Question

Q1. (Potential outcomes and causal effect) [10 points Consider the following table that shows the potential wages of Spencer and Shelby if they had gone to college and if they had not gone to college. Di = 1 if i did go to college, and 0 otherwise. Y. denotes wages of i, and the rest of the notation corresponds to the textbook and class notes (e.g., Yoi is the wage of i if they didn't go to college) Spencer Shelby Yoi Y1i 15 20 40 0 40 a. What is the causal effect of going to college on Spencer's wages? 1 point b. What is the causal effect of going to college on Shelby's wages? [1 point c. What is the average causal effect? 1 point d. Is there a difference between the potential wages of Spencer and Shelby when neither goes to college? If yes, what might be the potential causes of this difference? If no, why do you think there's no difference? 3 points] e. What is the difference between the observed wages of Spencer and Shelby? Decompose this difference into the true causal effect for Shelby and the se- lection bias. [4 points

Answer 1

Solution:

Let i = {Sp, Sh}, where Sp is for Spencer and Sh is for Shelby

a) Causal effect of going to college on Spencer's wages =

Y_{​​​​1Sp}​​​ - Y_​​​0Sp​​​ = 15 - 5 = 10

b) Causal effect of going to college on Shelby's wages =

Y_{​​​​​​1Sh}​​​ - Y_{​​​​0Sh} = 40 - 20 = 20

c) Average causal effect = E(Y_{​​​​​​1i} - Y_{​​​​​​0i}​| Di = 1)

Since, here Di=1 only for Shelby, this is same as what we obtained in (b). Thus, average causal effect = 20

d) Yes, there is difference in potential wages when neither goes to college. When Spencer doesn't goes to college, potential wage earning = Y_{​​​​​​0Sp}​​​ = 5, while when Shelby doesn't go to college, potential wage earning = Y_{​​​​​​0Sh}​​​ = 20.

Thus, the difference = 20 - 5 = 15

Potential causes for this difference could be that Shelby is more skilled and ability than Spencer (even without college) and thus, possess capability of being more competitive in work environment and so, can potentially earn higher. Also, Spencer might bear relatively more restrictions such as financial, than Shelby and so they might start with different capital investments and thus, Shelby will be able to perform different and better. (These could also act as potential reasons why Spencer actually didn't go to college, while Shelby did).

e) Difference between observed wages of Spencer and Shelby:

E(Yi|Di=1) - E(Yi|Di = 0)

This is same as E(Yi| i = Sh) - E(Yi| i = Sp) = 40 - 5 = 35 (notice the last row of the table)

This can be decomposed into true causal effect for Shelby and selection bias.

True causal effect for Shelby is the difference in potential wages of Shelby when she goes to college and when she doesn't. This captures the effect of difference in what is average outcome of the subject when it has been treated (that is average potential earning when Shelby has gone to college) and what would have been it's average outcome had the subject not been treated (that is average potential earning of Shelby had she not gone to college).

Thus, this equals what we obtained in part (c) = E(Y_{​​​​​​1Sh} - Y_{​​​​​0Sh}​​​) = 40 - 20 = 20

Selection bias occurs when the subjects have differences in the outcomes even without treatment of any. So, though Spencer didn't go to college and Shelby went to college, and thus there is difference in their observed earnings, but this might not be truly due to differences in their college attendence. It could be the case that even without college, both had difference in their potential earnings (which we have already seen that it exists, in part (d)). Thus, there might be bias in the selection on who should be and has been treated (Shelby here) and who shouldn't be and has not been treated (Spencer here). Thus, selection bias = E(Y_{​​​​​​0i}​​​​|Di = 1) - E(Y_{​​​​​0i} |Di =0)

Or E(Y0i|i = Sh) - E(Y0i| i= Sp) = 20 - 5 = 15 (as already obtained in part (d))

Finally, we have seen the decomposition as:

Difference in observed wages of Spencer and Shelby = True causal effect for Shelby (or average causal effect) + selection bias

35 = 20 + 15