A Result Concerning the Weak Consistency of LS Estimates of Linear Regression Coefficients

Suppose in linear model y_i=x'_iβ+e_i, 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,^β_n,is r-th mean consistent hence weakly consistent. In this note it is shown that for any constant sequence c_n such that lim inf（c_nn^-（2-r）/r）= 0, the condition S_n^-1=O（c_n^-1）is no longer sufficient for ^β_n to be weakly consistent.