计量经济学的各种检验.ppt

经济计量学的几种量学的几种检验多重共多重共线性性n n .Multicollinearity arises because we have put in too many variables that measure the same thing.n nAs the degree of multicollinearity increases,the regression model estimates of the coefficients become unstable and the standard errors for the coefficients can get wildly inflated.n nMeasure:vif,tol=1/vif,condition index;etc.多重共多重共线性的后果性的后果n n1.存在完全多重共线性时,参数的估计值无法确定,而且估计值的方差变为无穷大.n n2.存在不完全多重共线性时,可以估计参数值,但是数值不稳定,而且方差很大.n n3.多重共线性会降低预测的精度,甚至失效,增大零假设接受的可能性(t值变小).多重共多重共线性的性的检测方法方法(1)样本可决系数法本可决系数法n n如果如果样样本的可决系数本的可决系数R-square R-square 比比较较大大，且，且回回归归系数系数几乎没有几乎没有统计统计上的上的显显著性著性，则则可可认为认为存在多重共存在多重共线线性。性。n nTheil Theil 提出了一个指提出了一个指标标：多重共：多重共线线性效性效应应系数系数Theil test resultsn nSas 结果：n n结果表明有多重共线性。多重共多重共线性性检测方法方法（2）辅助回助回归检验法法n n若存在多重共若存在多重共线线性，性，则则至少有一个解至少有一个解释变释变量可精确或近似量可精确或近似地表示地表示为为其余皆是其余皆是变变量的量的线线性性组组合。合。n n相相应应的的检验统计检验统计量量为为：辅助回助回归检验结果果n nSas 结果：n nKlein经验法则：若存在一个i,使得n nR(i)-squareR-square,则认为多重共线性严重；本例中x1,x3有多重共线性。多重共多重共线性性检验方法方法（3）样本相关系数本相关系数检验法法FG test resultsn nfg=20.488013401 p=0.0001344625；n n拒绝零假设，认为存在多重共线性。n n具体那些变量之间存在多重共线性，除了上面提到的辅助回归的方法外，还有以下提到的条件数检验和方差膨胀因子法。多重共多重共线性性检验方法：方法：（4）特征）特征值分析法所用的分析法所用的检验统计指指标n n ;为第k各自变量和其余自变量回归的可决系数.VIF10,有多重共线性;TOL=1/VIF;n n条件指数:n n条件数条件数:;C20,共线性严重.多重共多重共线性的性的检验和和补救救n n例一例一:进进口口总额总额和三个自和三个自变变量之量之间间回回归归;n nSas Sas 结结果如下果如下:Pearson Correlation Coefficients,N=Pearson Correlation Coefficients,N=11 Prob|r|under H0:Rho=011 Prob|r|under H0:Rho=0n n x1 x2 x3 x1 x2 x3n nx1 1.00000 0.02585 x1 1.00000 0.02585 0.997260.997260.997260.99726n nGDP 0.9399 GDP 0.9399 .0001.0001.0001.0001n nx2 0.02585 1.00000 0.03567x2 0.02585 1.00000 0.03567n n存蓄量存蓄量 0.9399 0.91710.9399 0.9171n nx3 x3 0.997260.997260.997260.99726 0.03567 1.00000 0.03567 1.00000n n总消费总消费 .0001.0001.0001|t|VariableDFEstimateErrortValuePr|t|InflationInflationn nIntercept1-10.127991.21216-8.36.00010Intercept1-10.127991.21216-8.36.00010n nx11-0.051400.07028-0.730.4883x11-0.051400.07028-0.730.4883185.99747185.99747n nx210.586950.094626.200.00041.01891x210.586950.094626.200.00041.01891n nx310.286850.102212.810.0263x310.286850.102212.810.0263186.11002186.11002n n发现发现x1x1的系数的系数为负为负,和和现实经济现实经济意意义义不符不符,出出现现原因就是原因就是x1x1和和x3x3之之间间的的线线性相关性相关.补救措施救措施n n增加增加样样本本;岭回岭回归归或主分量回或主分量回归归;n n至少去掉一个具有多重共至少去掉一个具有多重共线线性的性的变变量量;对对具有多重共具有多重共线线性的性的变变量量进进行行变换变换.n n对对所有所有变变量做滞后差分量做滞后差分变换变换(一般是一一般是一阶阶差分差分),),问题问题是是损损失失观测值观测值,可能有自相关可能有自相关.n n采用人均形式的采用人均形式的变变量（例如在生量（例如在生产产函数估函数估计计中）中）n n在缺乏有效信息在缺乏有效信息时时,对对系数关系系数关系进进行限制行限制,变为变为有有约约束回束回归归(Klein,Goldberger,1955),(Klein,Goldberger,1955),可以降低可以降低样样本方差和估本方差和估计计系数的系数的标标准差准差,但不一定是无偏的但不一定是无偏的(除非除非这这种限制是正确的种限制是正确的).).n n对对具有多重共具有多重共线线性的性的变变量量,设设法找出其因果关系法找出其因果关系,并建立模并建立模型和原方程构成型和原方程构成联联立方程立方程组组.