IV. Difference-in-Differences (18 points total) (i) (2 points) Y}, is the average math test scores for school 3' in year t, Dit equals 1 if schoolt' has a free-lunch program in year t (zero otherwise). There are two groups: (Group A that implemented the free-lunch program in the second year but did not have it in the first year; Group 13 never implemented the free lunch program. (ii) (1 point) The differencein-differences (DID) estimate equals 11.4. (iii) (4 points) Difference-indifference figure similar to lecture notes. (iv) (3 points) The assumption required is the parallel trends assumption, which is stated mathematically as ELK\"; Y0,\" D11 = 0,1912 = 1) = Ell/,2 K1 D11 = 0, 9,2 = U], i.e. that Group A's counterfactual trend in the absence of the free-lunch program equals the average change in Group B's average math test scores. Under this assumption, the counterfactual for Group A in the second year would have been 92.8. (v) (2 points) In the multiple period, multiple group case, it is simplest to use the model with individual and year fixed effects K, ,5 Di, | or, | A, a\". Assuming strict exogeneity, this model is consistent with parallel trends. (vi) (3 points) To test the parallel-trends assumption, we would need to collect math test score data for the schools in the sample for multiple years before the free-lunch program was introduced. Comparing the trends in the average math test scores across the treated and untreated schools would allow us to check the parallel-trends assumption, i.e. in the absence of the treatment, average test scores of the schools should follow the same trend whether they ended up implementing the free-lunch program or not. (vii) (3 points) This criticism is valid if the confounding factor are time-varying school-level factors (1V ,,). However, it would not be valid if the confounding factors the specialist is referring to are time-invariant (W3)