial values assumption, and the Heckman’s reduced-form approximation method which is based on complicated econometric techniques. Various designs of Monte Carlo Experiments are provided using balanced and unbalanced panel data sets. We also provided two real data applications which concentrate on intertempo-ral participation and hours of work decisions of married women in Sweden between 1992 and 2001.
The evidence obtained from M CE and real data are in line with each other and con…rmed the fact that a misspeci…cation for the conditional distribution of initial values leads to serious bias on the magnitude of the true state dependence and the variance of unobserved individual-e¤ects. The exogenous initial values assumption is one of the these cases and leads to serious overestimation of the true state dependence and serious underestimation of the variance of the unobserved individual-e¤ects.
However, this is a syndrome for small samples and the bias decrease gradually as the duration of the panel data set increase.
We also obtain clear evidence on the performance of the Wooldridge method in comparison with the Heckman’s method. The key parameter to select one of them in the practice is mainly the duration of the panel data set. The Wooldridge method does not specify an explicit conditional probability distribution for the initial values, and the bias obtained with this method is behaviourally the same as the exogenous initial values assumption for very short panels. The persistence which is due to
structural reasons is overestimated whereas the persistence due to unobserved time-invariant individual characteristics is underestimated. However, the bias produced by the Wooldridge method is much smaller than the bias produced by the exogenous initial values assumption. The main message of our results is that the Wooldridge method can be used instead of Heckman’s method only for the moderately long panels, but for the short panels Heckman’s approximation is suggested. The other message of the paper is very intuitive. For the panels of longer durations, the relative importance of the initial period likelihood in the joint likelihood of all periods would be lower leading to a lower bias which is due to the initial values problem. This is what we observe in our M CE and real data applications that performance of all methods tends to be equal for the panels of long durations.
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Appendix
The Gaussian-Hermite Quadrature, that we implement to calculate the gral in the likelihood function (5), is based on approximating the Gaussian inte-gral, R1
1e v2h (v) dv PM
m=1wmh (vm); where v1; v2; :::; vM roots of the Hermite polynomial H(v); M (m = 1; 2; :::; M ) is the number of evaluation points in the approximation process; and wm is the corresponding weight for the root vm. The pairs of (vm; wm), for di¤erent M , can be easily obtained by using existing tables in the literature. Having assumed that the unobserved individual-e¤ects is normally distributed, and given the conditional distribution of initial values, the integral in the likelihood function of probit model can be calculated as follows:
