Problem 2: Consider the standard simple regression model y = 0 + 1x + with assumptions SR. Let di = { 1 if x 0 0 if x lt; 0 .

Problem 2: Consider the standard simple regression model y = β0 + β1x + ϵ with assumptions SR.1-SR.6. Let di = { 1 if x ≥ 0 0 if x < 0 . Define the estimator β¯ 1 = ∑ n i=1 yi(di − ¯d) ∑ n i=1 xi(di − ¯d) . Assume that ∑ n i=1 xi(di − ¯d) ̸= 0. (i) Show that E[β¯ 1] = β1