For a child i living in a particular a school district, let voucheri be a dummy variable equal to one if a child is selected to participate in a school voucher program, and let scorei be that child's score on a subsequent standardized exam. Suppose that the participation variable, voucheri, is completely randomized in the sense that it is independent of both observed and unobserved factors that can affect the test score.
(i) If you run a simple regression scorei on voucheri using a random sample of size n, does the OLS estimator provide an unbiased estimator of the effect of the voucher program?
(ii) Suppose you can collect additional background information, such as family income, family structure (e.g., whether the child lives with both parents), and parents' education levels. Do you need to control for these factors to obtain an unbiased estimator of the effects of the voucher program? Explain.
(iii) Why should you include the family background variables in the regression? Is there a situation in which you would not include the background variables?