I'm pondering about two test questions that I wasn't able to answer and I hope that someone can help me with that.
Let us assume a multiple linear regression model $Y=X \beta + \epsilon$ where $\epsilon \sim \mathcal N(0,\sigma^2)$. It is known that the maximum likelihood estimator for $\beta$ is given by $$\hat{\beta} = (X^T X)^{-1} X^T Y.$$
I was able to show that the estimator is unbiased and to calculate its variance as well. Furthermore, as a linear transformation of the normal random variable $\epsilon$ it also follows a normal distribution.
Then there were two questions I couldn't answer:
- How can we use the information about the distribution of $\hat \beta$ rather than just the estimate itself to gain information about the regression coefficients $\beta$?
- Say we add a new input $x$ and use the given model to estimate the new observation $\hat y = x \hat \beta$. Again, I was able to get the distribution of $\hat y$, but I couldn't say what it was good for.
Maybe someone can help me understand?