实证研究方法实例-4

1、1、等级相关系数检验的例子根据下表给出的10只共同基金的有关数据，估计组合证券理论中的资本市场线:等级相关系数为：T值为：在10%的显著水平下,10(100-1)0.3333,8=0.3333/=0.998,1-0.1110V1397,因此并不显著。其中，E为组合证券的预期回报率，仃为预期回报的标准差。共同基金平均年回报(%)Ei年回报标准差叼残差eeil的等级5的等级等级之差didi2A12.412.11.0394525B14.421.41.2410911C14.618.70.2047-39D16.021.70.22510-525E11.312.50.266511F10.010.40.597

2、2525G16.220.81.838800H10.410.20.103124I13.116.00.0626-416J11.312.00.0313-24Total0110故认为不存在异方差。2、GoldfeldQuandt检验的例子下表给出了30个家庭的收入X与消费Y的有关数据（已按X值重新排列）收入X消费Y80558570907510065105741108011584120791259013098140951451081501131601101651125180115185130190135200120205140210144220152225140230137240145245175250

3、189260180265178270191略去中间的4个观测值后，对两头的各13个观测值分别做OLS估计，并计算相应的残差平方和，结果如下：Y=3.40490.6968XiSer2=0.8887,RSS=377.17,df=11Y?u-28.02720.7941XjSe(30.6421)(0.1319)r2=0.7681,RS§=1536.8,df=11因此:、eQ(n12)'e12/(n2c-2)-2f1536.8-2=-e1j377.17=4.07在5%的显著水平下，F/1,11)=2.82因此，可以做出存在异方差的结论。3、Var(u)=仃：已知的情况的例子人均补偿(C

4、ompensationperEmployee)?就业规模(EmploymentSize)之间的关系(美国，1958年)原始数据如下表：Compensationperemployee($)innondurablemanufacturingindustriesaccordingtoemploymentsizeofestablishment,1958IndustryEmploymentSize(AverageNumberofEmployees)1-45-910-1920-4950-99100-249250-499500-9991000-2499Foodandkindredproducts299432

5、953565390741894486467649685342Tobaccoproducts172120573336332029802848307229693822Textilemillproducts360036573674343733403334322531633168Apparelandrelatedproducts349437873533321530302834275029673453Paperandalliedproducts349838473913413544454885513253425326Printingandpublishing361142064695508353015269

6、518253955552Chemicalandalliedproducts387546604930500551145248563058705876Petroleumandcoalproducts461651815317533754215710631664556347Rubberandplasticproducts353839844014428742214539472149055481Leatherandleatherproducts301631963149331734143254317733464067Averagecompensation339637874013401441464241438

7、745384843Standarddeviation743.7851.4727.8805.06929.91080.61243.21307.71112.5从表上数据可以看出，规模大的企业给雇员的补偿要比规模小的企业多，且不同规模的企业给雇员补偿的变化幅度差异较大，即出现了异方差。建立人均补偿与就业规模之间关系的回归模型如下：Yi=J，Xi3其中，Y代表人均补偿，X代表就业规模。由于异方差的存在，因而对模型进行变换如下：Y1XuCJ-CT-CT-CT.iiii变换后的模型为一古典模型(二元模型且不含常数项)。相应的数据如下表：Compensation(Y)EmploymentSize(X)1X%3

8、3961(1-4)743.74.56640.001337872(5-9)851.44.44800.002340133(10-19)727.85.51390.004141044(20-49)805.065.09780.005041465(50-99)929.94.45850.005442416(100-249)1080.63.92470.005543877(250-499)1243.23.52880.005645388(500-999)1307.73.47020.006148439(1000-2499)1112.54.35320.0081对变换后的模型进行OLS回归(变换后的模型不含常数项)，即

