Suppose gender and which place a person is born are both fully random and cannot be manipulated. For simplicity, suppose there are only two genders and we only consider people who are born in the north and people who are born in the south. Let FE {0, 1} indicate gender where F = 1 if and only if i is female. Similar, let's use NE {0, 1} to denote birthplace where N = 1 if and only if i was born in the north. Let Y be income. i) Let D = 1 if (F, N) = (1, 1), D = 1 if (F, N) = (1,0), D3 = 1 if (F, N) = (0, 1) and D = 1 if (F, N) = (0,0). In Lecture 5, we know that it is without loss of generality to write Y = 71D1 +72D2 + Y3D3 + 74D4+u with E(u D1, D2, D3, D4) = 0. Now verify that D = F x N, D = F x (1-N), D3 = (1-F) x N and D (1-F) x (1- N). ii) Given i), rewrite the regression in i) by replacing D to D4 with expressions of F and N. iii) What is the expected income for females born in the north? Female born in the south? Male in the north? Male in the south? iv) What is the difference in expected income between female and male who are both born in the north? How about the female-male difference for those who are both born in the south? v)What's the difference of the two difference in iv)? (This is called difference-in-difference.) vi) Can we consistently estimate the difference in v) by OLS?