Microstructural Foundations of Rough Noise

Recently, it has been proposed to model the microstructure noise in prices by a continuous-time process with continuous sample paths that are rougher than those of a standard Brownian motion. In this paper, we propose a microstructural model for the tick-by-tick price changes that explicitly separates the permanent price changes from the fleeting price changes due to noise. We show how this model converges to a standard semimartingale model for the permanent price process, plus a rough noise term originating from the fleeting price changes on the macro scale. This provides a microstructural foundation for the rough-noise model. We then develop a GMM estimation method applicable to tick-by-tick data, together with a formal test for rough noise. We show that the estimator and test work in finite samples through a simulation study, and apply them to tick-by-tick data on Dow Jones Industrial Average constituents in 2024. Because our estimator is designed for tick-by-tick data, we estimate roughness at the daily level, revealing substantial day-to-day variation. We find that rough noise, while present, is not universal: even when detected, the roughness index is typically close to zero, and it is most pronounced on days dominated by short-run price reversals.