Take-home examination for Econ 140A, Summer 2020, Week 3
• Your answers should be submitted in a single pdf document on GauchoSpace.
• You may either type or handwrite your answers, or some combination if you like. You
can take pictures of your handwritten responses, then include them in the single pdf
document that you submit. Regardless of what you decide, it is important that your
answers are as clear as possible. Your answers should appear in the order in which the
questions are asked. Please review your answers before submitting them to confirm that
they are easily readable.
• Be sure that you explicitly answer each question and explain each step, as if you were
writing solutions so that another student in the class would be able to follow your
thoughts. Part of your grade will depend on explaining each step of your answers.

1. (6 points) Conceptual questions. Briefly answer the following questions in no more than
   2 lines per question.
   (a) (1.5 points) What is the fundamental problem of causal inference?
   (b) (1.5 points) What is the average treatment effect on the treated (ATT)? Describe
   it in words and algebraically.
   (c) (1.5 points) False or true? Can we estimate the average potential outcome yi(1)
   for individuals in the control group? Explain.
   (d) (1.5 points) What is the attribution problem?
2. (10 points) Table 1 summarizes potential outcomes, treatment statuses, and observed
   outcomes for 6 individuals in a randomized experiment. Columns 2 and 3 show the
   potential outcomes when untreated (yi(0)) and treated (yi(1)), respectively. Columns
   4 and 5 show two possible treatment assignments Di and D0
   i
   . In the first treatment
   assignment, Di
   , individuals 1, 2 and 3 are selected into treatment, and individuals 4, 5
   and 6 are selected into the control. In the second treatment assignment, D0
   i
   , individuals
   2, 4 and 6 are selected into treatment, and individuals 1, 3 and 5 are selected into the
   control. Note that in both treatment assignments, Di and D0
   i
   , half of the individuals
   are treated, i.e., P6
   i=1 Di = 3 and P6
   i=1 D0
   i = 3. If the treatment status is Di
   , then you
   observe the outcome Yi(Di) = Diyi(1) + (1 − Di)yi(0) (column 6). Alternatively, If the
   treatment status is D0
   i
   , then you observe the outcome Yi(D0
   i
   ) = D0
   i
   yi(1) + (1 − D0
   i
   )yi(0)
   (column 7).
   (a) (2 points) Let AT E =
   1
   6
   P6
   i=1(yi(1)−yi(0)). Calculate the average treatment effect
   (ATE) using the data in Table 1.
   (b) (2 points) If you observe the sample [(Y1(D1), D1),(Y2(D2), D2), . . . ,(Y6(D6), D6)],
   what is the average treatment effect on the treated (ATT)?
   Table 1: Potential Outcomes and Treatment Statuses
   Individual (i) yi(0) yi(1) Di D0
   i Yi(Di) Yi(D0
   i
   )
   1 3 2 1 0 2 3
   2 7 1 1 1 1 1
   3 2 2 1 0 2 2
   4 1 3 0 1 1 3
   5 0 5 0 0 0 0
   6 0 4 0 1 0 4
   (c) (3 points) Assume that you observe the sample [(Y1(D1), D1),
   (Y2(D2), D2), . . . ,(Y6(D6), D6)]. Write down the formula for βb1, the regression coefficient of Di
   in the regression of Yi = Yi(Di) on Di
   . Compute βb1 for the observed
   sample.
   (d) (3 points) You would like to use βˆ
   1 to estimate the ATT, calculate the omitted variable bias if you observe the sample [(Y1(D1), D1),(Y2(D2), D2), . . . ,(Y6(D6), D6)].
   Using your answers to questions (b),(c) and (d), evaluate whether βb1 = AT T +
   “Omitted variable bias.”
3. (14 points) A randomized experiment on the effects of advertisement on purchases of a
   product selected a group of 40 customers. 20 of these customers were randomly assigned
   to the advertisement and the remaining 20 customers were randomly assigned to the
   control group with no advertisement. Let Yi be the observed outcome for customer i. If
   customer i bought the product then Yi = 1, otherwise Yi = 0. Similarly, let Di be the
   treatment status for customer i. If customer i was assigned to treatment then Di = 1,
   otherwise Di = 0. The sales department tells you that the proportion of customers who
   bought the product among those that were exposed to the advertisement was double the
   proportion of the customers that bought the product among those that were not exposed
   to the advertisement. You also know that Vb(by(1)) = 0.012 and Vb(by(0)) = 0.01050.
   (a) (7 points) What is the average observed outcome for the treated, by(1) = 1
   n1
   P
   i:Di=1 Yi?
   What is the average observed outcome for the untreated, by(0) = 1
   n−n1
   P
   i:Di=0 Yi?
   Let βb1 be the coefficient of Di
   in the regression of Yi on the treatment indicator
   Di
   . Calculate βb1.
   (b) (7 points) Compute a t-test to evaluate the null hypothesis that the average treatment effect is different from zero. Clearly state the null and alternative hypotheses,
   the test statistic, the test’s significance level and whether you reject or not the null hypothesis.

Sample Solution

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