The Partial Identification of a Dependence Operator: What an Observed Spectrum Recovers from Economic Comovement

How much can you learn about the hidden web of dependence in an economy by watching its measured series move together? This paper answers precisely. It shows that the pattern of comovement fixes a structure only up to a family of look-alikes, and that a quantity is pinned down exactly when it survives every member of that family. This test sorts every claim into firmly fixed, fixed only after a free constant is chosen, or open, with an ambiguity of measurable size that shrinks as data accumulate. In a network-game example, the data recover two aggregate channels and the shared adjustment speed but not the individual links. A worked example illustrates the theory rather than testing it, and the dependence measured is statistical, not a claim about the economy's stability.