A new test for unit roots with a partial quadratic trend.

This article introduces a new statistical test, called the DBF test, designed to distinguish between a unit root process with a partial quadratic trend (URQ process) and an explosive bubble process in asset prices, particularly when the break date of the quadratic trend is unknown. The URQ process models sharp price increases driven by fundamental improvements rather than irrational bubbles, capturing structural changes in long-run growth trends. The authors derive the asymptotic distributions of the test statistic under a local-to-unity framework and propose a trimmed version of the DBF test to handle unknown break dates, demonstrating through simulations that it effectively differentiates URQ processes from explosive bubbles with good finite sample properties. Empirical applications to Kweichow Moutai and Apple stock prices reveal that their rapid price rises are better characterized by URQ processes linked to fundamental improvements—supported by increases in earnings, returns, dividends, and fundamental scores—rather than by explosive bubbles, contrasting with conclusions drawn from existing bubble tests like the SADF test.