econometric: the simple regression model

Could you do step by step? please do V VI and VII

13 Let y be any response variable and x a binary explanatory variable. Let {(x, y): i = 1,..., n} be a sample of size n. Let no be the number of observations with = Xi 0 and n, the number of observations with x, 1. Let yo be the average of the y, with x; y; with x, = 0 and y, the average of the y, with x; = 1. Explain why we can write - (i) x = = no (1 x,), n, = Ex. Show that n/n and (1-x) = no/n. How do you interpret ? (ii) Argue that yo = no ' (1 x)y, y = n; ' 2x,y, Show that the average of y, in the entire sample, y, can be written as a weighted average: == - [Hint: Write y, = (1 x;)y; + x,y,.] (iv) Show that when x, is binary, [Hint: When x, is binary, x = x;.] y = (1-x)yo + x. nx (x) = x(1 x). i=1 - (v) Show that - n xy; xy = x(1 x) (y yo). - i=1 (vi) Use parts (iv) and (v) to obtain (2.74). (vii) Derive equation (2.73).