岭回岭回归n n岭回归估计:n nK=0,b(k)=b即为OLSE;n nK的选取:n n即使b(k)的均方误差比b的均方误差小.岭迹岭迹图岭回岭回归结果果Obs_MODEL_TYPE_DEPVAR_Obs_MODEL_TYPE_DEPVAR_RIDGE_k_RIDGE_k_PCOMIT_PCOMIT_ _RMSE_ _RMSE_Interceptx1x2x3y_Interceptx1x2x3y1MODEL1PARMSy0.48887-10.1280-0.0510.586950.287-11MODEL1PARMSy0.48887-10.1280-0.0510.586950.287-12MODEL1RIDGEVIFy0.002MODEL1RIDGEVIFy0.00方差膨方差膨方差膨方差膨胀胀因子因子因子因子185.997 1.01891 186.110 1 185.997 1.01891 186.110 1 3MODEL1RIDGEy0.000.48887-10.1280-0.0510.586950.28713MODEL1RIDGEy0.000.48887-10.1280-0.0510.586950.28714MODEL1RIDGEVIFy0.014MODEL1RIDGEVIFy0.01 8.599 0.98192 8.604 -18.599 0.98192 8.604 -15MODEL1RIDGEy0.010.55323-9.18050.0460.598860.14415MODEL1RIDGEy0.010.55323-9.18050.0460.598860.14416MODEL1RIDGE6MODEL1RIDGEVIFVIFy0.02y0.02 2.858 0.96219 2.859 -12.858 0.96219 2.859 -17 MODEL1 RIDGE y 0.02 0.57016 -8.9277 0.057 0.59542 0.127 -17 MODEL1 RIDGE y 0.02 0.57016 -8.9277 0.057 0.59542 0.127 -18MODEL1RIDGEVIFy0.031.5020.943451.502-8MODEL1RIDGEVIFy0.031.5020.943451.502-1 19MODEL1RIDGEy0.030.57959-8.73370.0610.590800.120-19MODEL1RIDGEy0.030.57959-8.73370.0610.590800.120-110MODEL1RIDGEVIFy0.040.9790.925320.979-10MODEL1RIDGEVIFy0.040.9790.925320.979-1 111MODEL1RIDGEy0.040.58745-8.55830.0640.585910.116-111MODEL1RIDGEy0.040.58745-8.55830.0640.585910.116-1主分量回主分量回归n n主分量回归是将具有多重相关的变量集综合得出少数几个互不相关的主分量.n n两步:(1)找出自变量集的主分量,建立y与互不相关的前几个主分量的回归式.(2)将回归式还原为原自变量结果.n n详见,方开泰;主分量回主分量回归结果果Obs_MODEL_TYPE_DEPVAR_PCOMIT_RMSE_Interceptx1x2x3yObs_MODEL_TYPE_DEPVAR_PCOMIT_RMSE_Interceptx1x2x3y1MODEL1PARMSy1MODEL1PARMSy0.488870.48887-10.1280-0.051400.586950.28685-10.1280-0.051400.586950.28685112MODEL1IPCVIFy12MODEL1IPCVIFy10.25083 1.00085 0.25038 0.25083 1.00085 0.25038 1133MODEL1 IPC y 1 MODEL1 IPC y 1 0.550010.55001 -9.1301 0.07278 0.60922 0.10626 -9.1301 0.07278 0.60922 0.10626 114MODEL1IPC4MODEL1IPCVIFVIFy2y20.249560.000950.249710.249560.000950.24971-1 15MODEL1IPCy25MODEL1IPCy21.052061.05206-7.74580.073810.082690.10735-7.74580.073810.082690.10735-1-1主分量回主分量回归结果果n n由由输输出出结结果看到在果看到在删删去第三个主分量（去第三个主分量（pcomit=1)pcomit=1)后的主分量回后的主分量回归归方程：方程：n nY=-9.1301+0.07278x1+0.60922x2+0.10626x3;Y=-9.1301+0.07278x1+0.60922x2+0.10626x3;n n该该方程的系数都有意方程的系数都有意义义，且回，且回归归系数的方差膨系数的方差膨胀胀因子均小于因子均小于1.11.1；主分量回；主分量回归归方程的均方根方程的均方根误误差差（_RMSE=0.55)_RMSE=0.55)比普通比普通OLSOLS方程的均方根方程的均方根误误差差（_RMSE=0.48887)_RMSE=0.48887)有所增大但不多。有所增大但不多。