Econometrics 7-10

1 Econometrics 第7讲 模型选择：标准与 检验 2011/6/7 2 第7讲模型选择：标准与检验 1. “好的”模型具有的特性 2. 设定误差：类型及后果 3. 设定误差：诊断及补救 2011/6/7 3 1.“好的”模型：标准(Harvey) I. 简约性(parsimony)：简单优于复杂 II. 可识别性(identifiability)：参数估计值存在且唯一 III. 拟合优度(goodness of fit)：比如R2 IV. 理论一致性(theoretical consistency)：符合理论背景 V. 预测能力(predictive power)：比较预测值与实际值 2011/6/7 4 2.模型设定误差(specification errors)：类型  关于解释变量选取的偏误  漏选相关变量  多选无关变量  关于模型函数形式选取的偏误  度量误差 2011/6/7 5 例7-1：U.S进口商品支出(Y)与个人可支配收入(X) 2686.300439.000019872066.600259.40001977 2640.900412.300019862001.000229.90001976 2542.800367.900019851931.700187.90001975 2469.800351.100019841896.900211.80001974 2331.900282.200019831916.300218.20001973 2261.500249.500019821797.400190.70001972 2248.600258.700019811728.400166.20001971 2214.300253.600019801668.100150.90001970 2212.600277.900019791599.800144.60001969 2167.400274.100019781551.300135.70001968 XYobsXYobs 2011/6/7 6 DL=1.100；DU=1.5370.000000Prob(F-statistic) 1.363316Durbin-Watson stat371.3761F-statistic 8.219718Hannan-Quinn criter.-78.90561Log likelihood 8.339921Schwarz criterion3128.836Sum squared resid 8.190561Akaike info criterion13.56647S.E. of regression 85.78792S.D. dependent var0.974992Adjusted R-squared 253.0800Mean dependent var0.977624R-squared 0.0000-5.4316014.270415-23.19519@TREND 0.00008.6803110.0745360.647000X 0.0000-7.602172116.1664-883.1170C Prob.t-StatisticStd. ErrorCoefficient Included observations: 20 Method: Least Squares Dependent Variable: Y 模型1：Y C X T 975.0 )432.5()680.8()602.7( 195.23647.0117.883ˆ 2    R t tXY 2 2011/6/7 7 模型2： Y C X 0.000000Prob(F-statistic)0.595073Durbin-Watson stat 276.0832F-statistic-88.96849Log likelihood 9.196423Schwarz criterion8558.709Sum squared resid 9.096849Akaike info criterion21.80559S.E. of regression 85.78792S.D. dependent var0.935392Adjusted R-squared 253.0800Mean dependent var0.938793R-squared 0.000016.615750.0147590.245231X 0.0000-8.33449531.32660-261.0914C Prob.t-StatisticStd. ErrorCoefficientVariable Included observations: 20 Dependent Variable: Y DL=1.100；DU=1.537 935.0 )616.16()334.8( 245.009.261ˆ 2    R t XY 2011/6/7 8 相关变量的遗漏(omitting relevant variables)及 后果 ttXYModel   210:1 tXYModel   10:2 1100 )ˆ(;)ˆ(.1   EE参数有偏性： 参数非一致性.2 参数估计量方差错误.3 错误随机误差项方差估计量.4 2011/6/7 9 变量遗漏偏差(omitting variables bias)：解释 tXXYModel   22110:1 tXYModel   110:2   2 1 1 1ˆ i ii x yx回归OLS iiii xxy   2211 2011/6/7 10              2 1 1 1 21 21 2 1 1 2 1 21 21 2 1 22111 2 1 1 1 )( ),( )( ˆ i ii i ii i ii i iiii i ii x x XVar XXCov x x x xx x xxx x yx    参数估计量的有偏性/非一致性：   2 1 1 1ˆ i ii x yx 代入 iiii xxy   2211 2011/6/7 11 参数估计量方差错误：说明 由 Y=0+ 1X1+得  21 2 1 )ˆ( ix Var   由 Y=0+1X1+2X2+得      )1()() ˆ( 22 1 2 2 21 2 2 2 1 2 22 1 21xxiiiii