Instrumental variables offer a way around confounding that nobody measured.
They exist because some causal questions have no full list of confounders and no possible trial.
The idea is to find a source of variation in the treatment that is unrelated to the confounders.
An instrument must move the treatment, be unrelated to confounders, and affect the outcome only through the treatment.
The first condition can be checked with data, and a weak instrument gives unreliable, biased estimates.
The other 2 cannot be checked with data and must be argued from knowledge of the setting.
The simplest estimate divides the instrument's effect on the outcome by its effect on the treatment.
For example, if living near a college adds 0.5 years of study and £1,000 of yearly earnings, each year is worth about £2,000.
2-stage least squares is the standard way to compute this with several instruments and controls.
In a trial where not everyone takes the treatment they were given, the random assignment can act as an instrument.
The estimate applies to compliers, the units whose treatment the instrument actually changed.
That group may differ from the population, so the estimate is not always the average effect for everyone.
Believable instruments are rare, and many proposed ones fail on close inspection.
In practice, instruments come from lotteries, rules, distance, timing and natural variation in supply.
The usual mistake is accepting an instrument because it is strongly related to the treatment, without arguing it has no other path.