Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behavior∗

Right-tailed unit root tests have proved promising for detecting exuberance in economic and financial activities. Like left-tailed tests, the limit theory and test performance are sensitive to the null hypothesis and the model specification used in parameter estimation. This paper aims to provide some empirical guidelines for the practical implementation of right-tailed unit root tests, focusing on the sup ADF test of Phillips, Wu and Yu (2011), which implements a right-tailed ADF test repeatedly on a sequence of forward sample recursions. We analyze and compare the limit theory of the sup ADF test under different hypotheses and model specifications. The size and power properties of the test under various scenarios are examined in simulations and some recommendations for empirical practice are given. Some new results on the consistent estimation of localizing drift exponents are obtained, which are useful in assessing model specification. Empirical applications to the Nasdaq and S&P 500 stock markets illustrate these specification issues and reveal their practical importance in testing.