= 1
The full log-likelihood function is,
Note that the solution methods for the initial values can also be easily adopted to the above procedure. The exogenous initial values assumption leads to ignoring f1(:), and it can be taken outside of (19). The likelihood function for the Heckman’s method is given as,
The same procedure can be easily implemented for the integral which will appear in the likelihood function of the dynamic random-e¤ects tobit model by simply using the same strategy given the above.
Table 1. Results of MCE 1: Normal regressor(True values:11β=,0.5γ=,1ασ=,1uσ=) Dynamic Probit ModelDynamic Tobit Model 0.5γ=1ασ=0.5γ=1ασ= TMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAE Known (non stochastic) initial values 3–0.0110.246–0.0180.163–0.0150.269–0.0420.1740.0120.092 0.0050.061–0.0140.115–0.0010.071 4 0.0080.140 0.0040.091 0.0000.187–0.0380.1290.0100.0600.0020.038–0.0090.094–0.0110.070 5 0.0210.125 0.0250.087 0.0010.1400.0180.0860.0090.0490.0130.031–0.0090.087–0.0160.063 8 0.0080.093 0.0170.072 0.0020.1020.0040.0640.0030.0290.0030.021 0.0020.076–0.0020.049 15 0.0140.063 0.0140.047–0.0120.073–0.0220.045 0.0020.020 0.001 0.014–0.0080.060–0.0060.036 20 0.0090.047 0.0160.040–0.0180.069–0.0190.041 0.0020.0170.0030.013–0.0090.060–0.0090.034 Exogenous initial values assumption 3 0.2640.2890.2490.249–0.2410.323–0.2430.2580.2450.249 0.2430.243–0.4890.512–0.4610.461 4 0.1750.2170.1530.153–0.1280.211–0.1540.1820.1870.1940.1890.189–0.3200.341–0.3240.324 5 0.1140.1560.1040.108–0.0790.156–0.0720.1240.1270.1350.1250.125–0.2070.227–0.2170.217 8 0.0530.105 0.0500.063–0.0210.093–0.0220.0690.0430.0540.0490.045–0.0600.092–0.0690.075 15 0.0230.072 0.0200.054–0.0110.078–0.010 0.057 0.0110.021 0.011 0.016–0.0220.067–0.0210.044 20 0.0170.054 0.0190.043–0.0180.066–0.0110.049 0.0090.0190.0110.015–0.0090.061–0.0100.038 Heckman’s reduced-form approximation 3 0.0130.1380.0120.0780.0030.262–0.0170.158 0.0090.0830.0190.061–0.0650.226–0.0890.171 4 0.0110.1340.0010.100–0.0050.184–0.0260.151 0.0030.070 0.0060.049–0.0280.162–0.0350.110 5 0.0000.1110.0020.074–0.0030.142–0.0010.0800.0080.057 0.0150.034–0.0190.138–0.0310.089 8 0.0040.090 0.0010.0560.0020.093 0.0000.0660.0050.039 0.0020.027–0.0170.126–0.0230.079 15 0.0060.068 0.0020.051–0.0040.077–0.0040.042–0.0040.029–0.0030.018–0.0140.094–0.0130.070 20 0.0090.052 0.0090.039–0.0180.066–0.0170.041–0.0020.024–0.0010.013–0.0090.093–0.0070.065 Wooldridge method 3 0.1680.3850.1910.273–0.1520.467–0.2210.3560.0980.182 0.0840.122–0.1390.228–0.1330.149 4 0.0990.2500.1130.195–0.0700.223–0.1070.1590.0350.1390.0400.102–0.0730.179–0.0540.123 5 0.0260.1820.0450.141–0.0210.166–0.0230.1040.0210.0870.0180.060–0.0600.154–0.0560.102 8 0.0200.115 0.0110.067–0.0100.099–0.0210.0610.0040.045 0.0020.034–0.0290.122–0.0260.074 15 0.0090.074 0.0050.052–0.0050.078–0.008 0.056–0.0060.030–0.0070.018–0.0120.099–0.0190.071 20 0.0010.051 0.0040.036 0.0010.069 0.0090.052 0.0010.022 0.0020.014–0.0010.065–0.0010.055
