3. Marco Teórico
3.6. El problema de la calidad de la educación
Results of the Monte Carlo studies on the just-identified model (equation 3.2) are presented in Tables 5.1 to 5.6. As mentioned in chapter 4, the Mean, Bias and Mean Squared Error are the comparison criteria, with more focus on the Mean squared error. For this model, 2SLS, 3SLS and LIML are equal for the coefficient of the endogenous explanatory variable in the first equation while all the classical estimators are the same as OLS for the second equation which contains no endogenous variable. The three runs on the just-identified model are again as follows;
(1). RUN ONE = 2.0, = 0.5, Xt : NID(0,1), negatively correlated residual terms
89
(2). RUN TWO = 2.0, = 0.5, Xt : NID(0,2), positively correlated residual terms (3). RUN THREE = 2.0, = 0.5, Xt : NID(0,9), positively correlated residual terms
Table 5.1: The mean, absolute bias and mean squared error for and in run 1
T Estimator
Mean Bias MSE Mean Bias MSE
20
Classical OLS 1.763 0.237 0.0664 0.506 0.006 0.243 2/3SLS/LI 5
ML
1.633 0.367 202.662 3
0.506 0.006 0.243 Bayesian BMPV 10 1.625 5
8
0.374 2
0.2695 0.294 5
0.205 5
0.168 BMPV 100 1.799 6
0
0.201 0
0.1672 0.268 9
0.231 1
0.167 BMPV1000 1.851 8
4
0.148 6
0.1678 0.238 4
0.261 6
0.171 4 40
Classical OLS 1.763 0.237 0.0611 0.501 0.001 0.126 2/3SLS/LI 2
ML
1.849 0.151 75.4489 0.501 0.001 0.126 Bayesian BMPV 10 1.731 2
6
0.268 4
0.2083 0.318 4
0.181 6
0.122 BMPV 100 1.889 0
1
0.110 9
0.1437 0.290 9
0.209 1
0.127 BMPV1000 1.936 2
4
0.063 6
0.1698 1.8891
0.262 3
0.237 7
0.136 1 60
Classical OLS 1.763 0.237 0.0594 0.497 0.003 0.078 2/3SLS/LI 3
ML
3.085 1.085 3185.14 0.497 0.003 0.078 Bayesian BMPV 10 1.838 3
7 0.161 3
0.161 3
0.1455 0.318 4
0.181 6
0.122 BMPV 100 1.966 0
5
0.033 5
0.1150 0.318 7
0.181 3
0.102 BMPV1000 2.005 4
8
0.005 8
0.1643 0.293 1
0.206 9
0.112 4 100
Classical OLS 1.762 0.238 0.0583 0.502 0.002 0.047 2/3SLS/LI 2
ML
2.078 0.078 2.1172 0.502 0.002 0.047 Bayesian BMPV 10 1.953 2
4
0.046 6
0.1420 0.387 2
0.112 8
0.065 BMPV 100 2.037 2
6
0.037 6
0.0997 0.363 3
0.153 1
0.073 BMPV1000 2.064 9
6
0.064 6
0.0837 0.342 5
0.157 5
0.083 0
In run 1 as shown in Tables 5.1 - 5.6, the Bayesian estimation method for with the three stated prior variances performed better than 2SLS both in terms of bias (except for T40 where 2SLS/3SLS/LIML slightly perform better than BMPV 10) and MSE in all sample size cases. It was also better than OLS in terms of bias except for T40where OLS was better than BMPV 10 only. This is consistent with results from literature, for example; Zellner
90
(1971), Gao and Lahiri (2001), Okewole, Olubusoye and Shangodoyin (2011). The 2SLS estimates were characterized with large variance while OLS estimates had the least variance as expected, though biased. The bias of OLS estimator is not affected by sample size. They are the same at all the levels of sample size. Estimates obtained through BMPV 100 are the best among the three prior variance levels for . The iterations in WinBUGS for the Bayesian estimates converged faster for in the just-identified model because it is a simple case involving only two endogenous variables and one exogenous regressor.
The situation is somehow different for parameter of the instrumental variable in the model as observed from Table 5.1 and 5.2. First, OLS, 2SLS, 3SLS and LIML are the same. This is due to absence of endogenous variable in the equation. The Bayesian estimates were better than those from the classical methods only in terms of MSE and in small sample sizes T20 The classical estimators had better performance than the Bayesian method in the large sample cases, being less biased and having lower MSE.
Table 5.2: Cases in which each estimator has minimum MSE in Run 1 for equation 3.2
T Estimator Total
20
Classical OLS 1 0 1
2/3SLS/LIML 0 0 0
Bayesian BMPV 10 0 0 0
BMPV 100 0 1 1
BMPV 1000 0 0 0
40
Classical OLS 1 0 1
2/3SLS/LIML 0 0 0
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 1 1
60
Classical OLS 1 1 2
2/3SLS/LIML 0 1 1
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
100
Classical OLS 1 1 2
2/3SLS/LIML 0 1 1
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
The Bayesian estimator of was slower in convergence than the estimates of . Varying the Bayesian Prior precision appears not to have much effect on the estimates obtained; BMPV
91
100 only performed a little better than other prior variance levels but could not bring the Bayesian estimates close to estimates from the classical methods. Most likely explanation for this is that for this parameter (), a lot more iterations are required to obtain estimates closer to the true value.
