On a topographic map, raindrops choose their path: falling on one slope, they flow into River A; on the other, into River B. An observer sees only the winding rivers (the response variables Y) and knows which slope the water came from (the instrument Z), but cannot see the drops themselves (the unobserved factors ε). This is the econometrician’s dilemma: using just a binary choice of slopes, one must reconstruct the shape of the terrain (the structural equation m) from two runoff maps.

The classic answer required either a rich continuous instrument or hard constraints on the model’s form. But rivers behave differently: water flows downhill, never looping back — mathematically, this is called cyclic monotonicity (a property that forbids closed loops of increments). The breakthrough is that this property alone is enough for even a simple slope switch to reveal the entire landscape. At its heart lies a dance of two distributions and their optimal matching.

In a multidimensional world, the classic control variable trick fails: you can’t uniquely order vectors. Instead, the author builds a bridge to optimal transport theory — a method for moving probability mass at minimal cost. For each raindrop U, its path X = h(Z,U) on slope Z follows a cyclically monotone mapping — a multidimensional analogue of non-decreasingness. Thanks to Brenier’s theorem, such a mapping is unique and invertible almost surely. Thus, one can define a 'rank map' $R_z$—the inverse mapping that sends points from one slope to points of the same rank on the other. The watershed is the set where the rank map leaves a point unchanged: $R_z$(x) = x.

The technical conditions require smooth densities, boundedness, and nondegeneracy: the Jacobian matrix of the mapping $R_z$(x)-x must have constant rank in a neighborhood of the watershed. Then an iterative process, switching slopes and applying the rank map, causes any starting point to slide down to the watershed, while the conditional distribution of the rain ε is preserved at each step. This is the key to identification. Moreover, the condition is testable: the author derived the asymptotics of the empirical process of the difference between two conditional distribution functions. It turns out that as the sample grows, this process converges to a Gaussian — meaning we can build confidence intervals and test for the existence of the watershed directly from the data.

For applied science, this lifts a long-standing limitation: contrived instruments are no longer needed. A simple binary factor is enough to nonparametrically estimate complex multidimensional effects. The work opens the door to more honest causal analysis in economics, epidemiology, and social sciences. Ahead lies the development of algorithms for high dimensions; for now, each iteration is like taking a blindfolded step in the mountains. But once those peaks are climbed, the watershed will become a precise tool. Now, gazing at the river map, the observer sees not just winding curves but hidden relief lines, traced by the flow of invisible drops.