Sas 程序程序n ndatadata ex01;ex01;n ninputinput x1 x2 x3 y;x1 x2 x3 y;n nlabellabel x1=x1=国内生国内生产总值产总值;n nlabellabel x2=x2=存存储储量量;n nlabellabel x3=x3=消消费费量量;n nlabellabel y=y=进进口口总额总额;n ncardscards;n n149.3 4.2 108.1 15.9149.3 4.2 108.1 15.9n n161.2 4.1 114.8 16.4161.2 4.1 114.8 16.4n n171.5 3.1 123.2 19.0171.5 3.1 123.2 19.0n n175.5 3.1 126.9 19.1175.5 3.1 126.9 19.1n n180.8 1.1 132.1 18.8180.8 1.1 132.1 18.8n n190.7 2.2 137.7 20.4190.7 2.2 137.7 20.4n n202.1 2.1 146 22.7202.1 2.1 146 22.7n n212.4 5.6 154.1 26.5212.4 5.6 154.1 26.5n n226.1 5.0 162.3 28.1226.1 5.0 162.3 28.1n n231.9 5.1 164.3 27.6 231.9 5.1 164.3 27.6 n n239.0 0.7 167.6 26.3239.0 0.7 167.6 26.3n n;n nrunrun;n nprocproc corrcorr datadata=ex01;=ex01;n nvarvar x1-x3;x1-x3;n nrunrun;n n*岭回岭回归归*;n nprocproc regreg datadata=ex01=ex01 outestoutest=ex012 graphics=ex012 graphics outvifoutvif;n nmodelmodel y=x1-x3/y=x1-x3/ridgeridge=0.00.0 to to 0.10.1 by by 0.010.01;n nplotplot/ridgeplotridgeplot;n nrunrun;n nprocproc printprint datadata=ex012;=ex012;runrun;n n*主分量回主分量回归归法法*;n nprocproc regreg datadata=ex01=ex01 outestoutest=ex103;=ex103;n nmodelmodel y=x1-x3/y=x1-x3/pcomitpcomit=1 1,2 2 outvifoutvif;*pcomit*pcomit表示表示删删去最后面的去最后面的1 1或或2 2个主分量个主分量,用前面用前面m-1m-1或或 m-2m-2各主分量各主分量进进行回行回归归*;n nrunrun;n nprocproc printprint datadata=ex103;=ex103;runrun;Sas 程序程序n n/*theil test*/*theil test*/;n nprocproc regreg datadata=ex01;=ex01;n nequation3:equation3:modelmodel y=x1 x2;y=x1 x2;n nequation2:equation2:modelmodel y=x1 x3;y=x1 x3;n nequation1:equation1:modelmodel y=x2 x3;y=x2 x3;n nrunrun;/*r-/*r-.9473;r3s=0.9828*/.9473;r3s=0.9828*/;n ndatadata theil;theil;n nrsq=rsq=0.99190.9919;r1s=;r1s=0.99130.9913;r2s=;r2s=0.94730.9473;r3s=;r3s=0.98280.9828;n ntheil=rsq-(theil=rsq-(3 3*rsq-*rsq-(r1s+r2s+r3s);(r1s+r2s+r3s);putput theil=;theil=;n nrunrun;n n/*/*辅辅助回助回归检验归检验法法*/;n nprocproc regreg datadata=ex01;=ex01;n nequation3:equation3:modelmodel x3=x1 x2;x3=x1 x2;n nequation2:equation2:modelmodel x2=x1 x3;x2=x1 x3;n nequation1:equation1:modelmodel x1=x2 x3;x1=x2 x3;n nrunrun;n n/*FG test*/*FG test*/;n nprocproc corrcorr datadata=ex01=ex01 outpoutp=corr=corr nosimplenosimple;varvar x1-x3;x1-x3;runrun;n nprocproc printprint datadata=corr;=corr;runrun;n ntitletitle 计计算相关矩算相关矩阵阵的行列式的行列式;n nprocproc imliml;n nR=R=1.0001.000 0.0260.026 0.9970.997,0.0260.026 1 1 0.0360.036,0.91520.9152 0.63060.6306 1 1;n nd=det(R);d=det(R);n nprint d;print d;n nrunrun;/*d=0.081371*/*d=0.081371*/;n ntitletitle 计计算算检验统计检验统计量及其量及其p p值值;n ndatadata fg;fg;n nn=n=1111;p=;p=3 