多重共线性案例.doc

多重共线性案例： 变量 Y，X1，X2，X3，X4，X5的数据年YX1X2X3X4X5197498.45560.2153.206.531.231.891975100.70603.11190.009.121.302.031976102.80668.05240.308.101.802.711977133.95715.47301.1210.102.093.001978140.13724.27361.0010.932.393.291979143.11736.13420.0011.853.905.241980146.15748.91491.7612.285.136.831981144.60760.32501.0013.505.478.361982148.94774.92529.2015.296.0910.071983158.55785.30552.7218.107.9712.571984169.68795.50771.1619.6110.1815.121985162.14804.80811.8017.2211.7918.251986170.09814.94988.4318.6011.5420.591987178.69828.731094.6523.5311.6823.37 资料来源：天津统计年鉴1988.用1974-1987年数据建立天津市粮食需求模型如下， Y = -3.49 + 0.13 X1 + 0.07 X2 + 2.67 X3 + 3.44 X4 4.49 X5 (-0.11) (2.12) (1.95) (2.13) (1.41) (-2.03) R2 = 0.97, F = 52.59, T = 14, t0.05(8) = 2.31, (1974-1987)其中Y：粮食销售量（万吨 / 年），X1：市常住人口数（万人），X2：人均收入（元 / 年），X3：肉销售量（万吨 / 年），X4：蛋销售量（万吨 / 年），X5：鱼虾销售量（万吨 / 年）。一、多重共线性的检验R2 = 0.97，而每个回归参数的t检验在统计上都不显著，这说明模型中存在严重的多重共线性。解释变量间的简单相关系数矩阵为：YX1X2X3X4X5Y1.0000X10.96171.0000X20.96970.86661.0000X30.92880.88230.94591.0000X40.89220.85240.96480.94051.0000X50.86550.82130.98250.94840.98201.0000显然两两简单相关系数均很高，X2与X5、X4与X5之间的系数达到了98以上，比回归方程的样本可决系数还要高，因此可以肯定模型存在严重的多重共线性。二、用逐步回归法检验和克服多重共线性 1、用每个解释变量分别对被解释变量做简单回归，从而决定解释变量的重要程度，为解释变量排序。 = -90.9 + 0.32 X1 (12) R2 = 0.92, F = 147.6, T = 14, (1974-1987) = 99.6 + 0.08 X2 (7.6) R2 = 0.82, F = 57.6, T = 14, (1974-1987) = 74.6 + 4.89 X3 (8.6) R2 = 0.86, F = 75.4, T = 14, (1974-1987) = 108.8 + 5.74 X4 (6.8) R2 = 0.79, F = 46.8, T = 14, (1974-1987) = 113.4 + 3.08 X2 (6.0) R2 = 0.75, F = 36.1, T = 14, (1974-1987)解释变量的重要程度依次为X1, X3, X2, X4, X5 。2、以Y = - 90.9 + 0.32 X1为基础，依次引入X3, X2, X4, X5 。（1）首先把X3引入模型， = -39.78 + 0.21 X1 + 1.91 X3 (4.6) (2.6) R2 = 0.95, F = 114, T = 14, (1974-1987)因为R 2从0.92增至0.95，且X3的系数通过显著性检验，所以在模型中保留X3。（2）再把X2引入模型， = 1.45 + 0.21 X1 + 0.009 X2 + 1.45 X3 (4.3) (0.5) (1.2) R2 = 0.95, F = 70.8, T = 14, (1974-1987)因为X2的引入没有使R 2得到改善，同时还使各回归系数的t值下降，所以应剔除X2。 （3）把X4引入模型， = -37.98 + 0.21 X1 + 1.75 X3 + 0.23 X4 (4.4) (1.5) (0.18) R2 = 0.95, F = 69, T = 14, (1974-1987)同理，应剔除X4。 （4）把X5引入模型， = -40.8 + 0.21 X1 + 2.14 X3 0.16 X5 (4.4) (1.6) (-0.2) R2 = 0.95, F = 69, T = 14, (1974-1987)同理，应剔除X5。最后确定的模型是 = 0.14 X1 + 2.8 X3 (14.0) (5.8) R2 = 0.94, F = 119.8, T = 14, (1974-