线性回归系数最小二乘估计弱相合性的一个结果
A Result Concerning the Weak Consistency of LS Estimates of Linear Regression Coefficients
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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^-\frac2-rr\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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