Table 5.3: The mean, absolute bias and mean squared error for and in run 2
T Estimator
Mean Bias MSE Mean Bias MSE
20
Classical OLS 2.225 0.225 0.0602 0.498 0.002 0.120 2/3SLS/LI 0
ML
2.210 0.210 179.595 9
0.498 0.002 0.120 Bayesian BMPV 10 1.793 0
9
0.206 1
0.1927 0.325 0
0.175 0
0.107 BMPV 100 1.952 1
9
0.047 1
0.1384 0.292 9
0.207 1
0.121 BMPV1000 1.999 8
3
0.000 7
0.1559 0.263 6
0.236 4
0.131 9 40
Classical OLS 2.227 0.227 0.0562 0.503 0.003 0.063 2/3SLS/LI 0
ML
1.947 0.053 7.6169 0.503 0.003 0.063 Bayesian BMPV 10 1.808 0
0
0.192 0
0.1841 0.362 3
0.137 7
0.075 BMPV 100 1.902 6
0
0.098 0
0.1418 0.338 6
0.161 4
0.088 BMPV1000 1.928 4
3
0.071 7
0.1556 0.314 5
0.185 5
0.098 9 60
Classical OLS 2.225 0.225 0.0536 0.500 0.000 0.039 2/3SLS/LI 1
ML
1.976 0.024 15.2951 0.500 0.000 0.039 Bayesian BMPV 10 1.838 1
9
0.161 1
0.1467 0.394 3
0.105 7
0.054 BMPV 100 1.888 7
0
0.112 0
0.1213 0.380 2
0.119 8
0.063 BMPV1000 1.899 3
3
0.100 7
0.1328 0.363 5
0.136 5
0.071 3 100
Classical OLS 2.226 0.226 0.0528 0.502 0.002 0.024 2/3SLS/LI 3
ML
1.964 -0.036
0.036 0.0593 0.502 0.002 0.024 Bayesian BMPV 10 1.889 3
1
0.110 9
0.0894 0.430 8
0.069 2
0.035 BMPV 100 1.906 1
8
0.093 2
0.0828 0.426 0
0.074 0
0.038 BMPV1000 1.911 9
2
0.088 8
0.1139 0.417 4
0.082 6
0.043 2
Going through tables 5.3 and 5.4 containing results from run 2, we could see that the same pattern of result as in run 1 (Tables 5.1 and5. 2) was recorded for both parameters of the model, and, except that MSE reduced for all the estimators. OLS performed better than the Bayesian approach
92
Table 5.4: Cases in which each estimator has minimum MSE in Run 2 for equation 3.2 Sample Size Estimator Total
20
Classical OLS 1 0 1
2/3SLS/LIML 0 0 0
Bayesian BMPV 10 0 1 1
BMPV 100 0 0 0
BMPV 1000 0 0 0
40
Classical OLS 1 1 2
2/3SLS/LIML 0 1 1
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
60
Classical OLS 1 1 2
2/3SLS/LIML 0 1 1
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
100
Classical OLS 1 1 2
2/3SLS/LIML 0 1 1
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
In run 3, where the variance of the exogenous variable Xt was increased to 9, the MSE for all the estimators was smaller than the other 2 runs. Also, the bias of 2SLS became smaller than that of the Bayesian. In all the three runs, BMPV 100 produced the smallest MSE among the three prior variances specified
93
Table 5.5: The mean, absolute bias and mean squared error for and in run 3
T Estimator
Mean Bias MSE Mean Bias MSE
20
Classical OLS 2.167 0.167 0.0354 0.499 0.001 0.0267 2/3SLS/LIML 1.945 0.055 2.5607 0.499 0.001 0.0267 Bayesian BMPV 10 1.8724 0.1276 0.1066 0.4195 0.0805 0.0391 BMPV 100 1.8973 0.1027 0.0956 0.4122 0.0878 0.0441 BMPV 1000 1.8999 0.1001 0.1136 0.4009 0.0991 0.0495
40
Classical OLS 2.172 0.172 0.0334 0.501 0.001 0.0140 2/3SLS/LIML 1.983 0.017 0.0219 0.501 0.001 0.0140 Bayesian BMPV 10 1.9452 0.0548 0.0378 0.4622 0.0378 0.0188 BMPV 100 1.9511 0.0489 0.0371 0.4633 0.0367 0.0193 BMPV 1000 1.9505 0.0495 0.0415 0.4601 0.0399 0.0203
60
Classical OLS 2.168 0.168 0.0304 0.500 0.000 0.0087 2/3SLS/LIML 1.991 0.009 0.0102 0.500 0.000 0.0087 Bayesian BMPV 10 1.9750 0.0250 0.0135 0.4788 0.0212 0.0102 BMPV 100 1.9777 0.0223 0.0130 0.4802 0.0198 0.0101 BMPV 1000 1.9779 0.0221 0.0130 0.4796 0.0204 0.0103
100
Classical OLS 2.169 0.169 0.0301 0.501 0.001 0.0054 2/3SLS/LIML 1.994 0.006 0.0059 0.501 0.001 0.0054 Bayesian BMPV 10 1.9860 0.0140 0.0066 0.4788 0.0212 0.0102 BMPV 100 1.9874 0.0126 0.0066 0.4894 0.0106 0.0058 BMPV 1000 1.9875 0.0125 0.0067 0.4893 0.0107 0.0058 For , Tables 5.5 and 5.6 show that estimates from the Bayesian approach have smaller MSE than those from 2/3SLS/LIML while OLS has the least MSE. The bias from OLS estimator is however more than that of the Bayesian as it is in runs 1 and 2.
Table 5.6: Cases in which each estimator has minimum MSE in Run 3 for equation 3.2 Sample size Estimator Total
20
Classical OLS 1 1 2
2/3SLS/LIML 0 1 1
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
Classical OLS 0 1 1
2/3SLS/LIML 1 1 2
94 40
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
60
Classical OLS 0 1 1
2/3SLS/LIML 1 1 2
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0
100
Classical OLS 0 1 1
2/3/SLS/LIML 1 1 2
Bayesian BMPV 10 0 0 0
BMPV 100 0 0 0
BMPV 1000 0 0 0