大连海事大学计量经济学Eviews实验课讲义-5序列相关与异方差-上机课

第五课序列相关与异方差模型的处理5.1序列相关模型一、首先，结合案例数据（5_1_1）研究天津市城镇居民人均消费与人均可支配收入的关系，分析一阶线性相关存在时模型的检验与处理。（1）案例数据：改革开放以来，天津市城镇居民人均消费性支出（CONSUM）,人均可支配收入（INCOME）以及消费价格指数（PRICE）数据（1978—2000年）见下表。（数据来源：晓峒，《计量经济学基础》P152，例6.1）（2）散点图考虑到价格指数的影响，将CONSUM和INCOME各自除以价格指数，形成被解释变量和解释变量：CONSUM/PRICE和INCOME/PRICE，并作散点图如下，分析散点图，CONSUM/PRICE和INCOME/PRICE呈现线性相关。14001200-tGBnp^usNOC1000-tGBnp^usNOC8006004002000400800120016002000INCOME/PRICE回归结果，Eviews输出结果报告，得到回归方程CONSUM/PRICE=111.4400081+0.7118287831*INCOME/PRICEDependentVariable:CONSUM/PRICEVariableCoefficientStd.Errort-StatisticProb.C111.440017.055926.5338040.0000INCOME/PRICE0.7118290.01689942.122210.0000R-squared0.988303Meandependentvar769.4035AdjustedR-squared0.987746S.D.dependentvar296.7204S.E.ofregression32.84676Akaikeinfocriterion9.904525Sumsquaredresid22657.10Schwarzcriterion10.00326Loglikelihood-111.9020F-statistic1774.281Durbin-Watsonstat0.598571Prob(F-statistic)0.000000«二0.05水平上，T=23条件下，k=1时，临界值Dl=1.26，由结果可知，DW=0.59〈Dl,因此原模型中存在序列正相关。LM检验LM统计量：Breusch-GodfreySerialCorrelationLMTest:F-statistic14.83210Prob.F(1,20)0.000996Obs*R-squared9.793792Prob.Chi-Square(1)0.001751

辅助回归：TestEquation:DependentVariable:RESIDIncludedobservations:23Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C3.17101513.268830.2389820.8136INCOME/PRICE-0.0046620.013177-0.3537810.7272RESID(-1)0.6789670.1762983.8512470.0010R-squared0.425817Meandependentvar-1.40E-13AdjustedR-squared0.368399S.D.dependentvar32.09156S.E.ofregression25.50424Akaikeinfocriterion9.436674Sumsquaredresid13009.32Schwarzcriterion9.584782Loglikelihood-105.5217F-statistic7.416052Durbin-Watsonstat2.005247Prob(F-statistic)0.003895可见’卡方统计量TR2=9・79,而八0.05水平下，xJ)-3W,暝=9.79〉XJ)-3W，因此’拒绝零假设,认为存在一阶序列相关。0.678967说明存在正相关。5)用广义最小二乘法估计参数DW0.6计算一阶相关系数p若令=1—°=1~~2=°7计算一阶相关系数p若令Y=CONSUM/PRICE,X二INCOME/PRICE，令GDY=Y-0.7Y,GDX=X-0.7X，则以ttttt-1ttt-1GDY和GDX为样本再次计算回归方程，GDY=45.24890183+0.6782321994*GDXttDependentVariable:GDYIncludedobservations:22afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C45.2489012.258623.6911910.0014GDX0.6782320.03398319.957990.0000R-squared0.952190Meandependentvar269.1295AdjustedR-squared0.949799S.D.dependentvar103.4908S.E.ofregression23.18764Akaikeinfocriterion9.211624Sumsquaredresid10753.33Schwarzcriterion9.310809Loglikelihood-99.32786F-statistic398.3214Durbin-Watsonstat2.308815Prob(F-statistic)0.000000DW值=2.3〈4-du=4-1.43=2.57，已经消除序列相关。由于0=0*/(1-P)=45.2489/(1—0.7)=150.8297，因此原模型的估计结果为:00F=150.8297+0.6782X。分析可知，天津市城镇居民人均消费性支出平均占人均可支配收入的67.82%。tt注意：对广义差分后模型与原模型的判定系数不可简单直接比较，因为其变量不同；两个模型的回归系数估计值可能有所不同，计量经济学理论认为，广义差分模型的估计量性质优于存在序列相关时模型的估计量。Eviews操作：生成新变量的方法：Quick——GenerateSeries——“X=C0NSUM/PRICE”、”INC0ME/PRICE”，但每次只能收入一个命令；LM（BG）检验方法：Equation——Views——ResidualTests——SerialCorrelationLMTest——OK。二、结合案例5_1_2，研究天津市保费收入和人口的回归关系，分析二阶序列相关存在时模型的检验与处理。（1）天津市保费收入和人口数据：1967—1978年天津市的保险费收入（Yt,万元）和人口（Xt,万人）数据见5_1_2,散点图见下图，Y与X呈指数关系，对Y对自然对数，LnY与X呈线性关系。（数据来源：晓峒，《计量经济学基础》P155,例6.2）2）散点图：通过散点图确定模型形式：LnY=卩+卩X+ut01tt(3)利用Eviews软件估计方程，得到LOG(Y)=-11.18098138+O.*X输出结果为：DependentVariable:LOG(Y)Includedobservations:32VariableCoefficientStd.Errort-StatisticProb.C-11.180980.534786-20.907400.0000X0.0254050.00068337.209290.0000R-squared0.978792Meandependentvar8.591552AdjustedR-squared0.978085S.D.dependentvar2.300249S.E.ofregression0.340525Akaikeinfocriterion0.743808Sumsquaredresid3.478727Schwarzcriterion0.835416Loglikelihood-9.900921F-statistic1384.531Durbin-Watsonstat0.363124Prob(F-statistic)0.000000

