1. (Vaccine and Death Rate) Researchers are interested in examining whether COVID-19 vaccines are e↵ective in reducing death rates among those who got the disease. To do so, researchers collected information from N cities. For each city, researchers observe the death rates because of COVID-19, and the vaccination rates, and estimated the following regression model:

deathi = 0 + 1vaccinei + ui.

deathi is the death rate among those who got COVID-19 (ranges between 0 and 1), and vaccinei is the fraction of individuals in a city who gets at least one COVID-19 vaccine shot (ranges between 0 and 1).

(a) Think the model from a descriptive perspective. How should we interpret 1? (5 points)

(b) Suppose we want to interpret the model causally. Explain the meaning of 1. (5 points)

(c) What assumption do we need to justify the causal interpretation? Can you think of one potential omitted variable that might bias the estimate? Explain why. (10 points)

(d) Take the one omitted variable you come up with in c). Do you think the likely bias is positive or negative due to this omitted variable? Please briefly explain. (5 points)

(e) Suppose researchers get more detailed measure on vaccine. Now they know the fraction of people with no vaccine shot (novac), one vaccine (onevac), two vaccine shots (twovac), and three or more vaccine shots (threevac). Is it okay to add all four variables into a single regression model? Why? (5 points)