Let ybe a response variable and xa binary explanatory variable. Consider a sample {(xi,yi) : i= 1,...,n}. Define n0 = #{i: xi = 0}, n1 = #{i: xi = 1}, and 1. Explain why and show that y0 = 1 n0 i:xi=0 yi, y1 = 1 n1 i:xi=1 n0 = n i=1 n i=1 n0 x= , 1 . n yi. xi. (1xi), n1 = n1 x= n How do you interpret x? 2. Show that y0 = n1 0 n (1xi) yi, y1 = n1 1 i=1 n xi yi. i=1 3. Show that the overall sample average of yi can be written as a weighted average: y= x y1 + (1 x) y0. 4. Show that when xi is binary, 1 n i=1 x2 i x2 = x(1 x). n 5. Show that 1 n n i=1 xi yi x y = x(1 x) y1 y0. 2 6. Suppose I'm running a regression of y on x, based on the above, prove that 0 = y0 1 = y1 y0 7. If in addition, suppose yi is also binary. For concreteness, let yi = 1, has a job, 0, does not have a job, and let xi = 1 denote participation in the job training program (and xi = 0 non- participation). Argue that the program and those who did not. 1 is the difference in employment rates between those who participated in