Variance of beta when estimating a censored variable using OLS
I have this statistical problem that has been bugging me for quite a while, and I would be extremly grateful if someone would be kind enough to help me.
Given this set-up:
y= c if y*>c
xiB+e if y*<c
where B= Beta
What is the variance of B when using OLS-estimation?
Note: I've attached a pdf with the equations below, where the numbering is the same, if that is preferred.
What I've been able to derive thus far is the following:
V(B)= sum[(xi^2/((sum(xi^2))^2) * V(yi) (1)
Using the law of total variance V(yi) should be:
V(yi)= V(E[yi|x]) + E(V[yi|x]) (2)
E(V[yi|xi])= V(c) * P(y*>c) + V(xiB+e) * P(y*<c) (3)
= Sigma^2 * P(y*<c) (4)
PROBLEM: Assuming my expression for E(V[yi|xi]) is correct, what does the expression for V(E[yi|x]) look like?