Instrumental Variables
By the end of this lesson, you should be able to: say what an instrument has to satisfy and which of those conditions you can check, compute an IV estimate and know what each stage does, say whose effect it measures, and recognise that a weak instrument returns a number rather than an error.
When adjustment can't help
Every method in Module 2 needs the confounders to be measured. Often the most important one isn't.
Do Vietnam veterans earn less than other men because they served? Studies comparing veterans with non-veterans gave mixed answers: many found losses, others found gains. The problem is who served. Men with fewer job prospects as civilians were more likely to enlist, and "job prospects" isn't a column you can adjust for.
eBay had the same problem with advertising. A regression of revenue on paid search spending across regions and days said 10% more spending raised revenue by 9%. Adding region and day controls cut that to 1.3%, still a large return. But eBay spends more on search ads when more people are searching, and people searching for eBay are already likely to buy. Purchase intent drives both spending and revenue, and it isn't measured.
An instrument gets around this. Instead of using all the variation in the treatment, which is mixed up with the confounder, it uses only the part that came from something random.
The draft lottery
From 1970 to 1972, the US decided who would be drafted by lottery. In a televised drawing, every birthday was given a random number from 1 to 365. Men whose number fell at or below a cut-off (195 for men born in 1950) were draft eligible.
Being eligible didn't decide who served. Some eligible men didn't serve, and some ineligible men volunteered. But it moved the chances a lot:
| White men born in 1950 | Served in the military |
|---|---|
| Draft eligible | 35.3% |
| Not eligible | 19.3% |
| Difference | 15.9 points |
Because birthdays were drawn at random, eligible and ineligible men were alike in everything else, including job prospects. So any difference in their later earnings comes from the difference in who served.
In 1981, eligible men earned on average $488 less than ineligible men. Only 15.9% of them were moved into service by the lottery, so the effect per man who served is:
That's about 15% of annual earnings, ten years after service. Angrist's estimates were similar across cohorts and years, though few were statistically significant on their own.