Timeline for The correct condition for OLS estimates to be unbiased?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13, 2023 at 2:22 | comment | added | Thomas Lumley | Condition on $X$. It's unbiased for every fixed $X$ and therefore it's unbiased for random $X$. | |
| Sep 13, 2023 at 0:51 | comment | added | Kay99 | Thanks a gain Thomas! I don't quite get how to derive assumption 3 when $\mathbf{X}$ is random. As $(\mathbf{X}^T \cdot \mathbf{X})^{-1}$ in $\mathbb{E}[(\mathbf{X}^T \cdot \mathbf{X})^{-1} \cdot \mathbf{X}^T \cdot\vec{\epsilon}]$ is also random, it seems we cannot take it out directly. How can we prove that $\mathbb{E}(\mathbf{X}^T \cdot \vec{\epsilon}) = \vec{0}$ will satisfy the unbiasness? | |
| Sep 12, 2023 at 22:18 | vote | accept | Kay99 | ||
| Sep 12, 2023 at 22:04 | comment | added | Thomas Lumley | No, neither assumption 1 nor assumption 2 is necessary: assumption 3 is weaker, and is necessary. | |
| Sep 12, 2023 at 19:34 | comment | added | Kay99 | Thx! This makes so much sense. But if we assume $\mathbf{X}$ to be random too: $$ \begin{align*} \mathbb{E}(\hat{\vec{\beta}}) = \vec{\beta} + \mathbb{E}_{X}[(\mathbf{X}^T \cdot \mathbf{X})^{-1} \cdot \mathbf{X}^T \cdot\mathbb{E}_{\vec{\epsilon}|X}(\vec{\epsilon}|\mathbf{X})] \end{align*} $$ When $\mathbb{E}_{\vec{\epsilon}|X}(\vec{\epsilon}|\mathbf{X}) = 0$, it is indeed unbiased (sufficient). But this isn't a necessary condition (non-zero inner expectations can cancel out in the outer one), right? | |
| Sep 11, 2023 at 5:53 | history | answered | Thomas Lumley | CC BY-SA 4.0 |