Nonlinearities, exponential trends, and Euler equations are three key features of standard dynamic volatility models of speculation, economic growth, or macroeconomic fluctuations with occasionally binding constraints and endogenous state-dependent volatility. A natural way to estimate a model with all such three features could be to use the observed nonstationary data in a single step without preliminary linearization, log-linearization, or preliminary detrending. Adoption of this natural strategy confronts a serious challenge that has been neither articulated nor solved: a dichotomy in the empirical model implied by the Euler equation. This leads to a discontinuity in the regression in the limit, rendering the approaches employed in available proofs of consistency inapplicable. We characterize the problem and develop a novel method of proof of consistency and asymptotic normality. Our methodological contribution establishes a foundation for consistent estimation and hypothesis testing of nonstationary models without resorting to preliminary detrending, an a priori assumption that any trend is exactly zero, linearization, or other restrictions on the model.