Table 2. Results of MCE2: Non-normal regressor(True values:11β=,0.5γ=,1ασ=,1uσ=) Dynamic Probit ModelDynamic Tobit Model 0.5γ=1ασ=0.5γ=1ασ= TMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAE Known (non stochastic) initial values 3–0.0030.2130.0050.1530.0130.2500.0060.181–0.0030.086–0.0050.043–0.0100.120–0.0050.071 4 0.0070.1440.0050.092 0.0110.161 0.0090.106–0.0010.047–0.0010.030–0.0020.085–0.0040.065 5 0.0030.1380.0040.0880.0010.1450.0100.092–0.0010.046 0.0010.032–0.0040.073–0.0070.055 8 0.0120.0950.0160.067 0.0150.112 0.0180.062 0.0040.025 0.0040.018–0.0030.072–0.0060.050 15 0.0040.0550.0020.043 0.0070.069 0.0090.043 0.0010.018 0.0010.014–0.0090.063–0.0030.036 20 0.0070.0460.0080.042 0.0160.062 0.0190.042 0.0000.015–0.0010.010–0.0070.060–0.0060.035 Exogenous initial valuesassumption 3 0.3090.344 0.3050.305–0.2640.360–0.2720.2850.2130.2180.2110.211–0.3950.419–0.3860.386 4 0.2300.262 0.2220.222–0.1590.271–0.1830.2070.1510.1600.1520.152–0.2420.275–0.2520.252 5 0.1480.192 0.1320.132–0.1260.144–0.1330.1440.0970.1070.1000.100–0.1530.178–0.1550.155 8 0.0690.116 0.0580.075–0.0290.083–0.0350.0490.0330.0430.0330.034–0.0450.083–0.0540.065 15 0.0160.059 0.0200.041–0.0120.072–0.0170.0500.0060.0180.0080.014–0.0170.062–0.0210.041 20 0.0110.054 0.0090.040–0.0130.065–0.0180.0490.0050.0160.0070.010–0.0080.055–0.0110.038 Heckman’sreduced-formapproximation 3–0.0160.167–0.0010.097 0.0340.3060.0250.183–0.0010.0810.0020.048–0.0520.217–0.0700.128 4 0.0130.134–0.0050.076 0.0200.217–0.0150.124 0.0030.053 0.0060.036–0.0290.142–0.0270.096 5 0.0040.120–0.0010.092 0.0140.121 0.0090.094 0.0050.053 0.0030.030–0.0280.137–0.0270.090 8 0.0090.091 0.0070.058 0.0000.078–0.0050.054 0.0010.034 0.0010.021–0.0190.105–0.0170.070 15–0.0010.056 0.0040.053–0.0080.070–0.0140.046 0.0000.028–0.0030.018–0.0050.095–0.0080.068 20 0.0020.053–0.0030.039–0.0240.065–0.0250.047 0.0000.014 0.0010.012–0.0010.091–0.0030.060 Wooldridge method 30.0760.4770.0750.235–0.0610.700–0.0360.2800.0980.1860.1100.139–0.1350.252–0.1420.163 40.0720.2350.0790.159–0.0390.284–0.0320.1570.0470.1110.0450.074–0.0530.180–0.0570.131 50.0220.1900.0230.120–0.0290.147–0.0250.0880.0220.0740.0230.051–0.0310.198–0.0370.107 80.0210.1120.0230.081–0.0180.082–0.0180.051 0.0010.0300.0020.021–0.0100.103–0.0100.062 15–0.0020.058–0.0010.039–0.0120.074–0.0200.052–0.0060.025–0.0060.016–0.0030.069–0.0050.043 200.0000.0560.0070.041–0.0080.064–0.0140.059–0.0020.015–0.0030.012 0.0010.053 0.0020.039
Table 3. Results of MCE3: Autocorrelated regressor(True values:11β=,0.5γ=,1ασ=,1uσ=) Dynamic Probit ModelDynamic Tobit Model 0.5γ=1ασ=0.5γ=1ασ= TMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAE Known (non stochastic) initial values 3 0.0080.2150.0000.1250.0330.589 0.0220.241–0.0150.140–0.0170.096–0.0340.200–0.0180.114 4 0.0000.1670.0060.1220.0010.220–0.0160.150 0.0010.082–0.0030.053–0.0080.132–0.0110.088 5–0.0110.130–0.0120.096–0.0090.141–0.0070.092–0.0030.065 