Week 1

Question 1

How to define counterfactual outcome. Use lead paint and test scores as example

Answer 1

Counterfactual is D=1 and observe Y0 or D=0 and observe Y1

For a school with 𝐷𝑖 = 1, 𝑦0𝑖 is the counterfactual outcome. We observe the IQ scores in the presence of leaded paint.
The counterfactual, unobserved, outcome is the IQ scores that would be observed if there were no leaded paint however nothing else differed.

Then vice versa for other counterfactual

Question 2

How to describe selection bias in english from the formula (use lead paint on test scores again)

Answer 2

𝐸[𝑦0𝑖│𝐷𝑖 = 1] − 𝐸[𝑦0𝑖|𝐷𝑖 = 0] is the formula
“The difference in average IQ scores that would exist without
leaded paint between schools with leaded paint and schools without leaded paint.”

i.e. the difference between average IQ scores that would exist between the control group and the treated group in the absence of treatment (had they not been treated)

Question 3

describe observed difference in outcomes in english using paint example

Answer 3

the difference between observed average differences in IQ scores between schools with leaded paint and schools without leaded paint

Question 4

describe average treatment effect on treated in english using paint example

Answer 4

the difference in average IQ scores with and without leaded paint restricted to schools with leaded paint

Question 5

Describe how assuming that selection bias is 0 allows us to estimate
the ATT

Answer 5

Assuming selection bias is 0 is assuming there is no difference between treated and control average potential outcomes in the absence of treatment (i.e. if in a parallel universe the control group was treated instead, it would have the same outcome as the treated one in the normal universe)

𝐸[𝑦0𝑖|𝐷𝑖 = 1] = 𝐸[𝑦0𝑖|𝐷𝑖 = 0] is assumed, so we can use the counterfactual as we say it is the same as the RHS

𝐸[𝑦1𝑖|𝐷𝑖 = 1] − 𝐸[𝑦0𝑖|𝐷𝑖 = 0]
= 𝐸[𝑦1𝑖|𝐷𝑖 = 1] − 𝐸[𝑦0𝑖|𝐷𝑖 = 1]
= 𝐴𝑇𝑇