Suppose that yi is the number of people that die swimming in a given beach i. Let zi be the number of people who went swimming in that beach. The true model is given by
yi = β0 + β1zi + εi
Let xi be the temperature at beach i, and Ai be the number of flowers at the trees in a 50-mile radius of beach i. For some α1, α2 > 0, it is true that zi = α1xi and Ai = α2xi. The first relation tells you that the number of people that go swimming in a given beach is a linear function of temperature. The second relation tells you that the number of flowers in trees around the beach is also a linear function of temperature.
You run the regression. yi = γ0 + γ1Ai + ui
expecting to find γ1 = 0. Suppose β1 > 0.
(a) What is γˆ1? Is it, in fact, 0?
(b) Explain what is going on.
