Problem 1: A government is considering making alcohol consumption illegal as part of a public health campaign. Let's think of making alcohol illegal as the treatment T. Write T = 1 if the government makes alcohol illegal and T = 0 if the government leaves alcohol legal.
We will think of a binary outcome for each person: either they drink alcohol or they do not. If person i drinks at treatment status T, we write her potential outcome as YTi = 1, and if she doesn't drink, we write it as Y Ti = 0. Suppose the society is made up of three groups: the always drinkers, the legal drinkers, and the never drinkers. The always drinkers will drink whether or not alcohol is legal. The legal drinkers will drink if and only if alcohol is legal. The never drinkers won't drink whether or not alcohol is legal.
(a) Write down, in potential outcomes notation and as a number (0 or 1), each of the two potential outcomes for each of the three groups.
(b) Write down, in both potential outcomes notation and as a number (0 or 1), the causal effect of making alcohol illegal on drinking for each of the three groups.
(c) Is there an effect, on average, of banning alcohol in this society?
(d) Suppose you are out to lunch with some friends and one of them says, "My uncle lives in a place where they banned alcohol and all of his friends kept drinking. So I don't think the ban does anything." Explain, in terms of our example, why this isn't a convincing argument.
Problem 2: The Republican National Committee (RNC) has hired three consultants and asked them to figure out the cause of their loss in the 2020 presidential election. The first consultant says that they didn't do enough television advertising. The second consultant reports that they should have encouraged more of their supporters to vote rather than criticizing voting by mail. The third consultant concludes that Donald Trump should have done a better job responding to the COVID-19 pandemic and should have shown more compassion on the campaign trail. Confused by the apparently conflicting information, the RNC hires you, a quantitative analyst, to adjudicate between these three possibilities. What would you tell them? How would you proceed?