Clarifying a sentence from Angrist and Krueger (2001) In Angrist and Krueger (2001), it is said, during the discussion of pitfalls in using IV, on page 79 that

Because the reduced form effects are proportional to the coefficient
  of interest, one can determine the sign of the coefficient of interest
  and guestimate its magnitude by rescaling the reduced form using
  plausible assumptions about the size of the first-stage
  coefficient(s).

The sentence is too long and too vague for me to understand what the authors are talking about. Can someone please explain with an example?
 A: This is because $$\beta_{IV}=\frac{dy/dz}{dx/dz},$$ where $dx/dz$ is the potentially unknown first-stage coefficient. You can also get this intuition from a simple Wald estimator.
Here's an example. Suppose when compulsory schooling law goes into effect in some state ($z=1$), students acquire 2 more years of schooling and earn 5000 more. This increase in earnings is a consequence of the indirect effect that increase in $z$ led to increase in schooling which in turn increases income. Then it follows that 2 years additional schooling are associated with a 5,000 increase
in earnings, so that a one year increase in schooling is associated with a
$5000/2 = 2,500 increase in earnings. The causal estimate of the effect if schooling is therefore 2,500 per year of education. 
Think about the schooling regression example, where the omitted variable is ability. People who go to school earn more because they are more able and have more schooling. We want to know about the effect of schooling adjusted for ability. It is possible that the effect of schooling is actually negative because it makes you hate learning if you are bored to death every day, and the ability effect is masking it. Solution: we get $dy/dz$ by running regression of earnings on whether the law applies. We can get $dx/dz$ by running a regression of schooling on the law or by making an assumption (the law will make everyone stay in school for 2 more years). Now I get the effect of schooling on earnings (which may have a different sign than the naive OLS regression of earning on schooling) by rescaling $dy/dz$ by $dx/dz$.