0.0020.047–0.0150.125–0.0140.072 8 0.0090.093 0.0040.0610.0100.087 0.0090.056–0.0050.041 0.0010.030–0.0090.094–0.0080.063 15 0.0100.068 0.0110.045–0.0050.069–0.006 0.047 0.0000.023 0.0010.015–0.0150.089–0.0150.061 20–0.0020.060–0.0030.037–0.0120.061–0.0090.042 0.0010.021 0.0000.014–0.0100.088–0.0130.060 Exogenous initial values assumption 3 0.2580.3070.2580.258–0.2220.761–0.2760.377 0.2070.214 0.2080.208–0.4380.487–0.4120.412 4 0.1500.2450.1490.168–0.1420.304–0.1390.206 0.1430.159 0.1500.150–0.2690.319–0.2660.266 5 0.1100.2090.1200.144–0.0570.258–0.0680.148 0.0880.104 0.0930.093–0.1540.207–0.1670.172 8 0.0600.141 0.0720.108–0.0310.132–0.0300.083 0.0320.049 0.0340.037–0.0480.119–0.0420.079 15 0.0250.1030.0150.067–0.0270.105–0.020 0.073 0.0050.028 0.004 0.022–0.0220.093–0.0240.066 20 0.0040.0810.0080.053–0.0140.097–0.027 0.068 0.0020.021 0.0020.014–0.0100.092–0.0120.058 Heckman’sreduced-formapproximation 3 0.0270.374 0.0450.234–0.0450.479–0.0380.276 0.0010.088–0.0020.062–0.0430.219–0.0520.150 4 0.0220.287 0.0230.194–0.0360.349–0.0170.190–0.0030.067 0.0010.042–0.0330.174–0.0370.119 5 0.0060.247–0.0120.146–0.0110.267–0.0130.151 0.0040.049 0.0030.033–0.0300.136–0.0230.081 8 0.0170.158 0.0060.097–0.0050.138–0.0080.091 0.0050.038 0.0020.024–0.0150.111–0.0140.072 15 0.0090.109 0.0050.063–0.0150.107–0.012 0.072 0.0010.025 0.0020.019–0.0110.095–0.0120.066 20 0.0000.089 0.0020.051–0.0080.099–0.009 0.068 0.0010.021 0.0000.015–0.0090.089–0.0100.061 Wooldridge method 3 0.0560.4090.0970.266–0.0830.451–0.0760.284 0.0440.181 0.0260.108–0.0480.290–0.0400.191 4 0.0420.3300.0440.208–0.0430.378–0.0610.231 0.0050.106 0.0040.074–0.0380.198–0.0380.140 5 0.0380.2370.0390.152–0.0310.273–0.0360.159 0.0090.076 0.0080.046–0.0270.148–0.0280.093 8 0.0290.162 0.0160.100–0.0270.138–0.0140.088–0.0010.042–0.0030.028–0.0170.109–0.0110.074 15 0.0160.109 0.0100.062–0.0230.108–0.018 0.074–0.0030.029–0.0020.019–0.0160.088–0.0100.062 20 0.0010.089–0.0070.053–0.0180.098–0.0250.072 0.0000.022–0.0010.016–0.0110.087–0.0090.059
Table 4.Dynamicprobit model withunbalanced panel data: autocorrelated regressor (True values11β=,0.5γ=,1ασ=,1uσ=) Dynamic Probit Model 0.5γ=1ασ= T21TT−rMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAE Exogenous initial values assumption 10-101.000.0340.1280.0390.085–0.0160.143–0.0200.104 9-110.990.0390.1230.0430.088–0.0110.135–0.0210.103 8-120.960.0360.1050.0320.072–0.0120.134–0.0120.096 7-130.910.0370.1200.0280.073–0.0150.131–0.0270.091 6-140.840.0610.1360.0520.097–0.0200.145–0.0360.096 5-150.750.0560.1300.0580.086–0.0310.130–0.0440.095 4-160.640.0510.1200.0520.076–0.0300.143–0.0390.092
10 3-170.510.0530.1240.0530.081–0.0370.147–0.0430.091 Wooldridge method 10-101.000.0210.1380.0080.089–0.0110.144–0.0180.095 9-110.990.0070.138–0.0010.084–0.0120.143–0.0160.102 8-120.960.0080.1240.0150.086–0.0100.141–0.0080.100 7-130.910.0120.1360.0170.100–0.0040.137–0.0120.086 6-140.840.0330.1480.0260.102–0.0110.142–0.0200.095 5-150.750.0100.1410.0090.090–0.0100.129–0.0090.075 4-160.640.0030.1260.0080.089–0.0070.148–0.0060.094 10 3-170.510.0080.1290.0110.084–0.0010.151–0.0040.011
