Lo peenaaay 7. Consider the following cross-tab The cells of the table record the sample averages of the outcome variable . Note that 3 of [6 ECON20222 the proportion of z = () individuals who are = = 1 is denoted p, so that %o = (1 p)oo + pifnn and is the overall sample average for all z = 0 individuals. Similarly, 7= (1 @) + qin is the overall sample average for all z = 1 individuals and where is the proportion of z = 1 individuals who are z = 1. In the population, the Wald/IV estimator is given by E(ylz = 1) E(y|= = 0) E(z|z = 1) E(z|z =0)" (a) (3 points) Using the analogy principle, write down an expression for the Wald/IV estimator in the sample. [Write down a formula in terms of the notation already defined.] (b) (6 points) In a treatment design, = is a dummy that records whether the individual is treated and z is a dummy that records whether the individual is eligible for treatment. Suppose being ineligible absolutely rules being treated. How does the Wald estimator change in the population and the sample, if at all? (c) (3 points) It is well-known that the Wald/IV estimator is the same as regressing y on using z as an instrument for z. Instead of using an IV estimator, consider the regression y on z using OLS. Write down an expression for the estimator in the population