摘要:
假定线性回归模型的误差列为i.i.d.有r阶矩, 1≤r＜2 [1]中证明:若$S_n^{-1}=\left(\sum_{i=1}^n x_i x_i^{\prime}\right)^{-1}=O\left(n^{-(2-r) / r}\right)$,则β的最小二乘估计^βn为r阶矩相合，因而也为弱相合, 本文证明了:这个阶不能有任何改进;对任何常数列｛cn｝,若$\liminf _{n \rightarrow \infty}\left(c_n n^{-(2-r) / r}\right)=0$,,则条件Sn-1＝O(cn-1)对^βn为弱相合不再是充分的。
Abstract:
Suppose in linear model yi=x'iβ+ei, 1≤i≤n, e1,e2,…are i.i.d.,Ee1=0,0＜E|e1| r＜∞,1≤r＜2. It is shown in [1] that if $S_n^{-1} \triangleq\left(\sum_{i=1}^n x_i, x_i^{\prime}\right)^{-1}=O\left(n^{-\frac{2-r}{r}}\right)$, then the LS estimate of β,^βn,is r-th mean consistent hence weakly consistent. In this note it is shown that for any constant sequence {cn} such that lim inf（cnn-（2-r）/r）= 0, the condition Sn-1=O（cn-1）is no longer sufficient for ^βn to be weakly consistent.
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