i rxxxxx x Var    2011/6/7 12 随机误差项方差估计量的有偏性：说明         22 2 22 110 2 )ˆˆ()ˆˆ)(ˆ(2ˆ)2( )ˆˆˆ( )ˆ()ˆˆ( ˆ)2( iiiiii iiii iiii YYYYYYn YYYY YYXY RSS n        3 2011/6/7 13 比较 984.0 )177.34)(929.24( 017.082.26ˆ 2    R t Xt 57.13ˆ 975.0 )432.5()680.8()602.7( )270.4()075.0()17.116( 195.23647.0117.883ˆ 2       R t std tXY 81.21ˆ 935.0 )616.16()334.8( )015.0()327.31( 245.009.261ˆ 2       R t std XY 2011/6/7 14 过度拟合：包括不相关变量  无关变量误差(irrevelant variables bias)  设定模型时，多选了非必须解释变量  设正确模型 Y=0+ 1X1+ (1)  但却估计了 Y=0+ 1X1+2X2+ (2)  X2：无关变量 2011/6/7 15 Y=0+1X1+2X2+  CLRM假定下：OLS估计量无偏且一致  E(0)=0； E(1)=1； E(2) =0；  正确地估计了2 /F检验有效  但的估计量非有效：方差比真实的要大  导致推断的精度下降 2011/6/7 16 过度拟合：OLS估计量不具有方差最小性 Y=0+ 1X1+  21 2 1 )ˆ( ix Var   Y=0+1X1+2X2+   )1()ˆ( 221 2 1 21xxi rx Var   2011/6/7 17 模型设定偏误的后果 增加共线性可能 扩大，易接受H0置信区间 下降假设检验过程有效性 模型误差项方差2 变大参数估计量标准差 参数一致性 参数无偏性 包括无关变量 (过度拟合) 略去相关变量 (过低拟合) 2011/6/7 18 例7-2：成本Y-产出X数据 10420 9350 8297 7274 6260 5257 4244 3240 2226 1193 XY($) 4 160 200 240 280 320 360 400 440 0 2 4 6 8 10 12 X Y 0.000000Prob(F-statistic) 2.700212Durbin-Watson stat1202.220F-statistic 5.372956Hannan-Quinn criter.-23.52865Log likelihood 5.626764Schwarz criterion64.74382Sum squared resid 5.505730Akaike info criterion3.284911S.E. of regression 65.81363S.D. dependent var0.997509Adjusted R-squared 276.1000Mean dependent var0.998339R-squared 0.000015.896770.0591060.939588X^3 0.0000-13.150050.985665-12.96154X^2 0.000013.283724.77860763.47766X 0.000022.236786.375322141.7667C Prob.t-StatisticStd. ErrorCoefficient Included observations: 10 Dependent Variable: Y 0.000098Prob(F-statistic) 1.038487Durbin-Watson stat45.37496F-statistic 8.970088Hannan-Quinn criter.-42.34834Log likelihood 9.160444Schwarz criterion2791.617Sum squared resid 9.069668Akaike info criterion19.97004S.E. of regression 65.81363S.D. dependent var0.907928Adjusted R-squared 276.1000Mean dependent var0.928389R-squared 0.02222.9245340.8690842.541667X^2 0.4403-0.8180859.809494-8.025000X 0.00009.46803723.48780222.3833C Prob.t-StatisticStd. ErrorCoefficient Included observations: 10 Dependent Variable: Y 0.000000Prob(F-statistic) 2.935953Durbin-Watson stat788.2650F-statistic 5.491928Hannan-Quinn criter.-23.28948Log likelihood 5.809188Schwarz criterion61.71970Sum squared resid 5.657895Akaike info criterion3.513394S.E. of regression 65.81363S.D. dependent var0.997150Adjusted R-squared 276.1000Mean dependent var0.998417R-squared 0.64160.4949630.0273740.013549X^4 0.33791.0594230.6055290.641511X^3 0.0632-2.3789424.531492-10.78016X^2 0.00704.39592813.0839557.51612X 0.000112.6130111.60839146.4167C Prob.t-StatisticStd. ErrorCoefficient Included observations: 10 Dependent Variable: Y 2011/6/7 23 比较 