对模型结果分析，判定系数较大，0.98，拟合较好，系数显著，但是DW值较小，怀疑有自相关。4）检验自相关查表，n=32,k=1,«二0.05，Dl=1.37,Du=1.50,而DW=0.36<1.37,存在正的序列相关。Eviews下的LM检验：Breusch-GodfreySerialCorrelationLMTest:F-statistic33.13129Prob.F(2,28)0.000000Obs*R-squared22.49464Prob.Chi-Square(2)0.000013辅助回归：TestEquation:DependentVariable:RESIDIncludedobservations:32Presamplemissingvaluelaggedresidualssettozero.C-0.0846620.315807-0.2680810.7906X0.0001160.0004060.2868600.7763RESID(-1)1.1732040.1740766.7396070.0000RESID(-2)-0.4421490.200364-2.2067230.0357R-squared0.702957Meandependentvar-4.66E-15AdjustedR-squared0.671131S.D.dependentvar0.334988S.E.ofregression0.192106Akaikeinfocriterion-0.345072Sumsquaredresid1.033330Schwarzcriterion-0.161855Loglikelihood9.521154F-statistic22.08752Durbin-Watsonstat1.956428Prob(F-statistic)0.000000VariableCoefficientStd.Errort-StatisticProb.从检验结果看，误差项存在二阶自相关。（5）广义差分法消除自相关依据残差自回归结果：DependentVariable:ETSample（adjusted）:19691998Includedobservations:30afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.ET(-1)1.1860050.1711456.9298240.0000ET(-2)-0.4666670.186755-2.4988160.0186R-squared0.706585Meandependentvar-0.016275AdjustedR-squared0.696106S.D.dependentvar0.339942S.E.ofregression0.187399Akaikeinfocriterion-0.446816Sumsquaredresid0.983312Schwarzcriterion-0.353403Loglikelihood8.702236Durbin-Watsonstat1.971666得到辅助回归方程为：ET=1.186004684*ET(-1)-0.46666712*ET(-2)进而得到二阶相关系数P1=1.186,P2=一0.467，对原变量做广义差分，若令GDY二LnY-1.186LnY+0.467LnY,GDX=X-1.186X+0.467X，则以GDY和GDX为样ttt—1t—2ttt—1t—2tt本再次计算回归方程，DependentVariable:GDLNYMethod:LeastSquaresSample(adjusted):19691998Includedobservations:30afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C-3.2462710.323473-10.035690.0000GDX0.0258660.00144117.944590.0000R-squared0.920002Meandependentvar2.525873AdjustedR-squared0.917145S.D.dependentvar0.649829S.E.ofregression0.187050Akaikeinfocriterion-0.450536Sumsquaredresid0.979660Schwarzcriterion-0.357123Loglikelihood8.758046F-statistic322.0083Durbin-Watsonstat1.993633Prob(F-statistic)0.000000从结果看，DW=1.99,序列相关消除。根据计算公式：B=0*/(1—P—P)=—3.246/(1—1.186+0.467)=—11.55，0012因此，原模型的广义最小二乘估计结果为：LnY=—11.55+0.0259Xtt与原估计结果LOG(Y)=-11.18098138+0.*X相比，稍有差别，计量经济学理论认为广义最小二乘估计量的特性优于误差项存在自相关条件下的最小二乘估计量的特性，即0.0259比0.0254更可信，其经济含义为：每增加1万人，保费收入的对数值增加0.0259.5.2异方差模型一、案例分析(1)数据：已知某地区的个人储蓄Y,可支配收入X的截面样本数据见5_2_1,建立它们之间的线性回归模型并估计(数据来源：晓峒，《计量经济学基础》P125,例5.1,该数据来源摘自【英】A.科扬尼斯著，许开甲等译《经济计量学理论一一经济计量方法概述》上册)。(2)建立模型根据经济理论确定计量经济学模型基本形式为Y二卩+卩X+u，i01ii估计方程为：Y=-700.4109607+O.*XEviews输出结果报告如下：DependentVariable:YMethod:LeastSquaresSample:131Includedobservations:31VariableCoefficientStd.Errort-StatisticProb.C-700.4110116.6679-6.0034580.0000X0.0878310.00482718.195750.0000R-squared0.919464Meandependentvar1266.452AdjustedR-squared0.916686S.D.dependentvar846.7570S.E.ofregression244.4088Akaikeinfocriterion13.89790Sumsquaredresid1732334.Schwarzcriterion13.99042Loglikelihood-213.4175F-statistic331.0852Durbin-Watsonstat1.089829Prob(F-statistic)0.0000002)异方差检验考虑到横截面数据的特点，怀疑会产生异方差问题，对其以各种方法进行检验①简单观察noXEnoXE18777193.51691724127159.308729210-3.513991825604105.58239954-83.86041926500-227.115410508-91.51882026760179.0492510979-141.8872128300414.7892611912-238.8342227430308.2024712747-13.17282329560209.12281349917.778192428150-172.036914269-121.8522532100131.03091015522-74.90422632500265.89841116730128.99572735250174.3627121766399.049252833500-521.9331318575-152.0532936000-561.5111419635-205.1543036200-379.077

