Some Remark about Consistency Problem of Parameter Estimation

Abstract

Let $Y_{i} = {x}^{\prime}_{i}\beta + e_{i}, 1 \leq i \leq n, n \geqslant 1$, be a linear regression model. Denote by λ _n and μ _n the smallest and largest eigenvalues of $\sum\limits^{n}_{i=1} x_{i}x^{\prime}_{i}$. Assume that the random errors e1, e2, ⋯ are iid, Ee ₁ = 0 and E ∣ e ₁ ∣ < ∞. Under the restriction that μ _n  = O(λ _n ), this paper obtains the necessary and sufficient condition for the LS estimate of β to be strongly consistent.