I'm currently studying multiple linear regression on my own, and I've come here with three questions. Photos are included, but I'll try my best to explain in LaTeX code as much as I possible. Please consider the photos for references.
Let $Y=(y_1, y_2, ... , y_n)^{T}$ the response variable and $X$ be the n $\times$ p matrix. Using the parameter vector $\beta = (\beta_1, \beta_2, ... , \beta_p)^{T}$, the following equation can be yielded: $Y = X\beta$ $+$ (n $\times$ 1 noise vector).
Question 1 https://i.stack.imgur.com/50IW4.jpg
So, I was in the middle of solving $\min_\beta ||Y - X\beta||^{2}$
I understand that the function is a quadratic function, but as you all know, to find the minimum value by finding the point where the derivative is 0, it must be ensured that the function is smooth and convex. However, I'm not sure how to prove that this function is smooth and convex.
Question 2 https://i.stack.imgur.com/8Ze2D.jpg
Let $\hat{\sigma}^{2}=\frac{1}{n-p} || Y - X\hat{\beta} || ^{2}$. I want to prove the unbiasedness i.e. $E[\hat{\sigma}^{2}]=\sigma^{2}$, but as you can see from my note, $E[\hat{\sigma}^{2}]$ is not equal to $\sigma^{2}$. Where did I go wrong?
Question 3 https://i.stack.imgur.com/BAVfT.jpg
I obtained a confidence interval for $\hat{\beta_j}$. $\hat{\beta_j} = \pm c \sqrt{\hat{Var} (\hat{\beta_j})}$. I'm not familiar with this subject at all, but that square root value is the standard deviation of the residual, not the error, right?
If you could take the time to answer, I would really appreciate it.