计量学实验报告 实验2-多元回归-多重共线性-预测问题

计量经济学讲课提纲PAGEPAGE1大连海事大学交通运输管理学院实例1——中国粮食生产函数根据理论和经验分析，影响粮食生产（Y）的主要因素有农业化肥施用量（X1）、粮食播种面积（X2）、成灾面积（X3）、农业机械总动力（X4）和农业劳动力（X5），其中成灾面积的符号为负，其余均应为正。下表给出了1983——2000中国粮食生产的相关数据，拟建立中国粮食生产函数。年份粮食产量Y（万吨）化肥施用量X1（万千克）播种面积X2（千公顷）成灾面积X3（公顷）农业机械总动力X4（万千瓦）农业劳动力X5（万人）198338728.01659.8114047.016209.318022.031645.1198440731.01739.8112884.015264.019497.031685.0198537911.01775.8108845.022705.320913.030351.5198639151.01930.6110933.023656.022950.030467.0198740208.01999.3111268.020392.724836.030870.0198839408.02141.5110123.023944.726575.031455.7198940755.02357.1112205.024448.728067.032440.5199044624.02590.3113466.017819.328708.033330.4199143529.02806.1112314.027814.029389.034186.3199244264.02930.2110560.025894.730308.034037.0199345649.03151.9110509.023133.031817.033258.2199444510.03317.9109544.031383.033802.032690.3199546662.03593.7110060.022267.036118.032334.5199650454.03827.9112548.021233.038547.032260.4199749417.03980.7112912.030309.042016.032434.9199851230.04083.7113787.025181.045208.032626.4199950839.04124.3113161.026731.048996.032911.8200046218.04146.4108463.034374.052574.032797.5（1）建立Y对所有解释变量的回归模型，结果如下：Y=-12815.75+6.213*X1+0.421*X2-0.166*X3-0.098*X4-0.028*X5VariableCoefficientStd.Errort-StatisticProb.

C-12815.7514078.90-0.9102800.3806X16.2125620.7408818.3853730.0000X20.4213800.1269253.3199190.0061X3-0.1662600.059229-2.8070650.0158X4-0.0977700.067647-1.4452990.1740X5-0.0284250.202357-0.1404710.8906R-squared0.982798

Meandependentvar44127.11AdjustedR-squared0.975630

S.D.dependentvar4409.100S.E.ofregression688.2984

Akaikeinfocriterion16.16752Sumsquaredresid5685056.

Schwarzcriterion16.46431Loglikelihood-139.5077

F-statistic137.1164Durbin-Watsonstat1.810512

Prob(F-statistic)0.000000

从计算结果看，R2较大并接近于1，而且F=137.11>F0.05=3.11，故认为粮食生产量与上述所有解释变量间总体线性相关显著。但是，同时，X4、X5前参数未通过t检验，而且符号的经济意义也不合理，故认为解释变量间存在多重共线性。为了进一步检验多重共线性，进行下面操作。（2）计算解释变量间的两两相关系数，得到简单相关系数矩阵如下：X1X2X3X4X5X11X20.0118231X30.640175-0.454911X40.960278-0.038480.6895651X50.545450.1823590.35573530.4541691从相关分析结果来看，部分解释变量间确实存在相关，尤其X1与X4之间高度相关。为了处理多重共线性，正确选择解释变量，进行逐步回归，首先选择最优的基本方程。（3）分别做粮食生产量对各个解释变量的回归，得A．Y对X1回归结果：VariableCoefficientStd.Errort-StatisticProb.

C30867.311206.36425.587060.0000X14.5761150.39819911.492020.0000R-squared0.891941

Meandependentvar44127.11AdjustedR-squared0.885187

S.D.dependentvar4409.100S.E.ofregression1493.984

Akaikeinfocriterion17.56072Sumsquaredresid35711799

Schwarzcriterion17.65965Loglikelihood-156.0465

F-statistic132.0666Durbin-Watsonstat1.855174

Prob(F-statistic)0.000000

B.Y对X2回归结果：VariableCoefficientStd.Errort-StatisticProb.

C-33822.4168409.15-0.4944140.6277X20.6988800.6132731.1395900.2712R-squared0.075073

Meandependentvar44127.11AdjustedR-squared0.017265

S.D.dependentvar4409.100S.E.ofregression4370.873

Akaikeinfocriterion19.70775Sumsquaredresid3.06E+08

Schwarzcriterion19.80668Loglikelihood-175.3698

F-statistic1.298665Durbin-Watsonstat0.118043

Prob(F-statistic)0.271231

C.Y对X3回归结果VariableCoefficientStd.Errort-StatisticProb.

