What Does Structural Noise Measure? Revealed-Preference Inconsistency and Model Misspecification

Empirical comparisons of preferences often treat the dispersion parameter in a random utility model as noise around a correctly specified utility function. We show that this interpretation can fail: dispersion absorbs systematic behaviour that the fitted preference family cannot span. Using portfolio choices from four studies, we estimate disappointment-aversion random utility models and compare the dispersion parameter with revealed-preference diagnostics. A money-metric decomposition separates axiomatic inconsistency from the additional loss due to imposing DA-CRRA preferences. Both components in dependently predict dispersion, and model-relative misspecification remains strongly predictive among subjects who satisfy revealed-preference consistency. In our sharpest test, coefficients estimated on three studies predict dispersion in a held-out representative sample. A calibrated simulation illustrates the distinction. The dispersion parameter is therefore not a model-free measure of decision quality. It is jointly informative about choice inconsistency and preference-model fit, with implications for comparisons of recovered preferences and stochastic precision across people and environments