1. Create an example like Example 16.2 in which α > 0 and θ 0.

2. Prove Theorem 16.4.

3. Suppose you are given data (X1, Y1),..., (Xn, Yn) from an observational study, where Xi ∈ {0, 1} and Yi ∈ {0, 1}. Although it is not possible to estimate the causal e?ect θ, it is possible to put bounds on θ. Find upper and lower bounds on θ that can be consistently estimated from the data. Show that the bounds have width 1.

Hint: Note that E(C1) = E(C1|X = 1)P(X = 1) + E(C1|X = 0)P(X = 0).

4. Suppose that X ∈ R and that, for each subject i, Ci(x) = β1ix. Each subject has their own slope β1i. Construct a joint distribution on (β1,X) such that P(β1 > 0) = 1 but E(Y |X = x) is a decreasing function of x, where Y = C(X). Interpret.