C35712.864926.5837.2490120.0000X30.3499780.2008021.7429060.1005R-squared0.159563

Meandependentvar44127.11AdjustedR-squared0.107036

S.D.dependentvar4409.100S.E.ofregression4166.457

Akaikeinfocriterion19.61196Sumsquaredresid2.78E+08

Schwarzcriterion19.71089Loglikelihood-174.5076

F-statistic3.037721Durbin-Watsonstat0.935587

Prob(F-statistic)0.100533

D.Y对X4回归结果：VariableCoefficientStd.Errort-StatisticProb.

C31918.721828.71517.454180.0000X40.3799670.0544486.9785870.0000R-squared0.752707

Meandependentvar44127.11AdjustedR-squared0.737252

S.D.dependentvar4409.100S.E.ofregression2260.060

Akaikeinfocriterion18.38861Sumsquaredresid81725964

Schwarzcriterion18.48754Loglikelihood-163.4975

F-statistic48.70067Durbin-Watsonstat1.109488

Prob(F-statistic)0.000003

E.Y对X5回归结果：VariableCoefficientStd.Errort-StatisticProb.

C-28260.0227240.49-1.0374270.3150X52.2396140.8423522.6587620.0172R-squared0.306429

Meandependentvar44127.11AdjustedR-squared0.263081

S.D.dependentvar4409.100S.E.ofregression3784.948

Akaikeinfocriterion19.41989Sumsquaredresid2.29E+08

Schwarzcriterion19.51882Loglikelihood-172.7790

F-statistic7.069018Durbin-Watsonstat0.357079

Prob(F-statistic)0.017160

（4）逐步回归，A、Y对X1、X4回归结果：VariableCoefficientStd.Errort-StatisticProb.

C31164.921137.21927.404510.0000X16.9259381.3315025.2015970.0001X4-0.2211780.120350-1.8377920.0860R-squared0.911800

Meandependentvar44127.11AdjustedR-squared0.900040

S.D.dependentvar4409.100S.E.ofregression1394.000

Akaikeinfocriterion17.46875Sumsquaredresid29148555

Schwarzcriterion17.61715Loglikelihood-154.2188

F-statistic77.53409Durbin-Watsonstat1.992572

Prob(F-statistic)0.000000

从回归结果看，拟合优度虽然上升，但X4的系数不显著，因此，存在共线性，而相比较而言，X1更重要，因此剔除X4（从相关分析也有助于这个结论）。B、Y对X1、X5回归结果：VariableCoefficientStd.Errort-StatisticProb.

C24133.8412406.481.9452610.0707X14.4315590.4858839.1206250.0000X50.2212890.4057060.5454420.5935R-squared0.894042

Meandependentvar44127.11AdjustedR-squared0.879914

S.D.dependentvar4409.100S.E.ofregression1527.902

Akaikeinfocriterion17.65219Sumsquaredresid35017273

Schwarzcriterion17.80059Loglikelihood-155.8697

F-statistic63.28281Durbin-Watsonstat1.839712

Prob(F-statistic)0.000000

拟合优度升高不显著，修正的拟合优度略微下降，且X5系数不显著，因此，剔除X5.C、Y对X1、X3回归结果：VariableCoefficientStd.Errort-StatisticProb.

C35065.011064.61232.936880.0000X15.6543300.31219918.111320.0000X3-0.3045460.056452-5.3948030.0001R-squared0.963248

Meandependentvar44127.11AdjustedR-squared0.958348

S.D.dependentvar4409.100S.E.ofregression899.8443

Akaikeinfocriterion16.59333Sumsquaredresid12145797

Schwarzcriterion16.74173Loglikelihood-146.3400

F-statistic196.5723Durbin-Watsonstat1.728340

Prob(F-statistic)0.000000从回归结果看，拟合优度提高，X1和X3的系数显著，因此接受X3.

D、Y对X1、X2、X3回归结果：VariableCoefficientStd.Errort-StatisticProb.

C-11978.1814072.92-0.8511510.4090X15.2559350.26859519.568280.0000X20.4084320.1219743.3485220.0048X3-0.1946090.054533-3.5686370.0031R-squared0.979593

Meandependentvar44127.11AdjustedR-squared0.975220

S.D.dependentvar4409.100S.E.ofregression694.0715

Akaikeinfocriterion16.11616Sumsquaredresid6744293.