0.64160.4949630.0273740.013549X^4 0.33791.0594230.6055290.641511X^3 0.0632-2.3789424.531492-10.78016X^2 0.00704.39592813.0839557.51612X 0.000112.6130111.60839146.4167C Prob.t-StatisticStd. ErrorCoefficient Dependent Variable: Y 0.000015.896770.0591060.939588X^3 0.0000-13.150050.985665-12.96154X^2 0.000013.283724.77860763.47766X 0.000022.236786.375322141.7667C Prob.t-StatisticStd. ErrorCoefficient 2011/6/7 24 错误的函数形式  错误函数形式偏误wrong functional form bias  eXAXY 21 21 vXXY  22110  5 2011/6/7 25 例7-3：U.S进口商品支出(Y)与个人可支配收入(X) 2686.300439.000019872066.600259.40001977 2640.900412.300019862001.000229.90001976 2542.800367.900019851931.700187.90001975 2469.800351.100019841896.900211.80001974 2331.900282.200019831916.300218.20001973 2261.500249.500019821797.400190.70001972 2248.600258.700019811728.400166.20001971 2214.300253.600019801668.100150.90001970 2212.600277.900019791599.800144.60001969 2167.400274.100019781551.300135.70001968 XYobsXYobs 0.000000Prob(F-statistic) 1.363316Durbin-Watson stat371.3761F-statistic 8.219718Hannan-Quinn criter.-78.90561Log likelihood 8.339921Schwarz criterion3128.836Sum squared resid 8.190561Akaike info criterion13.56647S.E. of regression 85.78792S.D. dependent var0.974992Adjusted R-squared 253.0800Mean dependent var0.977624R-squared 0.0000-5.4316014.270415-23.19519@TREND 0.00008.6803110.0745360.647000X 0.0000-7.602172116.1664-883.1170C Prob.t-StatisticStd. ErrorCoefficient Included observations: 20 Sample: 1968 1987 Method: Least Squares Dependent Variable: Y 0.000000Prob(F-statistic) 1.290975Durbin-Watson stat349.8538F-statistic -2.810411Hannan-Quinn criter.31.39568Log likelihood -2.690208Schwarz criterion0.050706Sum squared resid -2.839568Akaike info criterion0.054614S.E. of regression 0.335428S.D. dependent var0.973490Adjusted R-squared 5.480365Mean dependent var0.976280R-squared 0.0058-3.1540260.016684-0.052622@TREND 0.00006.4623280.6031153.897526LOG(X) 0.0001-5.3461494.447935-23.77933C Prob.t-StatisticStd. ErrorCoefficient Included observations: 20 Sample: 1968 1987 Method: Least Squares Dependent Variable: LOG(Y) 2011/6/7 28 度量误差：影响 误差项与自变量相关被误差项吸收 有偏无偏OLS估计量方差 有偏、不一致无偏OLS估计量 自变量度量误差因变量度量误差 2011/6/7 29 3.模型设定偏误的检验  检验是否含有无关变量  检验是否有相关变量的遗漏  检验是否有函数形式设定偏误 2011/6/7 30 是否存在无关变量：检验  基本思想  如果模型中误选了无关变量，则其系数的真值 应为零  对无关变量系数的显著性进行检验  t检验：检验某个变量是否应包括在模型中  F检验：检验多个变量（受限回归）  慎用逐步回归(stepwise regression)   3322110 XXXY 6 32260.438546.9212341125483827.9504541996 32334.536118222671100603593.7466621995 32690.333802313831095443317.9445101994 33258.231817231331105093151.9456491993 340373030825894.71105602930.2442661992 34186.329389278141123142805.1435291991 33330.42870817819.31134662590.3446241990 