经济计量学的几种检验多重共线性n .Multicollinearity arises because we have put in too many variables that measure the same thing.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. nMeasure :vif, tol=1/vif,condition index;etc.多重共线性的后果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.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.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).多重共线性的检测方法 (1)样本可决系数法n如果样本的可决系数R-square 比较大，且回归 系数几乎没有统计上的显著性，则可认为存在 多重共线性。nTheil 提出了一个指标：多重共线性效应系数Theil test resultsnSas 结果：n结果表明有多重共线性。多重共线性检测方法 （2）辅助回归检验法n若存在多重共线性，则至少有一个解释变量可精确或 近似地表示为其余皆是变量的线性组合。n相应的检验统计量为：辅助回归检验结果nSas 结果：nKlein经验法则：若存在一个i,使得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。多重共线性检验方法 （3）样本相关系数检验法FG test resultsnfg=20.488013401 p=0.0001344625；n拒绝零假设，认为存在多重共线性。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。多重共线性检验方法： （4）特征值分析法所用的检验统计指标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条件指数: n条件数: ;C20,共线性严重.多重共线性的检验和补救n例一:进口总额和三个自变量之间回归;nSas 结果如下:Pearson Correlation Coefficients, N = 11 Prob |r| under H0: Rho=0n x1 x2 x3nx1 1.00000 0.02585 0.99726nGDP 0.9399 |t| InflationnIntercept 1 -10.12799 1.21216 -8.36 ,方开泰;主分量回归结果Obs _MODEL_ _TYPE_ _DEPVAR_ _PCOMIT_ _RMSE_ Intercept x1 x2 x3 y1 MODEL1 PARMS y 0.48887 -10.1280 -0.05140 0.58695 0.28685 12 MODEL1 IPCVIF y 1 0.25083 1.00085 0.25038 13 MODEL1 IPC y 1 0.55001 -9.1301 0.07278 0.60922 0.10626 1 4 MODEL1 IPCVIF y 2 0.24956 0.00095 0.24971 -1 5 MODEL1 IPC y 2 1.05206 -7.7458 0.07381 0.08269 0.10735 -1主分量回归结果n由输出结果看到在删去第三个主分量（ pcomit=1)后的主分量回归方程：nY=-9.1301+0.07278x1+0.60922x2+0.10626x3;n该方程的系数都有意义，且回归系数的方差膨 胀因子均小于1.1；主分量回归方程的均方根 误差（_RMSE=0.55) 比普通OLS方程的均方根 误差（_RMSE=0.48887) 有所增大但不多。Sas 程序ndata ex01;ninput x1 x2 x3 y;nlabel x1=“国内生产总值“;nlabel x2=“存储量“;nlabel x3=“消费量“;nlabel y=“进口总额“;ncards;n149.3 4.2 108.1 15.9n161.2 4.1 114.8 16.4n171.5 3.1 123.2 19.0n175.5 3.1 126.9 19.1n180.8 1.1 132.1 18.8n190.7 2.2 137.7 20.4n202.1 2.1 146 22.7n212.4 5.6 154.1 26.5n226.1 5.0 162.3 28.1n231.9 5.1 164.3 27.6 n239.0 0.7 167.6 26.3n;nrun;nproc corr data=ex01;nvar x1-x3;nrun;n*岭回归*;nproc reg data=ex01 outest=ex012 graphics outvif;nmodel y=x1-x3/ridge=0.0 to 0.1 by 0.01;nplot/ridgeplot;nrun;nproc print data=ex012;run;n*主分量回归法*;nproc reg data=ex01 outest=ex103;nmodel y=x1-x3/pcomit=1,2 outvif;*pcomit表示删去最后面的1或 2个主分量,用前面m-1或 m-2各主分量进 行回归*;nrun;nproc print data=ex103;run;Sas 程序n/*theil test*/;nproc reg data=ex01;nequation3:model y=x1 x2;nequation2:model y=x1 x3;nequation1:model y=x2 x3;nrun;/*r- .9473;r3s=0.9828*/;ndata theil;nrsq=0.9919;r1s=0.9913;r2s= 0.9473;r3s=0.9828;ntheil=rsq-(3*rsq- (r1s+r2s+r3s);put theil=;nrun;n/*辅助回归检验法*/;nproc reg data=ex01;nequation3:model x3=x1 x2;nequation2:model x2=x1 x3;nequation1:model x1=x2 x3;nrun;n/*FG test*/;nproc corr data=ex01 outp=corr nosimple;var x1-x3;run;nproc print data=corr;run;ntitle “ 计算相关矩阵的行列式“;nproc iml;nR=1.000 0.026 0.997,0.026 1 0.036,0.9152 0.6306 1;nd=det(R);nprint d;nrun;/*d=0.081371*/;ntitle “计算检验统计量及其p值“;ndata fg;nn=11;p=3;d=0.081371;nfg=-(n-1- 1/6*(2*p+5)*log(d);df=p(p- 1)/2;np=1-probchi(fg,df);nput fg= p=;nrun;/*fg=20.488013401 p=0.0001344625,拒绝零假设*/;异方差的检验和补救n nOLSE unbiased,inefficient;t, F test invalid; forecast accuracy decreased.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.“ 异方差的检测nThere are graphical and non-graphical methods for detecting heteroscedasticity. A commonly used graphical method is to plot the residuals versus fitted (predicted) values. nExample :grade:educated years;potexp:working years;exp2=potexp2;union:dummy variable.收入方程回归的结果n Dependent Variable: LNWAGE n Analysis of Variancen Sum of Meann Source DF Squares Square F Value Pr Fn Model 4 12.42236 3.10559 14.06 |t|n Intercept 1 0.59511 0.28349 2.10 0.0384n GRADE 1 0.08354 0.02009 4.16 Fn Model 12 1.18881 0.09907 0.8