# Understanding how the mean shift outlier model is deleting observation

I am looking at the following notes. I understand the math but I do not understand why is adding $d$ the same as deleting the i-th observation and fitting the usual model with just $X$ as explanatory variables but with one observation less.

Question 1: Although I understand the math, I do not understand the rationale behind how it is deleting the i-th observation. Can someone kindly explain to me please ? Thanks.

Question 2: And what would the matrix $d$ be if I want to delete multiple observations ? Can someone show me an example matrix please ?

Question 3: I would like to know how is the following derived ?$$\hat{\sigma}^2 = \frac{y^TM_x y - \hat{\phi}^2(1-p_i)}{(N-K-1)}$$