32440.52806724448.71122052357.1407551989 31455.72657523944.71101232141.5394081988 308702483620393.71112681999.3402981987 3046722950236561109331930.6391511986 30351.52091322705.31088451775.8379111985 3168519497152641128841739.8407311984 31645.11802216209.31140471659.8387281983 X5X4X3X2X1YYear 劳动力农机成灾面积播种面积化肥粮食总产量年份 例7-4：无关变量检验 2011/6/7 32 参数估计及检验 tXX X XX X XX XX XY     32 5 4 32 4 3 322 32 1 10 lnln )ln(lnln X5X4X3X2X1YYear 劳动力农机成灾面积播种面积化肥粮食总产量年份 0.000000Prob(F-statistic)2.481731Durbin-Watson stat 88.53554F-statistic41.30481Log likelihood -4.958167Schwarz criterion0.002244Sum squared resid -5.186402Akaike info criterion0.015790S.E. of regression 0.083455S.D. dependent var0.964201Adjusted R-squared 10.65680Mean dependent var0.975216R-squared 0.7558-0.3206870.163558-0.052451LOG(X5/(X2-X3)) 0.2956-1.1103860.093783-0.104135LOG(X4/(X2-X3)) 0.00403.8382760.1834740.704222LOG(X2-X3) 0.00075.0910460.0797760.406146LOG(X1/(X2-X3)) 0.09001.8993312.0600723.912759C Prob.t-StatisticStd. ErrorCoefficientVariable Included observations: 14 Method: Least Squares Dependent Variable: LOG(Y) 2011/6/7 34 相关变量的遗漏/函数形式设定偏误：检验  模型判定的准则 1. R2、校正后的R2、AIC准则、SC准则 2. 与预期相比，估计系数的符号 3. 杜宾-瓦尔森d统计量 4. 预测误差  诊断方法  检验1：残差图示法  检验2：回归偏误设定检验 2011/6/7 35 预测误差 相对误差绝对植平均 Mean Absolute Percentage Error 绝对误差平均 Mean Absolute Error 误差均方根Root Mean Squared error    hT Tt tt yyh 1 2)ˆ(1    hT Tt tt yyh 1 |ˆ|1    hT Tt t tt y yy h 1 | ˆ |1 2011/6/7 36 Theil不等系数(Theil inequality coefficient)           nT Tt t nT Tt t nT Tt tt y n y n yy n 1 2 1 2 1 2 1ˆ1 )ˆ(1 7 2011/6/7 37 预测误差 Forecast: YF Actual: Y Forecast sample: 1 10 Included observations: 10 Root Mean Squared Error 16.70813 Mean Absolute Error 15.25333 Mean Abs. Percent Error 5.784881 Theil Inequality Coefficient 0.029538 Bias Proportion 0.000000 Variance Proportion 0.018574 Covariance Proportion 0.981426 Forecast: YF Actual: Y Forecast sample: 1 10 Included observations: 10 Root Mean Squared Error 2.544481 Mean Absolute Error 1.936410 Mean Abs. Percent Error 0.733252 Theil Inequality Coefficient 0.004495 Bias Proportion 0.000000 Variance Proportion 0.000416 Covariance Proportion 0.999584 3 3 2 210 XXXY   2210 XXY   2011/6/7 38 检验1：残差图示法 （a）趋势变化 ：模型 设定时可能遗漏了— —随着时间的推移而 持续上升的变量 （b）循环变化：模型 设定时可能遗漏了— —随着时间的推移而 呈现循环变化的变量 2011/6/7 39 •一元回归模型中，真实模型呈幂函数形式，但 却选取了线性函数进行回归。 2011/6/7 40 -.12 -.08 -.04 .00 .04 .08 .12 68 70 72 74 76 78 80 82 84 86 LOG(Y) Residuals -.15 -.10 -.05 .00 .05 .10 68 70 72 74 76 78 80 82 84 86 LOG(Y) Residuals 例7-3：比较残差图  左图：LOG(Y) = -23.78 + 3.9*LOG(X) - 0.053*@TREND  右图：LOG(Y) = -9.87 + 2.01*LOG(X) 2011/6/7 41 线性模型与对数线性模型：如何选择？  