152116363.640213138200145.26081622880392.8341通过对X与残差的观察，发现e似乎随着X变化而变化，怀疑有异方差，于是以各种方法对其进行检验。图示法分别绘制Y及残差与解释变量X的散点图，从散点图来看，随着可支配收入的增加，Y和残差的离散程度在增加，可200--200200--200戈德菲尔德-匡特(Goldfeld-Quandt,G-Q)检验将X的样本观察值按照升序排列，Y的观察值顺序与X观察值对应。略去中间的9个样本观察值，剩余的样本观察值平均分为两组子样本，每个子样本的样本观察值数量为11个。分别用两个子样本进行回归，得到各自的结果报告，从而得到各自的残差平方和。A.排序noYXnoYX1264.008777.00171578.0024127.002105.009210.00181654.0025604.00390.009954.00191400.0026500.004131.0010508.00201829.0026760.005122.0010979.00212017.0027430.006107.0011912.00221600.0028150.007406.0012747.00232200.0028300.008503.0013499.00242105.0029560.009431.0014269.00252250.0032100.0010588.0015522.00262420.0032500.0011898.0016730.00271720.0033500.0012950.0017663.00282570.0035250.0013779.0018575.00291900.0036000.0014819.0019635.00302100.0036200.00151222.0021163.00312800.0038200.00161702.0022880.00B.划分子样本并回归

