Content Assessment

You may use any notes, lectures, or your textbook. Do not work with anyone else on this assessment.

1. Open Rstudio Server Pro
2. Create a new project
3. Click File > New File > R Markdown
4. Delete everything except for

---
title: "Untitled"
output: html_document
---

5. Give your work a sensible title
6. When you’ve finished, upload the .html file to Canvas

• Show that

\[\textrm{var}(\epsilon')=\sigma^2\mathbf{I}\]

• Minimize \((y - \mathbf{X}\beta)^T\Sigma^{-1}(y-\mathbf{X}\beta)\) with respect to \(\beta\) to calculate the estimate for \(\hat\beta\)

• Using the estimate for \(\hat\beta\) that you previously calculated, take the expectation of \(\hat\beta\). Is this an unbiased estimate for \(\beta\)?

• Show that

\[\textrm{var}(\hat\beta)=(\mathbf{X}^T\Sigma^{-1}\mathbf{X})^{-1}\sigma^2\]