MWD检验(1983)  MacKinnon, James G. & White, Halbert & Davidson, Russell, 1983  H0：Y是X的线性函数  H1：lnY是X或lnX的线性函数 2011/6/7 42 例7-3：U.S进口商品支出(Y)与个人可支配收入(X) 2686.300439.000019872066.600259.40001977 2640.900412.300019862001.000229.90001976 2542.800367.900019851931.700187.90001975 2469.800351.100019841896.900211.80001974 2331.900282.200019831916.300218.20001973 2261.500249.500019821797.400190.70001972 2248.600258.700019811728.400166.20001971 2214.300253.600019801668.100150.90001970 2212.600277.900019791599.800144.60001969 2167.400274.100019781551.300135.70001968 XYobsXYobs 8 2011/6/7 43 MWD检验：思路步骤 iii ii i YYZ YY Y lnˆln.3 lnln.2 ;ˆ.1 1 计算对数差 的估计值到估计对数线性模型，得 估计线性模型，得到 01 1 HZ ZXYOLS 则拒绝的系数是统计显著的，如果 ；和对回归： iii YYZ ˆ)exp(ln.4 2 计算差 12 2lnln HZ ZXXYOLS 绝的系数统计显著，则拒如果 ；和或对回归： 2011/6/7 44 MWD检验及其说明 0.1741-1.4184770.002392-0.003392Z2 0.000020.499260.1011142.072771LOG(X) 0.0000-13.409100.771390-10.34365C Prob.t-StatisticStd. ErrorCoefficientVariable Dependent Variable: LOG(Y) 0.0204-2.556845126.9500-324.5915Z1 0.000017.655100.0149960.264753X 0.0000-9.49339731.90918-302.9265C Prob.t-StatisticStd. ErrorCoefficientVariable Dependent Variable: Y 2011/6/7 45 回归偏误设定检验： RESET检验  RESET检验(regression error specification test)  拉姆齐(Ramsey)于1969年提出  基本思想：  如果事先知道遗漏了哪个变量，只需将此变量 引入模型，估计并检验其参数是否显著不为零 即可；  未知遗漏了哪个变量，寻找一个替代变量 (proxy) Z，来进行上述检验  RESET检验中，采用所设定模型中被解释变量 Y的估计值Ŷ的若干次幂来充当该“替代”变量 2011/6/7 46 RESET检验：变量遗漏 tXYOLS   10.1 估计：先   322110 ˆˆ ˆ.2 YYXY Ye 比如： 定引入的阶数的图形表现的关系，决与通过残差 )1,(~ )1/( )/()(   URU UU RUUR knkkF knRSS kkRSSRSS F 利用F检验/t检验来判断是否增加 “替代”变量  H0：增加的变量系数为0 2011/6/7 47 例7-5：中国商品进口与国内生产总值 2436.195933.32001591.416909.21989 2250.989442.22000552.714928.31988 165782067.41999432.111962.51987 1402.478345.21998429.110202.21986 1423.774462.61997422.58964.41985 1388.367884.61996274.171711984 1320.858478.11995213.95934.51983 1156.146759.41994192.95294.71982 1039.634634.41993220.24862.41981 805.926638.11992200.24517.81980 533.521617.81991156.74038.21979 637.918547.91990108.93624.11978 商品进口MGDP年份商品进口MGDP年份 0.000000Prob(F-statistic)0.691459Durbin-Watson stat 394.3860F-statistic-154.3396Log likelihood 13.12647Schwarz criterion541345.2Sum squared resid 13.02830Akaike info criterion156.8649S.E. of regression 667.4365S.D. dependent var0.944763Adjusted R-squared 826.9542Mean dependent var0.947164R-squared 0.000019.859150.0010260.020381GDP 0.00343.28779846.64491153.3591C Prob.t-StatisticStd. ErrorCoefficientVariable Included observations: 24 Dependent Variable: M =0.05 DL=1.273 DU=1.446 9 2011/6/7 49 残差与预测值：图示 -400 -300 -200 -100 0 100 200 300 400 0 400 800 1,200 1,600 2,000 2,400 MF R E 0.000000Prob(F-statistic)1.445272Durbin-Watson stat 380.9182F-statistic-140.8727Log likelihood 12.26907Schwarz criterion176234.0Sum squared resid 12.07272Akaike info