Economics 706 - Problem Set 1
Consider the following data generation process:
We have 4 underlying types of people: A, B, C, and D and each comprise ¼th of the population
Assume that the conditional expected value of Y1i and Y0i take the following values in the data.
|
Type
|
E(Y0i|Type)
|
E(Y1i|Type)
|
|
A
|
1
|
1.5
|
|
B
|
0.5
|
1.5
|
|
C
|
1.5
|
1
|
|
D
|
1
|
1
|
Also assume that the variables are normally distributed with variance 1.
Problem 1 - First assume that the A's and B's receive the treatment but the C's and D's never receive it. Calculate
a) The Average Treatment Effect
b) The Average Treatment on the Treated
c) The Average Treatment on the Untreated
Problem 2 - Now suppose you run an experiment where you randomize everyone into treatment and controls
a) If everything goes smoothly and you just compare the treatment and controls what will you get?
b) Now suppose the type is observable in the data so that you can condition on it. If you estimate the average treatment effect for each type, what would you expect to find for each group
c) Suppose that you only have data on the A, B, and C groups because the D group does not answer your survey. If you look at the experiment only on the people who answer the question what would you expect to find.
d) Continue to assume that the D group never answers the survey and assume that the A and B answer whether they are part of the experiment or not. However, now suppose that the C group answers when they are in the treatment group but not if they are in the control group. What would happen if you just looked at the difference between treatment and controls for people who answer the survey?
e) Now suppose that if they were randomized into the programs the A and B types would always take the treatment but the C and D groups would not. (and if they are randomized out none of them take the treatment). What is the treatment on the treated parameter here? Show that if you use the formula we derived in class you would get an estimate of the treatment on the treated.
f) Now lets change things slightly by taking the same case as above but assume that the A's would take the treatment even if they were assigned to the control group. Thus the A's always take the treatment, the C's and D's never take the treatment, and the B's take the treatment when they are assigned the treatment and do not take the treatment when they are controls (they are typically referred to as the compliers). As in class let Ai represent assignment to the treatment group. Show that
(E(Yi| Ai = 1) - E(Yi| Ai = 0)/Pr(Ti = 1 | Ai = 1) - Pr(Ti = 1 |Ai = 0)) = E (?i| Type = B)
Problem 3 - Using whatever statistical package you would like simulate data using the model above and verify that your all of your answers to problems 1 and 2 are approximately right (where the approximately comes from the fact that you only have finite samples).
To do this in stata. You could use the following commands to start generating the data
- set obs 1000
- generate u=uniform()
- generate type="A" if u <= 0.25
- replace type="B" if u > 0.25 & u <= 0.5
- etc.
- generate y0=1+rnormal(0, 1) if type=="A"
- generate y1=1.5+rnormal(0, 1) if type=="A"
- replace y0=0.5+rnormal(0, 1) if type=="B"
- etc.
- generate A = uniform() > 0.5
- generate T1=A
- generate Y = T1 ∗ y1 + (1 - T1) ∗ y0
- etc.
- sort T1
- by T1: sum Y