3;d=;d=0.0813710.081371;n nfg=-(n-fg=-(n-1 1-1 1/6 6*(*(2 2*p+*p+5 5)*log(d);df=p(p-)*log(d);df=p(p-1 1)/)/2 2;n np=p=1 1-probchi(fg,df);-probchi(fg,df);n nputput fg=p=;fg=p=;n nrunrun;/*fg=20.488013401/*fg=20.488013401 p=0.0001344625,p=0.0001344625,拒拒绝绝零假零假设设*/;异方差的异方差的检验和和补救救n n n nOLSE unbiased,inefficient;t,F test invalid;forecast accuracy decreased.n nIf the model is well-fitted,there should be no pattern to the residuals plotted against the fitted values.If the variance of the residuals is non-constant,then the residual variance is said to be heteroscedastic.异方差的异方差的检测n nThere are graphical and non-graphical methods for There are graphical and non-graphical methods for detecting heteroscedasticity.A commonly used detecting heteroscedasticity.A commonly used graphical method is to plot the residuals versus fitted graphical method is to plot the residuals versus fitted(predicted)values.(predicted)values.n nExample:grade:educated years;potexp:working Example:grade:educated years;potexp:working years;exp2=potexp2;union:dummy variable.years;exp2=potexp2;union:dummy variable.收入方程回收入方程回归的的结果果n n DependentVariable:LNWAGEDependentVariable:LNWAGEn nAnalysisofVarianceAnalysisofVariancen nSumofMeanSumofMeann nSourceDFSquaresSquareFValuePrFSourceDFSquaresSquareFValuePrFn nModel412.422363.1055914.06.0001Model412.422363.1055914.06|t|VariableDFEstimateErrortValuePr|t|n nIntercept10.595110.283492.100.0384Intercept10.595110.283492.100.0384n nGRADE10.083540.020094.16.0001GRADE10.083540.020094.16FSourceDFSquaresSquareFValuePrFn nModel121.188810.099070.880.5731Model121.188810.099070.880.5731n nError879.830780.11300Error879.830780.11300n nCorrectedTotal9911.01958CorrectedTotal9911.01958n nRootMSE0.33615RootMSE0.33615R-Square 0.1079R-Square 0.1079n nDependentMean0.20989AdjR-Sq-0.0152DependentMean0.20989AdjR-Sq-0.0152n nCoeffVar160.15281CoeffVar160.15281n nParameterStandardParameterStandardn nVariableDFEstimateErrortValuePr|t|VariableDFEstimateErrortValuePr|t|n nIntercept1-0.077670.98580-0.080.9374Intercept1-0.077670.98580-0.080.9374n nGRADE1-0.012200.12502-0.100.9225GRADE1-0.012200.12502-0.100.9225n nPOTEXP10.077840.071881.080.2819POTEXP10.077840.071881.080.2819n nEXP21-0.003990.00409-0.970.3325EXP21-0.003990.00409-0.970.3325n nUNION10.648790.861600.750.4535UNION10.648790.861600.750.4535n ngrade210.002200.004250.520.6065grade210.002200.004250.520.6065n nexp41-3.34378E-70.00000151-0.22exp41-3.34378E-70.00000151-0.220.82560.8256n nexp310.000061700.000141920.43exp310.000061700.000141920.430.66480.6648n