You are working with a non-governmental organization (NGO) that provides free laptops to elementary school students in rural Malawi. You would like to estimate the impact of the laptops on student test scores (on an end-ofyear national exam). The NGO is considering running a pilot in a single village. The village comprises 80 percent poor households and 20 percent wealthier households. All wealthy households purchase laptops for their children (without any help from the NGO). Poor households cannot afford to buy textbooks for their children; however, one quarter of all poor households have secondhand laptops that have been given to them by (wealthy) neighbors. Suppose the true relationship between household wealth, laptop ownership, and student test scores is given by: score\": =a+b-W,:+c-L.: where scored is the test score of student i, W,; is a dummy variable equal to one if student i comes from a wealthy household, and L; is a dummy variable equal to one if student i owns a laptop (prior to the program). 1. What proportion of those with a laptop are wealthy? Suggestion: begin by drawing the following mama: on the board. not wealthy wealthy laptop no laptop Knowing the percentages in each cell, we can calculate the proportion of those who have a laptop who are wealthy. E[Willi = 1] = 2. What proportion of those without a laptop are wealthy? E [WilLi = 0] = 3. If a = 40, b = 20, and c = 20, what is the naive cross-sectional (i.e. difference in means) estimate of the impact of laptops on test scores? diff = E [score | Li = 1] - E [score; ]Li = 0] = a+ b . E[WilLi = 1] + c . E[Li|Li = 1] -(a+ b . E[Wi|Li = 0] + c . E[LilLi = 0])4. The naive cross-sectional estimate conflates the impact of laptops with selection bias (because households with textbooks are more likely to be wealthy). How much of this difference is due to selection bias? The "treatment" in this setting is having a laptop. Because those with laptops are, on average, wealthier than those without laptops, they'd do better on tests even if they didn't have laptops. Specifically, the selection bias term is given by bias = a + b . E [WilLi = 1] - (a + b . E [WilLi = 0]) =