Table 5.Dynamic tobit model withunbalanced panel data: autocorrelated regressor (True values11β=,0.5γ=,1ασ=,1uσ=) Dynamic Tobit Model 0.5γ=1ασ= T21TT−rMean BiasRMSEMedian BiasMAEMean BiasRMSEMedian BiasMAE Exogenous initial values assumption 10-101.000.0170.0400.0180.031–0.0240.121–0.0350.083 9-110.990.0160.0390.0170.029–0.0240.110–0.0290.069 8-120.960.0180.0380.0200.029–0.0280.106–0.0390.075 7-130.910.0180.0370.0200.025–0.0330.107–0.0470.086 6-140.840.0240.0420.0230.031–0.0360.108–0.0340.080 5-150.750.0230.0420.0240.030–0.0490.115–0.0570.076 4-160.640.0290.0440.0320.033–0.0500.113–0.0560.079
10 3-170.510.0290.0420.0310.034–0.0560.111–0.0470.075 Wooldridge method 10-101.000.0000.0380.0060.026–0.0090.113–0.0200.077 9-110.99–0.0040.036–0.0020.022–0.0090.104–0.0140.069 8-120.96–0.0030.032–0.0010.021–0.0050.094–0.0200.069 7-130.91–0.0040.035–0.0030.021–0.0070.102–0.0280.072 6-140.840.0000.0360.0020.025–0.0110.103–0.0130.067 5-150.750.0010.0320.0040.023–0.0150.088–0.0220.062 4-160.640.0020.0320.0000.022–0.0100.098–0.0090.059 10 3-170.51–0.0010.031–0.0010.022–0.0080.105–0.0130.061
Table6.Sample Characteristics Full SampleEmployed all 10 yearsEmployed 0 years
Single transition from work
Single transition to work
Multiple transitions Participation0.819 (0.395)--0.604 (0.417)0.713 (0.455)0.712 (0.456) Hours of Work1313.72 (809.24)1739.12 (418.26)-881.13 (831.21)775.30 (824.97)993.85 (764.99) Age43.38 (8.300)45.43 (7.181)46.12 (8.012)42.11 (8.557)41.37 (8.737)60.16 (7.966) Place of Birth (Born in Sweden=1)0.901 (0.303)0.928 (0.259)0.771 (0.420)0.892 (0.311)0.866 (0.341)0.853 (0.299) Education (Primary)0.200 (0.400)0.170 (0.375)0.487 (0.500)0.200 (0.400)0.245 (0.429)0.171 (0.400) Education (High-School)0.498 (0.500)0.478 (0.500)0.425 (0.494)0.517 (0.500)0.511 (0.500)0.554 (0.500) Education (University)0.302 (0.459)0.352 (0.477)0.088 (0.283)0.283 (0.450)0.245 (0.429)0.275 (0.460) #Kids(0-2)0.121 (0.362)0.045 (0.222)0.086 (0.314)0.156 (0.406)0.188 (0.442)0.292 (0.522) #Kids(3-5)0.188 (0.441)0.097 (0.325)0.135 (0.391)0.235 (0.481)0.278 (0.517)0.413 (0.604) #Kids(6-17)0.908 (1.006)0.842 (0.951)0.789 (1.064)0.946 (1.019)0.997 (1.072)0.998 (1.045) Husband’s earnings(SEK 100,000)2.353 (1.395)2.497 (1.803)1.711 (1.652)2.347 (1.855)2.193 (1.627)2.372 (1.627) Note:(Standard deviations in parentheses)
Table7. Estimation results, intertemporal participation decisions of married women in Sweden, 1992-2001. Dynamic Probit Model TPooleddynamic probitmodelExogenous initial values assumption Heckman’s reduced-form approximation