ngx210.000116830.000111021.05gx210.000116830.000111021.050.29550.2955n ngp1-0.003750.00494-0.760.4498gp1-0.003750.00494-0.760.4498n ngu1-0.051370.04430-1.160.2494gu1-0.051370.04430-1.160.2494n npu10.001930.060610.030.9746pu10.001930.060610.030.9746n neu1-0.000221850.00126-0.180.8605eu1-0.000221850.00126-0.180.8605n n残差残差项项平方平方对对所有一所有一阶阶,二二阶阶及交叉及交叉项项回回归归.n n1.1.由左由左边边的的结结果可知果可知:n n故同方差的假故同方差的假设设未被拒未被拒绝绝.n n2.Proc reg data=aa;2.Proc reg data=aa;n nModel y=x/Model y=x/specspec;n nRun;Run;n n可得到相同的可得到相同的结结果。果。布布罗施施-帕甘帕甘/戈弗雷戈弗雷检验怀特特检验的特例的特例（1 1）OLSOLS残差残差额额e et t和一个估和一个估计计的干的干扰误扰误差差 n n（2 2）用用OLSOLS将将 对对选选中的解中的解释变释变量量进进行回行回归归，并，并计计算解算解释释平方和平方和(ESS);(ESS);n n(3)(3)在零假在零假设设下，有下，有 n n(4)(4)一个更一个更简单简单且且渐进渐进等价的做法是直接利用残差平方等价的做法是直接利用残差平方对选对选中的中的解解释变释变量量进进行回行回归归.在零假在零假设设(同方差同方差)下下,Dependent Variable:rsqDependent Variable:rsqn n Sum of Mean Sum of MeanSource DF Squares Square F Value PrFSource DF Squares Square F Value PrFModel 12 1.18881 0.09907 0.88 0.5731Model 12 1.18881 0.09907 0.88 0.5731Error 87 9.83078 0.11300Error 87 9.83078 0.11300Corrected Total 99 11.01958 Corrected Total 99 11.01958 Root MSE 0.33615 R-Square 0.1079Root MSE 0.33615 R-Square 0.1079Dependent MeanDependent MeanDependent MeanDependent Mean 0.209890.209890.209890.20989 Adj R-Sq -0.0152 Adj R-Sq -0.0152BPG test results(1)BPG test results(2)n nDependent Variable:rsqadjustDependent Variable:rsqadjustn nAnalysis of VarianceAnalysis of Variancen n Sum of Mean Sum of Meann n Source DF Squares Square F Value Pr F Source DF Squares Square F Value Pr Fn nModel 3 Model 3 10.7041510.7041510.7041510.70415 3.56805 1.43 0.2386 3.56805 1.43 0.2386n nError 96 239.41116 2.49387Error 96 239.41116 2.49387 Corrected Total 99 250.11531 Corrected Total 99 250.11531 Root MSE 1.57920 R-Square 0.0428 Root MSE 1.57920 R-Square 0.0428 Dependent Mean 0.99997 Adj R-Sq 0.0129 Dependent Mean 0.99997 Adj R-Sq 0.0129n nCoeff Var 157.92443Coeff Var 157.92443n nESS=10.70415ESS=10.70415ESS=10.70415ESS=10.70415BPG test results(3)n n*ESS=*ESS=5.355.35 F Source DF Squares Square F Value Pr Fn nModel 3 0.47160 0.15720 1.43 0.2386Model 3 0.47160 0.15720 1.43 0.2386n nError 96 Error 96 10.5479810.5479810.5479810.54798 0.10987 0.10987n nRoot MSE 0.33147 Root MSE 0.33147 R-Square 0.0428R-Square 0.0428R-Square 0.0428R-Square 0.0428戈德菲戈德菲尔德德-匡特匡特(Goldfeld-Quandt)检验n n按potexp的值将数据从小到大进行排列.n n取前后个35个观测值分别回归.c=30;n n回归的主要结果:n nRSS1=6.39573;RSS2=7.2517;RSS2/RSS1=1.13;而 ;该比值不显著,不能拒绝同方差的原假设;n n去掉的中间观测值的个数要适中,否则会降低功效,一般取观测值个数的1/3.