Inference on Linear Models

Let $(Y, X, U)$ be a random vector where $Y$ and $U$ take values in $\mathbf{R}$ and $X \in \mathbf{R}^{k+1}$.

We already assume that

1. $E[X U]=0$

2. $E\left[X X^{\prime}\right]<\infty$

3. No perfect collinearity in $X$

4. $\operatorname{Var}[X U]<\infty$

Under these assumptions, we establish the asymptotic normality of the OLS estimator $\hat{\beta}$,

$$\sqrt{n}(\hat{\beta}-\beta) \stackrel{d}{\rightarrow} N(0, \mathbb{V})$$

with

$$\mathbb{V}=\left(E\left[X X^{\prime}\right]\right)^{-1} E\left[X X^{\prime} U^2\right]\left(E\left[X X^{\prime}\right]\right)^{-1}$$

We also described a consistent estimator $\hat{\mathbb{V}}_n$ of the limiting variance $\mathbb{V}$. We develop methods for inference under the assumption that $E\left[X X^{\prime} U^2\right]$ is non-singular.