子样本1：noYX1264.008777.002105.009210.00390.009954.004131.0010508.005122.0010979.006107.0011912.007406.0012747.008503.0013499.009431.0014269.0010588.0015522.0011898.0016730.00子样本2：noYX212017.0027430.00221600.0028150.00232200.0028300.00242105.0029560.00252250.0032100.00262420.0032500.00271720.0033500.00282570.0035250.00291900.0036000.00302100.0036200.00312800.0038200.00子样本1回归结果：DependentVariable:YMethod:LeastSquaresSample:111Includedobservations:11VariableCoefficientStd.Errort-StatisticProb.C-744.6351195.4108-3.8106140.0041X0.0882580.0157055.6196190.0003R-squared0.778216Meandependentvar331.3636AdjustedR-squared0.753574S.D.dependentvar260.8157S.E.ofregression129.4724Akaikeinfocriterion12.72778Sumsquaredresid150867.9Schwarzcriterion12.80012Loglikelihood-68.00278F-statistic31.58011Durbin-Watsonstat1.142088Prob(F-statistic)0.000326

子样本2回归结果:DependentVariable:YMethod:LeastSquaresSample:2131Includedobservations:11计算F统计量:iie计算F统计量:iie2i2966997150868=6.409557,在^=0.05的显著性水平下，查F分布表，F(9,9)二3.18,0.05VariableCoefficientStd.Errort-StatisticProb.C666.3811911.25850.7312760.4832X0.0457790.0278981.6409710.1352R-squared0.230295Meandependentvar2152.909AdjustedR-squared0.144772S.D.dependentvar354.4462S.E.ofregression327.7867Akaikeinfocriterion14.58557Sumsquaredresid966997.0Schwarzcriterion14.65791Loglikelihood-78.22063F-statistic2.692786Durbin-Watsonstat2.743586Prob(F-statistic)0.135222因此，F=6.41〉F0.(9,9)=3.18,因此，拒绝原假设，接受对立假设，原模型存在随机误差项的异方差。子样本2的差平方和较大，也可得出递增异方差的结论。White检验法根据White检验原理，得至到TR2统计量,White检验结果：WhiteHeteroskedasticityTest:F-statistic5.819690Prob.F(2,28)0.007699Obs*R-squared9.102584Prob.Chi-Square(2)0.010554辅助回归结果:TestEquation:DependentVariable:RESIDEMethod:LeastSquaresSample:131Includedobservations:31VariableCoefficientStd.Errort-StatisticProb.C19975.9882774.930.2413290.8111X-2.1986328.094419-0.2716230.7879XA20.0001460.0001760.8300460.4135R-squared0.293632Meandependentvar55881.73AdjustedR-squared0.243177S.D.dependentvar77875.67S.E.ofregression67748.39Akaikeinfocriterion25.17675Sumsquaredresid1.29E+11Schwarzcriterion25.31553

LoglikelihoodDurbin-Watsonstat5.8196900.007699LoglikelihoodDurbin-Watsonstat5.8196900.0076991.552875Prob(F-statistic)从检验结果看，TR2=9〉TR2=9>x2=6.0，可见，在0.05的水平上，可以拒绝原假设，即模型中存在异方0.05差。White检验的Eviews操作：EquationResidualTests\WhiteHeteroskedasticity(nocrossterms).纠正异方差由于异方差是由X引起的，可以以1/X作为权重来修正模型，得到加权最小二乘估计式：Y/X二B/X+P+u/X，估计结果如下：Y/X=0.0897509775-742.468372*1/Xtt0t1tt结果报告：DependentVariable:Y/XMethod:LeastSquaresSample:131Includedobservations:31VariableCoefficientStd.Errort-StatisticProb.C0.0897510.00434720.646960.00001/X-742.468471.91567-10.324150.0000R-squared0.786117Meandependentvar0.049255AdjustedR-squared0.778742S.D.dependentvar0.022178S.E.ofregression0.010432Akaikeinfocriterion-6.225537Sumsquaredresid0.003156Schwarzcriterion-6.133022Loglikelihood98.49583F-statistic106.5881Durbin-Watsonstat1.081175Prob(F-statistic)0.000000由结果Y/X=0.0897509775-742.468372*1/X，可以得到原模型估计结果：Y=-742.47+0.09X或者，可以直接应用加权最小二乘法，得到：DependentVariable:YSample:131Includedobservations:31Weightingseries:1/XVariableCoefficientStd.Errort-StatisticProb.C-742.468471.91567-10.324150.0000X0.0897510.00434720.646960.0000WeightedStatisticsR-squared0.936305Meandependentvar903.0766AdjustedR-squared0.934109S.D.depe