Correlated Signals, Precision Paradox, and Design of Official Disclosures

We develop an analytical model featuring two public signals on a terminal payoff: an early public signal (e.g., a social-media signal or an alternative-data signal) and a later official disclosure. Both signals arrive before a final payoff in an incomplete market characterized by investors with heterogeneous prior precisions. We demonstrate that error correlation across these public signals is a first-order driver of speculative surplus, price variance, and trading volume. This correlation determines the incremental informational value of the official disclosure relative to the early public signal, and the incremental precision is U-shaped in the error correlation. Moreover, a precision paradox might exist in the modern Big Data era in which early public signals (e.g., an aggregate alternative-data signal) can be more precise than official disclosure: increasing the official signal's precision under high error correlation can paradoxically reduce its marginal value. We investigate information design by studying how a regulator trades off investors' welfare (speculative surplus) against market stability (price variance). We discuss how to implement an optimal information design and show that changing the error correlation via, e.g., changes in disclosure timing, verification protocols, or measurement methodology, can be cheaper and more effective than simply increasing the official signal precision.