2. This problem is intended to test your understanding of the selection effect (
ID: 3304482 • Letter: 2
Question
2. This problem is intended to test your understanding of the selection effect (or selection bias). The relationship of interest is the effect of going to college (Xi) on earnings (Yi). The table below gives you information about each person's potential outcomes Alex Brett Chris David Edward S30,000.00 S31,000.00 S35,000.00 S31,000.00 S35,000.00 S39,000.00 Frank Yoi Y S38,000.00 S40,000.00 S45,000.00 $37,000.00 S45,000.00 S50,000.00 Xi RXi 0 0 0 2a. What is the true effect of college on earnings? (Hint: calculate the true effect for each person, then take the average of these true effects) 2b. What is the effect of college on earnings for the naive comparison (i.e. comparing the earnings of those who went to college to those who did not)? 2c. What is the selection effect associated with the naive comparison? Now, suppose that we were able to randomize who goes to college. This randomization is given by the row RXi, where RXi-1 means the person went to college and RXi-0 means no college 2d. Recalculate the effect of college on earnings as in 2b., this time using RXi as the treatment variable. What happened to the selection effect?Explanation / Answer
2a. True effect of college education on earnings:
effect for Alex = 38000 - 30000 = 8000 $
effect for Brett = 40000 - 31000 = 9000 $
effect for Chris = 45000 - 35000 = 10000 $
effect for David = 37000 - 31000 = 6000 $
effect for Edward = 45000 - 35000 = 10000 $
effect for Frank = 50000 - 39000 = 11000 $
True Effect for the poulation = average of true effects of all individuals in the population =
= (8000 + 9000 + 10000 + 6000 + 10000 + 11000) / 6 = 9000 $
2b. Average earning of those who did not go to college = (30000+31000+35000)/3 = 32000 $
Average earning of those who went to college = (37000+45000+50000)/3 = 44000 $
Thus a naive comparison results in giving effect of college on earnings as (44000 -32000) = 12000 $
2c. Selection Effect associated with naive comparison = (12000 - 9000) = 3000 $
2d. Effect of Randomised Selection on Earnings:
Average earning of those who did not go to college = (31000+31000+39000)/3 = 33667 $
Average earning of those who went to college = (38000+45000+45000)/3 = 42667 $
Hence effect of college education on earnings = (42667 - 33667) = 9000 $
This is the same as the True Effect calculated in 2a.
The Selection Effect has become zero.
Thus Ramdomization tends to eliminate or at least minimise the Selection Effect.
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