 Solution

No, not from the stated assumptions alone. The centered slope is a weighted sum,and the weighted errors need not be identically distributed. The ordinary central limit theorem therefore does not apply directly.

For a concrete counterexample, for every $n\ge 3$ chooseThese satisfy all three centring and orthogonality assumptions, and both explanatory columns are nonzero. Let the errors be independent with Rademacher distribution, taking values $\pm 1$ with equal probability. Thenwhich takes values $-1,0,1$ with probabilities $1/4,1/2,1/4$ for every $n$. Increasing the sample size adds no information about this slope, and the standardized estimator does not approach a normal distribution.

A useful additional assumption for asymptotic normality of a weighted regression slope is that no single design value dominates:Under this condition and the given finite error variance, the Lindeberg-Feller central limit theorem yieldsjustifying the approximation $N({\beta }_1,{\sigma }^2/S_{xx})$ for the slope as well.