Most relaxed assumptions to get consistency of linear regression?

Question

What are the most relaxed assumptions to get consistency of the linear regression estimates with $p$ variables?

The most basic assumptions that I know are in White (1984):

1) The model is correct

2) $X'\epsilon/n = op(1)$, with $\epsilon = Y - X\beta$

3) $X'X/n - M_n= op(1)$, with $M_n = Op(1)$ and uniformly positive definite

Assumption (3) implies that $(X'X/n)^{-1}$ exists asymptotically and is $Op(1)$. Then $$\hat{\beta} - \beta = (X'X/n)^{-1}(X'\epsilon/n) = Op(1)op(1) = op(1) $$

Did anyone since White came up with more general assumptions?

Answer 1

Maybe not more general, but different.

Let $A = [X'X]^{-1}X'$, and assume

$$E[\epsilon_k | X] = 0, k = 1,..., n$$

Then, $\hat{\beta} = \beta + op(1)$ if one of the following hold:

(1) $\epsilon_k$ are iid, and $\max_{i,j} | A_{ij}| = O(n^{-1})$

(2) $\epsilon_k$ are iid with finite variance, and $\max_{i,j} | A_{ij}| = o((n \log \log n)^{-1/2})$

(3) $\epsilon_k$ are iid with finite variance, and $(X'X)^{-1} = o(1)$

(4) $\epsilon_k$ are uniformly integrable, pairwise independent, $\max_{i,j} | A_{ij}| = o(1)$, $\sum_{i,j} | A_{ij}| = O(1)$

Note: Condition 4 is true if the probability of $X'X'$ being invertible tends to one, and $\limsup_{n\to\infty} n [\lambda_{min}(X'X)]^{-1} < \infty$, where $\lambda_{min}(A)$ is the smallest eigenvalue of $A$.

Proofs: Miao and Xu, 2012; Wang & Rao 1984