Law of Iterated Expectations
The law of iterated expectations says E[E[X | finer info]] = E[X | coarser info] — you cannot improve an expectation by forgetting information, and towers of forecasts must nest.
Definition
E[E[X | G] | F] = E[X | F] when F ⊂ G. This is why a martingale’s expectation is stable as time moves, and why you cannot have a forecast of a forecast that systematically disagrees with the forecast.
Why it matters
It is the consistency condition for term structures of expected returns and for sequential trading theorems. A model that violates the tower is internally broken.
Case
If your 1-year expected return is 4% and your path of 1-month expected returns averages to 12%, you violated LIE (or you changed the measure quietly).
How to read it
When two forecast systems disagree, check nesting. If they are not a tower, at least one is not a conditional expectation.