高级计量经济学课后习题参考答案

于，所以拒绝H:??0?j?2,3?；但??对应的t统计量的绝对值小于，在的显着性水平上不能拒绝H:??0的原假设。 （3）R?0.877022

（4）I在2003年的预测值为，置信度为95%的预测区间为（，） sf?105.9276

0j1012i

设一元线性模型为r (i=1,2,…..,n) 其回归方程为Y???????X，证明残差满足下式

23.1?i?Yi?Y??SXY(Xi?X)SXX

231如果把变量X,X分别对X进行一元线性回归，由两者残差定义的X，X关于X的偏相关系数r满足：

r?rr r?23123.123213123.1(1?r212)(1?r312)解答：

（1）对一元线性模型，由OLS可得

?X??Y????????X?X??Y?Y??SS??X?X?ii2iXYXX

所以，

???Y???i?Yi?Y??Xi??ii?SXY?S?Yi??Y?X??XiXYSXX?SXX?S?Yi?Y?XY?Xi?X?SXX

（2）偏相关系数是指在剔除其他解释变量的影响后，一个解释变量对被解释变量的影响。不妨假设X，X对X进行一元线性回归得到的回归方程分别为： ????X?eX??，X??????X?e

则，e,e就分别表示X，X在剔除X影响后的值。

所以X，X关于X的偏相关系数就是指e,e的简单相关系数。 所以，

?e?e??e?e? ?r?231212113121212231231121i12i223.1??e1i?e1???e2i?e2?1222因为e?0,er21?1i1i?0??X??，?22i221i?X1??X2i?X2?1i??X?X1???X?X??X?X???X?X???X?X?1212i21i,r31???X?X???X?X??X?X? ??X?X???X?X?1i11i13i3221i13i32??X，???21i?X1??X3i?X3?2

令X则???X1?x1i,X2i?X2?x2i,X3i?X3?x3i

2?r21?x?x2222i，??2?r311i?x?x223i ，所以e1i1i注意到X?1???2X1,X3???1???2X1???2x1i,e2i?x3i???2x1i?x2i??所以r23.1???e?e??e?e???e?e???e?e?1i12i221i12i22??ee?e?e1i2i21i2 2i其中，

?ee??x1i2i?2x1i??x3i???2x1i????x2i???2?x1ix3i???2??2?x1i2x???2?x2ix1i??22223i2i2i3i23122i1i2121i3i2123121i2i3i?x?xx?r?x?xx?r?xr?x?x??xx?r?x?x?x?x?xr?x?x?r?xr?x?x?r?r?x?x?r?x?x?r?x?x?rr?x?x?rr?x?x?rr?x?x?r?x?x?rr?x?x??r?rr??x?x2i3i1i1i1i1i22223i222i22232i3i31221222i1i212313i1i1i1i22222222223232i2i3i3i312131212i2i3i3i2131233i2i21313i3i2i2231212i2131r?x?x23i22i2同理可得：

?e?e21i2??1?r212??x2i2??1?r3123i2i ??x

2所以

r23.1??r23?r31r21??x2i2?x3i2?1?r??x?1?r??x22122i2312?r23?r31r213i?1?r??1?r?221231

考虑下面两个模型： Ⅰ：Y????X?L??X?L??X?? Ⅱ：Y?X??????X?L???X?L???X?? （1） 证明???????1,??????,j?1,2,L,l?1,l?1,Lk

（2） 证明模型Ⅰ和Ⅱ的最小二乘残差相等 （3） 研究两个模型的可决系数之间的大小

i122illikkiiili122illikkiilljj关系 解答： （

1）

??Xl1???1???1????1??????????????X222l2?,????,?????,????,X???l??M??M??M??M??????????????X?k??k??n???ln?设

?1,X21,L,Xk1?Y1????Y1,X22,L,Xk22Y???,X???LLLLLL?M????Y?n??1,X2n,L,Xkn则模型Ⅰ的矩阵形式为：Y?X??? 模型Ⅱ的矩阵形式为：Y?X?X????

取e??0,L,0,1,0,L,0??，其中1为e的第l个分量 则X?Xe

令Z?Y?X?Y?Xe，则模型Ⅱ又可表示为Z?X???? 又OLS得知，????X?X?X?Y，?????X?X?X?Z 将Z?Y?X?Y?Xe代入可得：

lllllll?1?1ll????X?X?X?Z??X?X?X??Y?Xe??l?1?1??X?X?X?Y??X?X?X?Xel??e??l?1?1

即

??????????0???1?11?????????MMM?????M???????????????????1???1?1??l????l??M??M??M??M????????????0??k????k?????k???????

（2）由上述计算可得：

????Z?X?e??Z?Z??e??Y?Xel??X?l??Y?Y??Y?X??e??