比较线性模型和Probit模型、Logit模型

研究生考试录取相关因素的实验报告
一，研究目的
通过对南开大学国际经济研究所1999级研究生考试分数及录取情况的研究，引入录取与未录取这一虚拟变量，比较线性概率模型与Probit模型，Logit模型，预测正确率。

二，模型设定
表1
定义变量SCORE ：考生考试分数；Y ：考生录取为1，未录取为0。

上图为样本观测值。

1． 线性概率模型
根据上面资料建立模型
i i i SCORE B B Y μ++=*21
用Eviews 得到回归结果如图：
Dependent Variable: Y Method: Least Squares Date: 12/10/10 Time: 20:38 Sample: 1 97
Included observations: 97
Variable Coefficient Std. Error t-Statistic Prob.
C -0.847407 0.159663 -5.307476 0.0000 SCORE
0.003297
0.000521 6.325970
0.0000
R-squared
0.296390 Mean dependent var 0.144330 Adjusted R-squared 0.288983 S.D. dependent var 0.353250 S.E. of regression 0.297866 Akaike info criterion 0.436060 Sum squared resid 8.428818 Schwarz criterion 0.489147 Log likelihood -19.14890 F-statistic 40.01790 Durbin-Watson stat
0.359992 Prob(F-statistic)
0.000000
参数估计结果为： i
Y ˆ-0.847407+0.003297 i SCORE Se=(0.159663)( 0.000521)
t=(-5.307476) (6.325970)
p=(0.0000) (0.0000)
预测正确率：
Forecast: YF Actual: Y
Forecast sample: 1 97 Included observations: 97
Root Mean Squared Error 0.294780 Mean Absolute Error
0.233437 Mean Absolute Percentage Error 8.689503 Theil Inequality Coefficient 0.475786 Bias Proportion 0.000000 Variance Proportion 0.294987 Covariance Proportion
0.705013
2.Logit 模型
Dependent Variable: Y
Method: ML - Binary Logit (Quadratic hill climbing) Date: 12/10/10 Time: 21:38 Sample: 1 97
Included observations: 97
Convergence achieved after 11 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -243.7362 125.5564 -1.941248 0.0522 SCORE
0.679441
0.350492 1.938536
0.0526
Mean dependent var 0.144330 S.D. dependent var 0.353250 S.E. of regression 0.115440 Akaike info criterion 0.123553 Sum squared resid 1.266017 Schwarz criterion 0.176640 Log likelihood -3.992330 Hannan-Quinn criter. 0.145019 Restr. log likelihood -40.03639 Avg. log likelihood -0.041158 LR statistic (1 df) 72.08812 McFadden R-squared 0.900282
Probability(LR stat) 0.000000
Obs with Dep=0 83 Total obs 97
Obs with Dep=1
14
得Logit 模型估计结果如下
p i = F (y i ) =
)
6794.07362.243(11
i x e
+--+ 拐点坐标 (358.7, 0.5)
其中Y=-243.7362+0.6794X
预测正确率
Forecast: YF Actual: Y
Forecast sample: 1 97 Included observations: 97
Root Mean Squared Error 0.114244 Mean Absolute Error
0.025502 Mean Absolute Percentage Error 1.275122 Theil Inequality Coefficient 0.153748 Bias Proportion 0.000000 Variance Proportion 0.025338 Covariance Proportion
0.974662
3.Probit 模型
Dependent Variable: Y
Method: ML - Binary Probit (Quadratic hill climbing) Date: 12/10/10 Time: 21:40
Sample: 1 97
Included observations: 97
Convergence achieved after 11 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -144.4560 70.19809 -2.057833 0.0396 SCORE
0.402868
0.196186
2.053504
0.0400
Mean dependent var 0.144330 S.D. dependent var 0.353250 S.E. of regression 0.116277 Akaike info criterion 0.122406 Sum squared resid 1.284441 Schwarz criterion 0.175493 Log likelihood -3.936702 Hannan-Quinn criter. 0.143872 Restr. log likelihood -40.03639 Avg. log likelihood -0.040585 LR statistic (1 df) 72.19938 McFadden R-squared 0.901672
Probability(LR stat) 0.000000
Obs with Dep=0 83 Total obs 97
Obs with Dep=1
14
Probit模型最终估计结果是
p i = F(y i) = F (-144.456 + 0.4029 x i) 拐点坐标(358.5, 0.5)
预测正确率
Forecast: YF
Actual: Y
Forecast sample: 1 97
Included observations: 97
Root Mean Squared Error 0.115072
Mean Absolute Error 0.025387
Mean Absolute Percentage Error 1.216791
Theil Inequality Coefficient 0.154476
Bias Proportion 0.000084
Variance Proportion 0.020837
Covariance Proportion 0.979080
预测正确率结论：线性概率模型RMSE=0.294780 MAE=0.233437 MAPE=8.689503 Logit模型 RMSE=0.114244 MAE=0.025502 MAPE=1.275122 Probit模型 RMSE=0.115072 MAE=0.025387 MAPE=1.216791 由上面结果可知线性概率模型的RMSE、MAE、MAPE 均远远大于Logit模型和Probit模型，说明其误差率比Logit模型和Probit模型大很多，所以正确率远远小于Logit模型和Probit模型。

而Logit模型和Probit模型的RMSE、MAE、MAPE相差很小，所以正确率相差不大。

综上所诉，此数据可以用Logit模型和Probit模型代替线性概率模型进行分析。