Testing for moderate explosiveness in the presence of drift

This paper considers a moderately explosive autoregressive(1) process with drift where the autoregressive root approaches unity from the right at a certain rate. We first develop a test for the null of moderate explosiveness under independent and identically distributed errors. We show that the t statistic is asymptotically standard normal regardless of whether the errors are Gaussian. This result is in sharp contrast with the existing literature wherein nonstandard limiting distributions are obtained under different model assumptions. When the errors are weakly dependent, we show that the t statistic based on a heteroskedasticity and autocorrelation robust standard error follows Student's t distribution in large samples. Monte Carlo simulations show that our tests have satisfactory size and power performance in finite samples. Applying the asymptotic t test to ten major stock indexes in the pre-2008 financial exuberance period, we find that most indexes are only mildly explosive or not explosive at all, which implies that the bout of the irrational rise was not as serious as previously thought.