The drift-diffusion model (DDM) is a model of sequential sampling with diffusion signals, where the decision maker accumulates evidence until the process hits either an upper or lower stopping boundary and then stops and chooses the alternative that corresponds to that boundary. In perceptual tasks, the drift of the process is related to which choice is objectively correct, whereas in consumption tasks, the drift is related to the relative appeal of the alternatives. The simplest version of the DDM assumes that the stopping boundaries are constant over time. More recently, a number of papers have used nonconstant boundaries to better fit the data. This paper provides a statistical test for DDMs with general, nonconstant boundaries. As a by-product, we show that the drift and the boundary are uniquely identified. We use our condition to nonparametrically estimate the drift and the boundary and construct a test statistic based on finite samples.
We provide an axiomatic analysis of dynamic random utility, characterizing the stochastic choice behavior of agents who solve dynamic decision problems by maximizing some stochastic process (U_t) of utilities. We show first that even when (U_t) is arbitrary, dynamic random utility imposes new testable restrictions on how behavior across periods is related, over and above period-by-period analogs of the static random utility axioms: An important feature of dynamic random utility is that behavior may appear history dependent, because past choices reveal information about agents’ past utilities and (U_t) may be serially correlated; however, our key new axioms highlight that the model entails specific limits on the form of history dependence that can arise. Second, we show that when agents’ choices today influence their menu tomorrow (e.g., in consumption-savings or stopping problems), imposing natural Bayesian rationality axioms restricts the form of randomness that (U_t) can display. By contrast, a specification of utility shocks that is widely used in empirical work violates these restrictions, leading to behavior that may display a negative option value and can produce biased parameter estimates. Finally, dynamic stochastic choice data allows us to characterize important special cases of random utility—in particular, learning and taste persistence—that on static domains are indistinguishable from the general model.