5. Now suppose you assign tutors randomly to students, but you cannot force anyone in the treatment group to actually get treated, i.e. tutored. You also cannot block any student in the control group from getting a tutor privately. The following is common terminology used by researchers: Always-takers units who would always receive the treatment, regardless of whether they are assigned to the treatment group. Compliers: units who would receive the treatment if they are assigned to the treatment group, but would not otherwise. Never-takers units who would not receive the treatment regardless of the assignment. In particular, suppose that anyone who has a potential score of 60 without tutoring and a delta of at least 12 always gets the treatment (always-takers). Additionally, anyone in the treatment group who gets a score of at least 30 without tutoring and a delta of at least 6 also gets the treatment (compliers). (a) Replace y to reflect who was treated and who wasn't. Regress y on whether someone is treated. Do you expect this coefficient to be biased? Why? (b) Run the so-called "reduced-form regression" to obtain the effect of being assigned to the treatment group on the test score. 3 (c) Run a regression to obtain the take-up rate, i.e. the effect of being assigned to the treatment group on actually being treated. (d) Now calculate the treatment effect as a function of the coefficients above. (e) Run the following regression, which is called an instrumental variables regression: ivregress 2sls y (treated=treatment_group) Is the coefficient identical to the one you calculated? Note that this coeffient is called a Local Average Treatment, Effect (LATE): it captures the causal effect of treatment on compliers only, i.e. those induced to take the treatment due to assignment to the treatment group. (f) Summarize delta for everyone who was treated. Then, summarize delta for compliers only. Why do the two differ? Which one is equivalent to LATE, defined above? Do you think LATE is an interesting parameter, from a policy perspective? Explain