补救措施救措施-已知方差的形式已知方差的形式n n1.广义最小二乘法(GLS);n n请参考讲义中的例子;n n2.模型变换法,适用于函数型异方差;已知方差的函数形式;n n3.加权最小二乘法(WLS);实质上是一种模型变换法;具体参见讲义中的例子;n n 采用面板数据,增加信息量.未知方差的形式未知方差的形式n nFurnival(1961)提出了一种拟合指数进行不断的修正,最后找出最佳的权重(使得该指数值最小).处理盲点理盲点-robust regressionn n1.迭代加权最小二乘法(IRLS),Neter提出了2中加权函数,Huber and Bisquare,但是不易操作.SAS v8中常使用Proc NLIN迭代.n n2.非参数回归.Proc Loess.n n3.SAS v9.0中有一个过程Proc robustregn nStata 中有一个比较好的命令:rreg直接进行鲁棒回归(robust),采用迭代过程.序列相关性序列相关性(serial correlation)n n n nOLSE OLSE unbiased,but inefficient and its standard error unbiased,but inefficient and its standard error estimators are invalidestimators are invalid;n nBLUE of the Gauss-Markov Theorem no longer holds.BLUE of the Gauss-Markov Theorem no longer holds.n nThe variance formulas for the least squares estimators The variance formulas for the least squares estimators are incorrect.are incorrect.n nAR,MA,or ARMA forms of serial correlation.AR,MA,or ARMA forms of serial correlation.n nTake the AR(1)for instance:Take the AR(1)for instance:Dw 检验需要注意的地方需要注意的地方n n假定了残差是服从正态分布,而且是同方差;自变量是外生的,如果包含了内生滞后变量,就需要用修正的dh检验(proc autoreg).n n只适用于一阶自相关,对高阶或非线性自相关不适用.n n样本容量至少为15.自相关自相关检验的的标准准n n德宾和沃森根据显著水平,n,k,确定了二个临界值du(上界),dl(下界);然后进行比较;n n(1)ddu,不拒绝零假设;n n(3)dlddu,无结论;n n直观:;d2,负自相关;d=2,无自相关;Eg:Ice cream demand(Hildreth,Lu(1960)n nCons:consumption of ice cream per head(pints);n nIncome:average family income per week($);n nPrice:price of ice cream(per pint);n nTemp:average temperature(in Fahrenheit);n nData:30 four-weekly obs from March 1951 to 11 July 1953;残差的散点残差的散点图回回归结果果n n Parameter EstimatesParameter Estimatesn n Parameter Standard Parameter Standardn nVariable DF Estimate Error t Value Pr|t|Variable DF Estimate Error t Value Pr|t|n nIntercept 1 0.19732 0.27022 0.73 0.4718Intercept 1 0.19732 0.27022 0.73 0.4718n nprice 1 -1.04441 0.83436 -1.25 0.2218price 1 -1.04441 0.83436 -1.25 0.2218n nincome 1 0.00331 0.00117 2.82 0.0090income 1 0.00331 0.00117 2.82 0.0090n ntemp 1 0.00346 0.00044555 7.76 .0001temp 1 0.00346 0.00044555 7.76 .0001n n Durbin-Watson D 1.021Durbin-Watson D 1.021Durbin-Watson D 1.021Durbin-Watson D 1.021n n Number of Observations 30 Number of Observations 30n n 1st Order Autocorrelation 0.330 1st Order Autocorrelation 0.3301.DW testn n查表可得:在0.05的显著水平上,dl=1.21(N=30,k=3);du=1.65;n n直接在回归的语句中加上一个dw选项;n nDw=1.021N*R-square=29*0.15=4.35 ;n n因此拒因此拒绝绝零假零假设设,认为认为有自相关有自相关;且且显显著一著一阶阶正相关正相关;n n Parameter Estimates Parameter Estimatesn n Parameter StandardParameter Standardn nVariable DF Estimate Error t Value Variable