?E(??2??2(ui?u)2)?2E?????(ui?u)xi?
?A?B?C
A?其中：
?xi2?2?xi22??2
2B?E?2ui2?nEu?n??nE2(n1?ui)
2
1?n??E(n?ui2???uiuj)i?j1?n?2?(n?2)?(n?1)?2 n?C?2E????xiui???ux?u?xi?2?ii?xi??
??2E?????xiui?2x?ixi22??2??2)2??2E(????xi2
?2?2??xi2
?2?2
故有：E?ei2?(n?1)?2
2??E??ei2??即有：
??n?2???， i2令
??2??en?2，则问题得证。
关于?ei2的计算：
?ei2??yi2???22?xi2??yi2???2?xiyi
关于R2?R2的证明：
R2?1??1?R2?n?1n?k?1?a??1?R2?，其中：当 k?1?a?1
R2?1??1?R2??n?1n?1?1??1?R2??R2 当k?1?a?1，当0?R2?1时，有：
R2?R2?R2??1??1?R2??a?
?R2?1?a?aR2
?a?1?R2?a?1? ??a?1??1?R2??0
a?1。
?R2?R2 Q.E.D.
关于R2可能小于0的证明。 设：Yt??2Xt?ut 则有：
J?min?e2t?min?2?2???J?0?2那么 ??
2??Yt??2Xt??
?2XtXt??Xtet?0 ??2?Yt???J?0?1但：?et?0，因为没有??存在。
??同时，还有：
?2X?e Y???2Xt?Y?et Yt?Y???2Xt???2X?e?et ???2?Xt?X???et?e? ??TSS?222??Y?Y?Y?nY?t?t
2??2?Xt?X???et?e?
?? ??? ??2??X?X??2??????e?e?2?2??Xt?X??et?e? ?2t?t其中：
??Xt?X??et?e???Xt?et?e??X??et?e?
???Xtet?e?Xt?0
n1??e?e?e?ne?e?n?t?t?t?et?0，
和 ?Xtet?0
???Xt?X??et?e???nXe
则：
222?2?2nXe ????TSS??X?X?e?e?2??t?t222222???2nXe ??X?n?X?e?ne?2??t 2?t2222222?????X?e?ne?2?nXe?n??t 2?t22X
222222?2????X?e?n?X?2?Xe?e?t?t 22??考虑到：
222??2?2Xe?e2 nY?n?2X?e?n?X?2?22222???Y??X?e??X?2?Xe?e?t?2tt?2tt?t 2?t2??????222???X?e?t 2?t 若定义
TSS??2?2Yt2?nY2???Xt2??22?2?2Xe?e2et2?n?X?2?
??2?2RSS?TSS???Xt2
??et2
222?2?2Xe?e2???2RSS?TSS?n?X?2????Xt2
?2?1???n???2?n ??2??2?Xe?e????2Xt??2?22???2?Xt2
??2?n?2??Xt??Xe?e2???2?n2?22???Xt2
?2?n?2????Xt2?t?s????Xe?e2???2XtXs??n2?22?????Xt2
?2??n?1??2??2Xt2?n?2t?s???Xe?e2XtXs?n2?2??
可能小于0。 参考书：
Dennis J. Aigner Basic Econometrics, Prentice-Hall, Englewood Cliffs, N. J. 1971,pp85-88