9、运用WLS,得到如下估计结果：Yixi-3406.639154.153二i二i二i_2Se(80.983)(16.959)R=0.9993T(42.066)(9.090)如果直接运用OLS进行估计，其结果如下：R2=0.9383Y?=3417.833148.767XiSe(81.136)(14.418)T(42.125)(10.318)4、Var(u。=仃2未知的情况的例子企业的R&D支出与销售额之间的关系(美国，1988)原始数据如下表：InnovationinAmerican:R&DExpenditureintheUnitedStates,1988(inMillionsof

10、dollars)IndustrygroupingSalesR&DExpenditureProfitsContainers&packaging6375.362.5185.1Nonbankfinancial11626.492.91569.5Serviceindustries14655.1178.3276.8Metals&mining21869.2258.42828.1Housing&construction26408.3494.7225.9Generalmanufacturing32405.61083.03751.9Leisuretimeindustries3510

11、7.71620.62884.1Paper&forestproducts40295.4421.74645.7Food70761.6509.25036.4Healthcare80552.86620.113869.9Aerospace95294.03918.64487.8Consumerproducts101314.11595.310278.9Electrical&electronics116141.36107.58787.3Chemicals122315.74454.116438.8Conglomerates141649.93163.89761.4Officeequipment&a

12、mp;computers175025.813210.719774.5Fuel230614.51703.822626.6Automotive293543.09528.218415.4首先建立R&D支出与销售额之间关系的回归方程：OLS:Y?=192990.0319XiSe(990.99)(0.0083)2T(0.1948)(3.8434)r=0.4783对于截面数据，很可能出现异方差，运用Glejser检验得到如下结果：ei=578,570,0119XiT(0.8525)(2.0931)e=-507.027.9270.XiT(0.5032)(2.3704)1ei=2273.7十19925

13、000XiT(3.7601)(1.6175)r2=0,21502-r=0.25992_r2=0,1405由此可以推断异方差的近似形式为:2Var(Ui)=，Xi2.e与Xi的散点图亦显不类似的规律:6QOODQOODQ=50000000:004aaoDDao.D0zp'tfl中a3000000000-20000000.00-1OOQDOOGDO-o.oo-a.OD5DOOO.DO1IODOQO.DO1SODOO-OD200000.00250000.003DOOOD.OOe与Xi的散点图:750000-500000-2500.000.00-2500.00-0000-0.005O0D0.0

14、D1100000JOOl5flMO.OO刘口前EM2SMM00anonofl.flD因此，对异方差进行矫正后，得到如下估计结果：Y?1WLS-1=-246.68+0.0368、XiK,XiSe(341.13)(0.0071)T(0.6472)(5.1723)R2=0.6258两边同乘以收,即可得到：Y?=-246.680.0368Xi比较OLS与WLS的估计结果可以看出，WLS的估计结果其解释能力明显增强，特别是回归系数的标准差降低了，说明在异方差存在的情况下，如果用OLS进行估计，将使估计精确度降低。运用SPSS得到的估计结果如下:OLS的估计结果:ModelSummaryModelRRSq

15、uareAdjustedRSquareStd.ErroroftheEstimate1.692(a).478.4462759.15326aPredictors:(Constant),XANOVA(b)ModelSumofSquaresdfMeanSquareFSig.1Regression111675211.3041111675211.30414.669.001(a)Residual121806827.541167612926.721Total233482038.84417aPredictors:(Constant),XbDependentVariable:YCoefficients(a)Mod

16、elUnstandardizedCoefficientsStandardizedCoefficientstSig.BStd.ErrorBeta1(Constant)192.993990.986.195.848X.032.008.6923.830.001aDependentVariable:Y101WLS的估计结果（权重W=1:XModelSummaryModelRRSquareAdjustedRSquareStd.ErroroftheEstimate1.791(a).626.6027.25713aPredictors:(Constant),XANOVA(b,c)ModelSumofSquaresdfMeanSquareFSig.1Regression1408.96511408.96526.753.000(a)Residual842.6561652.666Total2251.62117aPredictors:(Constant),XbDependentVariabl