criterion93.87065S.E. of regression 667.4365S.D. dependent var0.980219Adjusted R-squared 826.9542Mean dependent var0.982799R-squared 0.00006.3692521.35E-078.57E-07FITTED^3 0.0000-6.1360330.000455-0.002794FITTED^2 0.00007.8601670.0090550.071175GDP 0.9883-0.01482547.51244-0.704383C Prob.t-StatisticStd. ErrorCoefficientVariable Included observations: 24 Dependent Variable: M Test Equation: 0.000001Probability26.93387Log likelihood ratio 0.000013Probability20.71742F-statistic Ramsey RESET Test: =0.05 DL=1.1011 DU=1.656 0.000000Prob(F-statistic)1.973415Durbin-Watson stat 356.8875F-statistic-137.6365Log likelihood 12.13180Schwarz criterion134576.1Sum squared resid 11.88637Akaike info criterion84.16027S.E. of regression 667.4365S.D. dependent var0.984100Adjusted R-squared 826.9542Mean dependent var0.986865R-squared 0.02542.4251662.61E-106.34E-10FITTED^4 0.1063-1.6954251.19E-06-2.03E-06FITTED^3 0.41450.8342940.0018260.001524FITTED^2 0.30751.0485860.0215950.022645GDP 0.57820.56573343.8788724.82371C Prob.t-StatisticStd. Error Coefficien tVariable Included observations: 24 Dependent Variable: M Test Equation: 0.000000Probability33.40626Log likelihood ratio 0.000006Probability19.14311F-statistic Ramsey RESET Test: 2011/6/7 52 RESET检验：用于诊断  简单易行：无需设定备择模型  但：无助于选择正确模型 1 ECONOMETRICS 第8讲 多重共线性 2011/6/7 2 多元线性回归模型：若干假定 I. 解释变 量 II. 随机误 差项 III. 模型 1. 非随机（与扰动项不相关） 2. 无多重共线性：解释变量不完全线性相关 3. 零期望：E(i)=0 4. 同方差：var (i)= E(i2)=2 5. 序列不相关：cov(i, j)= E(i j)=0 6. 正态性： i~N(0, 2) 7. 参数线性 8. 设定正确 ikikiit XXXY   22110 2011/6/7 3 古典线性回归模型(CLRM)：违背情况 1. 多重共线性：解释变量问题 2. 异方差：误差项问题 3. 序列相关：误差项问题 2011/6/7 4 第8讲 多重共线性 1. 多重共线性：含义 2. 多重共线性：后果 3. 多重共线性：诊断 4. 多重共线性：克服 2011/6/7 5 1. 共线性  完全共线性  近似共线性 2011/6/7 6 多重共线性：概念  模型  Yi=0+1X1i+2X2i++kXki+i i=1,2,…,n  基本假设之一：X之间相互独立  多重共线性(Multicollinearity)  如果两个或多个解释变量之间：线性相关 2 2011/6/7 7 共线性(multi-collinearity)  完全共线性(perfect multicollinearity)  如果存在ci不全为0，使得 c1X1i+c2X2i+…+ckXki=0 i=1, 2, …, n  近似共线性(approximate multicollinearity)  如果c1X1i+c2X2i+…+ckXki +vi = 0 i=1,2,…,n 其中ci不全为0，vi为随机变量 2011/6/7 8 例8-1：对某种商品的需求 278.82801029 281.1282930 284.6284833 285.8286734 289.7288637 290.2290538 292.8292439 293.5294344 294.9296245 297.5298149 X3 每周收益/美元 X2 每周收入/美元 X1 价格/美元 Y 数量 2011/6/7 9 相关系数矩阵 10.98844-0.98844x3 1-1x2 1x1 x3x2x1 2011/6/7 10 模型1  模型1不能进行回归：  X1、 X2存在完全共线性：参数估计量不存在  舍去X2进行回归 iiii XXY   22110)1 0.000000Prob(F-statistic)2.051315Durbin-Watson stat 321.6650F-statistic-13.95996Log likelihood 3.252509Schwarz criterion9.551515Sum squared resid 3.191992Akaike info criterion1.092675S.E. of regression 6.613118S.D. dependent var0.972700Adjusted R-squared 37.80000Mean dependent var0.975733R-squared 0.000066.538110.74643949.66667C 