Wooldridge method Laggedemployment status 31.825 (0.120)[ 1.325]1.817 (0.128)[ 1.313]0.785 (0.241)0.877 (0.389)[ 0.117] 41.845 (0.091)[ 1.215]1.830 (0.105)[ 1.197]0.833 (0.166)0.894 (0.180)[ 0.073] 51.923 (0.082)[ 1.099]1.721 (0.063)[ 0.879]0.916 (0.135)0.946 (0.170)[ 0.032] 62.000 (0.079)[ 0.963]1.552 (0.180)[ 0.523]1.019 (0.126)1.026 (0.134)[ 0.007] 72.064 (0.074)[ 0.943]1.565 (0.101)[ 0.474]1.062 (0.119)1.057 (0.048)[–0.004] 82.136 (0.061)[ 0.789]1.451 (0.067)[ 0.215]1.194 (0.102)1.170 (0.096)[ 0.021] 92.178(0.058)[ 0.742]1.388 (0.124)[ 0.110]1.250 (0.090)1.249 (0.101)[–0.001] 102.212 (0.063)[ 0.698]1.359 (0.085)[ 0.043]1.303 (0.084)1.300 (0.080)[ 0.002] Variance of theunobserved individual-effects 3-0.197 (0.111)[–4.970]1.176 (0.264)0.916 (0.395)[–0.221] 4-0.190 (0.077)[–5.063]1.152 (0.141)0.864 (0.179)[–0.250] 5-0.205 (0.099)[–4.873]1.204 (0.121)1.023 (0.069)[–0.150] 6-0.606 (0.090)[–1.035]1.233 (0.139)1.080 (0.113)[–0.124] 7-0.758 (0.069)[–0.610]1.220 (0.127)1.053 (0.046)[–0.137] 8-0.882 (0.063)[–0.367]1.206 (0.108)1.078 (0.035)[–0.106] 9-0.941 (0.090)[–0.264]1.189 (0.106)1.099 (0.098)[–0.076] 10-1.021 (0.052)[–0.134]1.158 (0.097)1.129 (0.067)[–0.025] Note: All specifications include age, age-squared, educational status (elementary, Gymnasium–high school and university education;elementary schoolisthe base category), place of birth (immigrant is the base category), fertility variables, nun-labor income (transitory and permanent) and mean fertility variablesareused in the specification of the unobserved individual-effects. The nuisance parameters estimated with Wooldridge and Heckman’s method is not reported here.Standard errors are in parentheses. The figures in brackets (and bold) indicate therelativedifference betweensolution methods andHeckman’s reduced-form approximation (a negative value is an underestimation and a positive value is an overestimation).
Table8. Estimation results,hours of work decisions of married women in Sweden, 1992-2001. Dynamic Tobit Model TPooleddynamic probitmodelExogenous initial values assumption Heckman’s reduced-form approximation
Wooldridge method Laggedhours of work 30.992 (0.013)[ 3.724]0.988 (0.015)[ 3.705]0.210 (0.023)0.246 (0.033)[ 0.171] 40.989 (0.011)[ 2.818]0.978 (0.014)[ 2.776]0.259 (0.019)0.286 (0.023)[ 0.104] 50.998 (0.010)[ 2.251]0.910 (0.021)[ 1.964]0.307 (0.020)0.322 (0.018)[ 0.050] 61.009 (0.009)[ 1.516]0.820 (0.020)[ 1.045]0.401 (0.018)0.410 (0.015)[ 0.023] 71.011 (0.007)[ 1.128]0.760 (0.017)[ 0.600]0.475 (0.015)0.479 (0.014)[ 0.008] 81.003 (0.006)[ 0.791]0.731 (0.015)[ 0.305]0.560 (0.012)0.558 (0.013)[–0.004] 91.008 (0.005)[ 0.613]0.729 (0.015)[ 0.166]0.625 (0.012)0.624 (0.011)[–0.001] 101.120 (0.005)[ 0.580]0.727 (0.013)[ 0.025]0.709 (0.011)0.714 (0.011)[ 0.007] Variance of theunobserved individual-effects 30.623 (0.061)[–3.717]2.939 (0.157)2.231 (0.139)[–0.241] 40.844 (0.038)[–2.253]2.746 (0.141)2.296 (0.136)[–0.164] 51.029 (0.222)[–1.266]2.332 (0.138)2.166 (0.124)[–0.125] 61.326 (0.298)[–0.714]2.273 (0.120)2.041 (0.115)[–0.102] 71.448 (0.246)[–0.306]2.086 (0.114)1.909 (0.107)[–0.085] 81.533 (0.170)[–0.161]1.827 (0.101)1.687 (0.094)[–0.077] 91.490 (0.155)[–0.042]1.555 (0.092)1.491 (0.087)[–0.041] 101.376 (0.146)[ 0.035]1.329 (0.090)1.304 (0.082)[–0.019] Note:The variance of the unobserved individual-effects and its standard error are scaled by 100. See the note in Table 7)