A new test for unit roots with a partial quadratic trend

Abstract

This paper proposes a new test for unit root processes with a partial quadratic trend on an unknown break date, denoted as the URQ process herein. Such a process is extremely similar to the explosive bubble process and both can capture the sharp rise in prices. We develop the asymptotic distributions under the local-to-unity hypothesis which covers the URQ null and explosive root alternatives. Simulations show that the test has good finite sample performances and can differentiate explosive bubble processes from URQ processes. An application to the Kweichow Moutai and Apple stocks, which exhibit striking price rises during their respective sample periods, shows that both prices follow URQ processes. We further provide a fundamental analysis. The significant increases in earnings, returns, dividends, and fundamental score after the partial quadratic trend occurs provide evidence that a fundamental improvement rather than a bubble mainly drives such drastic price rises.