0.0000-17.935020.120300-2.157576X1 Prob.t-StatisticStd. ErrorCoefficientVariable Included observations: 10 Sample: 1 10 Method: Least Squares Dependent Variable: Y 2011/6/7 12 模型2  模型2进行回归  不完全多重共线性  尽管价格和收益并不完全线性相关，但两个变 量之间却存在高度的依存关系 iiii XXY   33110)2 3 0.000002Prob (F-statistic)2.560899Durbin-Watson stat 153.8192F-statistic-13.52557Log likelihood 3.395889Schwarz criterion8.756717Sum squared resid 3.305113Akaike info criterion1.118463S.E. of regression 6.613118S.D. dependent var0.971396Adjusted R-squared 37.80000Mean dependent var0.977752R-squared 0.26531.210747120.0622145.3650C 0.4516-0.7970890.400306-0.319080收益X3 0.0108-3.4443830.812185-2.797475价格X1 Prob.t-StatisticStd. ErrorCoefficientVariable Sample: 1 10 Method: Least Squares Dependent Variable: Y需求 0.000066.538110.74643949.66667C 0.0000-17.935020.120300-2.157576价格X1 Prob.t-StatisticStd. ErrorCoefficientVariable 0.26531.210747120.0622145.3650C 0.4516-0.7970890.400306-0.319080收益X3 0.0108-3.4443830.812185-2.797475价格X1 Prob.t-StatisticStd. ErrorCoefficientVariable 标准差变大 1.092675S.E. of regression 1.118463S.E. of regression 0.972700Adjusted R-squared 0.975733R-squared 0.971396Adjusted R-squared 0.977752R-squared 符号有误 2011/6/7 15 近似多重共线性：特性  OLS估计量：BLUE  程度问题，而非存在与否的问题  在解释变量是非随机的假设条件下出现  样本特性(sample specific)  非总体特征  无需：假设检验总体的共线性 2011/6/7 16 近似多重共线性：现实原因  经济变量相关的共同趋势  时间序列样本：简单线性模型  横截面数据  滞后变量的引入  例如，消费=f(当期收入, 前期收入）  样本资料的限制 2011/6/7 17 2.近似多重共线性：实际后果  OLS估计量方差/标准差变大：  置信区间变宽/估计精度下降/预测能力下降  变量的显著性检验失效  R2值较高，但t值则并不都显著  难以衡量各个解释变量对回归平方和(ESS)或 者R2的贡献  参数估计量经济含义不合理  回归系数符号可能有误 2011/6/7 18 近似共线性：OLS估计量方差  2 2 21 1 1 1)ˆvar( i xr          部分共线性 无线性相关 0; 1 1 0; )ˆvar( 2 2 2 2 2 2 2 2 2 1 11 1 r xxr r x ii i    Y= 0+1X1+2X2+ 的相关系数21: XXr 4 2011/6/7 19 方差膨胀因子Variance Inflation Factor, VIF 21 1 r VIF  10000.999 1 1000.99 500.98 33.330.97 250.96 200.95 100.9 50.8 20.5 10 VIFr 2 多重共线性使参数估计值 的方差增大  2 2 1 1 )ˆvar( i x VIF  存在近似多重共线性 参数估计值：方差与标准差变大 参数t检验更易不能通过，易接受H0 可能将重要的X剔出模型 2011/6/7 21 注意  多重共线性只是样本特性  除非完全共线，其并不违背CLRM假设；  OLS估计量仍BLUE  多重共线性未必必然不好  目的：系数估计/预测？提高拟合优度？ 2011/6/7 22 3.多重共线性：诊断 测度样本多重共线性的程度 综合统计判断法 简单相关系数法 存在与否 逐步回归法 判定系数法 存在范围 2011/6/7 23 判断样本多重共线性是否存在  简单相关系数法：两个解释变量的模型  求X1与X2的简单相关系数r  若|r|接近1，则说明存在较强的多重共线性  综合统计判断法：多个解释变量的模型  若 在OLS法下：R2与F值显著，但t检验不都显 著 2011/6/7 24 多重共线性范围诊断1：判定系数法  辅助回归  Xji=1X1i+2X2i+LXLi  判定系数Rj•2较大，则Xj与其他X存在共线性 )1,2(~ )1/()1( )2/( 2 . 2 .   knkF knR kR F j j j 5 2011/6/7 25 等价的检验  在模型中排除某一个解释变量Xj，估计模型；  如果拟合优度与包含Xj时十分接近，则说明Xj与 其它解释变量之间存在共线性 2011/6/7 26 多重共线性范围诊断2：逐步回归法  逐个引入解释变量，OLS回归；  如果拟合优度变化显著，则说明新引入的变量 是独立解释变量；  否则，新引入的变量与其它变量之间存在共线 性 2011/6/7 27 4、共线性的处理 I. 从模型中删掉不重要的解释变量 II. 获取额外的数据或者新的样本 III. 重新考虑模型 IV. 变量变换 V. 其他补救措施 2011/6/7 28 排除引起共线