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，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见数据表obsYSCOREobsYSCOREobsYSCORE114013403326702752140135033268027331392360332690273413873703317002725138438033071026761379390328720266713784003287302638137841032874026191376420321750260101371430321760256111362440318770252121362450318780252131361460316790245140359470308800243150358480308810242161356490304820241170356500303830239180355510303840235190354520299850232200354530297860228210353540294870219220350550293880219230349560293890214240349570292900210250348580291910204260347590291920198270347600287930189280344610286940188290339620286950182300338630282960166310338640282970123320336650282330334660278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用Eviews得到回归结果如图：Dependent Variable: YMethod: Least SquaresDate: 12/10/10 Time: 20:38Sample: 1 97Included observations: 97VariableCoefficientStd. Errort-StatisticProb.C-0.8474070.159663-5.3074760.0000SCORE0.0032970.0005216.3259700.0000R-squared0.296390Mean dependent var0.144330Adjusted R-squared0.288983S.D. dependent var0.353250S.E. of regression0.297866Akaike info criterion0.436060Sum squared resid8.428818Schwarz criterion0.489147Log likelihood-19.14890F-statistic40.01790Durbin-Watson stat0.359992Prob(F-statistic)0.000000参数估计结果为：-0.847407+0.003297 Se=(0.159663)( 0.000521) t=(-5.307476) (6.325970) p=(0.0000) (0.0000) 预测正确率：Forecast: YFActual: YForecast sample: 1 97Included observations: 97Root Mean Squared Error0.294780Mean Absolute Error0.233437Mean Absolute Percentage Error8.689503Theil Inequality Coefficient0.475786Bias Proportion0.000000Variance Proportion0.294987Covariance Proportion0.7050132.Logit模型Dependent Variable: YMethod: ML - Binary Logit (Quadratic hill climbing)Date: 12/10/10 Time: 21:38Sample: 1 97Included observations: 97Convergence achieved after 11 iterationsCovariance matrix computed using second derivativesVariableCoefficientStd. Errorz-StatisticProb.C-243.7362125.5564-1.9412480.0522SCORE0.6794410.3504921.9385360.0526Mean dependent var0.144330S.D. dependent var0.353250S.E. of regression0.115440Akaike info criterion0.123553Sum squared resid1.266017Schwarz criterion0.176640Log likelihood-3.992330Hannan-Quinn criter.0.145019Restr. log likelihood-40.03639Avg. log likelihood-0.041158LR statistic (1 df)72.08812McFadden R-squared0.900282Probability(LR stat)0.000000Obs with Dep=083Total obs97Obs with Dep=114得Logit模型估计结果如下 pi = F(yi) = 拐点坐标 (358.7, 0.5)其中Y=-243.7362+0.6794X预测正确率Forecast: YFActual: YForecast sample: 1 97Included observations: 97Root Mean Squared Error0.114244Mean Absolute Error0.025502Mean Absolute Percentage Error1.275122Theil Inequality Coefficient0.153748Bias Proportion0.000000Variance Proportion0.025338Covariance Proportion0.9746623.Probit模型Dependent Variable: YMethod: ML - Binary Probit (Quadratic hill climbing)Date: 12/10/10 Time: 21:40Sample: 1 97Included observations: 97Convergence achieved after 11 iterationsCovariance matrix computed using second derivativesVariableCoefficientStd. Errorz-StatisticProb.C-144.456070.19809-2.0578330.0396SCORE0.4028680.1961862.0535040.0400Mean dependent var0.144330S.D. dependent var0.353250S.E. of regression0.116277Akaike info criterion0.122406Sum squared resid1.284441Schwarz criterion0.175493Log likelihood-3.936702Hannan-Quinn criter.0.143872Restr. log likelihood-40.03639Avg. log likelihood-0.040585LR statistic (1 df)72.19938McFadden R-squared0.901672Probability(LR stat)0.000000Obs with Dep=083Total obs97Obs with Dep=114Probit模型最终估计结果是 pi = F(yi) = F (-144.456 + 0.4029 xi) 拐点坐标 (358.5, 0.5)预测正确率Forecast: YFActual: YForecast sample: 1 97Included observations: 97Root Mean Squared Error0.115072Mean Absolute Error0.025387Mean Absolute Percentage Error1.216791Theil Inequality Coefficient0.154476Bias Proportion0.000084Variance Proportion0.020837Covariance Proportion0.979080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