Schwarzcriterion16.31402Loglikelihood-141.0454

F-statistic224.0086Durbin-Watsonstat1.528658

Prob(F-statistic)0.000000

从回归结果看，拟合优度提高，X1、X2和X3的系数显著，因此接受X2.即，回归方程为：Y=-11978.18057+5.255935121*X1+0.408432175*X2-0.1946087795*X3实例2我国1988年-1998年的城镇居民人均全年耐用消费品支出、人均全年可支配收入以及耐用消费品价格指数的统计资料如下表，试建立城镇居民人均全年耐用消费品支出Y关于人均全年可支配收入x1和耐用消费品价格指数X2的回归模型，并进行回归分析。根据经验和对经济现实的分析，设定模型为二元线性回归模型，理论形式为：，（1）数据如下表，年份人均全年可支配收入X1（元）耐用消费品价格指数X2（1987）人均耐用消费品支出Y（元）19881181.4115.96137.1619891375.7133.35124.5619901501.2128.21107.9119911700.6124.85102.9619922026.6122.49125.2419932577.4129.86162.4519943496.2139.52217.4319954283140.44253.4219964838.9139.12251.0719975160.3133.35285.8519985425.1126.39327.26（2）Eviews的输出结果（下表）写出回归方程为：EXPENSEY=158.5398355+0.04940379666*INCOMEX1-0.911684216*PRINDEX2VariableCoefficientStd.Errort-StatisticProb.

C158.5398121.80711.3015640.2293INCOMEX10.0494040.00468410.547860.0000PRINDEX2-0.9116840.989546-0.9213160.3838R-squared0.947989

Meandependentvar190.4827AdjustedR-squared0.934986

S.D.dependentvar79.29127S.E.ofregression20.21757

Akaikeinfocriterion9.077982Sumsquaredresid3270.001

Schwarzcriterion9.186499Loglikelihood-46.92890

F-statistic72.90647Durbin-Watsonstat1.035840

Prob(F-statistic)0.000007

（3）检验①从经济意义来看，可支配收入前的系数为0.0494，正的，介于0和1之间，符号、大小与理论符合；价格指数前的系数为-0.91，大小和符号符合经济理论；②从统计角度看，R-squared=0.947989，AdjustedR-squared=0.934986，从多元回归修正的判定系数看，回归方程较好地拟合了散点，被解释变量的变异中有93%以上可以由方程来解释；从F统计量的结果来看，F=72.90647>F0.05(2,8)=4.46，而且F=72.90647>F0.01(2,8)=8.65，可见方程总体来看，无论在0.05还是0.01水平上都显著，即在我国城镇居民人均全年耐用消费品支出与人均全年可支配收入和耐用消费品价格指数之间存在显著的线性关系。这一点结论由F统计量的精确显著性水平Prob=0.000007也可得到。t统计量结果来看，可支配收入incomex1的系数，t值=10.54786>t0.05(8)=2.306，系数显著，可支配收入对耐用消费品支出有显著影响，变量x1保留；而对于耐用消费品价格指数prindex2的系数，t值=-0.921316，其绝对值小于t0.05（8），可以接受系数为零的原假设，剔除X2。以上结论由Eviews输出结果中系数的精确显著性水平Prob也可以直接得到。（4）预测：点预测和区间预测若已知2000年，我国城镇居民家庭人均可支配收入为5800元，耐用消费品价格指数为135，对2000年我国城镇居民家庭人均耐用消费品支出进行预测。①点预测，将x1=5800，X2=135代入估计方程，EXPENSEY=158.5398+0.0494*5800-0.9117*135，得到Y的估计值=321.9803（教材中是按照小数点后保留4位数字后的样本回归方程计算得到的，而Eviews软件的估计值是322.0045）。（也可可以在Eviews中调整扩大数据范围至2000，再将解释变量的数据输入，再利用Equation窗口下的forecast进行预测，估计值即保存在EXPENSEYf序列中。）②区间预测E(Y0)的区间预测Y的估计值的Y0的方差估计为，已知X2000矩阵为：，则X2000矩阵的转置矩阵为：，而解释变量X矩阵为：11181.4115.9611375.7133.3511501.2128.2111700.6124.8512026.6122.4912577.4129.8613496.2139.5214283140.4414838.9139.1215160.311×11×315425.1126.3